T 7.4.3 Microondas en guías de onda

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T 7.4.4MicrowavePropagation

in Waveguides

by Prof. H. J. Chaloupka

revised by Dipl.-Ing. Anton Oster

June 1998

LEYBOLD DIDACTIC GMBH . Leyboldstrasse 1 . D-50354 Hürth . Phone (02233) 604-0 . Fax (02233) 604-222 . e-mail: [email protected]

by Leybold Didactic GmbH Printed in the Federal Republic of GermanyTechnical alterations reserved



“The sensitive electronics of the equipment contained in the present experiment litera-ture can be impaired due to the discharge of static electricity. Consequently, electro-static build up should be avoided (particularly by utilizing appropriate rooms) or eliminated by discharging (e.g. at the panel frames or similar).”



3

MTS 7.4.4 Contents

Table of contents

Equipment overview ................................................................................................................... 4

Symbols and abbreviations ......................................................................................................... 5

Preface .......................................................................................................................................... 7

List of experiments performed in this training unit

Ex1 The Gunn oscillator ......................................................................................................... 11

Ex2 The direct modulation of the Gunn oscillator ...............................................................1 7

Ex3 The selective measurement amplifier ............................................................................ 19

Ex4 Attenuators ....................................................................................................................... 23

Ex5 The slotted measuring line.............................................................................................. 27

Ex6 The complex reflection coefficient ................................................................................ 37

Ex7 Matching and the Smith chart ........................................................................................ 43

Ex8 Reflection of a single-slot antenna ................................................................................ 53

Ex9 Measuring the permittivity..............................................................................................59

Ex10 The cross directional coupler .........................................................................................65

Ex11 The principle of the reflectometer..................................................................................71

Ex12 The cavity resonator ........................................................................................................77

Solutions ..................................................................................................................................... 81

Index ...........................................................................................................................................97



4

MTS 7.4.4 Contents

Equipment overview

Equipment

Gunn oscillator 737 01 1 1 1 1 1 1 1 1 1 1 1 1

Gunn power supply with SWR-Meter 737 021 1 1 1 1 1 1 1 1 1 1 1 1

Coax detector 737 03 1 1 1 1 1 1 1/(2) 1/(2) 1 1 1 1

Transition waveguide/coax 737 035 1 1 1 1 1 1 1 1 1

PIN modulator 737 05 (1) (1) (1) 1* 1* 1* (1) (1) (1)

Isolator 737 06 (1) (1) (1) 1* 1* 1* (1) (1) (1)

Variable attenuator 737 09 1 1 1 1

Fixed attenuator 737 095 1 1 1

Moveable short 737 10 1

Slotted measuring line 737111 1 1 1 1 1

Waveguide 200 mm 737 12 1 1 1 1

Slide screw transformer 737 13 (1) (1)

3-Screw transformer 737 135 1 1 (1)**

Waveguide termination 737 14 1 1 2 1 2 2 2

Cross directional coupler 737 18 1 1 1 1 1

Set of 4 diaphragms with slits and holder 737 22 1 1 1 1 1 1 1 1

Waveguide propagations accessories 737 29 1 1 1 1 1 1

E-Field probe 737 35 (1)

Set of thumb screws M4 737 399 1 1 1 1 1 1 1/2** 1 1 1 2 1

Accessories

Digital storage oscilloscope 575 292 1 1 1 (1) (1) (1) (1) (1)

Coax-cables with BNC/BNC plugs, 2 m 501 022 2 2 2 2 2 2 3 3 3 2 2 2

Stand base MF 301 21 2 2 4 2 2 2 2 2 2 3 3 2

301 26 1 1 2 1 1 1 1 1 1 2 2 1

Support for waveguide components 737 15 1 1 3 2 2 2 2 2 2 3 3 2

XY-Yt-recorder 575 663 (1) (1) (1) (1) (1) (1)

() = recommended ** = when using 737 13

* = required for reproducible results ()** = alternative intermediate piece for 200 mm waveguide

Stand rod L = 25 cm, ∅ = 10 mm

M T

S 7 . 4 . 4

E x p e r i m e n t s

E x 4 : A t t e n u a t o r s

E x 1 : T h e G u n n o s c i l l a t o r

E x 2 : T h e d i r e c t m o d u l a t i o n o f t h e G u n n o s c i l l a t o r

E x 5 : T h e s l o t t e d m e a s u r i n g l i n e

E x 6 : T h e c o m p l e x r e f l e c t i o n c o e f f i c i e n t

E x 7 : M a t c h i n g a n d t h e S m i t h c h a r t

E x 8 : R e f l e c t i o n o f a s i n g l e - s l o t a n t e n n a

E x 3 : T h e s e l e c t i v e m e a s u

r e m e n t a m p l i f i e r

E x 9 : M e a s u r i n g t h e p e r m

i t t i v i t y

E x 1 0 : T h e c r o s s d i r e c t i o n a l c o u p l e r

E x 1 1 : T h e p r i n c i p l e o f t h e

r e f l e c t o m e t e r

E x 1 2 : T h e c a v i t y r e s o n a t o

r



5

MTS 7.4.4 Contents

Symbols and abbreviations

a : Display value of the selective measurement amplifier

a' , s' : Geometric dimensions of the resonator cavitya, b : Geometric dimensions of the rectangular waveguide

aD : Directivity

aK : Coupling attenuation

B : Susceptance

c : Velocity of light

C : Capacitance

DUT : Device under test

E →

: Electric field vector

E : Electric field strength

E TH, U TH : Threshold field strength and Gunn voltage required to reach E TH

f : Frequency

f 0 : Resonance frequency of the Gunn oscillator f c : Cut-off frequency

f r : Frequency of the modulating wanted signal

G : Conductance

H : Magnetic field strength

H →

: Magnetic field strength vector

RF : Radio frequency

i(t), u(t) : Current and voltage of a Schottky diode

I Detekt : Direct current through the Schottky diode of the detector

I G : Gunn current through the Gunn diode

I S : Saturation voltage of a Schottky diode

J : Electrical dipole K : Constant

k : Coupling coefficient

L : Inductance

l , ξ : Linear measure

M : Magnetic dipole

n : Transformation ratio

LF : Low frequency

P in : Input or feed power

P rad : Radiated power

P rec : Received power

Q0 : Unloaded Q

r : Reflection coefficient (complex) R : Actual resistance

s : Standing wave ratio

S ij : Elements of the scattering matrix (S )

T : Period duration

t : Timeu : Mean value of a voltage

U 0 : Threshold value of a voltage-dependent switch

U 1(t ), U 2(t ), U CL(t ): Voltages in the equivalent circuit diagram of a lock-in amplifier

U D : Demodulated signal at the output of the detector (low frequency)

U G : Gunn voltage at the Gunn diode

U max, U min, ∆U : Maximum and minimum value of the demodulated receiving signal and theresulting voltage range

U n(t), U n,rms : Noise voltage



6

MTS 7.4.4 Contents

U T : Temperature voltage

û : Amplitude of a voltage

v1, v2 : Drift velocities

v ph : Phase velocity

vr : Radial velocity X : Reactance

x, z : Coordinates

Y : Admittance

Z : Impedance

Z 0 : Field characteristic impedance of free space, line impedance

α : Geometric angle

β : Phase constant

δ : Penetration depth

ε : Absolute error

ε r , ε r ' , ε r '' : Relative permittivity

λ 0 : Free space wavelengthλ g : Guided wavelength

φ : Phase-shifts

µr , µr ' , µr '' : Relative permeability

ω : Angular frequency



7

MTS 7.4.4 Preface

Preface

The experiments in the present training system

are intended to achieve various training objec-

tives in parallel:

(α) The understanding of the physical effects

which are of significance in microwave

technology (for example, diffraction and

interference of electromagnetic waves,

Gunn effect).

(β) Acquiring knowledge of the function of

important components and systems of mi-

crowave technology and the principles

behind the methods of realizing thesefunctions by exploiting physical phenom-

ena (such as the function of a waveguide

directional coupler in microwave circuits

in “black-box” representation and the re-

alization of the directional coupler using

effects in electromagnetic coupling

through holes).

(γ ) Acquiring skills in measuring techniques

and principles to determine the properties

of microwave devices (example: measur-

ing the reflection coefficient using the re-

flectometer principle).

(δ) Becoming familiar with the actual techni-

cal design of various components and us-

ing these in practical applications (exam-

ples: learning to correctly install a ferrite

waveguide isolator in a microwave circuit,

operation of a slide screw transformer for

matching of a load). The components for

carrying out experiments with the above

objectives are all designed in waveguide

technology. Dual-plate configurations

(unit MTS 7.4.3) are particularly suitable because they can be easily disassembled

and are also extremely robust. One dis-

crepancy, however, had to be considered

when designing this training system: on

the one hand, waveguide technology is the

most suitable of all technologies for exper-

iment purposes; on the other hand, in the

field of radio-frequency circuit technolo-

gy, it is with few exceptions, being in-

creasingly overtaken by microwave inte-

grated circuits (MIC) utilizing microstrip

line or coplanar line technology.

The result of this discrepancy is the concept of

a teaching and training system based primarily

on waveguide technology, in which the physi-

cal phenomena and the fundamental principles

of technical elements and measuring methods

are given priority, independent of the particular

transmission line forms (waveguide, coaxial

line, microstrip line, etc.). Thus, for example,

the knowledge gained in experiments on Gunnoscillators using waveguide technology will

enable students to understand an oscillator re-

alized using microstrip technology, as the fun-

damental principles of the interaction between

the various semiconductor elements and the

resonance circuit are the same. With the aid of

an extremely specialized waveguide element

such as the cross directional coupler, students

can become familiar with important effects in

electromagnetic fields, e.g. coupling through

small openings; in addition, the “black box”

behaviour of the cross directional coupler is

typical of a large class of different directional

couplers (hybrids in microstrip line technolo-

gy, coaxial-line couplers, etc.).

The target group of this teaching and training

system is students who have widely varying

levels of prior knowledge and/or experience.

Thus, the system is equally applicable in uni-

versity-level science instruction and in techni-

cal schools. One may assume that the manner

in which most of the experiments are presented

and carried out is suitable for the learning needsof this broad target group, but that the contents

and the interpretation of the experiment results

within the framework of a “theoretical struc-

ture” must be treated differently in each case.

For this reason, this handbook contains some-

times even advanced material. This is in-

tended for university-level students, and may

be omitted when working through the experi-

ments.



8

MTS 7.4.4 Preface

endangered! DO NOT TOUCH! Discharge any

long cables before connecting them to these

components. This is carried out by connecting

them to the power supply unit.

Components which operate with strong perma-nent magnets, e.g. isolators or circulators, must

be kept at a distance from magnetically con-

ductive materials. Avoid shaking or bumping

the equipment.

The flange surfaces should be treated with care.

Mechanically movable parts are to be carefully

lubricated from time to time. Do not allow any

oil or grease on electrical contacts or in the

waveguide.

Design of the microwave sourceWith the exception on experiment Ex1 a modu-

lated microwave source is required to perform

the microwave experiments in the MTS 7.4.4

training system. This source generates the mi-

crowave field needed for the load connected

downstream. Only through modulation is it

possible to perform frequency-selective detec-

tion of the demodulated receiving signal and

thus carry out effective noise suppression. For

the design of a modulated microwave source

there are two options shown in Fig. 0.3 and Fig.

0.5.1. Direct modulation of the Gunn oscillator

2. Modulation with external PIN modulator

1. The direct modulation of the Gunn

oscillatorDirect modulation of the Gunn oscillator is one

way of performing modulation without addi-

tional equipment but also without the best re-

sults. As is shown in Experiment 1 the emitted

microwave power (and the emitted spectrum) is

severely dependent on the Gunn voltage. Fur-thermore, these variables are also subject to

considerable manufacturer tolerance involving

the Gunn diode. Due to its dependency on the

Gunn voltage, the emitted microwave power

demonstrates discontinuities or irregularities.

In Experiment 2 it is shown that in the case of

amplitude modulation of the Gunn voltage

where the operating point (supply voltage) is in

the proximity of a discontinuity, a slight varia-

tion in the Gunn voltage leads to a considerable

The theoretical principles of the experiments

which are of a general nature may be found in

standard textbooks on the subject. However,

the experiment descriptions also contain some

of the theoretical principles in order to facilitatethe students’ application of general representa-

tions found in textbooks to the specific knowl-

edge required for each experiment.

This manual is structured as follows:

In the descriptions of the individual experi-

ments, the necessary theoretical principles are

discussed first. This is followed by a list of the

equipment required and a detailed description

of the individual experiment steps (experiment

procedure). The experiments are evaluated us-

ing a list of questions. Each experiment sectionincludes its own bibliography or, where suffi-

cient, a reference to the bibliography of another

experiment.

In order to facilitate evaluation and to provide

ideas for evaluating the results, empty tables

and diagrams are included in the list of ques-

tions.

Safety Instructions

Important!Read this before putting the equipment

into operation!Due to the low power level of the Gunn oscillator

(approx. ≈ 10 mW) there is absolutely no danger

for persons conducting the experiment, even in

the case of free space experiments. Neverthe-

less, the following rules should also be observed

in view of later work with stronger RF-sources:

– It is imperative that any direct viewing into

the radiating aperture be avoided in all of

the experiments in which RF-power is ra-diated. This applies for open waveguides,

for example, and especially all types of

antennas.

– Disconnect the supply voltage whenever

modifications are made to experiment set-

ups in which waveguide components are

interchanged

Active elements in microwave components can

be destroyed by electrostatic discharge. Detec-

tors and varactor components are especially



9

MTS 7.4.4 Preface

Fig. 0.1: Emitted spectrum for a Gunn voltageU G = 4 V

Fig. 0.2: Emitted spectrum for a Gunn voltageU G = 8 V

variation in the emitted power and thus to a

large demodulated signal.

However, if the operating point is selected

close to a discontinuity (approx. 4V), one con-

sequence is that the emitted spectrum contains

many different frequency components. But due

to the fact that all of the microwave compo-

nents are designed for a frequency of 9.4 GHz,

and normally demonstrate a very narrow fre-

quency bandwidth, the measurement results for

this case are frequently insufficient and require

explanation.

Fig. 0.1 and Fig. 0.2 provide examples of the

emitted spectrum of the Gunn oscillator for Gunn voltages of 4 V and 8 V respectively.

The emitted spectrum varies for discontinuities.

A purer spectrum (TE 101 mode at 9.4 GHz and

TE 202 mode at 18.8 GHz) is generally obtained

starting at a Gunn voltage higher than 8 V.

On its own the emitted microwave power

changes only very slightly in the range of

higher Gunn voltages (7 V up to 10 V) with the

variation of the Gunn voltage (flat characteris-

tic). This yields a demodulated signal which is

very small. This means that in the range where

the Gunn diode emits a pure spectrum (starting

at 8 V) direct modulation is possible with only

a very dimunitive amplitude deviation.

Add to this the further disadvantage that the

Gunn oscillator is subjected to reflections

which can alter its operating features, e.g. para-

sitic oscillations can be generated.

Due to the fact that the PIN-Modulator is only

included as a recommendation in the MTS 7.4.4

equipment set, the use of the PIN modulator is

dispensed with in the experiments whereever

possible. If you have a PIN modulator avail-

able, then we highly recommend that you use it

(see also subpoint 2) as this normally yields

better measurement results.

For the internal direct modulation of the Gunn

oscillator

– connect the Gunn oscillator to the GUNN

socket in the GUNN POWER SUPPLY

section of the basic unit.

– set the toggle switch in the PIN MODU-

LATOR section of the basic unit to

“GUNN INT”.

– set the Gunn voltage to approx. 8 V (up to

10 V) using the U G controller.The Gunn voltage is then superimposed by a

square-wave signal with a frequency of 976 Hz

and an amplitude of 300 mV, see Fig. 0.4.

GUNN-OSC.

737 01

737 01

Fig. 0.3: Connection sketch for direct modulation of theGunn oscillator. Switch set to “GUNN INT”.



1 0

MTS 7.4.4 Preface

Fig. 0.5: Connection sketch for modulation with externalPIN modulator and internal modulation signal.Switch set to “PIN INT”

Fig. 0.4: Principle characteristic of the Gunn voltage onthe basic unit (GUNN output socket) in the caseof Gunn-internal modulation. U G represents thesupply voltage, which is set using the rotary potentiometer U G.

2. Modulation with an external PIN

modulatorThe experiment setup is performed as depicted

in Fig. 0.5.

The Gunn oscillator generates a continuous wave

power, i.e. its power output is constant in time.

The modulation is performed by a PIN modula-

tor connected downstream. Depending on the

control voltage this can have a variably high re-

flection and transmission coefficient. In order to

prevent any undesired reflections from reflecting

back to the Gunn oscillator, this is isolated from

the rest of the circuit by an isolator. The unit con-

sisting of the Gunn oscillator, isolator and PIN

modulator constitutes a typical configuration fre-

quently used in practical applications and isknown for its easy handling. Thus you avoid the

disadvantages of direct modulation performed

along a Gunn diode's characteristic (subject to

discontinuity) which is severely dependent on

manufacturer's tolerance. The modulated signal

is considerably better than direct modulation

because in this case the amplitude range only

depends on the control voltage of the PIN

modulator. As the Gunn diode can operate at a

fixed operating point, the emitted microwavespectrum is not affected by the modulation. Ad-

ditional information can be taken from the in-

struction sheet for the PIN modulator.

If you have a PIN modulator at your disposal,

then we recommend using it for modulation to

obtain better measurement findings. The PIN

modulator can be modulated internally or ex-

ternally. For external modulation you need an

additional function generator. In this case a

measurement with the internal SWR meter in no

longer possible as this requires frequency and phase synchronization.



1 1

MTS 7.4.4 The Gunn Oscillator

The Gunn oscillator

PrinciplesMicrowave power, i.e. electromagnetic power

in the GHz frequency range, may be generated

using quite different physical phenomena.

Some examples of this are vacuum-tube oscil-

lators such as the klystron and the magnetron,

or semiconductor oscillators such as the FET

oscillator, the Gunn oscillator and the impatt

oscillator. Some simple experiments for under-

standing Gunn oscillators are described below.

Gunn effectIn some semiconductor materials, such as Gal-

lium Arsenide (GaAs), the mobility of the elec-

trons (= the quotient of the drift velocity v and

the electrical field strength E ) decreases above

a threshold value E TH of the electrical field

strength (see Fig. 1.1, left side). This is because,

as the field strength increases more and more

electrons “transfer” to a state in which their “ef-

fective mass” becomes greater, thus decreasing

their velocity. For field strengths where E > E TH

the electrons have a negative differential mobil-ity, i.e. an increase in the field strength results

in a decrease in the drift velocity.

When the electrical field strength in a homoge-

neously doped GaAs block (no barrier layer!)

is greater than the threshold value E TH “space

charge instabilities” occur as a result of the

negative differential mobility.

While any random local surplus or deficiency

of electrons will disappear by itself when a

positive differential mobility is present, this sur-

plus or deficiency will increase under a nega-

tive differential mobility. In the upper rightsection of Fig. 1.1, a random surplus of elec-

trons is assumed. The resulting increase in field

strength on the anode side leads to a decrease

in the drift velocity v2 on the anode side relative

to the drift velocity v1 on the cathode side (de-

crease in field strength). This causes “bunch-

ing” and yields a carrier enhancement layer.

The analogous effect occurs in the event of a

random deficiency of electrons, in which case

a depletion layer occurs (see Fig. 1.1, middle-

right section). When the enhanced and de-

pleted layers approach each other, they attract

each other and jointly pass through the diode in

the form of a “domain” (see Fig. 1.1, lower

right section). The field in the interior of the

domain can be so high that the field outside it

falls below the threshold value E TH. Thus no

new domain can be formed until the existing

one has disappeared at the anode.

If the Gunn element were not connected to a reso-

nator (tuned circuit), the frequency of the micro-

wave power generated would be determined by

the time it takes for the domains (velocity approxi-mately 107 cm/s) to pass through the diode (transit

frequency). If, however, the Gunn diode is oper-

ated with a resonator, the resonator frequency can

be “imposed” upon the Gunn diode. There are

several operating modes here (delayed mode,

quenched mode, LSA mode).

The delayed domain mode is briefly described

here as an example. It occurs when the resonator

frequency is lower than the transit frequency. At

the moment in which the domain reaches the

anode, the momentary value of the diode voltage

(= bias voltage + RF voltage) is less than the

threshold value. The formation of a new domain is

Fig. 1.1 Principle of a Gunn diodeleft: Drift velocity v of the electrons as afunction of the electrical field strength E for GaAs.right: Formation of a domin (cathode (–) left

and anode (+) right)



1 2


delayed until the voltage exceeds the threshold

value, thus “imposing” the oscillator frequency of

the resonator on the Gunn element.

Design of the oscillatorOne of the many possible resonator types in mi-

crowave technology is the rectangular cavity

resonator. Fig. 1.2, left side, shows a rectangu-

lar cavity enclosed on all sides by metal walls.

Just as in a resonant circuit built of a lumped

inductance and capacitance, oscillations of the

electrical field variables with a certain frequen-

cy (= resonance frequency f 0) can also exist in a

cavity resonator. In this case, the energy is

stored alternately in the electrical and the mag-netic field. While an (ideal) resonant circuit has

only one resonance frequency, the cavity reso-

nator has an infinite number of oscillation types

and resonance frequencies. Fig 1.2 (left) shows

the electrical and magnetic fields for the oscil-

lation type with the lowest resonance frequen-

cy (TE 101 resonance) at three different points in

time, at intervals of one quarter of the period

T = 1 / f 0. A side view of the cavity resonator

and the variations of the electrical field strength

as a function of the coordinate z are reproduced

in the middle of Fig.1.2. The resonance fre-

quency of this oscillation type is calculated (for

air) according to the formula:

f

a s

02 2GHz

151

( /cm)

1

( /cm)= ⋅ +

' '(1.1)

More specific details on cavity resonators are

dealt with in Experiment Ex12.

The electromagnetic oscillations of the cavity reso-

nator are attenuated due to losses occurring in the

metal walls. After installing the Gunn element,which transforms DC power into microwave

power, just enough microwave power is fed into

the resonator to compensate for the wall losses and

to achieve a continuous unattenuated oscillation.

To obtain a microwave oscillator, the resonator

must also have an “opening” through which

power can be fed to a “load”. In this case, the

Gunn element must generate not only enough

microwave power to compensate for the wall

losses, but also the greater amount of power

Fig. 1.2: Basic design of the Gunn oscillator left: TE

101oscillation in a rectangular cavity

resonator middle: Dependency of the electrical fieldstrength E on the longitudinal coordinate z for TE 101 resonanceright: Two possible configuratios for the Gunnoscillators (1) Gunn diode (2) apertureConfiguration B is used in these experimentsand configuration A in MTS 7.4.6.

generally required by the load. Developing a

Gunn oscillator from a cavity resonator re-

quires (I) that the Gunn element be coupled to

the resonator, and (II) that the load be coupled

to the resonator. Fig. 1.2, right section, shows

two possible configurations.

In configuration (B), the Gunn element is cou-

pled to the resonator using a round metallic

post, and the load is coupled using an aperture

(hole or slot). From the sketch of the longitudi-

nal electrical field distribution in configuration

(B), we can see that the planes of the post axis

and the aperture may be regarded as short-cir-

cuit planes for the purpose of estimating theresonance frequency.

In configuration (A) (Fig. 1.2, top right) the me-

tallic post fullfils both functions, i.e. the cou-

pling of the resonator to the Gunn element as

well as to the load.

In the present experiment, an oscillator con-

structed according to configuration (B) is used.

In contrast in experiments from MTS 7.4.6 a

mechanically adjustable oscillator according to

configuration (A) is assembled.



1 3


Required equipment

1 Basic unit 737 021

1 Gunn oscillator 737 01

1 Fixed attenuator 737 095

1 Transition waveguide/coax 737 0351 Coax detector 737 03

1 Thumb screws (2 each) 737 399

Additionally required equipment

1 Oscilloscope 575 29

1 XY recorder (optional) 575 663

2 Stand bases 301 21

1 Support for waveguide components 737 15

1 Stand rod 0.25 m 301 26

2 Coax-cables with BNC/BNC

plugs, 2 m 501 022

1 Slide caliper

Experiment procedure1. Visually study the design of the disassem-

bled Gunn oscillator (see also the instruc-

tion sheet to the device).

1.1 Disassemble the Gunn oscillator by loosening

the quick-release thumb screws. (Disconnect

the back panel of the housing, diaphragm and

waveguide terminating piece)

1.2 Consider the design of the Gunn element and

compare it to Fig. 1.2 (right). Determine thewaveguide width a’ and the distance s’

[see Fig. 1.2 right, configuration (B)] of

the post axis to the flange plane (= location

of the diaphragm) using a slide caliper.

Note:

Make sure that you DO NOT touch the Gunn

diode with the slide caliper.

2. Determine the dependency of the DC cur-

rent I G flowing through the Gunn element

and (in the case of oscillation) the micro-

wave signal generated by the supply volt-age U G (when the diaphragm is attached).

2.1 Reassemble the Gunn oscillator dismantled in

part 1 of the experiment, i.e. reattach the

back panel of the housing and the waveguide

to the Gunn element module.

2.2 Screw the fixed attenuator and waveguide/

coax transition onto the open end of the

waveguide (Fig. 1.3).

The attenuator keeps the detector in

the square-law characteristic range (see

MTS 7.4.5 and Experiment 4 of this book)

and attenuates undesired reflections]

2.3 Set up the equipment configuration on the

lab bench using the stand rods.

2.4 Connect the basic unit (set the rotary knob for the Gunn diode supply voltage U G to “0”)

to the Gunn oscillator using the coaxial

cable.

2.5 Attach the coax-detector to the

waveguide/coax transition (making sure

here that the thread does not get jammed)

2.6 Connect the detector to the oscilloscope

(not contained in the measurement sta-

tion). Set the oscilloscope to DC and the

measurement range for the following

measurements to approx. 2 to 100 mV.2.7 Switch on the Gunn power supply unit. In-

crease the supply voltage U G from 0 to 10 V

in steps of 0.5 V. At the same time read off

the Gunn element's DC current I G and the

receiving voltage U D proportional to the rai-

ated microwave power and enter the results

in columns 2 and 3 of Table 1.1. Draw the

findings in Diagram 1.1. Connect the points

representing the measurements from Table

1.1 with straight lines.

Note:

If an XY recorder is available, the function I G = I G (U G) can also be recorded directly.

Fig. 1.3: Experiment setup



1 4


In this case the appropriately designated

sockets X and Y of the Gunn power sup-

ply of the basic unit are connected to the X

and Y inputs of the XY recorder. Likewise

the function U D(U G) can be recorded. TheY-input of the recorder must be connected

to the coax detector (Since U D is negative,

interchange the “+”- and “–” input).

For the recording of the characteristic the

Gunn voltage is to be increased slowly by

turning the 10-turn potentiometer. You

can also use a digital storage oscilloscope

instead of the XY recorder. In this case an

additional triangular function generator (0

to 10 V) is required for wobbling the char-

acteristics. A very low wobble frequencymust be selected here, because the RE-

with diaphragm

with rear panel

without diaphragm

with rear panel

with diaphragm

without rear panel

0.0

0.51.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.05.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

U G

mV

I G

mA

U D

mV

I G

mA

I G

mAmVV

CORDER outputs X and Y have lowpass

filters. The advantage of a continuous in-

crease of UG as opposed to a step-by-step

increase (0.5 V steps) lies in the fact that

the “steps” (discontinuities) in the charac-teristics are easier to discern.

3. Repeat part 2 of the experiment but with-

out the diaphragm.

3.1 To remove the diaphragm loosen the thumb

screws, take out the diaphragm and then re-

assemble the waveguide adapter (including

the waveguide/coax transition).

3.2 Repeat the experiments in accordance with

point 2.7. Enter the results for I G and U D in

columns 4 and 5 of Table 1.1

Table 1.1 a'= mm s'= mm

U D U D



1 5


2. Assuming that the active GaAs layer has a

thickness of 10 µm, determine from the value

for the threshold voltage U TH

(= voltage

above which the differential mobility be-

comes negative) the threshold value E TH of

the electrical field strength in kV/cm. The

voltage drop outside the active layer may be

ignored here, and you may assume a ho-

mogenous spatial distribution of the field

strength.

3. Assuming a domain velocity of 107 cm/s,

determine the transit frequency of the Gunn

element.

4. Explain the different responses obtained in

experiment parts 2 (with diaphragm and rear

panel), 3 (without diaphragm, with housing

rear panel) and 4 (with diaphragm, but

without rear panel).

4. Repeat part 2 of the experiment, but without

the rear panel.4.1 Reinsert the diaphragm. Remove the rear

panel.

4.2 Repeat the experiment according to point

2.7. Enter the results for I G and U D in col-

umns 6 and 7 of Table 1.1

Questions1. Calculate approximately the oscillator fre-

quency according to the Equation (1.1) speci-

fied above for the resonance frequency of

a rectangular cavity resonator. For this use

the geometrical data determined in experi-ment part 1.2. Here you may assume a

TE 101 resonance with “effective short-cir-

cuit planes” at the location of the post axis

and the diaphragm (see also Fig. 1.2,

lower right).

Diagram 1.1: For the graphic representation of the measurements values from table 1.1



1 6


Bibliography

Simple introductory works:

[1] F. Nibler u. a. : Hochfrequenzschaltungstechnik (Abschnitt 1.5). Expert-Verlag,

Sindelfingen 1984

[2] G. A. Acket, R. Tijburg und P. J. de Waard: Die Gunn-Diode.

Philips techn. Rdsch. 32, 394-404 (1971/72)

[3] J. Magarshack: Gunn-Effekt-Oszillatoren und -Verstärker.

Philips techn. Rdsch. 32, 424-431 (1971/72)

[4] Jansen: Entwurf von Gunn-Effekt Oszillatoren. Mikrowellen Magazin, 330-357 (1977)

Tiefergehende Darstellungen wissenschaftlichen Charakters:

[5] H.-G. Unger und W. Harth: Hochfrequenz-Halbleiterelektronik. Hirzel-Verlag,

Stuttgart 1972

[6] K. Kurokawa: Microwave Solid State Oscillator Circuits. In: M. Howes, D. Morgan (Eds.):

Microwave Devices, Device Circuit Interactions. Wiley, London 1976

[7] J. E. Carroll: Hot Electron Microwave Generators. Edward Arnold, London 1970

[8] F. Sterzer: Transferred Electron (Gunn) Amplifiers and Oscillators for Microwave Applications.

Proceedings IEEE, 59, 1155-1163 (1971)

[9] C. P. Jethwa und R. L., Gunshor: An Analytical Equivalent Circuit Representation

for Waveguide- Mounted Gunn Oscillators. IEEE Trans. Microwave Theory Tech.,

MTT-20, 565- 572 (1972)

[10] J. F. White: Simplified Theory for Post Coupling Gunn Diodes to Waveguides. IEEE Trans.Microwave Theory Tech., MTT-20, 372-378 (1972)



1 7

MTS 7.4.4 Modulation of the Gunn Oscillator

Direct modulation of the Gunn oscillator

PrinciplesIn order to investigate direct modulation of the

Gunn oscillator, you first have to delve into

microwave detection. The waveguide/coax

transition used in Experiment 1 converts the

propagating wave into a TEM wave. The coax-

ial detector serves to determine the electrical

field strength of this TEM wave. To do this

there is a miniature antenna structure located

inside the detector. This antenna supplies a

voltage which is proportional to the sought-af-

ter electrical field strength.The voltage u(t) being applied to the terminals

of the antenna represents a microwave signal

(here approx. 9 GHz), which cannot be directly

displayed on a conventional oscilloscope.

A detector diode is used to generate a cohesive

LF signal which coincides with the envelope of

the microwave signal. Schottky diodes are par-

ticularly well-suited for this. They essentially

consist of a metal-to-semiconductor junction.

The following holds true for the relationship be-

tween the detector voltage u(t) and the detector

current i(t) in a Schottky diode

i t I u t

U ( )

( )= ⋅

−

S

T

exp 1 (2.1)

where U T stands for the temperature voltage (20

up to 30 mV) and I S stands for the saturation cur-

rent.

For low modulation levels | u(t) | << U T the

above equation can be approximated using the

two first terms of Taylor's series

i t I

U u t I

u t

U ( ) ( )

( )= + ⋅

S

T

S

T

2

1

2(2.2)

A microwave voltage

u(t) = û · cos ( ω · t) (2.3)

yields the following direct current based on the

square component of the above expression

I I

U uDetect S

T2

214

= ⋅ ⋅ ( )ˆ (2.4)

Due to the fact that û is, on the other hand, pro-

portional to the amplitude Ê of the electrical

field strength at the location of the probe, the

coax detector consisting of a dipole antenna and

detector diode supplies an output signal U D,

which for “sufficiently small” field strengths is

proportional to the square of the amplitude of the

electrical field:

U D = k · Ê 2

(2.5)

Here k is a constant with the dimensionm 2

V

The gain and measurement of small DC detec-

tor voltages is complicated by, among other

things, drift phenomena. For that reason it is

expedient to have a low-frequency AC signal

available at the output of the coax detector in-

stead of the DC voltage signal. This is made pos-

sible through amplitude modulation of the micro-

wave oscillator (in this case the Gunn oscillator).

Amplitude modulation of the Gunn oscillator's

microwave signal can be achieved by means of

a time-variable supply voltage. This becomes

clear from the results from Experiment 1 (Dia-

gram 1.1). There are no microwave oscillations

up to a supply voltage of approx. 4 V. Above

this threshold the amplitude of the microwave

signal is severely dependent on the supply volt-

age. Thus the amplitude of the microwave sig-

nal can be “modulated” as a function of time by

controlling the bias level U G of the Gunn oscil-

lator.

Required equipment 1 Basic unit 737 0211 Gunn oscillator 737 01


1 Transition waveguide/coax 737 035

1 Coax detector 737 03

1 Set of thumb screws (2 each) 737 399

Additionally required equipment 1 Oscilloscope 575 29


1 Support for waveguide

components 737 15

1 Stand rod 0.25 m 301 262 Coaxial cables with BNC/BNC

plugs, 2 m 501 022



1 8

MTS 7.4.4 Modulation of the Gunn Oscillator

Fig. 2.1: Basic characteristic of the Gunn voltage on the basic unit (GUNN output socket) for Gunn-internal modulation. U G represents here thesupply voltage, which is set via the rotary potentiometer U G.

Experiment procedure

1. Amplitude modulation of the Gunn oscilla-tor

1.1 Experiment setup as in Experiment 1 andspecified in Fig. 1.3.

1.2 In the “MODULATION” section of the

basic unit there is a switch setting (“GUNN

INT”) for the internal modulation of the

Gunn supply voltage. The Gunn voltage is

then superpositioned onto the square-wave

signal with the frequency 976 Hz. The am-

plitude of the signal has a value of 300 mVPP

(see Fig. 2.1).

1.3 Consider the receiving signal U D(t) of the

detector for an amplitude-modulated micro-

wave signal when changing the supply volt-age U G.

For this first connect the Gunn oscillator to

the basic unit (GUNN sockets) and set the

toggle switch to “GUNN INT”. Connect

the detector to the input of the oscillo-

scope (DC setting first).

1.4 Slowly turn the control knob for the Gunn

voltage from left to right (increasing the

supply voltage) and observe how the char-

acteristic of U D(t) (square-wave shaped

signal with DC component) changes.

1.5 Now set the oscilloscope to AC mode and

repeat point 1.4.

Questions1. What can be observed in subpoints 1.4

and 1.5? How do you explain this re-

sponse?



1 9

MTS 7.4.4 Amplifier

The selective measurement amplifier

PrinciplesFigure 3.1 shows the dynamic characteristic

U D(t) of the output signal of the E-field probe. It

results from the superposition of the square-

wave shaped “wanted signal” of the frequency

f r (here f r = 976 Hz) with the noise signal U n(t ).

If the rms value

u U t n,rms n2= ( ) (3.1)

of the noise signal is larger than the amplitude of

the voltage step ∆U = U max – U min to be deter-mined, than this determination is made consid-

erably more complicated, if not impossible

should no additional measures be taken on the

signal processing side. These considerations

lead to the sensitivity limits of the measurement

method.

In order to recognize which measures lead to an

increase of the sensitivity (lowering of the sen-

sitivity limits), it is advantageous to consider

the frequency spectrum of the signal U D(t). Fig-

ure 3.2 (above) depicts this frequency spectrum,

which consists of spectral lines at the frequencies

f r , 3 f r , 5 f r etc. belonging to the wanted periodic

signal and a continuous noise spectrum. The first

spectral line at f r belongs to the fundamental fre-

quency component.

U t u f t D rcos 2( ) ˆ ( )= ⋅ π (3.2)

whereby û is definitely related to ∆U via

û = 2 · ∆∆∆∆∆U /π π π π π (Fourier analysis).

If the received signal U D(t) is sent through a

narrow bandpass filter (see Fig. 3.2, middle)with the center frequency of f 0 ≈ f r , then the

first spectral line and thus the fundamental fre-

quency component U D(t) is almost completely

retained. However, only a “small” part of the

noise spectrum remains which is determined by

the effective bandwidth of the filter. The signal-

to-noise-ratio is considerably increased by

means of this form of selective frequency filter-

ing. And this increase is even greater the nar-

rower the bandwidth of the filter. In the

corresponding dynamic signal characteristic

obtained downstream from the bandpass filter,

the fundamental frequency component remainsnearly the same in comparison to the input sig-

nal, but the rms value of the noise signal has

been reduced in proportion to the bandwidth of

the filter. Based on the principle expounded

upon until now you could replace the bandpass

filter with a narrow-band low-niose amplifier

and supply its output signal to an AC voltme-

ter. Thus, a frequency selective measurement

amplifier is obtained; a principle often applied

in the measurement of small signals.

However, if a still greater increase in the sensi-

tivity is desired, then we must also make it clear

that there is a limit to the reduction of the filter’s

Fig. 3.2: Frequency spectrum of the signals according toFig. 3.1 upstream (above) and downstream

(below) from a narrow-band bandpass filter.1 Wanted signal2 Noise3 Frequency response of the filter

Fig. 3.1: Dynamic characteristic of the output signal of theE-field probe with superpositioned noise voltage

( f r = 976 Hz)

umin

umax

U D(t)

∆U



2 0

MTS 7.4.4 Amplifier

Fig. 3.3 Design of the lock-in amplifier 1 Clock generator 2 Bandpass filter for suppesing the

harmonics3 Amplifier 4 Phase-sensitive rectifier

(Synchronuos rectifier)

5 Low-pass filter

Fig. 3.4: Voltage controlled switch as phase-sensitiverectifier (PSD). Switch set to position A, if voltage U Cl(t) of clock generator is smaller thanthreshold value U 0. Otherwise in setting B.

for the case that U 1(t) is in phase with the clock

signal U CL(t), is shown in Fig. 3.5 (right). If a

very narrow-band low-pass filter (e.g. band-

width 2 Hz) is arranged behind the PSD, this

filter supplies the mean value u2 [DC voltage

component, see Fig. 3.5 (right)] to its output,

which is clearly in conjunction with û1. In order

to conduct a more exact analysis of the lock-in

amplifier features, you can observe its “re-

sponse” to an input signal of any given fre-

quency f and phase ϕ in accordance with the

equation

U t u f f t D cos 2( ) = ( ) ⋅ +( )ˆ π ϕ (3.3)

Only if f equals f r or a multiple of f r in whole

numbers, is there a DC voltage value u2 other

than zero at the output. A very high frequency

bandwidth using the principle dealt with up to

now. The narrower the bandwidth of the filter

is, the better its center frequency f 0 must coin-

cide with the clock frequency f r , so that the

wanted signal is not significantly attenuated by

the filter. Due to drift phenomena in the filter,

e.g. of a thermal nature or due to fluctuations in

the clock frequency, deviations between f 0 and

f r must be tolerated to a certain extent. Therefore,

the bandwidth of the filter cannot be reduced to

arbitrarily low values.

A solution to this problem can be found when

the clock signal, on which the voltage U D(t) to

be measured is based, is available and can thus

be used for the “synchronization” ( f 0 joined

with f r ) of the bandpass filter.

This basic idea is utilized in the so-called “lock-

in” amplifiers, the fundamental principle of

which is explained in the following paragraphs.

First a simple bandpass filter (with no extreme

demands on the bandwidth) eliminates the re-ceived signal U D(t) from harmonics of a higher

mode and spectral noise components and then

the filtered signal can be amplified in a low-

noise, narrow-bandwidth amplifier. The result-

ing signal U 1(t) (downstream from the

amplifier (3), see Fig. 3.3) is supplied together

with the signal U CL(t) of the clock generator to a

phase sensitive detector (PSD), sometimes

called a synchronous detector. Figure 3.4

shows a simple design for a PSD. Here the sig-

nal of the clock generator controls a switch so

that the output signal is alternatively identical to

the input signal U 1(t) or the negative input sig-

nal – U 1(t). The resulting output voltage U 2(t)

Fig. 3.5: Voltage characteristics of the single phase-sensitive rectifier Clock generator voltage U Cl(t) (above),Input voltage U 1(t) (bottom left),Output voltage U 2(t) (bottom right)



2 1

MTS 7.4.4 Amplifier

selectivity is achieved by selecting a very nar-

row bandwidth for the low-pass filter (= pro-

longed duration of the integration time). This

yields a considerable improvement in the sig-

nal-to-noise ratio (see above). Thus, extremelyweak signals can be detected in measurement

systems which are based on the lock-in ampli-

fier principle.

If it is true that f = f r , but ϕ ≠ 0 (phase-shift be-

tween the clock and receiving signal), then

uu

2 = ⋅2

cos1ˆ

π ϕ (3.4)

Thus, we obtain not only frequency- but also

phase-selectivity.

In the lock-in amplifier version considered up

until now the receiving signal U D(t) appears inits baseband. Such systems are referred to as

homodyne and is the type of system integrated

into the basic unit of the existing training sys-

tem. It is designated here (equipment designa-

tion) as “SWR Meter”. The designation SWR

meter (“standing wave ratio”) comes from its

frequent use in determining the standing wave

ratio on transmission lines.

In improved systems of greater complexity the

received signal is first converted into a (fixed)

intermediate frequency (IF) and supplied to thePSD (Heterodyne System).

Required equipment



1 Waveguide 200 mm 737 12




Additionally required equipment 1 Oscilloscope 575 29


3 Supports for waveguide

components 737 15

2 Stand rods 0.25 m 301 26

2 Coax cables with BNC/BNC

plugs, 2 m 501 022


1. Operation of the selective measurement

amplifier

1.1 Set up the experiment as specified in Fig.

3.6. Position the two waveguide ends op-

posite each other at a distance of approx.


4 cm (The waveguide axes are in perfect

alignment without transverse shift).

1.2 Connect the coax-detector to the input of

the SWR meter. Set the toggle switch for

modulation to “GUN-INT”. Connect the

Gunn oscillator to the GUNN socket of the

basic unit.

1.3 Presetting of the selective measurementamplifier:

Set the gain selection switch of the selec-

tive measurement amplifier (SWR meter)

to “0 dB”.

Turn the GAIN ZERO to far left stop.

1.4 Set U G for maximum pointer deflection on

the SWR meter (see Experiment 2).

(Note: in this experiment the spectral mode

distribution is not decisive, in this context

see also “Design of the Microwave

Source” in the preface). In the process

vary the gain using the selection switchuntil the display is in the range from 0 to

5 dB. Afterwards set the SWR meter to

maximum pointer deflection by setting the

ZERO control knob to “0 dB”.

2. Transverse shift of the waveguide end in

0.5 cm steps.

2.1 As shown in Fig. 3.7 shift the end of the

waveguide in a transverse manner in ac-

cordance with the values given in Table

3.1 and read off the respective values (in

dB) displayed on the selective measure-

ment amplifier and enter these measured



2 2

MTS 7.4.4 Amplifier

values into column 2 of the table. During

this measurement select an appropiate set-

ting for the gain factor of the selective

measurement amplifier and take this into

account in the result.2.2 Now connect the detector to the oscillo-

scope (AC setting) and repeat experiment

point 2.1. This time determine the voltage

range ∆U ( x0) = U max( x0) – U min( x0) of the

receiving signal and enter it into column 3

of the Table.

Note: If you have a BNC-T adapter (cat.

no. 501 091) and an additional RF cable

(cat. no. 501 022) at your disposal, you

can do experiment points 2.1 and 2.2 in

parallel.3. Demonstrating the sensitivity gain when

using the selective measurement amplifier

3.1 Move one of the waveguides until noise

makes it impossible to recognize the

square-wave signal on the oscilloscope.

Covering one waveguide aperture with

your hand should now have no effect on

the signal.

3.2 While maintaining the transverse shift as it

is, now connect the detector to the meas-

urement amplifier (INPUT socket) and setthe display of the SWR meter to a value

between –5dB und 0 dB by selecting a

suitable gain factor. Now cover one

waveguide aperture with your hand and

observe the display.

Questions1. Based on column 3 of the table (∆U ) deter-

mine the respective values

Fig. 3.7: Transverse shift of the waveguide axes (viewfrom above)

∆U x 0( )

mV

a

dB10 log

(0)

0⋅

∆

∆

U x

U

( )

0 0.0 0.0

0.5

1

1.5

2

2.5

3

x 0

cm

Table 3.1

10 log(0)

10 log(0) (0)

0 max 0 min 0

max min

⋅ = ⋅−

−

∆

∆

U x

U

U x U x

U U

( ) ( ) ( )

(3.1)

for the given transverse distances x0 and

enter these into column 4.

2. Compare the findings in columns 2 and 3

of the table.

Comments:

• If under point 2.2 you are already unableto determine ∆U ( x0) in the range around

x0= 0 up to 2 cm (due to too much noise),

reduce the longitudinal distance from

4 cm to a smaller value, and repeat the en-

tire experiment.

• However, if you have too high a signal

level you can also increase the longitudi-

nal distance and repeat the entire experi-

ment.



2 3

MTS 7.4.4 Attenunators

AttenuatorsCalibrating the attenuator, measuring relative microwave power

side, or slide the attenuator vane into the field

from the side. The latter possibility is imple-

mented in the attenuator 737 09. The shift

movement of the attenuator vane is performed

backlash-free using a micrometer screw. In or-

der to attain low insertion loss the vane is insert-

ed in such a manner that the absorbing layer

lies against the waveguide wall ( E = 0) for a

shift of x = 0 mm. Because of scarfing the re-

flection coefficient caused by the vane is kept

as small as possible. The attenuating vane itself

can consist of coated fiber glass, mica or plas-tics like mylar or Kapton. Nickle-chrome or

tantalum alloys are deposited as layered coat-

ings in vacuum metallization processes. These

metal layers have inherent absorbtion properties

(as opposed to reflecting ones) as long as their

layer thicknesses are smaller than the penetrat-

ing depth δ of the electrical field at the desired

operating wavelengths. Resistance film cards

are manufactured with varying resistance per

square unit (surface resistance). Standard values

lie between 50 and 377 Ω/square (Square means

that the resistance is the same for any random

square surface and is given a specific value. Thus

the resistance is dependent on the geometry, not

on the surface area). The surface resistance in an

attenuator in conjunction with the field distribu-

tion of the mode in question (often TE 10) deter-

mines the attenuation characteristic as a function

of the shift x. Fig. 4.2 contains a cross-section

through the adjustable attenuator 737 09.

The calibration of the attenuator is performed in

the experiment using the power scale of the

SWR meter. The microwave power is con-

Definition and propertiesAttenuators are among the linear, reciprocal

components of electrical lines (four-pole). They

are frequently realized like reflection-free

waveguide terminals in the form of dissipating

resistances. As such the operating principle

comprises the transformation of RF power into

thermal energy. With the exception of a few

high-load attenuators reciprocity always exists.

Only in the former is the input designed to take

more power than the output, which is why any

interchanging of the gates is not permitted. A dis-tinction is drawn between fixed attenuators and

variable attenuators. Variable attenuators can be

adjusted mechanically or manufactured with

electronically controllable line components. The

electronic attenuators are designed using, for ex-

ample, PIN diodes. The PIN diode is used here

as an electrically controllable resistance for mi-

crowaves. For that reason a PIN diode attenua-

tor has variously high transmission and reflection

coefficients depending on the control voltage.

The function of the PIN diode is explained in

greater detail in Experiment Ex1 “Principle of

the PIN modulator” in MTS 7.4.5, for that rea-

son we will only focus here on passive attenuat-

ing elements. Fig. 4.1 presents 2 conventional

principles for the assembly of mechanically tune-

able attenuators.

The basic idea involves inserting an absorbing

medium into the waveguide. The rectangular

waveguide depicted in Fig. 4.1 guides the funda-

mental mode (TE 10). You can either insert a

vane attenuator into the waveguide through a

middle slot along the waveguide's longitudinal

Fig. 4.1: On the principle of attenuation by means of insertion with an attenuating vane

Fig. 4.2: Cross-section through the attenuator 737 09



2 4


verted into a LF signal with the coax detector.

As long as the coax detector is operated in the

range of square-law characteristic there is a

proportional relationship between its output

signal and the incidenting microwave power. A

correct reading of the dB scale is only possible

under these conditions. Generally speakingwhen it comes to detectors, low power levels

are the precondition for the square law charac-

teristic range. Fig. 4.3 shows the principle char-

acteristic curve of the output signal versus the

microwave power.

Required equipment




1 Variable attenuator 737 09







components 737 15

1 Stand rod 0.25 m 301 26

2 Coaxial cable with BNC/BNC

plugs, 2 m 501 022

Recommended

1 PIN modulator 737 05

1 Isolator 737 06

Experiment procedure Note:

When using the PIN modulator and isolator

complete the experiment setup as specified in

Fig. 0.5 (Preface).

1. Calibration of the attenuator

– Experiment setup as specified in Fig. 4.4

Note:

The fixed attenuator is required to attenu-

ate the microwave signal present at the

coax detector by approx. 10 dB. This is

how the detector is supposed to be operat-

ed in its square-law characteristics range.

– Connect the coax detector to the SWR re-

ceiver “INPUT”.

Modulate the microwave signal (generally

performed by means of direct modulationof the Gunn oscillator). Set the variable

attenuator to x = 0.00 mm. Calibrate the

display a of the homodyne SWR meter to

Fig. 4.3: Detector output voltage as a function of themicrowave power

1 Saturation range2 Linear range3 Quadratic range4 Noise




2 5


0 dB using the gain selection switch V/dB

and the ZERO control knob. Do not make

further changes to the ZERO control knob

in the course of the experiment.

First check the validity of the square law(the detector operates in the linear range).

Now increase the attenuation to 3 dB by

turning the micrometer screw (set x to

value specified on the attenuator). The dis-

play a of the power meter should now lie

in the range from –2.5 dB to –3.5 dB. You

will hardly be able to reach the ideal value

of –3 dB due to the unavoidably large dis-

persion of the detector's microwave diode.

– Reset the attenuator to x = 0.00 mm. Now

set the attenuation to the values specifiedin the table. Enter the measured values x

into Table 4.1 and plot the dependency

|a (x)| in a graph.

Questions1. Which other components can be used to

reduce the microwave power to avoid

overloading the detector?

Table 4.1: Calibration of the attenuator

a / dB x / mm

0 0.00

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

–12

–14

–16

–18

–20

Diagram 4.1: Plot of the values from Table 4.1



2 6




2 7

MTS 7.4.4 The Measuring Line

The slotted measuring lineSampling the field in front of a short-circuit plate and a waveguide termination

PrinciplesVarious line types for electromagnetic waves

and their technical function

Lines fulfill various functions in radio fre-

quency technology:

(a) They serve in the transmission of radio fre-

quency signals between remote locations.

Examples: radio link between the transmit-

ter and a remote antenna system, broadband

cable for signal distribution of a satellite re-

ceiver system to the individual subscribers

and underwater cables. Transmission linesand directional radio links in free space are

frequently considered potential alternatives

in this function as a medium for communi-

cations transmission

(b) They also serve as circuit elements (in-

stead of capacitors and inductors) and in-

terconnecting lines for the realization of

passive microwave circuits (e.g. transmis-

sion-line filters).

In general, transmission lines can be consid-

ered as structures for guiding electromagnetic

waves. For this function a multitude of various

transmission line types are suitable.

The coaxial and two-wire line depicted in Fig.

5.1a) guides transverse electromagnetic waves

(TEM waves) and can also be used for any ar-

bitrarily low frequency.

In other transmission line types, however, the

wave propagation is dependent on the condi-

tion that the cross-sectional dimensions of the

line are at least about as large as a half free-

space wavelength (λ 0/2). This leads to the result

that in the range of frequencies above approx.1 GHz there are a larger number of transmis-

sion line types available than for low frequen-

cies. This is because it is only at the higher

frequencies where the dimension λ 0/2 results in

“handy” cross-sections.

Figure 5.1 b) shows various planar transmission

line types, which are particularly well-suited for

the design of microwave integrated circuits

(MIC).

Transmission lines can be realized without any

metal conductors, see Fig. 5.1 d). The dielectric

Fig. 5.1: Various line types for electromagnetic waves(1 = metal conductor, 2 = isolator). Numbering from left to right.(a) Coaxial and two-wire lines(b) Planar line structures:

Microstrip line,coplanar line, slotted line

(c) Waveguide: Rectangular, circular andridged waveguides

(d) Dielectric waveguides: Round andrectangular-shaped surface waveguide;optical waveguide

waveguides shown there are based on the prin-

ciple of total reflection of electromagnetic

waves at the boundary surfaces between the

insulators with “higher” to “lower” relative per-

mittivity ε r .

Necessary fundamentals drawn from elemtary

transmission line theoryFor the description of line-bound wave propa-

gation we can begin our investigation with a



2 8


simple two-wire line on which the voltage and

current can be specified at any given location.

Subsequently the results can be generally ap-

plied as a model for any other homogenous

transmission line.Fig. 5.2 shows a homogenous transmission line

structure represented symbolically by two dou-

ble lines. The electromagnetic state on this line

can be expressed by specifying the spatial and

time dependency of the instantaneous voltage

u(x,t) and current i (x,t).

First we will consider the case in which a single

wave propagates on the line in the +x direction.

The following holds true with ω = 2π f as the an-

gular frequency and λ g as the guided wavelength

(“ g ” = guide) and where the attenuation on theline is disregarded

u x t û t x

, cos( ) = ⋅ ⋅ −

ω π λ

2

g

(5.1)

The phase velocity resulting here is

v f ph

g

g=⋅

= ⋅ω λ

π λ

2(5.2)

If Z 0 expresses the line's characteristic imped-

ance, then the following holds true for the corre-

sponding voltage u (x,t) and current i (x, t)

i x t u x t

Z ,

,( ) =

( )

0

(5.3)

In the description of time-harmonic processes

you can also make use of complex amplitudes.

As such an AC voltage of the form

u (t) = û cos (ω t + φ ) can be expressed by a cor-responding complex amplitude U = û exp ( jφ )and the following applies

u(t) = Re U · exp ( j ω t) (5.4)

Complex numbers are now specially marked

by an underline. Equations (5.1) and (5.3) are

given complex numbers with the form

U x U x j x( ) = ( ) ⋅ ( )[ ]exp φ u (5.5)

where U x û( ) =

Fig. 5.2: Wave propagation on a line(a) Spatial dependency of the voltage u (x,t)

at two different time points(b) Spatial dependency of the current i (x,t)

at two different time points(c) Spatial dependency of the magnitude |U (x)|

and phase Φ u (x) for the complex voltageamplitude (phasor)

(d) Spatial dependency of the magnitude |I(x)|and phase Φ I (x) for the complex currentamplitude (phasor).

(a)

(b)

(c)

(d)2p

Z 0

t= t d

i(x,t)

û

û0

0

p

p

2p f

u(x)

f I(x)

û

d t

t= t d

t = 0

t = 0

Z 0

i(x,t)

|U(x)|

|U(x)|

|I(x)|

u(x,t)

u(x,t)

x

x

2

l g

û Z

0

and φ π λ u

g

( ) xx

ß x= − = − ⋅2 (5.6)

and as such the following is true for the current

I x I x j x( ) = ( ) ⋅ ( )[ ]exp φ I (5.7)

where

I xû

Z x x ß x( ) = ( ) = ( ) = − ⋅

0

und I uφ φ

(5.8)

Figures 5.2 c) and d) show the spatial depend-

ency of the magnitudes |U (x)| and |I (x)| as well

as the phases Φ u (x) and Φ I (x) of the complex

voltage and current amplitude. In equations

(5.6) and (5.8) the following phase constant has been introduced



2 9


ßv

= =2

g ph

π λ

ω (5.9)

as a second parameter to express line-boundwave propagation.

Description of the field in front of a short-

circuit

If there is a short at the end of the transmission

line ( x = 0) (see Fig. 5.3, above), the wave can

be described from the superposition of one

component travelling to the load u+(x, t) and

one reflected component travelling back from

the load u –

(x, t).

As the total voltage at x = 0 must be identical to

zero (short-circuit!), it follows that:

u (x, t) = u+ (x, t) + u – (x, t)

= û cos ( ω t – ßx) –û cos( ω t + ßx)

= 2 · û · sin ( ω t) · sin (ßx) (5.10)

The current of the reflected wave is related to

u – by virtue of ( –Z 0) and thus it follows (see Fig.

5.3 b) that

i x t û Z

t x( , ) cos= ⋅ ⋅ ⋅( ) ⋅ ⋅( )20

cos ω β

(5.11)

The current and voltage now have distinct re-

sponses with respect to time and space. We are

dealing with a standing wave [see Fig. 5.3 c)

and d)]. The nodes and maxima of the u and i

characteristic are at locations unchanged with

respect to time. The nodes of the voltage are

shifted by λ g/4 with respect to the nodes of

the current. Moreover, there is a phase shift of

+ π /2 between the voltage and current. Thismeans that u exhibits its extreme values exactly

when i (for each x) is zero and vice versa. If

these relationships are represented for the mag-

nitude and the phase of the complex amplitude,

then we obtain the results depicted in Figures

5.3 e) and f).

Wave propagation in a rectangular

waveguide

Generally a waveguide is a “hollow tube” inwhich electromagnetic waves can propagate. A

Fig. 5.3: Standing wave on a line shorted on one end:a) Voltage characteristic u+(x, t) of the wave

travelling to the load and u – (x, t) of thereflected wave travelling back at the time point t = T /8.

b) i+(x,t) and i – (x,t) at t = T /8.c) Characteristic of the total voltage

u (x, t) = u+ (x, t) + u – (x, t)at 4 different points in time

d) i (x, t) = i+ (x, t) + i – (x, t)e) Spatial dependency of the magnitude | U (x)|

and phase Φ u (x) of the complexvoltage amplitude

f) Spatial dependency of the magnitude |Ι (x)| and phase Φ I(x) of the complex currentamplitude.

rectangular waveguide as shown in Figure 5.4is a special form of waveguide. This rectangu-

lar waveguide is the object of a series of experi-

ments and should therefore be considered more

closely in theoretical terms. This is also be-

cause many of the results found for rectangular

waveguides also apply to other forms of

waveguides (e.g. circular waveguide).

If you consider the zig-zag reflection of a plane

uniform wave between the side surfaces (dis-

tance a) of a waveguide, you obtain a critical

frequency called the cut-off frequency, as of

which wave propagation becomes possible



3 0


f c c

ac

c,0

= =λ 2

(5.12)

For the phase velocity a relationship is yielded

υ

λ ph

0

2

12

=

−

c

a

(5.13)

A calculation of the guided wavelength λ g and

the phase constant ß amounts to

λ λ

λ

g

ph 0

0

2

12

= =

−

v

f a

(5.14)

ßa

= = −

2 2π

λ

π

λ

λ

g 0

0

2

12

(5.15)

For more information regarding the derivation

of these relationships we wish to refer to the

specified sources in the bibliography.

Another approach used for describing propa-gation originates from the solution of Maxwell's

equations with the boundary conditions exist-

ing at the metal surfaces taken into account. But

do not be alarmed. Here only the result of this

analysis is reproduced for the fundamental

mode (TE 10):

The electrical field has only one Cartesian com-

ponent. It points in the y-direction and so it fol-

lows that:

E x za b

U a

x ey jßz, sin( ) =

⋅⋅ ⋅

⋅ −2

0π

(5.16)

It is evident that E y is not only dependent on the

propagation coordinate z , but also on the trans-

verse coordinate x. As the tangential compo-

nents of the electrical field must vanish at the

surface of (ideal) conductors, it is true that

E y (x = 0 , z) = E y (x = a, z) = 0.

Fig. 5.4: Rectangular waveguide

The field strength is maximum in the middle of

the waveguide at x = a/2, and it holds true that

E a b

U y 02, max = ⋅ ⋅

whereby U 0 constitutes an arbitrarily intro-

duced voltage.

Its corresponding magnetic field has both a

transverse component H x( x, z ) as well as a lon-

gitudinal component H z( x, t ).

The transverse component H x has the same

phase as E y and also the same spatial depend-

ency, and so it follows that:

H x zß

abU

a x ex

0

0 jßz2

sin( , ) = − ⋅ ⋅ ⋅

−

ωµ

π

(5.17)

At every cross-sectional point z = const. the ra-

tio of E y to (–Hx) is equal and given by the char-

acteristic impedance

Z E x z

H x z ß

a

0

y

x

0

1

=−( )

= =⋅

−⋅

( , )

( , )

ωµ π

λ

120

02

2

(5.18)

The following applies for the longitudinal com-

ponent H z:

H x z ja ab

U a

x e zz

00

jß1 2cos ,( ) = ⋅

⋅ −π

ωµ π

(5.19)



3 1


E-plane

E-PlaneT-plane

.. H →

–E

λ g

2

(a)

(b)

.. H →

I

Fig. 5.6: Existence zones for various wavetypes capableof propagating in the rectangular waveguide(a = 2.25 b).

T-plane

III

II

I

Fig. 5.5: Field pattern of the fundamental mode(TE 10-wave) in the rectangular waveguide(a) E and T fields(b) Current density distribution on the

metal walls

Thus it is phase-shifted by π /2 with respect to

the transverse components of E →

and H →

. In con-

trast to these it assumes at x = 0 and x = a its

maximum value (in terms of magnitude) at the

metal sidewalls. The spatial dependencies of

the field components reproduced in Equations

(5.16) to (5.19) are shown in Fig. 5.5. The wave

type under consideration till now is also re-ferred to as “TE 10-mode”. Here the TE stands

for transverse electric and “1” means that the

number of halfwaves in the x-direction is 1

while the index 0 means that the field is con-

stant in the y-direction. At higher frequencies

higher wave modes, namely TE mn-waves and

TM mn-waves are capable of propagation.

According to Fig. 5.6 the TE 20 mode is capable

of propagation starting from a frequency of

f c2 = 2 f c,TE10, when there is a side ratio of

a/b = 2.25 so that a “frequency octave” is avail-

able for single-mode operation.

Fig. 5.7: Principle of the slotted measuring line (explodedview)1 Slit in the waveguide2 Field probe = short rod-type antenna

3 Detector diode

Slotted measuring line

As will be demonstrated with particular care in

experiment 6, a reflection at the end of thetransmission line has the effect that maxima

and minima are formed in the spatial distribu-

tion of the field strength along the line. Based

on the ratio of the amplitude values (maximum/

minimum) and the locations of the maxima and

minima you can draw conclusions as to the

magnitude and phase of the reflection coeffi-

cient.

If to this purpose you wish to measure the dis-

tribution of the field strength along the trans-

mission line (slotted line), the following must

be taken into consideration:(a) You should interfere as little as possible

with the electromagnetic field in the

waveguide. This requirement is met, if a

“narrow” slit is added in the center of the

wide section of the waveguide (see

Fig. 5.7 and compare to Fig. 5.5 below).



3 2



(b) The characteristic impedance Z 0 (see

Equation [5.19]) is slightly altered by this

modification. This can be compensated for

by slightly increasing the waveguidewidth a in the region of the slit.

(c) A short probe (“electrical dipole”) as

shown in Fig. 5.7 supplies a voltage to its

output which is proportional to the trans-

verse component | E y| of the electrical field

strength. Thus behind the detector probe

(square-law rectification) you obtain a

voltage

U K E yD = ⋅2

(5.20)

Here K is a constant with the dimension

m 2

V

Low reflection waveguide termination

By inserting a wedge-shaped absorber material

the power of the incident wave is nearly com-

pletely absorbed thus suppressing any reflec-

tion almost totally.

Required equipment



1 Diaphragm with slit

2 x 15 mm 90° 737 22

1 Slotted measuring line 737 111


1 Short-circuit plate, from

accessories 737 29

1 Waveguide termination 737 14



1 Oscilloscope (optional) 575 29

1 XY recorder (optional) 575 6632 Stand bases 301 21


components 737 15

1 Stand rod 0.25 m 301 26

2 Coaxial cables with BNC/BNC

plugs, 2 m 501 022

Recommended


1 Isolator 737 06



3 3



If you are using a PIN modulator and isolator

the experiment setup in Fig. 5.8 is supple-

mented as explained in the preface.

1. Set up the experiment arrangement in ac-

cordance with Fig. 5.8 (perhaps modify

according to Fig. 0.5)

Note:

The diaphragm slit serves as a frequency-

selective component (filter) to improve the

spectral purity of the guided wave.

2. Measurement with short-circuit plate

2.1 Attach the short-circuit plate at the open

end of the slotted measuring line.2.2 Set the range switch V/dB of the SWR me-

ter to the most insensitive range.

Set the Gunn voltage to between 8 V and

9 V (Tip: first increase it to 10 V and then

adjust it back to the desired value), set the

modulation switch to GUNN-INT.

2.3 Place the slotted measuring line probe at

one end position. Now slowly push theslide to the other end position and at the

same time search for the maximum detec-

tor voltage (always adjusting the gain to a

suitable level). Calibrate the maximum

value to “0” dB using the “ZERO” con-

troller. The maximum position is desig-

nated x0 (read off the scale and note down)

and functions as the reference position.

2.4 Now shift the position of the probe in

2 mm steps (in the direction of the end po-

sition which is farthest removed from x0),in other words to the corresponding posi-

tions | xn – x0| = n · 2 mm. Enter the values

measured for the detector voltage into the

second column of Table 5.1.

= ⋅−( )

200

log

max

U x x

U

U x x

U

−( )0

max

cos2

0

π

λ g⋅ −

x x

Display in dB

02

4

6

8

10

12

14

16

18

20

22

24

26

28

Minimum at

Table 5.1

Probe position

x x− 0 in mm

Distance of the Minima

∆ x / mm=_________ λ g=_________ mm



3 4


You can conclude the measurement after

exceeding the position of the 2nd maxi-

mum (Display/dB ≈ 0).

2.5 Determine the distance ∆ x/mm between

two minima and enter the result in the last

line of Table 5.1. (Tip: By successively

increasing the gain (V/dB) in the minima,

you can determine the positions more pre-

cisely).

3. Measurement with reflection-free

waveguide termination.3.1 The short-circuit plate is replaced by the

“reflection-free” waveguide termination.

3.2 Slide the probe along the slotted measur-

ing line over the entire range and note

down by which value (in dB) there is de-

viation in the characteristic, i.e. by which

value the level (∆a) has dropped in com-

parison to the short-circuit experiment.

3.3 As in point 2.3 move the probe over the

entire range and search for the location of

the maximum detector voltage and againre-calibrate this value to 0 dB.

Display in dB

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

Probe position

x x− 0 in mm = ⋅−

( )200

logmax

U x x

U

U x x

U

−( )0

max

Table 5.2

3.4 Position the probe in 2-mm steps in the

same locations as in Experiment 2.4 (see

also Table 5.1), and enter the display val-

ues in Table 5.2.

Note:If you have an XY recorder, a digital stor-

age oscilloscope or a CASSY interface at

your disposal you can also directly record

the measurement curve. For this purpose

the slotted measuring line is equipped with

an integrated displacement sensor.

Ž Connect the IN socket of the slotted meas-uring line to the X output (supplying |U G|)

of the basic unit (or with an external power

supply with 10 V DC).

Ž Connect the X socket of the slotted meas-

uring line to the X input of the XY re-

corder (oscilloscope or CASSY).

Ž Connect the Y input of the XY recorder

(oscilloscope or CASSY) to the AMP-OUT

socket of the basic unit.

The AMP-OUT socket supplies a linear outputsignal whereby 0 V is about –20 dB and

approx. 4.5 V corresponds to 0 dB.



3 5


Furthermore, it is important to point out that the

integrated displacement sensor is not com-

pletely backlash-free (it does possess hyster-

esis), for that reason the curve should only be

plotted once in one direction.

Questions1. Based on the distance between two

neighboring zero points (distance to the

minima) ∆ x/mm (see Table 5.1) determine

the wavelength λ g of the guided wave in

mm.

2. Using the value for λ g determined above

under point 1 with the waveguide width of

a = 22.9 mm taken into consideration, de-termine the free-space wavelength λ 0 and

frequency f of the guided wave.

3. What is the mathematical value resulting

for the phase velocity v ph and the phase

constant ß according to the Equations

(5.13) and (5.15).

4. What is the cut-off frequency f c of the TE 10

mode in the given rectangular waveguide,

and from which frequency is the TE 20

wave capable of propagation? For thisalso refer to Fig. 5.6.

5. In addition to the values from the measur-

ing instrument displays determined

through experimentation in experiment

2.4 calculate the ratio of the respective

voltages (field strength levels) to the max-

imum value and enter these in column 3 of

Table 5.1.

Note:

As described in Experiment 2 the detector voltage is proportional to the square of the

received field strength (or here the voltage

from the transmission line model accord-

ing to Fig. 5.2). You take the logarithm of

the value indicated in the SWR meter dis-

play in accordance with the expression

10 · log(U D / U D,ref ). From this you obtain

the relationship

Display

dB

where

= ⋅−( )

= ( )

200

0

log

max

max

U x x

U

U U x

6. According to Fig. 5.3 e) the voltage (field

strength) responds accordingly.

U x x

U

x x

−( )= ⋅ −

0

0

2

max

cosπ

λ g

[ Note:

In Equation 5.10 the sinusoidal term is de-

fined as sin( ßx). However, here it is per-

mitted to convert to the cosine term be-

cause it only concerns a phase-shift and a

maximum is assumed to exist at x – x0 = 0].

The values resulting from this equation are

entered into column 4 of Table 5.1 and

then they are compared to the values in

column 3. Discuss any possible causes if

you notice regular deviations between thevalues in the two last columns.

7. In conjunction with the values determined

experimentally under point 3.4 from the

measuring instrument display, calculate

the ratio of the voltage (field strength) to

the voltage maximum at the respective lo-

cation of the slotted measuring line. Enter

the values into the 3rd column of Ta-

ble 5.2.

8. Discuss the value ∆ obtained under experi-

ment point 3.2.

By what magnitude in dB do the measured

values from subpoint 3.4 vary?



3 6


Bibliography

[1] G. Megla: Dezimeterwellentechnik, Berliner Union, 1962

[2] Altmann: Microwave Circuits, Van Norstrand, 1964

[3] F.E. Gardiol: Introduction to Microwaves, Artech House, Dedham 1984

[4] M. Sucher, J. Fox: Handbook of Microwave Measurements. Polytechnic Press,

Brooklyn (NY) 1963

[5] C.G. Montgomery: Techniques of Microwave Measurement. McGraw Hill,

New York 1947

[6] R.E. Collin: Field Theory of Guided Waves. McGraw Hill, New York 1947

[7] R.F. Harrington: Time Harmonic Electromagnetic Fields. McGraw Hill,

New York 1961

[8] S. F. Adam: Microwave Theory and Applications. Prentice Hall, Englewood Cliffs, 1969

[9] H. Meinke, F. W. Gundlach: Taschenbuch der Hochfrequenztechnik. Springer - Verlag

1986

[10] E. Meyer, R. Pottel: Physikalische Grundlagen der Hochfrequenztechnik. Vieweg Verlag,

Braunschweig 1969

[11] H. Groll: Mikrowellen - Meßtechnik. Vieweg - Verlag, Braunschweig 1969



3 7

MTS 7.4.4 The reflection Coefficient

The complex reflection coefficientDetermining the reflection coefficient according to magnitude and phase

PrinciplesVoltage curve for random termination impedance

In Experiment 5 two special cases were studied,

namely the case where a line is terminated in a

short-circuit (r = –1) and a line which is termi-

nated with total absorbtion (r = 0). Now we will

describe in more detail the standard case where

the transmission line is generally terminated

with any given complex impedance Z (see Fig.

6.1(a)).

In this case we have the superpositioning of

two waves, a reflected wave U 0,– · e+jßx travel-ling in the (– x) direction and the wave U 0,+ · e –jßx

travelling in the (+ x) direction, whereby the

(complex) reflection coefficient r is defined as

the ratio of the (complex) amplitudes of the

wave being sent back and the wave travelling

to the load at the point of reference (here x = 0):

r r eU

U = ⋅ = −

+

jdef

0,

0,

φ (6.1)

This results in a spatial dependency for the

complex amplitude of total voltage U(x) andtotal current I(x) according to

U x U e r e( ) = ⋅ + ⋅[ ]+−

0 , jßx jßx (6.2a)

I xU

Z e r e( ) = ⋅ − ⋅[ ]

+ −0

0

, jßx jßx (6.2b)

But due to the fact that the ratio U(x)/ I(x) at

x = 0 must be equal to the impedance Z , it is

true that

Z U

I

r

r Z = ( )

( )= +

−⋅

0

0

1

10 (6.3)

Solving equation (6.3) for r yields the signifi-

cant relationship

r r e

Z

Z

Z

Z

= ⋅ =

−

+

jφ 0

0

1

1

(6.4)

l

l'

Z 0

Z 0

~

(a)

(b)

(c)

I(x)

|U(x)|

|U(x)|

|r|=r ~

|U(x)|

Z 0

xo

R

U maxU min

D X

l g

2

Z

x

Fig. 6.1: Voltage distribution along a transmission line(a) Random complex termination impedance Z (b) Equivalent circuit for Z ( R and line with a

length of l )(c) Shorted transmission line

If we consider the distribution of the magnitude

|U(x)| along the transmission line, then it fol-

lows from equation (6.2 a) that

U x U r r ß x( ) = ⋅ + + ⋅ ⋅ +( )+0

21 2 2, cos φ

(6.5)

Part a of Fig. 6.1 shows the voltage distribu-tion, where it is true that

U U r

U U r

max 0,

min 0,

1

1

= ⋅ +

= ⋅ −

+

+

( )

( )

and

(6.6)

The voltage standing wave ratio (VSWR = s) is

given by:

s

U

U

r

r = =

+

−

def max

min

1

1 (6.7)



3 8


and

r s

s=

−+

1

1(6.8)

Determining the complex reflection coefficient

from the measurements performed with the

slotted measuring line

According to Eq. (6.8) the magnitude |r | of the

reflection coefficient can be calculated from the

standing wave ratio s determinable using the

slotted measuring line.

The phase φ of the reflection coefficient can be

determined by comparing the location x1 of the

minimum value of |U(x)| to the location x 0 of

the zero point or minimum when the transmis-

sion line is shorted.

In this context first consider Fig. 6.1 b). Here

the voltage distribution is extrapolated beyond

location x = 0 up to the next maximum value.

In conjunction with this maximum value there

is a positive real resistance R > Z 0 where

Rr

r Z =

+

−⋅

1

10 (6.9)

The line segment with the length l , terminated

with R, has the same input impedance Z as the

original circuit. Thus, for the phase φ it is true

that

φ π

λ = − ⋅ = − ⋅4

2g

l ß l (6.10)

On the other hand, due to l + l’ = λ g/4 (distance

of the minimum to the maximum) and

∆ x + l’ = λ g/2 (generally: ∆ x + l’ = n · λ g/2) the

following relationship applies for l

l x n= − −( )∆ 2 14

gλ (6.11)

and therewith

φ π λ

π / rad

g

4 2 1= − + −( )∆ x

n or

φ λ

= − ° + °

= ± ±

720 180

0, 1, 2, ...

g

∆ x

n( ) (6.12)

Fig. 6.2: For the determination of s and |r| from themeasurement of the “node width” ∆l .

To determine φ you can proceed as follows:

(a) You determine a location x0 for the zero

point or minimum when the transmission

line is shorted.

(b) Afterwards you determine a location x1 for

the minimum when the line is terminated

with an unknown impedance.

(c) ∆ x constitutes the shift from x1 with re-

spect to x0. For this ∆ x is counted as posi-

tive, if you must shift from the minimum

in the direction of the termination ( Z ) (see

also Fig. 6.1).

The determination of the magnitude |r| out of s,i.e. the ratio of the maximum to the minimum

amplitude, becomes inexact as soon as the am-

plitude ratio (= s) becomes somewhat greater

than 5. It is here namely that slight measure-

ment errors arise through the overload of the

measuring amplifiers, deviation in the diode

characteristic when measuring the maximum

and on account of noise during the measure-

ment of the minimum. These errors can be

avoided to a great extent, if instead of

measuring the voltage quotients s = U max

/U min

you determine the width of the voltage

minimum (node width). If according to Fig. 6.2

you express the so-called node width at

U x U ( ) / min = 2 (corresponding to 3 dB) with

∆l , the exact result is:

1

sin

1

g

2

g

s

l

l

=

+

π λ

π λ

∆

∆

sin

(6.13)



3 9


Fig. 6.3: Complex number plane

(a) the reflection coefficient r = | r | exp( jϕ ) and(b) for the normed impedance Z / Z 0 = R/ Z 0 + j X / Z 0.

: Short-circuit : Matching

Fig. 6.4 Smith chart

For ∆l /λ g<< 1 this can be approximated with

sin α ≈ α << 1 for α << 1

yielding

1

gs

l≈ π

λ

∆ (6.14)

The value |r | is extrapolated from s using Equa-

tion (6.8).

Smith chartThe reflection coefficient r and the normalized

impedance Z / Z 0 are complex numbers depicted

accordingly in separate number planes in Fig.

6.3. If according to Equation (6.3) the corre-

sponding values of Z / Z 0 for the reflection coef-

ficient r are entered into the complex number plane (Fig. 6.3 a) and this is done in the form of

lines of the constant real component R/ Z 0 and

constant imaginary component X / Z 0, you ob-

tain the Smith Chart depicted in Fig. 6.4. It is

well suited for the rapid determination of Z / Z 0for a given value of r (and vice versa). Fur-

thermore, it is very useful – as explained in

more detail in Experiment 7– for describing

various properties of transmission line net-

works. In Fig. 6.4 the reflection coefficient is

entered for Z / Z 0

= 0.5 + j 1.4. By dimensioning

the length |r | relative to the radius of the outer

circle you obtain |r| ≈ 0.72 and by measuring

the angle you arrive at φ ≈ 67°. The determina-

Z

Z 0

Re Z

Z

R

Z 0 0

=

jim Z

Z

j x

Z 0 0

=

tion of Z / Z 0 at a given value of r is performed in

a similar manner.

The Smith chart can also be used for admit-

tances Y Z 0 = Z 0/ Z . The corresponding complex

number is then (– r ). Then if it is true that

Y /Y 0 = Y Z 0 = 0.5 + j 1.4, the corresponding re-

flection coefficient is:0.72 · exp [ j (67°–180°)] = 0.72 · exp (– j 113°).



4 0


Note:

As is shown in Experiment 3 of MTS 7.4.5,

“any” random reflection coefficient (load) can

be realized using the combination (series con-

nection) of a variable attenuator, a variable phase-shifter and a short-circuit plate. Here the

attenuator primarily influences the magnitude

and the phase-shifter affects the phase-angle of

the reflection coefficient.

Required equipment




2 x 15 mm 90° 737 22


1 Coax detector 737 031 Short-circuit plate,

from accessories 737 29

1 Sample holder, from accessories 737 29

1 Sample made of graphite(*),

from accessories 737 29






2 Coaxial cables with BNC/BNCplugs, 2 m 501 022


2 Supports f. waveguide components 737 15

1 Stand rod 0.25 m 301 26

Recommended


1 Isolator 737 06

Experiment procedure1. Set up the experiment configuration as

specified in Fig. 6.5.

Note:

The diaphragm with slit again serves as a

filter to improve spectral purity.

2. Calibration

Screw on the short-circuit plate to the end

the slotted measuring line to function as

the device under test (DUT). Determine

the location x0/mm of the first minimum

(zero point) in front of the short-circuit

plate and enter the findings in Table 6.1.

3. Measurement of the sample

3.1 Assemble a measurement object A con-

sisting of a sample holder with graphite

material sample (see Fig. 6.5) and a reflec-

tion-free waveguide termination. This

measurement object is fastened onto the

open end of the slotted measuring line.

3.2 Determine the standing wave ratio(VSWR) s with the help of the scale on the

frequency-selective voltmeter. To do this,

calibrate in the voltage maximum to

“0 dB” using the range switch and the

(*) : In order to be able to compare the measured values of this experiment with those from Experi-

ment 11, the samples used here should be stored so as not to mix them up with any other exist-

ing samples.

Fig. 6.5 Experiment setup



4 1


“ZERO” controller. Shift the position of

the slotted measuring line probe to the first

minimum in front of the load. This is

where you read off the VSWR value and

enter it into Table 6.1.3.3 Determine the location x1 of the minimum

and enter this value into Table 6.1

3.4 Preliminary evaluation:

Determination of |r| from s with the aid of

Equation (6.8) and extrapolation of φ from

∆ x = ± ( x1 – x0) and Equation (6.12). Enter the

values into Table 6.1.

3.5 Determining |r| by means of the node

width ∆l . For this proceed with measure-

ment techniques corresponding to Fig. 6.2

to determine the node width (3 dB beyondU min). Calculate s according to Equation

(6.13) and |r| according to Equation (6.8).

Enter the findings into Table 6.2

Tip:

Adjust the value of the minimum to –3 dB

using the gain selection switch and the“ZERO” controller, and find the positions

(to the left and right of the minimum),

where the display assumes a value of 0 dB.

Questions1. Determine the value of the normalized im-

pedance Z / Z 0 for the measurement object

A (see Table 6.1) using the Smith chart

(approximately with Fig. 6.4 or more ac-

curately with Fig. 7.7 of the subsequentexperiment).

Table 6.2

∆l /mm s |r|

Measurement

object A

Table 6.1

DUTLocation of the

Minimum/mm s |r|

Short-circuit plate 1 180

Measurement

object A

∞

φ / degree



4 2


Bibliography

[1] S.J. Algeri, W.S. Chiung, L.A. Stark: Microwaves Made Simple -The Workbook.

Artech House, Norwood 1986

[2] G.J. Wheeler: Introduction to Microwaves. Prentice Hall, New Jersey

[3] T. Moreno: Microwave Transmission Design Data. Dover Publ., New York

[4] C.G. Montgomery: Technique of Microwave Measurements. McGraw Hill

[5] S. F. Adam: Microwave Theory and Applications. Prentice - Hall, Englewood Cliffs, 1969

[6] T. S. Laverghetta: Modern Microwave Measurements and Techniques.

Artech House, Norwood 1988

[7] T. A Johnk: Engineering Electromagnetic Fields and Waves. Wiley & Sons,

New York 1975



4 3

MTS 7.4.4 The Smith chart

Matching and the Smith chart

Principles

Preliminary considerations

If the input impedance Z of a one-port is not in

agreement with the characteristic impedance Z 0of the transmission line this results in a reflection

characterized by the reflection coefficient

r r e

Z

Z

Z Z

= ⋅ =

−

+

jφ 0

0

1

1

(7.1)

For various reasons this reflection is undesirable

for most applications. Some of the reasons are

listed here:

(a) If the transmission line is fed from a gen-

erator with the available power P av and this

generator is matched to the line (see Fig-

ure 7.1, part 1), then the real power ab-

sorbed by the load is

P = P av (1 – |r |²). (7.2)

This shows that the maximum power is

only supplied to the load when it is per-

fectly matched; (|r | = 0).

(b) Because of non-linear effects, the exist-

ence of a wave returning to the generator

can lead to changes in the operation char-

acteristics of the generator, e.g. to a fre-

quency shift and to parasitic oscillations at

different frequencies (fulfilling the condi-tion for self-excitation through reflection).

(c) Compared with the case of |r | = 0 an en-

hanced field strength (standing wave) is

caused by the interference of the reflected

wave with the one travelling forward.

Thus, the danger of disruptive discharges

(high electric field strengths) is associated

with the transport of relatively high power

levels.

The danger described in point (c) does not exist in

this training system because only low power lev-

els are used here.

Basic function of a lossless matching element

and the matching condition

For a given frequency one can generally match

any one-port with |r | < 1 to the characteristic

impedance Z 0 by positioning a two-port (=

matching network) upstream in series between

the reflecting one-port and the transmission line.

If a lossless two-port (ideally) is used for the

matching network, then its general function is to

compensate the given reflection coefficient r to

zero by adding additional reflections. Section 2 of

Fig. 7.1 initially demonstrates the general situa-tion. By connecting the network N (two-port) in

series upstream, the reflection coefficient r is

transformed into the reflection coefficient ~r .

The aim is to achieve correct matching by select-

ing the right parameters of the lossless linear

two-port, i.e. to achieve ~r = 0.

The condition to be fulfilled here (matching con-

dition) can be deduced in general, that means

without any special knowledge regarding the con-

struction of the two-port N. Here a consideration

of only one parameter of the two-port is suffi-cient; namely the backward reflection coefficient

Γ , the definition of which is shown in section 3 of

Fig, 7.1. Here, it is assumed that port 1 (1-1') of

the two-port is terminated reflection-free (i.e.

the internal impedance of the source is Z 0). Γ is

then the effective reflection coefficient “felt” by

a wave travelling to port 2 (2-2') from right to

left. (Bear in mind that r applies to a wave arriv-

ing from the left.)

You can demonstrate that it is sufficient for

matching when Γ is chosen to be the conjugate

complex value of of the reflection coefficient r to be matched (see below):

Γ = r * = |r | e –jφ (matching condition)

(7.3)

Section 4 of Fig. 7.1 shows the resulting situation

for this specific case. At port 1 the reflection re-

sults in ~r = 0 (matching). Between port 2 and the

mismatched (|r | ≠ 0) one-port you obtain the

superpositioning of a wave with the power

P av/(1 – |r |²) travelling to the one-port while the

reflected wave has the power P av · |r |²/(1 – |r |²).



4 4


The total power results from the difference of

the power of both waves and equals P av. Thus, as

desired, the total available power is provided to

the one-port (for a lossless matching network).

Matching element according to the principle of

the slide screw transformer

According to the findings from the previous

section there are two different ways to explain

how a matching element works:

(A) The explanation dealing with the transfor-

mation of a given reflection coefficient

r = |r | e jφ to the value ~r = 0 using a match-

ing element.

(B) The explanation involving the setting

Γ = |r | e –jφ

of the matching element con-nected reflection-free at the input-side.

According to Figure 7.2 a slide screw trans-

former consists of a homogeneous waveguide

section designed with an “obstacle” which can

be adjusted in terms of location (variable posi-

tion setting ξ 0) and “magnitude”. If the “obsta-

cle” is in the form of a metal post as in Figure

7.2 (left) with an adjustable penetration depth of

h < b, this can be represented in the equivalent

circuit diagram by a shunt capacitance C (see

Fig. 7.2, right) as long as h is sufficiently smallcompared to the height b of the waveguide. C is

zero for h = 0 and increases with h. If h is only

“slightly” smaller than b, one obtains a series

resonant circuit, whereas, if the post touches the

opposite side, it responds like a shunt inductance.

In the case of the slide screw transformer here

we can always assume that we are dealing with a

shunt capacitance.

Now we shall first consider the transformation of

a random reflection coefficient r = |r | e jφ in the

Fig. 7.1: General considerations regarding the problem of matching1 Transmission line with matched generator

and mismatched load.Corresponding complex amplitudes fromwave travelling to and reflected from load.

2 Series connected two-porttransforms reflection coefficient r into the value ~r .

3 On the definition of the backwardsreflection coefficient G of the two-port

4 Ratios for matching, i.e.two-port N is the ideal non-dissipativematching network.

f : Phase displacement through Z g : Phase displacement through thematching network

Fig. 7.2: Technical construction of a slide screw transformer (left) andcorresponding equivalent circuit diagram (right).

Pav

r e P jav

φ

Pav

~r P⋅ av

j Pe e

r av

j

2

j2

1

−

−

φγ

j Pr e e

r av

j2 j

21

⋅

−

−φ

γ

Pav



4 5


matching point ~r = 0 (Explanation A according

to above case distinction).

In Fig. 7.3 point 1 specifies the random reflec-

tion coefficient r = |r | e jφ (in the example:

|r | = 0.605 and φ = 210°). To be able to readthe admittances out of the Smith chart (expedi-

ent for parallel circuits), the transition to – r

(point 2) is carried out through inversion at

the matching point (r = 0). In the example

Y /Y 0 = 2 + j 1.9 of the normalized admittance

belongs to the value of – r (for practice please

verify in Fig. 7.3).

Based on this preliminary step it is now easy to

explain the function of the slide-screw trans-former. Only the phase of the reflection coeffi-

cient but not the magnitude is changed by the

waveguide section with the length ξ 0 (phase

rotation by the angle, 720° ξ 0 / λ g in clockwise

Fig. 7.3: Matching of a random reflection coefficient r = |r | e jφ (where |r | < 1)with the aid of a slide screw transformer:1 Reflection coefficient r to be matched2 Point for – r (admittance representation)3 Negative reflection coefficient – r 1 after transformation by the waveguide

section of the arbitrary length ξ 0.4 Negative reflection coefficient – ~r (resp.

~Y /Y 0) after parallel connection of a

positive susceptance B of random magnitude.Matching case:5 and6 represent – r 1 and – ~r for the correct selection ξ 0 = ξ

0

and B = B for the line length and susceptance.

2 . 0

3 . 0

0 , 3 5

0 , 2

3

0 , 2

2

0

,1 9

0 ,1

8

0 ,1 7

0 ,16

0 ,15

0 ,140,130,12

0,11

1 .

0

4 . 0

5 . 0

2 .

0

3 .

0

4 .

0

5 .

0

0

0 ,

2 5

1 0

0 , 2 4

2 0

3 0

4 0

5 0

6 0

7 0

8 0 901 0 0

1 1 0

0, 1 0

1 2 0

0, 0 9

1 3 0

1 4 0

1 5 0

1 6 0

1 7 0

1 8 0

0

1

9 0 j

j

0 ,

4 9

0 ,

0 0

0 ,

0 1

0 , 0

2 G e n e r a t o

r

0 , 0 3

0 , 0 4

0 , 0 5

0 , 0

6

0, 0 7

0, 0 8

2 0 0

2 1 0

2 2 0

2 3

0

2 4 0

2 5 0

2 6 0 2 7 0

0 , 3 8 0 , 3 9 0 , 4 0

0 , 4 1 0 , 4 2

0 , 4

3

0 , 4

4

0 , 4

5

0 , 4

6

0 , 4

7

L o a d

0 , 4

8

2 8 0

2 9 0

3 0 0

3 1 0

3 2 0

3 3 0

3 4 0

0 , 2 8

0 , 2 9

0 , 3

0

0 , 3

1

0 , 3 2

0 , 3 3 0 , 3

4 0 , 3 5

0 , 3 7

3 5 0 0

, 2 7

0

, 2 6

0 . 2 1

0 , 2

0

l

Smith Chart

0 .

8

0 .

6

0 . 5

0 .

4

0 .

3 0

. 2

0 .

1

2 0

1 0

1 0

2 0

1 .

5

+

oZ

o

R

0.2

0. 4

0. 6

1, 0

0 . 8

0 . 2

0 . 2

0 . 4

0 , 6 0 , 8

1 , 0

0 , 4

0 , 6

0 , 8

1 , 0

0.4

0.2

0 .6

0 .8

1,0

1 . 5

1 . 5

10

2 0

0 . 1

0 . 1

0 . 2

0 . 2

0 . 4

0 . 4

0 . 5

0 . 5

0 . 6

0 . 6

0 . 7

0 . 7

0 . 8

0 . 8

0 . 9

0 . 9

1 , 0

1 ,

0

0 . 3

0 . 3

2 . 0

3 .0

4 .0

5 .0

X Z

o

-

l

F

5

3

l g

x 0·720°

l g

x 0·720°

Y 0

B

–r

–r

r

Y 0

B

Y 0

Y

4

6

1

2

–r

~

1

j

j



4 6


direction). Thus, one obtains the value of – r 1given by point 3 in Figure 7.3. The parallel con-

nection of j · B corresponds to the addition of an

imaginary admittance and consequently the

value of the reflection coefficient changes into~r (point 4) along the locus of a constant real

component (in this example Re (Y /Y 0) = 0.4).

In order to achieve matching, i.e. r = 0, the

length ξ 0 = ξ 0

must be selected in accordance

with Figure 7.3 so that – r 1 is located on the locus

Re (Y /Y 0) = 1 (point 5 where ξ 0 ≈ 0.118 · λ g).

Under this prerequisite there exists a value

B = B for which – ~r is located at the matching

point (point 6) (in the example: B /Y 0 ≈ 1.5).

Predicated on the explanations and descriptions

found in the previous section an alternative explanation suggests itself for the function of the

slide screw transformer (explanation B accord-

ing to the case distinction above). In this pres-

entation we will be referring to the circuit found

on the left of the reference plane (Symbol Γ ) and

not to the circuit found on the right – as above. If

the slide screw transformer is terminated reflec-

tion-free at port (1- 1' in Fig. 7.4), then~Γ = 0

(see Fig. 7.4) also applies to the left of the shunt

capacitance. Γ 1 is obtained to the right of the

shunt capacitance C , whereby the magnitude

|Γ €1| can be altered by varying C (penetrationdepth of the post) between 0 and values “approx-

imating” 1. Here the phase of Γ €1 also changes,

but as a function of the magnitude |Γ €1|. A phase

variation independent of the magnitude is possi-

ble by changing the location ξ 0. As such the phase

condition for matching specified above

Γ € = r * = |r | e –jφ

can always be fulfilled by varying C and ξ 0.

Matching element according to the principle of

the multi-screw transformer (2- or 3-screw

transformer)There are two variable parameters in the slide

screw transformer dealt with above, namely the

position ξ 0 and the penetration depth h of the

post (screw). In the case of multi-screw trans-

formers we dispense with an adjustment of the

screw position ( x-coordinate). To realize this

the number of screws is greater than 1 and each

of the two or three screws can be adjusted inde-

pendently of each other in terms of their penetra-

tion depth.

The lower part of Fig. 7.5 shows the equivalentcircuit diagram for a 2-screw transformer. The

positions (ξ 0 and ∆ξ ) of the screws are fixed but

the penetration depth and therefore both

susceptances ( B1 and B2) can be adjusted inde-

pendently.

The negative value – r of the reflection coeffi-

cient at the end of the transmission line (r ) is

given by point 1 (bear in mind: the same value

as in Fig. 7.3). The waveguide section of the

fixed length ξ 0 transforms – r into point 2 (same

magnitude, phase rotated by [ξ 0/λ g] · 720° in

the clockwise direction). By means of a parallelconnection of j B1 one obtains a value corre-

sponding to point 3. The waveguide section of

fixed length ∆ξ transforms point 3 into point

4 (same magnitude, rotation of the phase by

(∆ξ /λ g) · 720° in the clockwise direction) and fi-

nally the parallel connection of j B2 transforms

point 4 into point 5. In order for point 5 to be

located in the matching point (r = 0), B1 must be

Fig. 7.4: How the slide screw transformer works (alternative explanation to Fig. 7.3).

Γ Γ =−

1

4 0

0e j π

ξ

λ

= r * =|r | e –jφ



4 7


selected precisely so that point 3 is located on

the auxiliary circle K’ . The auxiliary circle K’ is

obtained from circle K (Re Y /Y 0 = 1) by rota-

tion around the matching point (rotation angle

[[∆ξ /λ 0] · 720° counterclockwise). As you canconclude from Figure 7.5, not every reflection

coefficient |r| < 1 can be matched using the 2-

Fig. 7.5: Matching using a 2-screw transformer.

screw transformer.

However, this can be achieved using the 3-

screw configuration. But in this case there are

several solutions for B1, B2 and B3 for a given

value of r . Consequently, we shall dispensewith a more detailled theoretical investigation

of the 3-screw transformer.

l g

x 0·720°

l g

Dx ·720°

G e n e r a t o

r

L o a d

2 . 0

3 . 0

0 , 3 5

0 , 2

3

0 , 2

2

0 ,1

9

0 ,1

8

0 ,1 7

0 ,16

0 ,15

0 ,140,130,12

0,11

1 .

0

4 . 0

5 . 0

2 .

0

3 .

0

4 .

0

5 .

0

0

0 ,

2 5

1 0

0 , 2 4

2 0

3 0

4 0

5 0

6 0

7 0

8 0 901 0 0

1 1 0

0, 1 0

1 2 0

0, 0 9

1 3 0

1 4 0

1 5 0

1 6 0

1 7 0

1 8 0

0

1 9 0 j

j

0 ,

4 9

0 ,

0 0

0 , 0

1

0 , 0

2

0 , 0 3

0 , 0 4

0 , 0

5

0 , 0 6

0, 0 7

0, 0 8

2 0 0

2 1 0

2 2 0

2 3

0

2 4 0

2 5 0

2 6 0 2 7 0

0 , 3 8 0 , 3 9

0 , 4 0 0 , 4 1

0 , 4 2

0 , 4

3

0 , 4

4

0 , 4

5

0 , 4

6

0 , 4

7

0 ,

4 8

2 8 0

2 9 0

3 0 0

3 1 0

3 2 0

3 3 0

3 4 0

0 , 2 8

0 , 2 9

0 , 3

0

0 , 3

1

0 , 3 2

0 , 3 3 0 , 3 4

0 , 3 5

0 , 3 7

3 5 0 0

, 2 7

0 , 2 6

0 . 2 1

0 , 2 0

l

0 .

8

0 .

6

0 . 5

0 .

4

0 .

3 0

. 2

0 .

1

2 0

1 0

1 0

2 0

1 .

5

+

oZ

o

R

0.2

0. 4

0. 6

1, 0

0 . 8

0 . 2

0 . 2

0 . 4

0 , 6 0 , 8

1 , 0

0 , 4

0 , 6

0 , 8

1 , 0

0.4

0.2

0 .6

0 .8

1,0

1 . 5

1 . 5

10

2 0

0 . 1

0 . 1

0 . 2

0 . 2

0 . 4

0 . 4

0 . 5

0 . 5

0 . 6

0 . 6

0 . 7

0 . 7

0 . 8

0 . 8

0 . 9

0 . 9

1 , 0

1 ,

0

0 . 3

0 . 3

2 . 0

3 .0

4 .0

5 .0

X Z

o

-

l

j Y 0

B1

jY 0

B2

1

2

3

4

5

–r

K

K'

l g

Dx ·720°



4 8


Required equipment




2 x 15 mm 90° 737 221 Isolator 737 06




1 Cross directional coupler 737 18

1 Transition waveguide / coax 737 035


1 3-screw transformer 737 135

1 Short-circuit plate, from

accessories 737 29

1 Sample holder, from accessories 737 291 Graphite sample, from accessories737 29


2 Waveguide terminations 737 14



1 Oscilloscope (optional) 575 291 XY recorder (optional) 575 663

3 Coaxial cables with BNC/BNC

plugs, 2 m 501 022



components 737 15

1 Stand rod 0.25 m 301 26

Recommended

1 Slide screw transformer 737 13




4 9


Notes:

· In the experiment a cross directional coupler

is used (Coupling diaphragm with 2 cross-

shaped holes) to measure the reflected wave.

The cross directional coupler is an arrange-

ment made up of 2 waveguides, which are

connected together vis-à-vis coupling holes.

A portion of the travelling and reflecting

wave can be detected at the output ports of

the coupling waveguide. The exact function

of the cross directional coupler is explained

in Experiment 10.

· The coax detector is connected alternately

to the slotted measuring line and the cross

directional coupler. Naturally, it is also

possible and even recommended to usetwo detectors.

· Since this experiment responds with par-

ticular sensitivity to parasitic modes you can

only expect reproducible results when using

a PIN modulator (see also “Design of the

Microwave Source” from the preface).

Experiment procedure1. Calibration1.1 Set up the experiment in accordance with

Fig. 7.6. First attach the short-circuit plateto the open end of the line.

1.2 Use the slotted measuring line to deter-

mine the location x0 of the first (“right”)

field strength minimum.

Enter this value into Table 7.1.

1.3 Replace the short-circuit plate with the

measurement object A from Experiment 6,

consisting of a sample holder with graph-

ite sample and waveguide termination and

determine the location of the first (again

from the “right”) minimum x1/mm and en-

ter the value into Table 7.1 a).

2. Matching using the 3-screw transformer

2.1 Insert the 3-screw transformer between the

reflecting one-port (measurement object A

from Experiment 6) and the “200 mm

waveguide”.

2.2 Connect the coax-detector to the cross di-

rectional coupler to measure a portion of

the reflected microwave. Here be sure to

select a suitable gain factor V/dB.

2.3 Make an attempt to achieve matching by

adjusting all three screws one after the

other in succession. This means a mini-

mizing of the reflected microwave power

(ideally: no reflected wave). Here the best

procedure is to find the screw which has

the greatest effect on the reflected wave. To

do this turn one of the three screws (in-creasing the penetration depth) while

keeping the other two respective screws

set to a penetration depth of 0. Then set the

screw with the greatest effect to its opti-

mum setting (local minimum of the re-

flected wave) and then turn the other

screws in succession to further diminish

the reflected wave. At the same time you

have to suitably adapt the gain factor to be

able to determine the optimum setting of

the matching element.Note: If you are unable to obtain a distinct

response for any of the screws, then it

might prove useful to turn the 3-screw trans-

former in the configuration (i.e. interchang-

ing the ports), because it is designed slightly

asymmetrically. This gives you a different

phase angle for the screws.

2.4 Now replace the coax detector again with

the slotted measuring line and check the

transforming effect on the standing wave

ratio (reset the maximum to 0 dB). The

standing wave ratio should tend towards thevalue 1 ( s → 1, i.e. |r | → 0, ideally:

s = 1, |r | = 0). You have attained good

matching if you reach a standing wave ratio

under s < 1.1. In order to rule out any in-

fluence of the cross directional coupler on

the reflection coefficient, you can also con-

nect the matching transformer plus the

measurement object directly to the slotted

measuring line (i.e. remove the cross direc-

tional coupler and the waveguide section).

2.5 Afterwards measure the backward reflec-tion coefficient Γ of the 3-screw trans-

former without any further adjustment to

the screws. To do this

(α) Remove the reflecting single-port.

(β) Reverse the connection of the 3-screw

transformer (reversing the ports) to the

slotted measuring line.

(γ ) Equip the open end of the 3-screw

transformer with the reflection-free

waveguide termination (see also

Fig. 7.6).

Read off the standing wave ratio s and the

value of x1. Enter the findings in Table 7.1 b).



5 0


3-screw transformer

Table 7.1

· Short-circuit plate x0 = ________ mm

a) Measurement object without 3-screw transformer

VSWR

s

b) Calibrate to s → 1, i.e. |r | → 0 by successively turning all the screws

Backward reflection coefficient Γ (see Fig. 7.1)

VSWR s

3. Matching using the slide screw transformer ( optional )

Minimum at

x1/mmr

s

s=

−+

1

1

φ λ

= °− °⋅−

180 7201 0 x x

g

3.1 In accordance with Fig. 7.6 insert the slide

screw transformer between the reflecting

one-port (measurement object A from Ex-

periment 6) and “waveguide 200 mm”.Connect the coax detector to the cross di-

rectional coupler.

3.2 Minimize the reflected wave by succes-

sively adjusting the longitudinal position

x’ and the penetration depth h. In general

there are several combinations you can use

to achieve matching.

3.3 Again mount the coax detector on the slot-

ted measuring line and check the effect on

the standing wave ratio (reset the maxi-

mum to 0 dB). You have again reached

good matching if s < 1.1 (i.e. r €< 0.05). Torule out any influence from the cross di-

rectional coupler on the reflection coeffi-

cient, you can once again connect the

matching transformer plus measurement

object to the slotted measuring line (i.e.remove the cross directional coupler and

the waveguide section).Use a table similar

to table 7.1a.

3.4 The backward reflection coefficient Γ of

the slide screw transformer is determined

for matching. The measurement procedure

is analogous to that shown in experiment

point 2.5. Determine the standing wave

ratio s and x1. Use a table similar to Table

7.1b.

φ λ

Γ = °− °⋅ −180 720 1 0 x x

g

Γ = −+

ss

11

Minimum at x1/mm



5 1


Questions1. For measurement object A calculate the val-

ues of r = |r | e jφ corresponding to the meas-

urement data.

2. Determine the complex value Γ of the

backward reflection coefficient of the 3-

screw transformer based on the measure-

ment data in Table 7.1 b). Check the

validity of Equation (7.3).

(Optional)

3. Determine the complex value (|Γ | and φ Γ )of the backward reflection coefficient from

the measurement data. Test the validity of

Equation (7.3) by comparing the value de-

termined for Γ with the value of r . Com-

pare the value determined for Γ using the

slide screw transformer with the respective

value determined using the 3-screw trans-

former.

Bibliography

See Experiment 5



5 2


Fig. 7.7: Smith chart



5 3

MTS 7.4.4 Reflection

Reflection of a single-slot antenna

Fundamentals

Preliminary remarks

In experiments 6 and 7 the partially absorbing

and reflecting one-port is used as an example

for an incorrectly matched “load” and its

matching with the aid of a matching network.

In the present experiment we will consider a

case which frequently occurs in practice,

namely the mismatched antenna.

An antenna can generally be considered a spe-

cial form of a wave-type converter. In the caseof a transmitting antenna, the power P in sup-

plied by the generator arrives in the form of a

guided wave at the input of the antenna. The

function of the antenna is to convert the power

P in as completely as possible into a free-space

wave (with a given directional dependency =

directional pattern). An incomplete conversion

( P rad < P in) results from a reflection at the an-

tenna input (r) and/or dissipation (efficiency

η < 1) within the antenna (conversion into ther-

mal energy). The radiated power P rad resulting

is

P r Prad in= ⋅ −( ) ⋅η 12

(8.1)

In the case of a receiving antenna, it supplies

receiving power to the “load” or consumer

equalling

P r Prad rec,max= ⋅ −( ) ⋅η 12

(8.2)

whereby P rec, max is the maximum available re-

ceiving power.

From equations (8.1) and (8.2) you can discern

that incorrect matching (|r | > 0) leads to both a

reduction in the radiated power as well as a

drop in the power actually received (= reduc-

tion of the signal-to-noise ratio).

In the following section we will look into this

special type of antenna, namely the slot an-

tenna.

Principle of the slot antennaFor reasons of a didactic nature we shall now

expound on the principle of the slot antenna

step by step.

Here we will first consider a plane metal plate

of infinite extent, into which a slot (width h, in-

finitely long) has been mounted as specified in

Figure 8.1. Such a structure represents a special

form of a double transmission line (slotted

transmission line), on which guided waves can

propagate in both directions (parallel to the

slot). The field pattern of the guided wave of the

slot is depicted in part 1 of Figure 8.1.

It is assumed in the next step of our explanation

that the slot according to Fig. 8.1 (part 2, left)

only has a finite length w. Thus, it represents a

transmission line resonator short-circuited on

both ends. According to theory, without any

excitation a field can only exist on a transmis-

sion line of the length w, when w is an integer

multiple of λ g/2 (here equal to a free-space wave-

length of λ 0). Hence, the following results for

H

E h

h

w

|E 0||E|

E

(1) (2) (3)

Fig. 8.1: On the principle of the slot antenna:(1) Field pattern of the wave in a slotted line(2) Slot guide resonator (= at the end of the shorted slotted guide)

and corresponding distribution of the electric field strength along the slot(3) Slot antenna fed via waveguide

left: Slot perpendicular to the electrical field lines of the TE 10 fundamental moderight: Slot at an oblique angle to the electrical field lines of the TE 10 mode



5 4


the lowest resonance frequency (w = l 0/2, f = c/

l 0):

f c

w0

2

=

⋅

(8.3)

The part on the right of Fig. 8.1, part 2, shows

the distribution of the electrical field strength

along the slot. The maximum at the middle of

the slot is denoted→

E 0 .

In the case of external excitation of the electro-

magnetic field in the slot, there results a non-

vanishing field for each excitation frequency f

where

→=

⋅

+( ) + ⋅ ⋅ ⋅−

⋅→

E

j Q f f

f

E 0

00

0

1

2

1 2

β

β

c

c

(8.4)

Here→

E 1

stands for the amplitude of the ex-

cited field (presupposing a suitable definition). ßc is the coupling coefficient and Q0 is the “un-

loaded Q” of the transmission line resonator. ßc

and Q0 are among other things functions of the

slot width h.

The dependency of interest here is that for a

given coupling coefficient ßc (degree of exci-

tation)→

E 0 reaches maximum at f = f 0 and is se-

verely reduced, if f deviates radically from f 0.

There are various possibilities for feeding the

slot antenna. Frequently a coaxial line is con-

nected in a suitable form to the slot (see the rel-evant literature in the bibliography).

Here we will study the case of a waveguide

feed. Fig. 8.1 (part 3) demonstrates this princi-

ple. The dominant wave (TE 10 mode) propa-

gates in the rectangular waveguide; its

electrical field is polarized parallel to the nar-

row sides of the waveguide. The electrical field

of the guided wave is excited in the slot by this

field. If the exciting frequency f is in agreement

with f 0, then we find a slot width w, so that it is

true that ßc = 1 for f = f 0 and thus P rad = P in and|r | = 0.

In the slot antennas used in this experiment the

conditions f = f 0 and ß = ßc are more or less se-

verely violated so that it is true that |r | ≠ 0.

Therefore, these antennas require the use of a

matching element.

Required equipment 1 Basic unit 737 021


1 Isolator 737 06





1 Transition waveguide / coax 737 035



1 3-screw transformer 737 135

1 Diaphragms with slits incl. holder 737 221 Set of thumb screws (6 each) 737 399



1 XY-recorder (optional) 575 663

3 Coax cable with BNC/BNC

plugs, 2 m 501 022



1 Stand rod 0.25 m 301 26

Recommended 1 E-field probe 737 35

1 Slide screw transformer 737 13

1 Digital oscilloscope 575 292

1 XY recorder 575 663

Notes:

· As in experiment 7 a cross directional cou-

pler is used here (coupling diaphragm

with 2 cross-shaped holes) to measure the

reflected wave. For more detailled infor-

mation please refer to Experiment 7 or Ex- periment 10.

· Instead of the 3-screw transformer (with

200 mm waveguide) the slide screw trans-

former can also be employed. The experi-

ment points can be carried out in like

fashion.

· To be able to obtain reproducible findings,

you need to use the PIN modulator with

isolator. If you only use the oscillator with

internal modulation, normally the Gunn

oscillator is so severely affected by reflec-

tions that it generates high-frequencymodes with the result that the experiment

objective can no longer be achieved.



5 5



1. Set up the experiment arrangement in ac-

cordance with Fig. 8.2. Set the screws of

the 3-screw transformer to a penetration

depth of approx. 0 (i.e. no effect from the

matching transformer).

(Optional: Set up the E-field probe as re-

ceiving antenna approx. 30 cm in front of

the open end of the waveguide. Keep the

intermediate space free of objects which

might cause scattering.)

2. Investigating mismatching using the three

different slot antennas (versions A, B and

C according to Fig. 8.3).

2.1 Attach slot antenna A to the open end of

the waveguide.

2.2 Connect the measurement amplifier with

detector to the slotted measuring line.

2.3 Determine the standing wave ratio s and

enter the value into Table 8.1. Calculate

the magnitude |r | of the reflection coeffi-

cient and enter the value into Table 8.1 a).2.4 Repeat measurements 2.1 to 2.3, but re-

place slot antenna A with the slot antenna


Fig. 8.3: Dimensions and alignment of the three slotantennas used here(1) Metal cover with slit (= slot antenna)(2) Waveguide flange

(C)

(B)

(A)

(2)(1)

210

215

2

1 5

45°

versions B resp. C (see Fig. 8.3). Enter the

findings for s and |r | in Table 8.1 a).



5 6


Note:

Large standing wave ratios should be deter-

mined according to Fig. 6.2 and Equations 6.13

resp. 6.14. However, the applicability of these

methods are limited by the fact that ∆l becomesincreasingly smaller with increasing s and con-

sequently more and more difficult to determine

using measuring techniques. Alternatively the

standing wave ratio can also be computed from

the level at the minimum amin (bear in mind the

adjustment of the gain selection switch V/dB).

With amax = 0 dB (maximum level set to 0 dB)

the standing wave ratio can be calculated ac-

cording to the following equation:

s E

E

U

U s

a a

a

= = ⇒ =

=

−

max

min

max

min

max min

min

D

D

dB

dB

10 20

2010

Optional (when using the E-field probe) :

2.5 Establish the connection of the measure-

ment amplifier to the E-field probe. Work

through points 2.6 and 2.7 using the slot

antennas A, B and C.2.6 Rotate the E-field probe around its own

axis (α = 0, ±45° and ±90°) (Attention:

leave the receiving dipole of the E-field

probe on the symmetrical axis of the

waveguide). At the same time observe the

receiving voltage. Determine α = α opt. for

maximum receiving voltage. Enter the

value of α opt. in Table 8.1 b).

2.7 Select a “calibrated” (clear) setting for the

measurement amplifier e.g. “ZERO” to the

far right limit. Read off the receiving volt-age (relative unit in dB). Enter the result

into Table 8.1 b) together with the setting

of the range switch V/dB.

3. Matching the slot antennas A and deter-

mining the radiated power (relative meas-

urement).

3.1 Reattach the slot antenna A to the open

end of the waveguide.

3.2 Connect the coax-detector to the cross di-

rectional coupler to measure the reflected

wave.3.3 Successively adjust the 3-screw trans-

former (see Experiment 7 point 2.3, to

minimize the reflected wave (i.e. match-

ing). Set to the achievable maximum

matching (a value of s < 1.5 should be

reached; very good matching is reached

when s < 1.15).

3.4 Reconnect the detector to the slotted meas-uring line and determine the standing

wave ratio (here again set the display to 0

dB at the maximum). Enter the values for sand |r | in Table 8.2 a). (If necessary you

can remove the cross directional coupler

and waveguide section again, see Experi-

ment 7.)


3.5 Establish the connection of the measure-

ment amplifier to the E-field probe.

3.6 Verify the optimum angle α = α opt from point 2.6. Transfer the value of α opt into

Table 8.2 b).

3.7 Select the calibrated setting for the meas-

urement amplifier (“ZERO” to the far right

limit). Read off the receiving voltage (rela-

tive unit in dB). Enter the result into Table

8.2 b) together with the setting of the range

switch.

4. Matching of the slot antenna B and deter-


urement).4.1 Replace slot antenna A with slot

antenna B.

4.2 Repeat experiment points 3.2 to 3.4 for

this case.


4.3 Repeat the experiment points 3.5 to 3.7 for

this case.

5. Matching the slot antenna C and deter-


urement).5.1 Replace slot antenna B with slot

antenna C.


this antenna type. At the maximum attain-

able matching you should achieve s < 2.



this antenna type.


6. Calculate the “absolute” level a’ = a – V,(here a and a’ are negative and V > 0). De-

termine the maximum level a’ max. Then

compute the value for a’ – a’ max



5 7


Table 8.1

a) Standing wave ratio and reflection coefficient without matching.

Antenna type s |r|

A

B

C

Optional :

b) Receiving level at the E-field probe without matching.

Antenna type

A

B

C

Table 8.2

a) Standing wave ratio and reflection coefficient

with maximum matching

Antenna type s |r|

A

B

C

Optional :

b) Receiving level at the E-field probe with maximum matching

Antenna type

A

B

C

The “absolute” level computed is: a’ = a – V

The maximum level is a’ max = ________ dB

α opt V/dB a/dB a’ /dB (a’– a’ max)/dB




5 8


Questions1. Why is there a considerably higher reflec-

tion coefficient resulting for version C of

the slot antennas (see Fig. 8.3) than for

versions A and B?

2. Why is the matching of version C with the

aid of a matching transformer less sensible

from a technical point of view?

Optional:

3. In parts 2.6, 3.6 etc. of the experiment pro-

cedure the polarization of the electrical

field was experimentally determined for

the radiated waves (α opt). What conclusion

could be drawn from this result?

4. Compare the magnitude of the receiving

signal for versions A and B. Why do sig-nals of nearly the same magnitude result,

although the reflection coefficients are

considerably different (measurements 2,

Table 8.1)?

5. Assess the matching based on the level dif-

ference a’ – a’ max while keeping in mind

for which case a’ max is reached.

BibliographySee Experiment 5



5 9

MTS 7.4.4 Measurment of the permittvity

Measurement of the permittivityDetermining the complex dielectric constant of material samples from reflection coefficient measurements

FundamentalsThe interaction of the transparent materials is

characterized by complex permittivity

ε r = ε r ’ – jε r ’’= ε r ’ (1 – j tan δ ε) (9.1)

(ε r ’ = real component of ε r , tan δ ε = dielectric loss

factor) and the complex permeability

µ r = µ r ’ – jµ r ’ = µ r ’ (1 – j tan δ µ). (9.2)

These material parameters are generally de-

pendent on the frequency.For various technical problems and their solu-

tions knowledge of these material parameters

are required:

(a) If the materials are used for the construction

of microwave components, then the exact

values of ε r and µ r are required as input data

for computer codes in computer aided design

(CAD). Examples: low-loss dielectrics as

substrate materials for planar circuits, for

example, in microstrip technology and for

the realization of dielectric resonators for os-cillator stabilization (DRO), and strongly ab-

sorbing materials for attenuating elements.

(b) Materials for the construction of radomes for

antenna systems and absorber materials for

reducing the radar cross-section of targets.

(c) The complex dielectric constant can fre-

quently be used for the determination of an-

other physical quantity. Thus the moisture

content of a material, for example, (sand,

coal, chipboards, tobacco etc.) modifies the

real as well as the imaginary part of the ef-

fective dielectric constant. Consequently, itis possible to continuously monitor moisture

content using microwaves.

(d) Should the material be heated with micro-

waves (cooking, drying, etc.), the complex

dielectric constant and its change must be

known, in order to optimize the correspond-

ing microwave system.

(e) In physics and chemistry one can draw

conclusions as to material composition

from the position and width of the “ab-

sorption spectra” in the frequency re-sponse of ε r or µ r . If a material sample of a

given shape is placed in a waveguide, the

desired information on the value of ε r or µ r is generally contained in the measurable

reflection coefficient and transmission co-

efficient. For this the following considera-

tions apply:

α ) Only for a geometrically “simple”

sample does a simple mathematical

expression apply for the relationship

between the measurable (reflection

coefficient and eventually the transmis-

sion coefficient) and the sought after quantities ε r and µ r

β ) The dimensions of the samples and

the selection of the measurement

quantities must ensure a sufficient

“sensitivity” on behalf of the meas-

urement quantities to changes in the

material parameters.

γ ) If both ε r as well as µ r are unknown,

at least two independent (complex)

measurement quantities must be

determined. It is frequently known

beforehand that the material has a

value of µ r = 1. In this case one

measurement quantity would be

sufficient for the determination of ε r .

In the following we shall presuppose that

µ r = 1, so that only ε r needs to be determined.

Figure 9.1a shows the possible measurement con-

figuration in which a material sample with the

length L with an unknown value of the (complex)

relative permittivity ε r fills up the rectangular

waveguide completely.

The short-circuit plane coincides with the back-side of the material sample for the position l = 0

of the moveable short , and thus the equivalent cir-

cuit diagram (b1) of the transmission line applies.

Thus the following is true for the normalized input

impedance:

Z

Z

a

a

j L

a

Α

0

0

2

0

2

0

0

212

2

2

2=

−

−

⋅ −

λ

ε λ

π

λ

ε λ

r

rtan

(9.3)



6 0


Z A/ Z 0 can be determined from the reflection

measurements and λ 0/2a as well as L/λ 0 are

known variables. Consequently, the sought-

after value of ε r can be determined from Equa-

tion (9.3). However, Equation (9.3) is a complextranscendental equation, which can only be

solved numerically (“zero position search pro-

gram”) for ε r .Thus, a different approach should be taken here

and, in addition, the normalized input impedance

should be used for the open-circuit case. We then

obtain the open-circuit case for the position

l = λ g/4 of the moveable short, and with the

equivalent circuit according to (b2) in Figure 9.1

the result is

Z

Z

a

a j

L

a

Α

0

0

2

0

2

0

0

2

12

2

1

2

2

=

−

−

⋅

−

λ

ε λ π

λ ε

λ r rtan

(9.4)

If you form the product from Z A and Z B, then the

tan-function cancels out, and you obtain an expression which can be solved directly for the de-

sired value ε r .

ε λ λ

r =

+ ⋅ −

0

2

0

2

2

1

2a a

Λ (9.5)

whereby Λ is used as an abbreviation for the ex-

pression

Λ =⋅

Z

Z Z

0

2

A B

(9.6)

Z A and Z B can be calculated from the values r Aand r B of the reflection coefficient determined

using the slotted measuring line via

Z

Z

r

r

Z

Z

r

r

Α Β

0 0

1

1

1

1

=+

−=

+

−

A

A

B

B

and

(9.7)

or determined using the Smith chart.

4

l gl =

(3)(3)

r a

Z B Z d

Z 0

Lr A

Z A Z d

Z 0

L

l = 0

(b2)(b1)

r 00e e m

l

L(a) (2)

(1)

(3)

Fig. 9.1: (a) Measurement configuration withmaterial sample (1) with the length Land moveable short (2) connecteddownstream.Distance of the short-circuit plane fromrear side of material sample is l .Reference plane for measurement is (3).

(b) Equivalent circuit diagram for trans. linefor computation of the input impedances Z A and Z B for the short-circuit (l = 0) andopen-circuit case (l = λ g/4).

Required equipment




2 x 15 mm, 90° 737 22

1 Isolator 737 06




1 Short-circuit plate 737 29

1 Sample holder 737 29

1 Material sample of polystyrene 737 29

1 Material sample of graphite 737 29

1 Moveable short 737 101 Set of thumb screws (2 each) 737 399





plugs, 2 m 501 022



1 Stand rod 0.25 m 301 26

1 Caliper



6 1


Note:

· To be able to obtain reproducible results,

you need to use the PIN modulator with iso-

lator, because in this experiment monomode

operation is indispensible. Reproducible re-sults can only be guaranteed with a PIN

modulator.


1. Setup and calibration measurement

1.1 Set up the experiment according to Fig.

9.3. Arrange the measurement object ac-

cording to Fig. 9.1 and Fig. 9.2.

1.2 Attach the short-circuit plate to the open end

of the slotted measuring line.1.3 Determine the position x0 of the first mini-

mum (“from the left”) on the slotted measur-

ing line and enter the value into Table 9.1.

Determine the position x1 of the 3rd mini-

mum and enter it into Table 9.1.

Fig. 9.3: Experiment setup(1) Short-circuit plate(2) Meas. object corresponding to Fig. 9.2 (see also Fig. 9.1)(3) Moveable short-circuit

Fig. 9.2: Measurement object for Experiment 9

1.4 Calculate the guided wavelength

λ g = − x x0 1 .

2. Determining short-circuit and open-cir-

cuit input reflection coefficient for

sample I (polystyrene)

2.1 Insert sample I (polystyrene, black) into thesample holder. On the positioning of the sam-

ple in the holder please refer to Fig. 9.2. Use a

caliper to measure the distance l H to the “right-

hand” edge of the sample holder (l H should

amount to just about 10.0 mm exactly).



6 2


2.2 Attach the sample holder (with sample) to

the open end of the slotted measuring line

(see Fig. 9.3).

2.3 Attach the moveable short to the open side

of the sample holder.2.4 Adjust the moveable short ( xk ) so that the

position of the short-circuit plane coincides

with the rear side of the sample (l = λ g/2, i.e.

xk = λ g/2 – l H). Note: Read xk from the mi-

crometer of the variable short.

2.5 Based on the standing wave ratio sA and the

position of the minimum xA (first from the

“left”), determine the short-circuit reflec-

tion coefficient:

r r e''

A A j= + φΑ

according to magnitude and phase (refer-

ence plane = front surface of the sample).

Enter into Table 9.1. (Note: see Table 9.1

for the individual mathematical steps)

2.6 Shift the moveable short further by one

quarter of a guided wavelength (λ g/4) so

that the open-circuit plane coincides with

the rear side of the sample.

2.7 Based on the standing wave ratio and the

position of the minimum determine the

open-circuit reflection coefficient

r r e' ''

B B j

=+ φ Β

according to magnitude and phase and

enter the values into Table 9.1.

3. Determining the short-circuit and open-

circuit input reflection coefficient for sam-

ple II. (non-magnetic absorber material

made of synthetic resin with graphite,

color: anthracite )

Analogous to 2.1 to 2.7 determine

r r e' ' ' '''

A A j= + φΑ

and

r r e' ' ' '''

B B j= + φΒ

Enter the values into Table 9.1.

General notes

· To avoid big mistakes when determining

the dielectric constant or permittivity, the

phase measurements have to be performed

with great care (precise values of xA, xB

and x0). Furthermore, the guided wave-

length must be known precisely (deter-

mine using the slotted measuring line) and

the moveable short ( xk ) must be set pre-

cisely (so that l = λ g/2 or 3 λ g/4, see Fig.

9.1).

· Tip: For the determination of the mini-

mum, the gain should be successively in-

creased to find the location with more

precision. An even finer setting is possible

by a slight tapping, i.e. moving the car-riage with a pen.

Questions1. In accordance with the equations

Z

Z

r

r

Z

Z

r

r

Α Β

0 0

1

1

1

1=

+

−=

+

−

A

A

B

B

and

calculate the normalized short-circuit and

open-circuit impedances for samples I

and II and enter the values into Table 9.1.

(Alternatively: Determine using the Smith

chart, see Fig. 9.4).

2. Calculate the expression (see also Equa-

tion 9.6)

Λ =⋅

Z

Z Z

02

A B

for samples I and II and enter the valuesinto Table 9.1.

3. Determine the complex dielectric con-

stants from sample I and II according to

ε λ λ

r =

+ ⋅ −

0

2

0

2

21

2a aΛ

(λ 0= free-space wavelength,

a = waveguide width = 22.8 mm).

Enter the final results into Table 9.1.



6 3


Sample I (Polystyrol) Sample II (Absorber material)

from the Smith chart

or mathematically


or mathematically

VSWR sA

xA/mm

r s

sA =

−

+

Α

Α

1

1

φ λ

AA

g

= °− °⋅−

180 7200 x x

r r jA A A A= ⋅ + ⋅( )cos sinφ φ

VSWR sB

xB/mm

φ λ

BB

g

= °− °⋅−

180 7200 x x

r s

s

B =−

+

Β

Β

1

1

r r jB B B B= ⋅ + ⋅( )cos sinφ φ

Z

Z

r

r

B B

B0

1

1

=+

−

Λ =⋅

Z

Z Z 0

2

A B

ε r = ε ' r – ε r ''

Z

Z

r

r

A A

A0

1

1

=+

−

tan δ ε = ε r ''/ε r '

Table 9.1

x0 = ________ mm x1 = ________ mm

λ g = ________ mm ⇒ λ 0 = ________ mm



6 4


Fig. 9.4: Smith chart

Bibliography

[1] M. Sucher, J. Fox: Handbook of Microwave Measurements. Polytechnic Press,

Brooklyn NY, 1963

[2] A. R. von Hippel: Dielectric Materials and Applications. J. Wiley & Sons, New York 1954

[3] F. E. Gardiol: Introduction to Microwaves. Artech House, Dedham MA, 1984



6 5

MTS 7.4.4 The cross directional coupler

The cross directional coupler

Fundamentals

General properties of waveguide (directional)

couplers

The cross directional coupler is a special type of

directional coupler. Thus, it makes sense to follow

with a general explanation applicable to the func-

tion of all types of waveguide couplers and their

most important parameters. An (unconnected)

waveguide coupler is a reciprocal four-port, which

is also ideally loss-free. Fig. 10.1 illustrates the re-

sponse of the ideal directional coupler. If a wave isfed exclusively into port 1, its effective power P 1is distributed to port 2 (primary path) and port 4

(coupling path). If the power exiting port 4

amounts to k 2 P 1, then the power must be (1 – k 2)

P 1 at port 2 because of the lossless property of

waveguide port 2 (conservation of energy). Ideally

there is no power exiting port 3, i.e. it is “decou-

pled” (isolation path). If, on the other hand, a wave

is fed into port 2, port 4 is decoupled and the power

fed is distributed to port 1 (primary path) and port 3

(coupling path). The magnitude of k (coupling

coefficient) can vary in amplitude depending on

the design of the directional coupler. The following:

ak = –20 log k [dB] (10.1)

is referred to as coupling loss. If you consider the

definitions provided above and furthermore the

relationships between the S-parameters resulting

from the loss-free property of the waveguide (see

Experiment 5 from MTS 7.4.5 or the bibliography

provided there), we obtain the following scattering

matrix for the ideal waveguide coupler.

S e

k jk e

k jk e

jk e k

jk e k

( )=

−−

−−

j

j

j

j

j

ψ

ϕ

ϕ

ϕ

ϕ

0 1 0

1 0 0

0 0 1

0 1 0

2

2

2

2

(10.2)

Here, in addition to the coupling coefficient k , thereare also the phase angles ψ and ϕ as parameters

(ψ is the phase rotation in the primary guide and ψ

+ ϕ are the phase rotations in the coupling guide).

Fig. 10.2 shows one of the primary applications of

waveguide couplers, namely the separate

measurement in front of a one port (e.g. antenna)

of the wave propagating to and reflected from the

load.

The signal exiting at port 3 is only proportional to

the wave P 2,in = |r |² P 2,out reflected by the load

(here the antenna), while a signal proportional tothe wave propagating to the load appears at port 4.

The following applies:

P

P

k P

k Pr k

3,out

4,out

22,in

21,in

2 2=⋅⋅

= −(1 )

(10.3)

When the value of k 2 is known, the reflection co-

efficient |r | can be derived from the relationship

between P 3,out and P 4,out. The configuration shown

Fig. 10.2: Use of the directional coupler as a reflectometer Fig. 10.1: Principal response of a directional coupler

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4



6 6


in Fig. 10.2 is referred to as a reflectometer. In

Experiment 11 the cross directional coupler is usedas a reflectometer.

Contrary to the statements above, in real direc-

tional waveguide couplers decoupling via an isola-

tion path (e.g. from port 1 to port 3 in Fig. 10.1) is

not total. If a wave is only fed in at port 1, power is

still obtained at port 3; P 3,out > 0. The ratio of

undesired power at port 3 ( P 3,out) to desired power

at P 4,out at port 4 is one of the ways of determining

the quality of a waveguide coupler. This is called

the directivity factor. The following applies for

directivity:When feeding at port 1

a P

P

D 4,out

3,outdB10 log= − ⋅ (10.4)

When feeding at port 2

a P

P

D 4,out

3,outdB10 log= − ⋅

High quality directional couplers are expensive

and can have a directivity of over 50 dB, whereas

simple waveguide couplers average around

20 dB.

The cross directional waveguide coupler as a

special form of directional waveguide coupler

Up until now the response of a directional

waveguide coupler has only been dealt with as a

kind of “black box”, i.e. only its operation within a

circuit. However, nothing has been mentionedabout the “physical effects” which are needed to

exploit this behavior. There are directional

couplers for all conventional transmission lines inmicrowave engineering (coaxial lines, microstrip

lines, hollow waveguides). These couplers employ

a wide variety of different principles to realize

directivity (the separation of the incident and

reflected wave). Examples for this include the

directivity employed in coupling TEM lines running

parallel and the interference of waves in hollow

waveguides, which are coupled together via holes.

In the following the principle of the cross

directional coupler is studied in more detail. In

accordance with Fig. 10.3 the cross directionalcoupler is made up of two hollow waveguides

arranged at a 90° angle as specified in Fig. 10.3, so

that they have a common wall (“overlapping

square surface”). Coupling is carried out via one

or two coupling holes in this common wall. For a

closer examination of the theoretical aspects of

this phenomena please refer to the literature

specified in the bibliography. Only a few important

findings are dealt with here.

Electric and magnetic coupling through a

hole in a metal wall In the upper part of Fig. 10.4 a cutaway section of

a closed metal wall is shown, on whose underside

an electrical ( E ) and a magnetic field ( H ) is found.

This means that:

1. An electric and magnetic field exist on the

underside of the closed metal wall.

2. Penetration of the electric and magnetic field

through a coupling hole.

3. Description of the coupling via an electric ( J )

and a magnetic ( M ) dipole.

Fig. 10.5: gives a schematic depiction of the crossdirectional coupler with a cylindrical coupling hole.

Where:

Fig. 10.3: Principal design of a waveguide (cross direc-tional) coupler and including port numbering

Fig. 10.4: Explanation of the hole coupling



6 7


Fig. 10.5: Cross coupler with a circular coupling hole witha diameter of 2 R

Fig. 10.6: Wave excitation when feeding wave into port 1

321

1 2

3

4

d

a

J

M X

M Y

1 Transversal hollow waveguide section

2 Coupling hole3 Reference planes (port 3) for the phases of

the S-parameters

We are assuming that the TE 10 wave propagates

from port 1 to port 2. If you now consider sepa-

rately the effect of the x- and y-components of the

magnetic dipole and the effect of the electric di-

pole, see Fig. 10.6, you arrive at the following con-

clusions:

The partial magnetic waves at port 3 completely

cancel each other out, while the magnetic waves

at port 4 superposition each other constructively.Thus, if only magnetic coupling were present, the

four-port would respond like an ideal directional

coupler. However, there are still the partial waves

excited by the electric dipole (“electric coupling”).

Here, two partial waves of the same magnitude

are formed for port 3 and port 4. The complete

decoupling of port 3 thus fails due to the partial

waves caused by the electric coupling.

Measures to increase directivity over that

attained with a single round coupling hole

Deviation from this ideal directional coupler resultswhen electric coupling is added. If an increase in

directivity is desired, the electric coupling must be

reduced. One suitable way of realizing this is by

substituting the round coupling hole with the cross-

shaped hole. Compared to round-shaped holes, in

cross-shaped coupling holes (see Fig. 10.7 centre)

the “magnetic penetration” predominates more

than the “electric penetration”. As the experiment

will show a substantial increase in directivity is

achieved by exchanging the round coupling hole

with the cross-shaped hole. An added improve-ment is achieved by using two cross-shaped holes

instead of one (see Fig. 10.7, right). Due to the in-

teraction between the two holes, the electriccoupling is weakened.

The cross coupler included in the training system is

comprised of several dismountable parts and there

are various options between different alternative

hole configurations for coupling. Therefore, the

improvements attained when going from one round

hole to one cross-shaped hole and from two round

holes to two cross-shaped holes can be verified ex-

perimentally.

Required equipment



1 Diaphragm with slots

2 x 15 mm, 90° 737 22





2 Waveguide terminations 737 14




plugs, 2 m 501 022

3 Standbases 301 21

3 Supports for waveguides 737 15

2 Stand rods 0.25 m 301 26

Fig. 10.7: Measures to increase directivity



6 8



2.1 Install the diaphragm with round hole into the

cross coupler. Here it is important to focus on

the installation direction (e.g. “4” to “4”) (see

Fig. 10.9, above).

2.2 Connect port 1 of the cross coupler to the

open end of the variable attenuator (instead

of the measurement head in Fig. 10.8,

above).

2.3 To measure the magnitude of the transmis-

sion coefficient (|S 21|) to port 2 you must

equip ports 3 and 4 with a reflection-free

waveguide termination and connect a meas-

urement head (transition waveguide/coax

with coax detector) to port 2. The display of

the SWR meters supplies |S 21| in dB. Enter

the result in Table 10.1.

2.4 For the measurement of |S 31| connect meas-

urement head to port 3 and reflection-free

terminations to ports 2 and 4. Enter result in

Table 10.1.

2.5 For the measurement of |S 41| connect meas-

urement head to port 4 and reflection-freeterminations to port 2 and port 3. Enter result

into Table 10.1

Recommended:


1 Isolator 737 06


When using the isolator and PIN modulator modify

the experiment setup in Fig. 10.8 according to the

preface!

1. Calibration of the experiment setup

1.1 First set up the measuring system as speci-

fied in Fig. 10.8, above, without the cross

directional coupler. Set “ZERO” to the far

right, the gain level V/dB to approx. 10 dB

(resp. 15 dB). With the aid of the variable

attenuator calibrate the display of the SWR

meter to 0 dB. This setting is no longer

changed during the experiments.

2. Measurement of some S-parameters of thecross directional coupler with 1 round

coupling hole.



6 9


Fig. 10.9: On the numbering of the ports of the cross-coupler (here, for example, port 4 is marked)

2.6 Now make connections for wave feed via

port 2, i.e. connect port 2 to the open end of

the variable attenuator. The transmission co-

efficients |S 12|, |S 32| and |S 42| are determined

as in 2.3 to 2.5. Enter the results in Table 10.1.

3. Measurement of some S-parameters of the

cross coupler with 1 cross-shaped hole.

3.1 Exchange the diaphragm with round hole for

the diaphragm with cross-shaped hole.

3.2 Measure the transmission coefficients |S 21|,

|S 31|, |S 41|, |S 12|, |S 32| and |S 42| as in 2.3 to 2.6.Enter the values in Table 10.1

4. Measurement of some S-parameters of the

cross directional coupler with 2 round

holes.

4.1 Exchange the current diaphragm for the dia-

phragm with 2 round holes.

4.2 Measure the transmission coefficients as set

forth under 2.3 to 2.6. Enter the results in

Table 10.1

5. Measurement of some S-parameters of thecross directional coupler with 2 cross-

shaped holes.

5.1 Exchange the current diaphragm for the dia-

phragm with 2 cross-shaped holes.

5.2 Measure the transmission coefficients as in

2.3 to 2.6. Enter the results in Table 10.1

Exercises1. In Fig. 10.2 the coupling loss attenuation of

the directional coupler is assumed to be 20 dB

(“20 dB-coupler”) and the power fed into the port is P 1,in = 1 W.

Specify the values for P 1,out, P 3,out and P 4,out

as well as P 3,out / P 4,out, if

α) |r| = 0.0

ß ) |r| = 0.5.

2. Determine the directivity for the 4 different

configurations of the cross directional coupler

and enter these into Table 10.1.

Note

· Directivity:

When feeding from port 1

aS

SD log= − ⋅

2031

41

or when feeding from port 2

aS

SD log= − ⋅

2042

32

Here the S -parameters are not to be used

logarithmically.

· If when determining the S-paramters of the

isolation path you reach the limits of themeasurement amplifier, then try to enhance

the modulated signal by varying the Gunn

voltage (see Experiment 2. But take care:

Monomode operation must be maintained).

Otherwise, we would highly recommed using

a PIN modulator (with isolator).



7 0


Table 10.1

|S 21| / dB|S 31| / dB

|S 41| / dB

|S 12| / dB

|S 32| / dB

|S 42| / dB

|aD| / dB

|aD| / dB

1 Round hole1 Cross-shaped

hole2 Round holes

2 Cross-shaped

holes

Feedvia

port 1

Feed

via

port 2

Port 1 :

Port 2 :



7 1

MTS 7.4.4 Principle of the reflectometer

Principle of the reflectometer

As demonstrated in Experiment 10, incident and

reflected waves can be separated with the aid of a

directional coupler (here specifically the cross di-

rectional coupler). Thus the directional waveguide

coupler constitutes an important component in the

measurement of reflection coefficients. This

special measuring technique is called

reflectometry as opposed to the reflection coeffi-

cient measurement using the slotted measuring

line. The greatest advantages over the slotted

measuring line method are; one, it takes considera-

bly less time to conduct the measurement and two,it can be used in automatic sweep measurements.

Various measurement principles of

reflectometry

(1)Use of only one coupling guide and compar-

ing the result with reflection at the short

In the measurement configuration according to

Fig.11.1, port 4 remains terminated reflection-free

throughout the entire measurement procedure.

Thus, reflection-free termination can be an integral part of the directional coupler in the practical

design of a reflectometer. The wave travelling via

the feed-through guide from port 1 to port 2

(weakening of the amplitude by 1 2− k ) is re-

flected at the measurement object with the un-

known reflection coefficient r . A portion of the

wave incidenting port 2 with the amplitude:

a r b r k a2 22

11= ⋅ = ⋅ − ⋅

(11.1)

is transmitted to port 3 via the coupling guide sothat the magnitude of the wave exiting at port 3 is

given by:

b k a k k r a3 22

11= ⋅ = ⋅ − ⋅ ⋅

(11.2)

The coupling coefficient k of the directional cou-

pler is known in advance, but not the amplitude |a1|

of the wave incidenting at port 1. For that reason

this measurement only suffices to determine |r |.

Thus a reference measurement is performed with

a short (r = – 1) at port 2. For this you obtain from

Eq. (11.2)

Fig. 11.1 Measurement configuration 1

r' = 04

3

1 2

a = r b2 2

b2

b3

b1

a1

DUT

r

~b k k a3

211= − ⋅ (11.3)

and consequently by forming the ratio we obtain

the desired magnitude of the reflection coefficient

b

br

3

3

~ = (11.4)

However, the Equation (11.4) only applies under

the assumptions made above; i.e.

(α) Ideal directional coupler with infinitely high

directivity and free of reflection at its ports.

(ß) The wave b1 exiting port 1 is not reflected bythe circuit connected at port 1.

If condition (ß) is violated, we have a situation

(due to multiple reflection) where in the reference

measurement with short circuit, the wave entering

port 1 is different from the wave entering port 1 in

the measurement of the device under test (DUT).

Under the assumption that an ideal directional cou-

pler is present, a precise mathematical analysis

supplies the equation

b

br

k r e

k r e3

3

2

2

1 1

1 1~ = ⋅

+ − ⋅ ⋅

− − ⋅ ⋅G

j

G j

ψ

ψ

(11.5)

instead of Eq. (11.4). Here r G is that reflection co-

efficient which is present for wave b1 leaving port

1. It can be seen from equation (11.5) that |r G|

must be kept as small as possible to avoid meas-

urement errors. This can be accomplished by, for

example, connecting an isolator or an attenuator

with a ≈ 10 dB in series. If it is assumed that

r G

= 0 but the directivity factor is finite – as in the

case of all real waveguide couplers, then using

equation (11.4) we obtain a maximum error of



7 2


ε

def

=

Db

br

a3

3

2010~

max

−

≈−

(11.6)

which for a directivity of aD = 60 dB is smaller than

0.001, but at a directivity of only 20 dB amounts to

approx. 0.1.

(2) Utilizing two coupling paths

In the measurement configuration according to

Fig. 11.2 a) with one directional coupler, the waves

exiting from port 3 and port 4 are measured andtheir amplitudes compared. The following holds

true for ideal directional couplers:

|b4| = k · |a1|

and

b k k r a32

11= ⋅ − ⋅ ⋅

and thus

b

bk r 3

4

= − ⋅1 2 (11.7)

Since k is unknown, Eq. (11.7) can serve to deter-

mine |r |. In this procedure a reference measure-

ment with a short is not needed and a reflecting

circuit (r G ≠ 0) does not create any fault source (as

|a1| is the same for the measurement of |b3| and

|b4|). In the case of less than ideal directional cou-

plers the use of Eq.(11.7) also leads to errors in the

determination of |r |. When using two separate

directional couplers in the configuration according

to Fig. 11.2 b) coupling errors have less of an

impact on the final result (naturally the coupling

coefficients of the two directional couplers must

be known).

(3) Prospects for more modern measurement

methods of reflectometry

In the previous investigations it was assumed that

only the magnitudes of the waves exiting ports 3 or

4 could be measured and that furthermore only the

magnitude of r could be determined. Nevertheless,

not only the amplitudes but also the phases of the

waves can be compared as, for example, in the

configuration according to Fig. 11.2.

For this you can use a so-called network analyzer,which is a special kind of dual channel receiver, in

which the signals in both channels are converted to

a lower frequency range while maintaining their

amplitudes and phase relationships. This frequency

conversion can be carried out with mixing or

sampling. In the low frequency range the ampli-

tudes and phase relationships of the signals can be

determined by electronic means. One of the great

advantages of using the network analyser is the

possibility of correcting errors arising from compo-

nents with less than ideal characteristics (e.g. finite

directivity of directional couplers) in the mea-

surement circuit. For this several calibration

measurements are first conducted on known com-

ponents and the results entered into a computer.

Since the “faulty” characteristics of the measure-

ment circuit are also contained in the results of the

calibration measurements, they can be corrected

using suitable algorithms.

Required equipment


1 Gunn oscillator 737 011 Diaphragm with slit

2 x 15 mm, 90° 737 22

Fig. 11.2: Measurement configuration 2a) Measurement with a directional coupler b) Measurement with two directional couplers



7 3







2 Sets of thumb screws (12 each) 737 399

For the assembly of the measurement object

or DUT the following is required:

1 Sample holder 737 29

1 Absorbing material sample

(graphite) 737 29



or

3 screw transformer (*) 737 135



plugs, 2 m 501 022


3 Supports for waveguide 737 15


(*) : The 3-screw transformer serves only as an intermediate piece to permit attachment. For this all screws must

be set to a penetration depth of 0. In this case the thumb screw requirement is reduced to 8 each (1 set).

3

1

24

737 14

737 14737 29737 12or

737 135

737 29

Short circuit plate

737 18

737 14737 14

7 3 7 1

8

737 09737 22737 01

GUNN-OSC.

737 01

2 x 15 mm

90°Gunn

WG

coax

SC

ATT

Holder +sampler



7 4


2 Stand rods 0.25 m 301 26

Recommended:


1 Isolator 737 06


When using the isolator and PIN modulator the

experiment setup specified in Fig. 11.3 must be

changed in accordance with the preface!

1. Set up a reflection-free one port as the de-

vice under test, DUT).1.1 In Experiment 6 the reflection coefficient of

a DUT was determined using the slottedmeasuring line. Use the same DUT to permit

a comparison of the reflectometer measure-

ment with the slotted measuring line measure-

ment. In accordance with Fig. 11.4 use the

sample holder 1 , the absorbing graphite sam-

ple 2 and a reflection-free waveguide termi-

nation 3 for set up of the DUT.

2. Measurement of the reflection coefficient

using only one coupling path and com-

parison with reflection at the short-circuit

plate.

2.1 Set up the measurement circuit as specified

in Fig. 11.3. Use the coupling plate with 2

cross-shaped holes in the cross coupler.

Note:

The attenuator set to approx. 10 dB should be

used to reduce the reflection coefficient for

waves propagating out of port 1 and re-

flected backwards to the microwave source.

Port 4 is terminated reflection-free and the

measurement head (transition waveguide/

coax with coax detector) is connected to port

3.2.2 Calibration measurement with short-circuit

plate.

For this attach the short-circuit plate to port

2. After the power supply voltages have been

switched on, calibrate the display of the SWR

meter (port 3) to 0 dB using the “ZERO”

control knob. (Set the modulation to Gunn-

Int., U G to 8 V)

2.3 Replace the short-circuit plate with the DUT.

Read off the display a (port 3) in dB (a < 0).

Fig. 11 .4: Assembly of the device under test (DUT)1 Sample holder 2 Graphite sample3 Waveguide termination

According to Eq. (11.4) a represents thereflection coefficient in dB, i.e.:

a / dB = 20 log |r |.

Calculate |r | and enter the value into Table

11.1.

3. Measurement of the reflection coefficient

when using 2 coupling paths

3.1 The DUT remains connected to port 2.

3.2 First terminate port 3 reflection-free by re-

placing the measurement head (transition

waveguide/coax to the coax detector) withthe reflection-free waveguide termination.

Attach the measurement head to port 4. Set

the display a of the SWR meter (port 4) to a

value of 0 dB using the “ZERO” control knob

and keep this setting steady.

3.3 Reverse the connection configuration of ports

3 and 4, i.e. connect reflection-free termina-

tion to port 3. Read display a of the SWR

meter off in dB (a < 0). According to Eq.

(11.7) the following applies:

a/dB = 10 log (1 – k 2) + 20 log |r |

≈ 20 log |r |

Enter the values determined for |r| in Table

11.1.

Table 11.1



7 5


Experiment

point 2Experiment

point 3

Questions1. Compare all the values determined for |r| to

each other and with the results from Experi-

ment 6 (measurement principle with slotted

measuring line).

2. Compute the ratio of |b4| / |b3| from the con-

figuration in Fig. 11.2 b). The coupling coeffi-cients are k 1 and k 2. What is the ratio for the

special case k 1 = k 2 = k ?

a/dB |r|



7 6




7 7

MTS 7.4.4 The cavity resonator

The cavity resonator

FundamentalsFirst of all a completely enclosed cavity resona-

tor is considered with (theoretically) an ideally

conductive metallic surface. In this resonator

unattenuated electromagnetic oscillations can ex-

ist at discrete frequencies (= resonance frequen-

cies). Fig.12.1 shows the TE 101 resonance in a

rectangular waveguide resonator with the fre-

quency:

f

l

0

2 2GHz15

1

a

cm

1

cm

= ⋅

+

(12.1)

Due to the losses in the metallic walls of real reso-

nators, natural oscillations are attenuated. One

measure for attenuation in a cavity resonator is its

unloaded Q value:

QW

P0

0

V

=

⋅ω

(12.2)

Here W is the stored energy, ω 0 = 2 π f 0 is the

(angular) resonance frequency and P V is the

power dissipation. Fig. 12.1 shows the electro-

magnetic field in the cavity resonator at various

points in time (T = 1/ f 0). In order to couple a

resonator to a microwave circuit, you can use a

diaphragm with aperture, for example, as illus-

trated in Fig. 12.3a. Fig. 12.3b shows the corre-

sponding waveguide equivalent circuit diagram

in which the resonator is represented by a short-

circuited waveguide and the diaphragm with ap-

erture by a shunt inductance jω L = jX . Fig. 12.4shows Z 0/ X as a function of the relationship of the

aperture diameter d to the waveguide width a

(22.86 mm). If you mathematically convert the

parallel connection of the resistance Z 0 and jX

into a series connection, the result is an equiva-

lent circuit diagram for X << Z 0 as specified in

Fig. 12.2.

In accordance with the transformation ratio:

n Z

X

=0

(12.3)

the resistance Z 0 is transformed to lower resist-

ance Z ’0 = Z 0/n². Thus, the diaphragm with aper-

ture can also be understood as a transformer. If we

suppose that the reactance jX is taken (or ignored)

Fig. 12.1: TE 101 resonance in a rectangular cavity resona-tor

Fig. 12.2: Series equivalent circuit diagram for resonatorswith coupling diaphragm



7 8


Fig. 12.3: Development of an equivalent circuit diagramfor the cavity resonator coupled via a diaphramwith aperture

Fig. 12.4: Normalized transformation ratio for thecoupling diaphragm Z 0/ X . λ correspondshere to the free-space wavelength λ 0

Required equipment



1 Diaphragm with slits

2 x 15 mm, 90° 737 22






1 Accessories waveguide propagation 737 29

1 Moveable short 737 10




plugs, 2 m 501 022


2 Supports for waveguides 737 15

1 Stand rod 0.25 m 301 26

Recommended:


1 Isolator 737 06

together with the waveguide, we obtain the

equivalent circuit depicted in Fig. 12.3 c). At ap-

proximately the resonance (angular) frequency

2 π f 0 the λ g/2 long waveguide can be designed by

a resonance circuit with:

1

2

1

2

0

0 0

02

c2

0

c

0

π

π ω

ω ω

π

ω ω

⋅

=

⋅

≈ ⋅

⋅

−

≈ =

f L C

LZ Z X 0

and

R X

QV

c

0

= (12.4)

Finally, we obtain for resonance ω = ω 0 the

equivalent circuit diagram according to Fig.

12.3 e).



7 9




Note:

When using the isolator and PIN modulator

modify the experiment setup shown in Fig. 12.5 in

accordance with the preface!

1. Set up the experiment according to Fig.

12.5. Use the coupling plate with 2 cross-

shaped holes in the cross directional coupler. Set the attenuator to approx. 10 dB

(see Experiment 11, subpoint 2.1).

2. Calibration

Determination of the reflection coefficient is

performed like in Experiment 11, point 1,

i.e. using only one coupling path and then

comparing the results to the reflection

ocurring during short-circuit.

Thus for calibration insert the moveable

short with the diaphragm d = 6 mm into the

experiment setup and set the moveable short

to 0 mm (i.e. implement short in front of port 2). The voltage displayed for this at the

SWR meter corresponds to the reflection



8 0


coefficient |r | = 1 and can thus be used for

calibration. Set the SWR meter to 0 dB us-

ing the “ZERO” control knob. As a further

reference for the matching case you can in-

sert a reflection-free waveguide termina-tion instead of the moveable short.

3. Resonator measurementsInsert the moveable short with diaphragms

(one after the other d = 6, 7, 9, 10 mm).

Change the position of the moveable piston

and observe how the reflection coefficient is

dependent on the setting of the micrometer.

Determine the minimum of |r | = |r |min for

each diameter of the diaphragm aperture.

Enter the values into Table 12.1.

Using Fig. 12 .4 carry out the exact deter-mination of the value n corresponding to

each aperture diameter d and enter it into

the table. (First compute the parameter

value λ 0/a, ( f ≈ 9.4 GHz, a = 22.86 mm),

but bear in mind the logarithmic scaling).

Sketch the measured values of |r | as a

function of n/n0. Here n0 is the value cor-

responding to |r | = 0. Assume that you

achieve matching at an aperture diameter

of d = 8 mm (standard diaphragm aperture

of the Gunn oscillator).

Questions

1. Compute the value of the transformation ra-

tio n = n0 using the equivalent circuit dia-

gram according to Fig. 12.2 e) so that the

reflection coefficient becomes r = 0. For

this use approximately X c = π · Z 0 and enter

the result in the form n0 = f (Q0).

2. Compute and plot the characteristic magni-tude of the reflection coefficient |r | in dB as a

function of n/n0 in the range 0.1 to 2.5.

3. Estimate the unloaded Q0 of the resonator

using the experiment results according to 3

and the relationship between the aperture

diameter d and the transformation ratio n in

Fig. 12.3.

Table 12.1

n0 = ___________

6

7

9

10

r min

dBr

mind / mm

nn

n0



8 1

MTS 7.4.4 Solutions

Solutions

Experiment 1 sample solution

1.3 Waveguide width: a’ = 23 mm

s’ = 21.8 mm

2.7 See Table 13.1.1 and subsequent

diagrams

3.1-4.2 see Table 13.1.1

Table 13.1.1

with diaphragmwith rear panel without diaphragmwith rear panel with diaphragmwithout rear panel

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.5 0.0 44.3 0.0 47.3 0.0 46.5

1.0 0.0 90.0 0.0 90.0 0.0 90.5

1.5 0.0 126.6 0.0 126.0 0.0 126.2

2.0 0.0 153.3 0.0 152.9 0.0 152.7

2.5 0.0 169.2 0.0 169.1 0.0 168.73.0 0.0 174.3 0.0 174.0 0.0 174.1

3.5 0.2 156.8 0.0 168.0 0.0 167.4

4.0 19.6 152.9 0.0 163.7 0.0 162.3

4.5 40.1 148.4 0.0 159.6 0.0 159.8

5.0 55.0 145.8 0.0 158.0 0.0 154.5

5.5 66.5 144.5 33.5 151.4 0.0 152.6

6.0 76.0 143.0 30.0 149.3 0.0 151.1

6.5 82.1 142.2 19.7 147.9 0.0 150.3

7.0 86.5 141.0 7.2 147.2 63.2 144.6

7.5 90.1 140.0 0.0 147.3 69.2 143.7

8.0 94.3 138.6 0.0 147.2 74.8 143.1

8.5 95.6 137.3 0.0 146.4 79.8 142.0

9.0 97.6 135.8 0.0 145.1 84.7 140.4

9.5 99.4 134.6 0.0 142.8 90.4 138.4

10.0 100.0 133.1 0.0 141.0 94.7 136.4

U G

mV

I G

mA mV

I G

mA

I G

mAmVV

U D U D U D



8 2

MTS 7.4.4 Solution

Diagram 13.1.1:Gunn current and detector voltage as a function of theGunn voltage in the case with diaphragm and with rear wall.

A component which emits power can be consid-ered as a negative resistance in the equivalent cir-

cuit diagram. This can also be discerned in the

characteristic of the Gunn oscillator. Starting with

the voltage at which power is emitted, the charac-

teristic of the Gunn current demonstrates a nega-

tive differential resistance (negative slope).

Diagram 13.1.2:Gunn current and detector voltage as a function of theGunn voltage in the case without diaphragm and with rear wall.

Diagram 13.1.3Gunn current and detector voltage as a function of theGunn voltage with diaphragm but without rear wall.

Notes :

The characteristics are subject to strong compo-

nent tolerances (primarily the Gunn diode). Con-

sequently the characteristics shown here are only

meant as examples for orientation.

Sudden discontinuities can arise in the detector

voltage. During these discontinuities the emitted

spectrum of the oscillator varies. For small Gunn

voltages a broad spectrum of frequencies are emit-

ted, while normally only two frequencies exist [a

dominant mode at approx. 9.4 GHz (TE 101) and a

mode normally attenuated in amplitude by

20 dB at approx. 18.8 GHz (TE 202)] for higher

voltages. (See also “Design of the Microwave

Source” from the preface).

Frequently differences can be distinguished as towhether the characteristic is taken for rising Gunn

voltage or for falling Gunn voltage U G. The above

mentioned discontinuities or sudden steps only

arise in the characteristic of falling U G in the

range of low Gunn voltage “hysteresis effect”.

This can be attributed to the fact that more energy

is needed to generate mode distribution (higher

U G), than is required to maintain it.

If high reflections are allowed to have an

unattenuated impact on the Gunn diode, then its

general operating response is severely altered. A

measurement curve which has been recorded

without the attenuator has been depicted in Dia-

gram 13.1.4. Extremely disadvantageous in this

case was the fact that a pure spectrum only set in

right under 10 V after the last discontinuity. If af-

terwards you reduce the Gunn voltage again,

mode distribution is maintained until just above

5 V. For that reason it is always favorable to

work with high Gunn voltages, i.e. first increase

the Gunn voltage to 10 V then subsequently set

it to the desired value (e.g. 8 V).

Diagram 13.1.4:Gunn current and detector voltage as a function of theGunn voltage for the case of strong reflections



8 3

MTS 7.4.4 Solutions

Answers

1.

f

a s

0

2 2GHz 15

1

cm

1

'

cm

GHz= ⋅

+

='

.9 48

2.

U TH = 4 V ⇒ E TH = 4 · 10 –3 kV/10 –3 cm

= 4 kV/cm

3.

f s

s

= = =−

10

10

101

107

3

10cm

cm

GHz /

4. In the current experiment we will use config-

uration B from Fig. 1.2 for the resonator op-

eration of the Gunn element. The resonance

field is mainly located between the stud for

the Gunn element (part 1 in Fig. 1.2) and the

diaphragm with aperture. This means that if

only the housing's rear wall is removed, the

characteristics compared to the case with

rear wall and diaphragm do not change that

much. The situation is different if the diaphragm is removed, as now there is no cav-

ity resonator coupled to the Gunn element.

In the case without the diaphragm (experi-

ment part 3) there are only relatively small

discontinuities in the falling part of the

I G (U G) curve and the radiated microwave

power is also only relatively low. In the case

with the diaphragm (Experiment 2 and 4) a

cavity resonator exists. Here the radiated mi-

crowave power is considerably higher than

in experiment part 3. The onset of the oscil-lation can be noticed by the powerful and

abrupt drop (discontinuity) in the I G (U G)

characteristic at U G ≈ 4 V.


Answers

1. In the case of direct modulation of the Gunn

diode, modulation is carried out along the

static characteristic U D (U G) from Experi-

ment 1, Diagram 13.1.1 (i.e. Table 13.1.1).If the characteristic has sudden discontinui-

ties, a small change in U G results in a con-

siderable change in the emitted microwave

power. This can be observed in points 1.4

and 1.5 (in this context refer also to “Design

of the Microwave Source” from the pref-ace).

Normally the maximum receiving signal lies

at a supply voltage of approx. 4 V, meaning

the voltage at which the Gunn diode begins

emitting microwave power and where the

static characteristic begins showing the

greatest discontinuity.

Note:

At a voltage of approx. 4 V the Gunn diode

emits a whole series of frequencies (see pref-

ace). Because spectral purity is normally de-sired in the following experiments, the

working point must be set to approx. 8 V (or

higher) for Gunn internal modulation. There

the static characteristic is normally very flat

and thus the modulated signal (AC compo-

nent) only very small.

If you have a PIN modulator at your dis-

posal, use it instead as it provides a consid-

erable improvement for generating a mod-

ulated microwave signal.

Diagram 13.2.1 graphically demonstrates the rela-tionship of the receiving voltage as a function of

the Gunn voltage during direct modulation. For

this the detector was connected to the input of the

SWR meter and the output signal U Amp Out of the

SWR meter observed. The maximum value was

calibrated to approx. 0 dB on the scale of the

SWR meter. The diagram confirms the technical

explanation given above.

The maximum from Diagram 13.2.1 coincides

with the discontinuities from Diagram 13.1.1.

Diagram 13.2.1

Output signal of the SWR meter as a function of the Gunnvoltage during direct modulation of the Gunn oscillator



8 4

MTS 7.4.4 Solution


Table 13.3.1


Table 13.4.1: Calibration of the attenuator

a / dB x / mm

0 0.00

–1 1.28

–2 1.75

–3 2.05

–4 2.32

–5 2.55

–6 2.74

–7 2.91

–8 3.04

–9 3.17

–10 3.3

–12 3.53

–14 3.74

–16 3.97

–18 4.23

–20 4.59

— : Noise

Responses

1. see column 4 from Table 13.3.1

2. A comparison between columns 2 and 4

shows real good agreement.This shows that in the current level range a

description of the display a can be obtained

with the relationship

aU U

U U

= ⋅−

−( )

10 logmax min

max min ref

0 0.0 25.6 0.0

0.5 –0.9 21 –0.86

1 –3.0 13 –2.94

1.5 –6.2 6.2 –6.16

2 –11.0 2 –11.07

2.5 –16.5 — —

3 –22.0 — —

x 0

cm

a

dB

∆U x 0( )

mV

10 log(0)

0⋅

∆

∆

U x

U

( )

Diagram 13.4.1Graphic display of the values from Table 13.4.1(1): Measurement curve recorded with a microvoltmeter from Rohde&Schwarz(2) and (3): Measurement curves recorded with twodifferent coax detectors

The reference values on the attenuator were deter-

mined using a microvoltmeter.



8 5

MTS 7.4.4 Solutions

Deviations from these values are based on the

characteristics of the respectvie detector.

1) Verifying the square law

Specified value on the attenuator used:3 dB at x 0 = 2.12 mm

→ Meas. value at x0: 3.2 dB

Answer

1. The power can be further reduced using

• The out coupling via a slotted measur-

ing line (coupling attenuation of approx.

20 dB). The slotted measuring line is

terminated with low-reflection wave-

guide termination.

• The use of a circulator (see MTS 7.4.5

Experiment 8): Feeding at port 1, con-nect e.g. a 3-screw transformer plus

waveguide termination to port 2 and

couple this out to port 3. With the 3-

screw transformer (+ waveguide con-

nection) a reflection coefficient can be

set. The smaller it is, the lower the

power coupled out at port 3.

• The use of a 3-screw transformer as an

inserted two-port to generate a larger reflection coefficient (around 1 in terms

of magnitude) so that only a small

portion of the power is transmitted.

Disadvantage:

The reflections return to have an

impact on the microwave source.



2.4 see Table 13.5.12.5 ∆ x = 22 mm

3.2 ∆ x ≈ – 6 dB

3.4 see Table 13.5.2

= ⋅−( )

200

log

max

U x x

U

U x x

U

−( )0

max

cos2

0

π

λ g

⋅ −

x x

Display in dB

0 0 0.997 1.000

2 –0.4 0.956 0.959

4 –1.5 0.837 0.841

6 –3.7 0.656 0.655

8 –10.4 0.303 0.415

10 –16.6 0.149 0.142

12 –11.1 0.279 0.142

14 –5.5 0.529 0.415

16 –3.1 0.702 0.655

18 –1.7 0.826 0.841

20 –0.7 0.925 0.959

22 –0.1 0.987 1.000

24 –0.8 0.915 0.959

26 –2.1 0.786 0.841

28 –5.1 0.556 0.655

Minimum at 10.34 –17.7 0.131 0.093

Probe position

x x− 0 in mm

Distance of the minima

∆ x / mm= 22

λ g = 44 mm

Table 13.5.1



8 6

MTS 7.4.4 Solution

Table 13.5.2

Display in dB

0 –0.5 0.940

2 –0.2 0.979

4 0.0 1.000

6 –0.3 0.971

8 –1.0 0.894

10 –1.3 0.861

12 –1.5 0.845

14 –1.6 0.835

16 –1.8 0.812

18 –1.9 0.804

20 –1.7 0.820

22 –1.4 0.852

24 –1.2 0.867

26 –1.3 0.865

28 –1.3 0.856

Probe position

x x− 0 in mm = ⋅−( )

200

logmax

U x x

U

U x x

U

−( )0

max

Answers

1. From ∆ x = 22 mm there follows λ g = 44 mm,

because the maxima and minima have a dis-

tance of λ g / 2.

2. Relationship between λ g and the free space

wavelength λ 0 according to Equation (5.14):

λ λ

λ

g0

0

2

12

=

−

a

Resolving the equation with respect to λ 0at a = 22.9 mm results in

λ λ

λ 0

2

44

144

45 8

31 7=

+

=

+

=g

g

2

12

mmmm

a .

.

and

f c

= = =0

0

300

31 79 46

λ ..GHz GHz

There is very good agreement with the

base frequency of the Gunn oscillator

(9.4 GHz).

3.

υ

λ

ph

02

12

s

=

−

= ⋅ = ⋅c

a

cm0

081 386 4 16 10. .

ß = = =− −2

0 143 1 431 1π

λ g

mm cm. .

4.



8 7

MTS 7.4.4 Solutions

f f c

a

f f

c c

c c1

m / s

mGHz

GHz

= =⋅

=⋅

⋅ ⋅≈

= ⋅ ≈

−1

08

3

2

2

3 10

2 22 9 106 55

2 13 1

..

.

5. See 3rd column from Table 13.5.1

6. See 4th column from Table 13.5.1

The measured values deviate from the calcu-

lated ones (sometimes they are larger, some-

times they are smaller). Possible causes lie in

the field distortions caused by the probe.

7. See Table 13.5.2

8. According to Figures 5.2 and 5.3 in the case

of short-circuit termination U max is twice as

high as the voltage of the interfering waves,

i.e. ∆a = –6 dB, this corresponds to a volt-

age ratio of 0.5.

If the measured value deviates from this,

the reason is that due to multiple reflections,

the strength of the wave incidenting at the

reflection-free termination differs from the

wave incidenting at the short.

In this measurement example the voltagevaries by 1.9 dB according to the measure-

ment values from Table 13.5.2.

Note:According to the standing wave ratio

sU

U

=max

min

,

at 1.9 dB this yields a standing wave of

s = =10 1 24

1 9

20

,

.

In the best case you can expect a variation of

approx. 0.4 dB. This corresponds to a stand-

ing wave of s = 1.05, i.e. a reflection coef-

ficient of r = 0.025.

Fig. 13.5.1Sample measurement to point 2.4 and 3.4, detector voltagelogarithmically plotted

Fig. 13.5.2Sample measurement to points 2.4 and 3.4, detector voltage plotted linearly

Fig. 13.5.3Sample measurement to points 2.4 and 3.4, detector voltage plotted logarithmically.Spectrum with TE 10 and TE 20 modes.



8 8

MTS 7.4.4 Solution

Fig. 13.5.4Sample measurement to points 2.4 and 3.4, detector voltage plotted logarithmically.Experiment setup with PIN modulator and isolator.

Fig. 13.5.5

Sample measurement to points 2.4 and 3.4, detector voltage plotted logarithmically.Use of the diaphragm with slit 2 x 10mm 90°. Dominantfrequency at 14.1 GHz.

The curves have been recorded using the CASSY

program and except for curve 13.5.2 the logarithm

taken subsequently.

Fig. 13.5.1 and 13.5.2 (plotted logarithmcially

and linearly) are examples where the spectrum is

dominated by the TE 10 mode, meaning we obtain a

quasi-pure spectrum. This cannot always be at-

tained due to equipment tolerances and local dis-continuities in the experiment setup. In the rela-

tively compact setup where the total length of the

installation lies in the area of only approx. 5 λ g,the local discontinuities (interference) have a di-

rect impact because they can influence the com-

ponents of the experiment setup almost unattenu-

ated (Gunn oscillator, detector etc.).

Fig. 13.5.3 shows the case where besides the

TE 10 mode also the TE 20 mode predominates. A

prediction of the precise characteristic is hardly

possible as the amplitude relationship of the in-

dividual modes is not exactly known and the phase angles of the modes are also unknown.

The function characteristic shows a periodicity,

but due to the short measurement line, only a

small portion of this period can be seen. Which

section this is depends primarily on the unpre-

dictable phase shift between the individual

modes, which constantly vary from experiment

setup to experiment setup.

You can expect less problems with the spectral

purity when using the PIN modulator to modulate

the microwave source (in this context also refer to“Design of the Microwave Source” from the Pref-

ace). Fig. 13.5.4 shows a measurement curve re-

corded using the measurement setup by adding

the PIN modulator and isolator.

By using the diaphragm with slit 2 x 10 mm 90°

instead of the diaphragm with slit 2 x 15 mm 90°,

a higher frequency can be separated for a suitable

setting of the Gunn voltage (U G ≈ 4 V, i.e. at a dis-

continuity of the function U D (U G) from

Expriment 1). Here the Gunn oscillator must be

internally modulated (i.e. without the PIN modu-

lator). The diaphragm with slit 2 x 10 mm 90°represents a filter for frequencies between 13 to

17 GHz. The success of the experiment primarily

depends on which frequencies the Gunn oscilla-

tor generates at the “discontinuities”. It is required

that only a dominant mode is generated in the

transmission band of the diaphragm with slot. If

this is not the case, you will not obtain similar re-

sults as found in Fig. 13.5.5.

Experiment 6 sample solutionto 2.) See the first line of Table 13.6.1

to 3.2) and 3.3) See the second line in Table 13.6.1

to 3.4) See Table 13.6.1

to 3.5) See Table 13.6.2



8 9

MTS 7.4.4 Solutions

∞

DUTLocation of the

Minimum/mm s |r|

Short-circuit plate 69.3 1 180

Measurement

object A76.8 4.2 0.615 5.73

∆ x = 7.5 mm

r s

s=

−+

1

1

Table 13.6.1

Note:

The phase primarily depends on the position of

the sample in the sample holder. It becomes

smaller when the sample is moved closer to the

measurement line.

φ λ

= − ° + °720 180

g

∆ x

∆l /mm s |r|

Measurement object A 3.4 4.28 0.621

Table 13.6.2

λ g = 44 mm Answers

to 1.

Starting from the complex reflection coefficient

r ≈ 0.62 · e j57°

the related termination impedance results at

Z

Z

j

0

0 85 1 48≈ +. .

read off the Smith-Chart (similar to Fig. 6.4)

mathematically there results a value of

Z

Z

r

r

j

0

1

10 868 1 467=

+

−≈ +. .

x U

x U

s

l

l

1

2

2 75 1

2 78 5

min

min

.

.

sin

( ) =

( ) =

=

+

mm

mm

1

sin

2

g

g

π

λ

π λ

∆

∆

φ /degree



9 0

MTS 7.4.4 Solution


3-screw transformer

Table 13.7.1

Short-circuit plate x0 = 59.8 mm

a) Measurement object without 3-screw transformer

VSWR

s

4 67.5 0.6 54°

b) Calibrate to s → 1, i.e. |r | → 0 (ideally:

s ≈ 1, |r | ≈ 0) by successively turning allthe screws.

(in the test measurement the optimum of the

calibration was at s = 1.06, i.e. |r | = 0.03)

c) Backwards reflection coefficient Γ (see Fig.

7.1)

VSWR

s

3.9 74.8 0.6 –65.4°

Minimum at

x1/mmΓ =

−+

s

s

1

1φ

λ Γ = °− °⋅

−180 720

1 0 x x

g

The value of φ Γ

corresponds approximately to

the negative value of f = 54° from the above

measurement of the measurement object

Slide screw transformer

Table 13.7.2

x0 = 59.8 mm (see above)

a) Matching ( s ≤ 1.1)

VSWR

s

2.25 71.8 0.385 –16.4°

Minimum at

x1/mmΓ =

−+

s

s

1

1φ

λ Γ = °− °⋅ −

180 7201 0 x x

g

The value of φ Γ

is only in partial agreement in

the sample solution (here too the magnitude of

Γ is not the same). This can be attributed to the

fact that in this case the optimum has not been

reached, although matching is very good. Ac-

cording to theory the slide screw transformer has

several constellations in which matching is

guaranteed. Furthermore, matching can only

be achieved by setting the metal stud to a very

deep penetration depth in the waveguide. The

result is that the stud no longer functions like a

simple shunt capacitance.

φ λ

= °− °⋅−

180 7201 0 x x

g

Minimum at

x1/mmr

s

s=

−+

1

1



9 1

MTS 7.4.4 Solutions

Fig. 13.7.1Recording of the detector voltage via the slotted measuringline when terminated using(a) short, (b) measurement object A from Experiment 6

Fig. 13.7.2Detector voltage recorded via the slotted measuring line

when using the 3-screw transformer (a) no matching, (b) “matching” at s = 1.06

Fig. 13.7.3Detector voltage recorded via the slotted measuring linewhen using the slide screw transformer (a) no matching (b) “matching” at s = 1.05

It is shown that the 3-screw transformer and the

slide screw transformer are well suited to match a

load to a source. However, finding the optimum

setting for the transformers is not always easy and

there are several combinations particularly with

the slide screw transformer. A calculation of the

parameters needed for matching is not possible

without some effort because in practice the metal

studs do not act like simple shunt capacitances.


Table 13.8.1

a) Standing wave ratio and reflection coefficient. No matching.

Antenna type s |r|

A 1.8 0.286

B 4.2 0.615

C 14.1 0.868

Optional :

b) Receiving level at the E-fiel probe in unmatched case. No matching.

Antenna type

A 0° 20 –3 –23 0

B +45° 25 –2 –27 –4

C 0° 35 0 –35 –12




9 2

MTS 7.4.4 Solution

Table 13.8.2

a) Standing wave ratio and reflection coefficient for matching.

Antenna type s |r|

A 1.2 0.091

B 1.3 0.13

C 1.8 0.286

Optional :

b) Receiving signal level at the E-field probe for matching.

The “absolute” level is calculated at: a’ = a – V

Antennentyp

A 0° 20 –3.8 –23.8 –0.8

B +45° 20 –3.8 –23.8 –0.8

C 0° 25 –1.5 –26.5 –3.5


ated. As a consequence of this, most of the

power must be reflected.

2. With a (theoretically) ideal matching net-

work it is possible to compensate for any re-

flection coefficient |r | < 1 (see also

Fundamentals from Experiment 7).

However, in the case of compensation (i.e.

matching) of a one-port with a reflection co-

efficient in “close” proximity to 1 you have

the following disadvantages:

(a) Compensation has “extremely nar-

row” bandwidth.

(b) The loss mechanisms (tranforming

power into thermal energy) which are

always present in real matching net-

works have even greater impact, the

higher the reflection coefficient |r |

needing compensation.Thus, in accordance with fact (b), although

the input reflection coefficient is zero when

matching for |r | ≈ 1, the power of the

incidenting wave is to a great extent not

transmitted further to the load (|r |), but in-

stead converted into heat in the matching

network.

3. The polarisation of the radiated field is not

determined by the direction of the E-field of

the guided wave, but by the direction of the

E-field in the slit. In the case of the slit ro-tated by 45° (version B), also the direction of

the E-field and thus the direction of the

equivalent magnetic current are rotated

The maximum level here is a’ max = –23 dB (an-

tenna version A, unmatched)

Answers

1. The resonance frequency f 0 of the slit con-

figuration is

f c

w f c

w0

0

00

2 2

= ≈ ⇒ ≈⋅λ

λ where

Thus you obtain the following resonance fre-

quency for cases A and B where w = 15 mm

f

w

0 300

210

(A,B)

GHz mm

≈

⋅

=

/

whereas in case C with w = 10 mm you ob-

tain

f 015

(C)

GHz≈

When the frequency of the exciting wave

is f ≈ 9.4 GHz, the resonance frequency in

cases A and B lies in “close” proximity to

f , resulting in “strong excitation” (reso-

nance step-up) of the field in the slit. In

contrast, case C has a very weak field ex-

cited in the slit due to the considerable de-

viation between f and f 0(c). Because the

radiated power is proportional to thesquare of the magnitude of the electric

field strength in the slit, only a very small

portion of the incidenting wave is radi-



9 3

MTS 7.4.4 Solutions

(see Fig. 8.1 and 8.2).

4. In both cases, matching measures are car-

ried out to compensate for the reflection so

that more power is available to the antenna

for radiation.

5. It is demonstrated that when matching an-

tenna version A (Table 13.8.2 b) ) the receiv-

ing signal level drops, i.e. the advantages

attained through matching (here from s =

1.8 to s = 1.2) are cancelled out again be-

cause the transformer does not operate

loss-free. On the other hand, the matching

of antenna B results in the desired effect.

In the case of antenna C it is demonstratedthat the matching described in question 2, is,

on the one hand, not easy to set (narrow

bandwidth) and the losses in the trans-

former can be significant.

Sample I (Polystyrol) Sample II (Absorber material)

158 19.4

22.1 25.7

0.987 0.9

212° 154.3°

–0.84 – j 0.52 –0.813 + j 0.39

from the Smith chartor mathematically

0.007 – j 0.29 0.054

178 11.2

30.6 21.7

0.989 0.836

75.5° 218.6°

0.25 + j 0.96 –0.654 – j 0.521


or mathematically

0.015 + j 1.29 0.1 – j 0.347

2.69 – j 0.033 11.83 – j 0.55

1.86 – j 0.017 6.524 – j 0.281

0.009 0.043

VSWR sA

xA/mm

r s

sA =

−

+

Α

Α

1

1

φ λ

AA

g

= °− °⋅−

180 7200 x x

r r jA A A A= ⋅ + ⋅( )cos sinφ φ

VSWR sB

xB/mm

φ λ

BB

g

= °− °⋅ −180 720 0 x x

r s

sB =

−

+

Β

Β

1

1

r r jB B B B= ⋅ + ⋅( )cos sinφ φ

Z

Z

r

r

B B

B0

1

1

=+

−

Λ =

⋅

Z

Z Z

02

A B

ε r = ε ' r – jε r ''

Z

Z

r

r

A A

A0

1

1=

+

−

tan δ ε = ε r ''/ε r '


Table 13.9.1

x0 = 24.1 mm x1 = 68.9 mm

λ g = 44.8 mm ⇒ λ 0 = 32 mm



9 4

MTS 7.4.4 Solution

Equation for determining the free-space wave-

length

λ λ

λ 0

g

g

2

12

=

+

a

It is shown that the absorber material has a

greater dissipation factor than the polystyrene

sample, and a considerably larger relative per-

mittivity ε r ’ as well.

Note :

•

The values obtained for ε r ’ and ε r ’’ (i.e. tanδ e) can vary due to manufacturer toleranc-

es. The values provided in the sample so-

lution are only meant as examples.

• When measuring the standing wave ratio

of sample I the problem could arise (de-

pending on the coax detector) that the

level cannot be detected in the minimum

because the detector supplies a signalwhich is too small and the gain of the

measurement amplifier does not suffice to

make it measurable.

• Furthermore, in the case where standing

waves exist with a considerable magni-

tude, it proves difficult to determine the

position and the level of the minimum.

The measurement is then also burdened

with even greater inaccuracies.

• If excessive measurement errors arise, an

indicator for this is when you obtain nega-tive values for ε r ’’. (Keep an eye on the

sign in ε r = ε r ’ – jε r ’’ ⇒ ε r ’’= – Im(ε r ) ).


Table 10.1

1 Round hole

1 Cross-shaped

hole 2 Round holes

2 Cross-shaped

holes

Feed »0 »0 »0 »0

via –38.5 –43 –36.5 –38.8

port 1 –27.3 –24.4 –22.1 –18

Feed »0 »0 »0 »0

via –26.7 –24.4 –21.9 –17.8

port 2 –36 –41.5 –41 –44

Port 1 : 11.2 18.6 14.4 20.8

Port 2 : 9.3 17.1 19.1 26.2

|S 21| / dB

|S 31| / dB

|S 41| / dB

|S 12| / dB

|S 32| / dB

|S 42| / dB

|aD| / dB

|aD| / dB

Note: directivity

aS

S

a S S

S S

D

D

log

log

dB dB

= − ⋅

⇒

= − ⋅ + ⋅

= −

20

20 20

31

41

31 41

41 31

log

It is demonstrated that when the ports are inter-

changed the coupling coefficients are compa-

rable (|S 41| ≈ |S 32|), while the levels at the

isolation paths differ considerably. This is due

to the fact that the level at the isolation path

(relatively low) responds sensitively to

asymmetries. Even slight asymmetries cause

differences in the levels.The absolute measure-

ment error becomes larger, the smaller the sig-

nal being measured becomes (i.e. the greater

the gain of the measurement amplifier be-comes).



9 5

MTS 7.4.4 Solutions

Exercises

1. ak = 20 dB ⇒ k = 0,1 (Eq. 10.1)

P 2,out = (1 – k 2) P 1,in

= [1 – (0.1)2] · 1 W

P 2,out = 0.99 W

α) and ß)

P 4,out = k 2 P 1,in = 0,01 W

α) |r | = 0 ⇒ P 1,out = 0, P 3,out = 0

P 3,out / P 4,out= 0

ß) |r | = 0.5 ⇒ P 2,in = |r |2 P 2,out

= 0.52 · 0.99 W

P 2,in = 0.2475 W

P 1,out = (1 – k 2) P 2,in = 0.245025 W

P 3,out = k 2 · P 2,in = 0.002475 W

P 3,out / P 4,out= 0.2475

Sample (energy balance): Dissipated power ab-

sorbed by the four-port

P diss = P 1,in + P 2,in – ( P 1,out + P 2,out

+ P 3,out + P 4,out)

= [1 + 0.2475 – (0.245025

+ 0.99 + 0.002475

+ 0.01)] W

= (1.2475 – 1.2475) W = 0 W


Table 13.11.1

Experiment

point 2 –3.6 0.66

Experiment

point 3 –4.2 0.616

Supplement to experiment point 3

If you want a more accurate calculation and not

disregard k , then k can be computed out of | s41|

from Experiment 10. It holds true that:

k

S

= 10

41

20 .The result out of the sample solution

from Ex10 is k = 0.126 ;thus our calculation is

r

a k

= =− ⋅ −( )

10 0 62

10 1

20

2log

.

Answers

1. Sample solution from Experiment 6 re-

sulted in |r | = 0.615 resp. 0.621.

The values measured here from 0.66 resp.0.616 are in very good agreement, but

even larger deviations are acceptable due

to the limited measurement accuracy.

2.

b k a

b k k a r k

4 1 1

3 12

22

1 21 1

= ⋅

= − ⋅ − ⋅ ⋅ ⋅

from this it follows that

b

br

k

k k k

b

br k k k k

3

4

2

1

12

22

3

4

21 2

1 1

1

= ⋅ ⋅ − −

= ⋅ −( ) = =for

a/dB |r|



9 6

MTS 7.4.4 Solution

Experiment 12 sample solutionλ 0 / a ≈ 32 mm / 22.86 mm ≈1.4

d/mm d/a n

6 0.262 21

7 0.306 12.5

8 0.35 8

9 0.394 5.5

10 0.437 3.8

d = 8 mm and d /a = 0.35 ⇒ n0 = 8

Table 13.12.1

6 –1.5 0.84 21 2.63

7 –6.4 0.48 12.5 1.56

9 –9.5 0.33 5.5 0.6910 –5.75 0.52 3.8 0.48

d / mmn

n

n0

r min

dBr

min

1.

r Z Z

Z Z

n R Z

n R Z

nQ

nQ

r n nQ

=−

−=

−

+=

−

+

= = =

0

0

20

20

2

0

2

0

00

1

1

0

V

V

for

π

π

π

2.

r

n

n

n

n

=

−

+

0

2

0

2

1

1

3. Q0 = π · n0²

where n0 ≈ 8⇒ Q0 ≈ 201

Diagram 13.12.1

Graphic representation of | r(n/n0) | in dB and measurement points from Table 13.12.1



97

MTS 7.4.4 Index

Index

Coplanar line ..................................................... 27

Coupling attenuation ......................................... 65

Coupling coefficient ................................... 54, 65

Coupling path .................................................... 65

Cross directional coupler ........................... 65, 71

Short-circuit plane ...................................... 12, 59

Moveable short .................................................59

L

Transmission line equivalent circuit diagram ....59

Transmission line resonator .............................. 53

Line wavelength ................................. 28, 34, 54Leitungswellenwiderstand .................. 28, 39, 43

Optical waveguide ............................................ 27

Lock-in amplifiers ............................................. 20

M

Material parameter ........................................... 59

Multi-screw transformer ...................................46

Slotted measuring line .......................................27

Metal semi-conductor junction.......................... 17

N

Network analyzer ............................................. 72

Zero position search program ...........................60

Wanted signals ..................................................19

O

Surface waveguide ........................................... 27

P

Phase velocity ...................................................28

Phase constants ................................................ 28

Phase shift ........................................................ 40

Phasen selectivity ............................................. 21

R

Space-charge instabilities ................................. 11

Noise .................................................................19

Rectangular waveguide ...................... 23, 29, 54

Reflectometer ............................................ 65, 71

Reflection coefficient ............ 23, 37, 43, 59, 71

Directivity .........................................................66

Directivity factor ............................................... 66

Directional waveguide coupler .................. 65, 71

Backwards reflection coefficient ..................... 43

S

A

Amplitude modulation ....................................... 17

Matching condition............................................ 43

Matching network ...................................... 43, 44

Matching point .................................................. 44

Enrichment layer ............................................... 11

Antennas .................................................... 17, 53

C

Cut-off frequency ............................................. 21

DAttenuation........................................................23

Attenuator ......................................................... 23

Detector probe .................................................. 31

Dielectric waveguide ........................................ 27

Relative permittivity ................................... 27, 59

Differential mobility .......................................... 11

Dipole ......................................................... 31, 66

Dipole antenna ................................................ 17

3-screw transformer ......................................... 46

Drift velocity .....................................................11

Primary path ..................................................... 65

E

Effective mass .................................................. 11

F

Enhanced field strength .................................... 43

Free space wavelength .............................. 27, 54

Frequency-selective measurement amplifier ....19

Frequency selectivity ........................................ 21

G

Slide-screw transformer ................................... 44Critical frequency ............................................. 29

Gunn effect ....................................................... 11

H

Guided wave .....................................................17

Cavity resonator ............................................... 12

I

Integrated microwave circuit MIC ...................27

Isolation path.....................................................65

K

Node width ........................................................38

Coaxial line................................................. 54, 66



MTS 7.4.4 Index

Slot antenna ...................................................... 53

Slot guide .................................................... 27, 53

Schottky diode ..................................................17

Signal-to-noise ratio ................................... 19, 53

"Single-mode" operation .................................... 31Smith chart ................................................. 39, 43

Ridged waveguide ............................................ 27

Standing wave ratio ................................... 38, 49

Scattering matrix ............................................... 65

Synchronous rectifier ........................................ 20

T

TEM wave ................................................. 17, 27

Temperature voltage ......................................... 17

Transit frequency ............................................. 11

Transmission faktor .....................................9, 59

Unloaded Q .......................................................54

V

Depletion layer ..................................................11

Delayed domain mode ...................................... 11

Four-port, reciprocal .........................................65VSWR .............................................................. 38

W

Wall losses ........................................................ 12

Wavetype converter ......................................... 53

Real power.......................................... 32, 43, 65

Z

Two-wire line ................................................... 27

Documents

T 7.4.3 Microondas en guías de onda