Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Ch. 3. The Smith Chart

Mapping of the reflection

coefficient in the complex

domain

ljir

L

L ejZZ

ZZ 000

0

00

d

d = l 0

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Normalized impedance

irdj jed 2

0

Generalized reflection coefficient:

ir

irin

in

j

j

d

djxrz

Z

dZ

1

1

1

1

0

Normalized impedance:

2

2

2

22

22

1

1

1

1

1

rr

r

r

ir

ir

ir

Real part: Imaginary part:

22

2

22

111

1

2

xx

x

ir

ir

i

d

d = l 0

Γ(d)

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Representation of normalized resistance

2

2

2

1

1

1

rr

rir

01

2

1

2

11

10

22

2

2

2

22

ir

ir

ir

r

r

r

Examples:

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Representation of normalized reactance

01

1111

00

1111

01

22

22

22

22

ir

ir

i

ir

ir

x

x

x

x

x

Examples:

22

2 111

xxir

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Combined diagram: Smith Chart

1

1

jxr

jxrjxrz

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Repetition

Reflection coefficient:

ljir

L

L ejZZ

ZZ 000

0

00

Generalized reflection coefficient:

irdj jed 2

0

ir

irin

in

j

j

d

djxrz

Z

dZ

1

1

1

1

0

Normalized impedance:

22

22

1

1

ir

irr

221

2

ir

ix

Normalized

resistance:

Normalized

reactance:

d

d = l 0

Γ(d)

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

How to use the Smith Chart

• Normalize load impedance: ZL zL

• find reflection coefficient: zL Γ0

• rotate reflection coefficient: Γ0 Γ(d)

• find normalized input impedance: zin(d)

• de-normalize input impedance: zin(d) Zin(d)

Example: determine

input impedance Zin(d)d

d = l 0

Γ(d)

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Impedance transformation using

Smith chart

1) Normalize load impedance:

ZL zL = ZL/Z0

2) Find reflection coefficient:

zL Γ0 = (zL-1)/(zL+1)

3) Rotate reflection coefficient:

Γ0 Γ(d) = Γ0 exp(-j2βd)

4) Find normalized input impedance:

Γ(d) zin(d) = (1+ Γ(d))/(1- Γ(d))

5) De-normalize input impedance:

zin(d) Zin(d) = Z0 zin(d)

zL=0.6+j1.2 Γ0 =0.2+j0.6

zin=0.3-j0.53

Γ=-0.32-j0.55

ZL=30+j60 Ω

Z0=50 Ω

d=l=2cm

f=2GHz vp=c/2

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Generalized standing wave ratio

1

1)(

)(1

)(1)(

)( 20

SWR

SWRd

d

ddSWR

ed dj

Can determine SRW for a

given Γ(d) by drawing circle

with center at Γ = 0 through

Γ(d) in the Smith chart.

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Open circuit TL as a reactive element

djdjZZ

djZZz

L

Lin

cot

tan

tan

0

0

Note: ZL = corresponds to r = 0

(outer circle of Smith chart)

Upper half-circle: inductive

Lower half-circle: capacitive

Example: zin = jx = 1/jωCZ0

n

CZd

0

1 1cot

1

f = 3GHz

β = 81.6m-1

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Short circuit TL as a reactive element

djdjZZ

djZZz

L

Lin

tan

tan

tan

0

0

Note: ZL = 0 also corresponds to

r = 0 (outer circle of Smith chart)

Upper half-circle: inductive

Lower half-circle: capacitive

Example: zin = jx = jωL/Z0

n

Z

Ld

0

1tan1

f = 3GHz

β = 81.6m-1

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Admittance transformation

)(1

)(1

)(1

)(11

)(1

)(111

0

de

de

d

d

zy

d

d

ZZY

j

j

in

in

in

in

e-jπ Γ(d) corresponds to 180º

rotation of Γ(d) in Smith chart.

This converts impedance to

admittance

Alternatively: Rotate Smith chart

by 180º : Admittance Smith chart

zin = 1+j

yin = 0.5-j0.5

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Admittance Smith chart

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

ZY Smith chart

2222,

11

xr

xb

xr

rg

jxrzjbgy

in

in

Use original Smith chart

to display impedances

and rotated chart to

display admittances.

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Frequency dependence: parallel R and L

L

Zj

R

Zjbgy oo

in

Example:

Z0 = 50 Ω, L = 10 nH

g = 0.3, 0.5, 0.7, 1

f = 500 MHz to 4 GHz

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Frequency dependence: parallel R and C

CjZR

Zjbgy o

oin

Example:

Z0 = 50 Ω, C = 1 pF

g = 0.3, 0.5, 0.7, 1

f = 500 MHz to 4 GHz

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Frequency dependence: series connections

CZ

j

Z

Lj

Z

R

jxrz

ooo

in

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

A

B

C

D

E

A: gA = Z0/RL = 1.6

B: yB = gA + jZ0ωCL = 1.6 + j1.2

zB = 0.4 - j0.3

C: zC = zB + jωL1/Z0 = 0.4 + j0.8

yC = 0.5 – j1.0

D: yD = yC + jZ0ωC = 0.5 + j0.5

zD = 1 – j1

E: zE = zD + jωL2/Z0 = 1

Z0 = 50 Ω f = 2 GHz

Zin = Z0 = 50 Ω : Match at 2 GHz

Example: T-type network

Institutt for InformatikkIFI5480: RF kretser, teori og design

Tor A Fjeldly

Simulation of Zin for T-network

CAD simulation of Zin for

frequences 0.5 – 4 GHz

Note that C behaves as a short

at the highest frequencies and

the network will be dominated

by L2 (purely inductive)