� � � �

Chapter 11 Oscillator design

11.1 Feedback and basic concept

Hartley and Colpitts oscillators

11.2 Crystal oscillators

11.3 Electronic tuning of oscillators

varactor diode

11.4 Phase-locked loop

closed-loop transfer function

11.5 Frequency synthesizers

11.6 One-port negative resistance oscillators

11.7 Microwave transistor oscillators

CB and CG with feedback structure, 3-port S-parameter and its reduced 2-port

S-parameter relation

� � � �

11.1 Feedback and basic concept

Basics

1. Oscillator concept noise trigger an unstable circuit feedback and active device

amplifies near wo nonlinear active device at steady state oscillation at wo

without power

�

AloopclosedYXY

1X

YTgain)(

A: forward path gain, A: loop gain

A=1: oscillation condition

If - , A=-1: oscillation condition

    ¡ ¢

Discussion

1. Hartley and Colpitts oscillators from feedback point of view

£

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2. Hartley and Colpitts oscillators from negative resistance point of view

oscillatorHartley:)(

1)(

1,

oscillatorColpitts:,

noscillatioforconditionnecessarythe

)(

1

)(1

,

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11.2 Crystal oscillators

± ² ± ³ ± ´ µinductive:2

)11(w:figurepull

1,1

wsCp

Cs

Cp

Cswswswp

Cp

Cswswp

LsCsws

¶ ¶ · ¸

11.3 Electronic tuning of oscillators

Basics

Varactor diode

(1) basic structure: heavily doped PN diode

(2) operating principle:

in reverse bias a large depletion zone variable capacitance with reverse bias

voltage

(3) abrupt junction varactor: high Q, hyperabrupt junction varactor: low Q with

linear tuning characteristics

(4) applications: VCO, frequency multiplier

¹ ¹ º »

11.4 Phase-locked loop

Basics

¼

K

sV

s

sVKssdt

dVKwwwwVKwVCO

free

d

ddssd

00

000

000

)(

)(

)()(:

2)(,

VeKd:detectorphase

When the VCO is locked with reference signal, fo=fr and r d to be a constant

½ ½ ¾ ½ ¿

Discussion

1. Ex.11.3 VCO free-running frequency at 10MHz, Ko=2 x107, PD is a multplier with

Km=0.5, Vosc and Vin peak value 0.75V, find phase difference of Vosc and Vin for

9MHz and 11MHz

7.1307.4090,7.4010)119(2

)9(

3.497.4090,7.4010)1011(2

)9(

/14.075.05.05.05.0

5.0sin5.0

)]2sin([sin5.0

)cos(sin

)cos(,sin

0

60

6

00

00

2

KdKMHzd

KdKMHzd

KdK

ww

Kd

Vdd

K

wwVwwVKw

radVEEKd

VdKd

EiEKEEKVd

wtEiEK

wtwtEiEViK

VVKV

dwtEVwtEiVi

sinsin

dinsd

nioscm

dnoscmdnioscm

dnoscm

dnoscnm

oscinmPD

oscoscnn

À À Á À À

2. Closed-loop transfer function

)()(0

)(0

)(

)(T(s)function transfer loop-closed

)()(0)()(

)()(0)()(sin0

)()(sin00

)(2)(

),(sin)(sin5.0)(2

)]2sin()[sin(5.0

)cos()sin()(1

)](cos[)(

)](sin[)(

0

0

0

sFseKdKs

sKdFK

sr

so

sFseKdKsrsses

tfteKdKdt

rdtfteKdK

dt

rd

dfeKdKdt

d

dt

rd

dt

d

dt

rd

dt

ed

dvKto

oretfeKdtfeKmArAotv

orwtorKmArAo

rwtrwtKmArAotv

towtAotVo

trwtArtVr

t

t

  à  Ä

11.5 Frequency synthesizer (indirect synthesis technique)

Å =1KHz, switching time 25ms, N=105~1.1x105

(2) fr=100KHz, switching time 0.25ms, M=100, N=105~1.1x105,

VCO :10GHz~11GHz

Æ Æ Ç Æ È

Ex.11.6 Design a synthesizer to have 1MHz~50MHz with 1kHz resolution

(1) fr=1KHz, switching time 25ms, 1KHz resolution

(2) fr=100KHz,switching time 50us, 1KHz resolution

É É Ê É Ë

11.6 One-port negative resistance oscillators

Ì circuit unstable RG(I,w)+RL(w) < 0

feedback and amplify near wo

at steady state RG(Io,wo)+RL=0, XG(Io,wo)+XL(wo)=0

oscillation at wo with out power Po=1/2Io2R

2. ZG+ZL=0 G L=1

3. Ex.11.7 design a 6GHz oscillator with diode G=1.25 40 @6GHz

ZG=-43.5+j124 , ZL =43.5-j124

LZZL

ZZL

ZZL

ZZL

ZZG

ZZGG

RGRLXGXLRGRL

ZGZLI

IZGZL

1

0

0

0

0

0

0

00,0,0

00

0)(KVL

noscillatioofstatesteady@

Í Í Î Í Ï

11.7 Microwave transistor oscillators

Ð = -Rin/3, XG = -Xin

@ steady state in G = 1 out L = 1

Discussion

1. At steady state

111

,1

1

1

11

22

11

211222

21122211

22

11

22

211211

outL

G

G

G

Gout

G

GL

L

Lin

G

S

S

S

SSS

SSSSS

S

S

SSS

Ñ Ñ Ò Ñ Ó

2. Ex. 11.8 A CG GaAs FET with a 4nH inductor to provide positive feedback, and

give S11=2.18 -35 , S12=1.28 18 , S21=2.75 96 , S22=0.52 155 at 4GHz,

design an oscillator

91.19.2791.13

67.8391.167.83

4.296.31

1040.59Lselect

67.0,1.3308.1,77.4,8.14167.4

121.02

1,134.2

22

211211

2112

22

22

2

11

21122211

jjZGjZin

S

SSSin

rLCLrsCs

SS

SSkSSSS

L

L

Ô Ô Õ Ô Ö

3. Ex. 11.9 A CB BJT with S11=0.9 150 , S12=0.07 120 , S21=1.7 -80 , S22=1.08

-56 at 2.75GHz, design an oscillator

35.24835.243

8.2335.248.23

8.11817.28.1675689.0L

41.4,1.33024.5,132.1,96.4226.1

54.0,26.1,3.10113.168.1389755.1

71.946945.01.996733.1]'[

509.11459.09067.03747.1

5779.00323.0j0.77021337.0][][][seriesin)3(

5011.05011.0

5011.05011.0][inductornH45.1)2(

0101.21459.0484.13747.1

0768.00323.0j0.26911337.0][

4.7Table

][,6447.0,907.0,BJT)1(

jjZGjZin

inselect

rLCLrscs

kS

jj

jLZZZ

jj

jjLZ

jj

jZSk

BJT

BJT

× × Ø × Ù

4. A 3-port network, its 9 elements are not independent, i.e.,

3

1

3

1

3

1

3

13

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3,2,1,1

3213aa2a1

currentinputno voltagesame thehasport)(

3,2,1,1

13(1)0,a2a1,12(1)0,a3a1

1101(1)0,a3a2

)1...(0)(,0)(,0)(

)(22

1

2

)(22

1

2

2

2

3

2

1

333231

232221

131211

3

2

1

3,2,1,1and3,2,1,1

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i

ii

ii

i jii

ref

in

inref

inin

refin

refin

ji

iSijaSijaSijajbi

abbbaif

aibieveryb

jSij

SiifajSiif

SiajSiaiif

SijajaibiaiIia

IiZoiZoi

Vi

Zoi

Vibi

IiZoiZoi

Vi

Zoi

Viai

ZoiIiViVi

ZoiIiViVi

Zoi

ViViIi

ViViVi

a

a

a

SSS

SSS

SSS

b

b

b

iSijjSij

Ú Ú Û Ú Ü

5. Find the S-parameter of a 3-port network from its 2-port Scattering

parameters with port 3 terminal common

)2

3'3'-a'2(22)

2

3'3'-a'1(212221212

2

3'3'-b'2

)2

3'3'-a'2(12)

2

3'3'-a'1(112121111

2

3'3'-b'1

2

1

2221

1211

2

1,

2'2'2'222

1'1'1'111

32'2'222

31'1'111

2

3'3'-b'2

2

V3-b'2b2

2

3'3'-a'2

2

V3-a'2a2

2

3'3'-b'1

2

V3-b'1b1

2

3'3'-a'1

2

V3-a'1a1

3portat voltagenormalized:332

33'3'

)(22

1

2'

)(22

1

2'

'

3'

2'

1'

333231

232221

131211

3'

2'

1'

baS

baSaSaSb

ba

baS

baSaSaSb

ba

a

a

SS

SS

b

b

IababI

IababI

VbabaV

VbabaV

ba

ba

ba

ba

VZo

Vba

IiZoiZoi

Vi

Zoi

Viib

IiZoiZoi

Vi

Zoi

Viia

a

a

a

SSS

SSS

SSS

b

b

b

ref

in

Ý Ý Þ ß à

3

1

3

1

3

1

3

1

3,2,1,1'and3,2,1,1'

2221122,1211111

2212121,2111112

22122111

3'

2'

1'

4

2

4

212

4

1224

222

4

212222

4

122221

4

112

4

211112

4

121111

3'

2'

1'

3'4

22'

4

2121'

4

122

2'1'3'2'1'3'2'2'1'1'3'0)''(0

2

3'3')22211(2'221'21b'2

2

3'3')12111(2'121'11b'1

ji

ii

iijSjijS

SSSS

SSSS

SSSS

a

a

a

SS

SS

b

b

b

aaa

bbaaaabababibiaIi

baSSaSaS

baSSaSaS

á á â ã á

6. Find the 2-port S-parameter of a terminated 3-port network

parametersSport-2 toreducedisitZ0, with trminatedis3portIf

2'

1'

33'

32'23'22

33'

31'23'21

33'

32'13'12

33'

31'13'11

2'

1'

)2'33'

32'1'

33'

31'(23'2'22'1'21'3'23'2'22'1'21'b'2

)2'33'

32'1'

33'

31'(13'2'12'1'11'3'13'2'12'1'11'b'1

2'33'

32'1'

33'

31'3'3'3'33'2'32'1'31'3'3'a'3

with3portteminate

3'

2'

1'

333231

232221

131211

3'

2'

1'

11

11

11

11

11

1

a

a

S

SSS

S

SSS

S

SSS

S

SSS

b

b

aS

Sa

S

SSaSaSaSaSaS

aS

Sa

S

SSaSaSaSaSaS

aS

Sa

S

SaaaSaSaSbb

a

a

a

SSS

SSS

SSS

b

b

b

ä ä å æ æ

7. Procedures to convert S-parameters from CS to CG with

ç3'

2'

11'

13'31'33

11'

12'31'32

33'

13'21'23

11'

12'21'22

3'

2'

with1portTerminate)2(

2221122,1211111,2212121,2111112

22122111

3'

2'

1'

4

2

4

212

4

1224

222

4

212222

4

122221

4

112

4

211112

4

121111

3'

2'

1'

11

11

a

a

S

SSS

S

SSS

S

SSS

S

SSS

b

b

SSSSSSSS

SSSS

a

a

a

SS

SS

b

b

b

è è é ê ë

stablenallyunconditio,1

21122211,211222*11

22'1factor-define

11'1

13'31',

11'1

*11'233'C withplane-S33nincircle

11'1

12'21',

11'1

*11'122'C withplane-S22nincircle

11,portatloadreactive

13'31'33'11'2,23311'

33'33

11'

13'31'33'33

12'21'22'11'1,12211'

22'22

11'

12'21'22'22

3'

2'

3332

2322

3'

2'

11'

13'31''

'

'''

33'

13'21''

'

'''

'

'

Design)3(

2

2222

2121

1

1

133111

123132

123111

122122

3

2

nnSSnnSSnnnSSSnnSnS

nS

S

SSR

S

SSa

S

SSR

S

SSa

SSSSnSS

SnS

S

SSSnS

SSSSnSS

SnS

S

SSSnS

a

a

nSnS

nSnS

a

a

S

SSS

S

SSS

S

SSS

S

SSS

b

b

ì ì í î ï

8. Ex. 11.9 A CB BJT with S11=0.9 150 , S12=0.07 120 , S21=1.7 -80 ,

S22=1.08 -56 at 2.75GHz, design an oscillator

baseat theinductor1.45nH theaddingby126.0674.0

631.02,6.84587.02:circle22

947.01,84837.01:circle11)3(

5.102163.14.136083.2

8.94715.01.100722.1

inductor1.45nHa1withportTerminate)2(

4.115468.09.34425.08.1187.0

60163.17.54534.01.79582.0

5.43852.03.290.3934.840.396

parameter-Sport-3Determine)1(

RCnS

RCnS

Hw #10 (due 2 weeks)

11, 14