전자파 연구실

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Chapter 12. Oscillators and mixersChapter 12. Oscillators and mixers

전자파 연구실

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12.1 RF oscillators12.1 RF oscillators

Figure 12-1 (p. 578)Block diagram of a sinusoidal oscillator using an amplifier with a frequency-dependent feedback path.

)()()()( oio AVHAVV

)()(1

)(

io VAH

AV

0

Nyquist criterion : If the denominator becomes zero, it is possible to achieve a non-zero output voltage, thus forming an oscillator.

1)( AH

전자파 연구실

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Figure 12-2 (p. 579)General circuit for a transistor oscillator. The transistor may be either a bipolar junction transistor or a field effect transistor. This circuit can be used for common emitter/source, base/gate, or collector/drain configurations by grounding either V2, V1, or V4, respectively. Feedback is provided by connecting node V3 to V4.

Three types of topology :

•Common emitter : V2 = 0

•Common base : V1 = 0

•Common collector : V4 = 0

General circuit for a transistor oscillatorGeneral circuit for a transistor oscillator

1Y

2Y

3Y

1V

2V

3V

4V

전자파 연구실

4

0

0)(

0)(

)()(

0)()(

4

3

2

1

00

3223

020211

3131

V

V

V

V

GgGg

YYYY

GYgGGYYgGY

YGYGYY

mm

mimi

ii

)(,0)(

)(4 3

1

323

331or

m

i VVV

V

YYYg

YGYY

Oscillators using a common emitter BJT 02 V

43 VV

332211 ,, jBYjBYjBY

0)(

)(

323

331

BBjjBg

jBBBjG

m

i

Kirchhoff’s current law on voltage nodesKirchhoff’s current law on voltage nodes

Feedback signal is provided from the collector (node 3) to node 4.

If the feedback network consists only of lossless capacitors and inductors, Y’s must be imaginary.

To have non-trivial solution, the determinant must be zero.

전자파 연구실

5

01

11

,0111

23321

BG

g

BBBB i

m

33

22

11

1,

1,

1

BX

BX

BX

21

321 0

XG

gX

XXX

i

m

If the inverse of B’s are represented by reactances,

Both of X1 and X2 are either capacitive or inductive.

(1) Both of X1 and X2 are capacitive →X3 is inductive : Colpitts oscillator(2) Both of X1 and X2 are inductive. →X3 is capacitive : Hartley oscillator

The real and imaginary part of the determinant must be zero separately.

전자파 연구실

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Colpitts oscillatorColpitts oscillator

Both of X1 and X2 are capacitive →X3 is inductive : Colpitts oscillator

30320

210

1 ,1

,1

LXC

XC

X

0111

30210

LCC

Figure 12-3 (p. 581)Transistor oscillator circuits using a common-emitter BJT. (a) Colpitts oscillator.

21

21

30

1

CC

CC

L

21

321 0

XG

gX

XXX

i

m

i

m

G

g

C

C

1

2

전자파 연구실

7

Hartley oscillatorHartley oscillator

Both of X1 and X2 are inductive →X3 is capacitive : Colpitts oscillator

303202101

1,,

CXLXLX

01

)(30

210 C

LL

Figure 12-3 (p. 581)Transistor oscillator circuits using a common-emitter BJT. (a) Colpitts oscillator.

)(

1

213

0LLC

21

321 0

XG

gX

XXX

i

m

i

m

G

g

L

L

2

1

전자파 연구실

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Practical considerationsPractical considerations

The previous analysis is based on very idealized assumptions. Practical oscillator design requires attention on the following factors.

1. Reactances associated with the input and output transistor ports.2. Variation of transistor properties with temperature3. Transistor bias and decoupling circuitry4. Effect of inductor losses

Example : Common emitter BJT Colpitts oscillator with the inductor impedance

Z3=1/Y3=R+jωL3.

0)(

)(

323

331

YCjYg

YGYCj

m

i

21312130

111111

CCLC

RG

CCLi RG

CC

i

11

1

1

3

2120

/1

C

L

CC

Gg

G

R im

i

Oscillation frequency:

Condition for oscillation:1

3

2120

/1

C

L

CC

Gg

G

R im

i

전자파 연구실

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Crystal oscillatorsCrystal oscillators

1. Battery. 2. Electric stepping motor. 3. Microchip. 4. Circuit connects microchip to other

components. 5. Quartz crystal oscillator. 6. Crown screw for setting time. 7. Gears turn hour, minute, and second

hands at different speeds. 8. Tiny central shaft holds hands in place.

Inside of a wrist watch

전자파 연구실

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Inductive

LCs

1

CCCC

L

p

0

0

1

Figure 12-4 (p. 584)(a) Equivalent circuit of a crystal. (b) Input reactance of a crystal resonator.

• The resonant frequency of an oscillator is determined from the condition that a 180° phase shift occurs between the input and output of the transistor. If the resonant feedback circuit has a high Q, there is a very rapid change in the phase shift with frequency and the oscillator will have good frequency stability.Unloaded Q of quartz crystals ~ 100,000. temperature drift < 0.001%/C°.Further stability can be obained by controlling the temperature of the quartz crystal.Crystals may be used in place of the inductor in a Colpitts or Pierce oscillator.

전자파 연구실

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Figure 12-5 (p. 585)Pierce crystal oscillator circuit.

전자파 연구실

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12.2 Microwave oscillators12.2 Microwave oscillators

Figure 12-6 (p. 585)Circuit for a one-port negative-resistance oscillator.

),(),(),( jIjXjIRjIZ ininin

LLL jXRZ Applying Kirchhoff’s voltage law gives

0)( IZZ inL

If oscillation is occurring, such that the RF current I is nonzero, then the following conditions must be satisfied:

0

0

inL

inL

XX

RR

Load impedance

Rin should be negative since RL>0.

전자파 연구실

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Tunnel diode I-V characteristics

전자파 연구실

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0 inL ZZ 1 inL

The process of oscillation depends on the nonlinear behavior of Z in.

0),( Lin RjIR 0),( 00 Lin RjIR

0),( 00 Lin XjIX

Initially, Steady state

inin

in

in

in

L

LL ZZ

ZZ

ZZ

ZZ

ZZ

ZZ

1

0

0

0

0

0

0

전자파 연구실

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For stable oscillation, any current and frequency perturbation should be damped out.

)(),(),( sZsIZsIZ LinT

0),(0000 ,,

II

Zs

s

ZsIZ

Is

T

Is

TT

IZ

ZIZjI

sZ

IZjs

T

TT

IsT

T

2

*

, /

)/)(/(

/

/

00

0Im*

TT Z

I

Z

0

TTTT R

I

XX

I

R

0

LLL R

I

X

I

R

For a passive load

0)(

inin

inLin R

I

XXX

I

R

0 Usually, I

Rin

0)(

Lin XXI

High Q circuit will be stable.

0

Stable oscillation conditionStable oscillation condition

전자파 연구실

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Example 12.2Example 12.2

12344 jZ in

12344 jZL

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Transistor oscillatorsTransistor oscillators

Figure 12-8 (p. 587)Circuit for a two-port transistor oscillator.

1. Typically, CE(common emitter), CB(common base) types are used.2. Output power port can be on either side of the transistor.

inLin

L XXR

R ,3

Typical load impedance.

전자파 연구실

18

1 inL

T

T

T

Tin

L S

S

S

SSS

22

11

22

211211 11

1

L

LT S

S

22

111

L

L

L

Lout S

S

S

SSS

11

22

11

211222 11

1 inT

Load 가 연결되어 있는 단자에서 oscillation 이 생기는 조건을 만족하는 경우

위의 관계식에서 ΓT 와 ΓL 과의 관계식을 얻을 수 있다 .

Terminating network 가 연결되어 있는 단자에서 반사 계수는 다음과 같다 .

ΓT 와 ΓL 과의 관계식을 이용하면

outT ZZ

Bidirectional oscillation conditionBidirectional oscillation condition

2 단자소자에서 한 단자만 oscillation condition 이 만족되면 다른 단자도 동시에 oscillation condition 이 만족됨 .

전자파 연구실

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Example 12.3 Transistor oscillators designExample 12.3 Transistor oscillators design

Figure 12-9a (p. 589)Circuit design for the transistor oscillator of Example 12.3. (a) Oscillator circuit.

15552.0

1826.1

9675.2

3518.2

22

12

21

11

S

S

S

S

전자파 연구실

Figure 12-9b (p. 589)(b) Smith chart for determining T.

ΓT 정하기 Γin 정해짐ZL 정해짐 .

15552.0

1826.1

9675.2

3518.2

22

12

21

11

S

S

S

S

665.0

3308.1

2

22

2

2112

22

22

*11

*22

S

SSR

S

SSC

T

T

4.296.31 22

211211

L

Lin S

SSS

9.1283

jjXR

Z inin

L

전자파 연구실

Dielectric resonator equivalent circuitDielectric resonator equivalent circuit

Figure 12-10 (p. 590)(a) Geometry of a dielectric resonator coupled to a microstrip line; (b) equivalent circuit.

0

21

2

/21

11

Qj

RN

LjCj

RNZ resonator

)2(,2)/(

)/(0

0

2

020 ZR

Z

RN

LNR

LR

Q

Qg L

Le

12)(

)(2

0

2

02

0

02

0

g

g

RNZ

RN

ZRNZ

ZRNZ

전자파 연구실

Tuning range 가 넓음 . 설계 간단함 .

oscillator using parallel feedback oscillator using series feedback

전자파 연구실

Example 12.4 DRO designExample 12.4 DRO design

1306.0L

L

Lout S

SSS

11

211222 1

설계 순서 :

(1) Load 방향의 반사계수가 최대가 되는 경우 가정하여 out 설정 .

0)1( 11 LS 이 되도록 ΓL 선택(2) L 설정 .

(3) ΓL 로부터 Γout 구하고 이로부터 Zout 구하고 다시 ZT 구함 .

1327.101 11

211222

L

Lout S

SSS

1.67.431

10 jZZ

out

outout

전자파 연구실

1.65.53

jjXR

Z outout

T

(4) ZT 가 되도록 matching 회로 만들기 .

1306.0L

(4) ΓL 이 되도록 matching 회로 만들기 .

1306.0L

Dielectric resonator 바로 옆에서 반사계수는

1806.0)1306.0( 22 rr ljljLL ee

위 식에서 Γ′L 의 위상은 0 도 또는 180 도이어야 한다 . (resonator 가 공진인 경우 저항 성분만 있으므로 . Over coupled 인 경우 0 도 , under-coupled 인 경우 180 도 )

전자파 연구실

)(,)/(

)/(0

020 ZRLNR

LR

Q

Qg L

Le

25.0

50

5.12g

5.121

10

L

LL ZZ

dB3

0

02

LN

RQ Le

120

2

02

02

g

g

RNZ

RN

ZRN

ZRN

전자파 연구실

전자파 연구실

12.3 Phase noise12.3 Phase noise)](cos[)](1[)( 00 tttAVtOSC

0)( tA Control 하기 힘듦 .

Phase variations 1. discrete : spurious mixer product or harmonics2. Random : thermal or other random noise sources

Oscillator output spectrum

Phase nose 의 영향 1. SNR 이 나빠짐 .2. Selectivity 가 나빠짐 .

전자파 연구실

ttf

ft mpm

m

sinsin)(

Small changes in the phase can be represented as a frequency modulation of the carrier with the modulation frequency ωm.

])cos()cos([2

cos

]sinsin[cos

)]sinsin(sin)sincos([cos)(

0000

000

000

tttV

tttV

ttttVt

mmp

mp

mpmpOSC

Small phase or frequency variations result in modulation side band at ω0 ±ωm.

)2/(44

2

221

)(22

20

2

0

prmsrmsp

p

c

n

V

V

P

PfL

Phase noise 단위 : (XX) dBc/Hz at (offset YY)kHz

전자파 연구실

Leeson’s model for oscillator phase noiseLeeson’s model for oscillator phase noise

Figure 12-14 (p. 596)Feedback amplifier model for characterizing oscillator phase noise.

)(1

)()(

H

VV io

00

0

/21

1

1

1)(

jQ

jQ

H )( 0

)(1)(4

1

)(/4

/41)(

)(1

1)(

2

2

22

20

20

22

20

222

SSQ

SQ

QS

HS

h

)]2/([ 0 Qh

Half power bandwidth

Oscillator output power :

전자파 연구실

Figure 12-15 (p. 596)Noise power versus frequency for an amplifier with an applied input signal.

Fricker noise which is caused by random fluctuations of the carrier density in the active devices

Figure 12-16 (p. 597)Idealized power spectral density of amplifier noise, including 1/f and thermal components.

K

P

kTFS 1)(

0

ωα : Corner frequency of the 1/f noiseSi FET : 50~100HzBJT : 5kHz~50kHzGaAs FET : 2MHz~ 10MHz

전자파 연구실

144

14

1)(

22

20

32

20

0

22

20

0

K

QQ

K

P

kTF

K

QP

kTFS

Figure 12-17 (p. 597)Power spectral density of phase noise at the output of an oscillator. (a) Response for fh > f (low Q). (b) Response for fh > f (high Q).

전자파 연구실

Figure 12-18 (p. 598)Illustrating how local oscillator phase noise can lead to the reception of undesired signals adjacent to the desired signal.

Example 12.5 GSM receiver phase noise requirementsExample 12.5 GSM receiver phase noise requirements

)dBc/Hz(),log(10)dBm()dB()dBm()( BISCfL mphase

Maximum allowable phase noise level :

전자파 연구실

)log(10 BLPP phaseOSCOSC

I C

cOSCIF LCPP desired

cOSCIF LIPP undesired

OSCP

SBLIC

LIPLCPPP

phase

cOSCcOSCIFIF

10

undesireddesired

log10

)(

)dBc/Hz(log10 10 phaseLBISC

Selectivity

S

전자파 연구실

System BandUplink (MHz)

Downlink (MHz)

Channel number

T-GSM-380 380380.2–389.8

390.2–399.8 dynamic

T-GSM-410 410410.2–419.8

420.2–429.8 dynamic

GSM-450 450450.4–457.6

460.4–467.6 259–293

GSM-480 480478.8–486.0

488.8–496.0 306–340

GSM-710 710698.0–716.0

728.0–746.0 dynamic

GSM-750 750747.0–762.0

777.0–792.0 438–511

T-GSM-810 810806.0–821.0

851.0–866.0 dynamic

GSM-850 850824.0–849.0

869.0–894.0 128–251

P-GSM-900 900890.0–915.0

935.0–960.0 1–124

E-GSM-900

900880.0–915.0

925.0–960.0975–1023, 0-124

R-GSM-900

900876.0–915.0

921.0–960.0955–1023, 0-124

T-GSM-900 900870.4–876.0

915.4–921.0 dynamic

DCS-1800 18001710.0–1785.0

1805.0–1880.0

512–885

PCS-1900 19001850.0–1910.0

1930.0–1990.0

512–810

GSM frequency bandsThere are fourteen bands defined in 3GPP TS 45.005, which succeeded 3GPP TS 05.05:

RFP

RF

Frequency offset fm (MHz)

Interfering signal level (dBm)

Phase noise (dBc/Hz)

3.0 -23 -138

1.6 -33 -128

0.6 -43 -118

-23dBm

-33dBm

-43dBm

-99dBm

GSM minimum spec.

전자파 연구실

12.6 Mixers12.6 MixersA mixer is a 3-port device that uses a non-linear or time-varying element to achieve frequency

conversion.

tft LOLO 2cos)(

tft IFIF 2cos)(

(1) Up-converter

])(2cos)(2[cos2

2cos2cos)()()(

tfftffK

tftfKttKt

IFLOIFLO

IFLOIFLORF

IFLORF fff

전자파 연구실

(2) Down-converter

tft RFRF 2cos)(

])(2cos)(2[cos2

2cos2cos)()()(

tfftffK

tftfKttKt

LORFLORF

LORFLORFIF

LORFIF fff

전자파 연구실

Image frequency Image frequency 문제문제

RF

RF

LO

LO

IF

IF

IF IF

Lower side LO

Upper side LO

RF

RF

LO

LO

IF

IF

IF IF

ωLO 로부터 같은 주파수 차이를 갖는 두 신호가 동시에 ωIF 로 conversion됨 .

LO 의 fractional tuning range 가 작으므로 더 많이 쓰임 .

전자파 연구실

Image reject filterImage reject filterSuper heterodyne 수신기의 단점인 image 신호와 원신호가 중간주파수 같은 신호로 합쳐지는 것을 막기 위해 image reject filter 를 사용함 .

전자파 연구실

Conversion lossConversion loss

dB0poweroutput IF available

powerinput RF availablelog10 CL

Noise figureNoise figure

SSB 인 경우가 DSB 보다 NF 가 3dB 크다 .

전자파 연구실

Single side band modulatorSingle side band modulator

전자파 연구실

])(cos)([cos)( ttAt IFLOIFLODSB

)(cos)(cos2

)(cos2

)( tAKtAK

tAK

t IFIFIFIF

222

22A

AASi

2

22KASo

BkT

N

LKNS

NSNF added

Coo

iiDSB

02

12

/

/

addedC

o NBkTL

N 0

1

DSB NF

SSB NF

tAt IFLOSSB )(cos)(

2

2ASi 8

22KASo

)(cos2

)( tAK

t IFIF

BkT

N

LKNS

NSNF added

Coo

iiSSB

02

14

/

/

DSBSSB NFNF 2

전자파 연구실

Single-ended diode mixerSingle-ended diode mixer

tVt RFRFRF cos)(

tVt LOLOLO cos)(

200 )]()([

2cos)]()([ tt

GtttGII LORF

dLORFd

전자파 연구실

tVVtVV

tVtVG

tVttVVtVG

tVtVG

ti

LORFLORFLORFLORF

LOLORFRFd

LOLOLORFLORFRFRFd

LOLORFRFd

)cos(2)cos(2

)2cos1()2cos1(

4

coscoscos2cos2

]coscos[2

)(

22

2222

2

tVVG

ti IFLORFd

IF cos2

)(

LORFIF

전자파 연구실

Single-ended FET mixerSingle-ended FET mixer

Figure 12-32 (p. 622)Variation of FET transconductance versus gate-to-source voltage.

전자파 연구실

1

00 cos2)(n

n tnggtg

Local oscillator 의 출력이 매우 크므로 g (FET trans-conductance) 는 LO 에 비례한다 .

2

2

22

2

2

4

4RF

IFD

L

Lg

g

RF

L

LIFD

availRF

availIFconv

V

V

Z

RR

R

V

Z

RV

P

PG

Conversion gain

)(1 gigsRF

RF

gsRFgigsRF

RFRFC ZRCj

V

Cj

ZRCj

VV

ttVgtVgttg LORFRFCRF

RFC

RFCm coscos2cos)()( 10

tVgttg IFRFC

RFCm

IF

cos)()( 1

Ld

Ld

gigsRF

RF

Ld

LdRFC

IFD ZR

ZR

ZRCj

Vg

ZR

ZRVtgV

)(1)( 1

1

전자파 연구실

222

2

2

1

)()(

2

LLd

L

gsRFggi

g

gsRF

DRFC

matchednotC XRR

R

Cj

XZR

R

C

RgVG

igsRF

DRFCC RC

RgVG

22

21

4

전자파 연구실

Single balanced mixerSingle balanced mixer

200 2

1)( dd GGIiIi

Diode current :

]coscos[)(1 tVtVt RFRFLOLO

]coscos[)(2 tVtVt RFRFLOLO

]coscos4[2

1]cos2[

2

1

)]()([2

1)()()( 2

221212211

tVtVGtVG

ttGGiii

LOLORFRFdLOLOd

ddo

1

2

1i

2i

•RF port 에서 perfect matching 됨 .•Even order inter-modulation term 들이 모두 0 됨 .

LO

RF

전자파 연구실

Balanced mixerBalanced mixer

•RF port 에서 perfect matching 됨 .•Even order inter-modulation term 들이 모두 0 됨 .

전자파 연구실

tVt RFRFRF cos)(

tVt LOLOLO cos)(

]cossin[2

1

)]180cos()90cos([2

1)(1

tVtV

tVtVt

LOLORFRF

LOLORFRF

]sincos[2

1

)]90cos()180cos([2

1)(2

tVtV

tVtVt

LOLORFRF

LOLORFRF

]2cossin22cos[2

)()()()( 2222

2121 tVtVVtV

KKtititi LOLOIFLORFRFRFt

)( LORFIF tVVKti IFLORFIF sin)(

Balanced mixer with 90 deg. HybridBalanced mixer with 90 deg. Hybrid

2

2

2

1

2

1

RF

RF

VVV

VjVV

RFRFRFLO

RFRFRF

VjVjVjjVV

V

VVVjV

V

2

1

2

1

22

02

1

2

1

22

21

21

전자파 연구실

Image reject mixerImage reject mixer

tVtVt RFLRFURF )cos()cos()( 2/2/

])sin()sin([2

1

)]90cos()90cos([2

1)(

2/2/

2/2/

tVtV

ttVttVt

RFLRFU

RFLRFUA

])cos()cos([2

1

)]180cos()180cos([2

1)(

2/2/

2/2/

tVtV

ttVttVt

RFLRFU

RFLRFUB

전자파 연구실

]sinsin[2

)( 2/2/ tVtVKV

t LULOA

IF

]coscos[2

)( 2/2/ tVtVKV

t LULOB

IF

]cos[2

]coscoscoscos[4

)]180()90([2

1)(

2/

2/2/2/2/

1

tVKV

tVtVtVtVKV

ttt

ULO

LULULO

BIF

AIF

]sin[2

]sinsinsinsin[4

)]90()180([2

1)(

2/

2/2/2/2/

2

tVKV

tVtVtVtVKV

ttt

LLO

LULULO

BIF

AIF

전자파 연구실

Double balanced mixerDouble balanced mixer

•Good isolation between all three ports•Reject even order harmonics of the RF and LO signals.

Virtual ground for LO signal

Virtual ground for RF signal

전자파 연구실

Differential FET mixerDifferential FET mixer

전자파 연구실

Frequency doubler

Sub-harmonically pumped mixerSub-harmonically pumped mixer