Ch 13 Oscillators 1ECES 352 Winter 2007

Oscillators

* Feedback amplifier but frequency dependent feedback

* Positive feedback, i.e. βf () A () < 0

* Oscillator gain defined by

* Oscillation condition at ω = ωo (Barkhausen’s criterion) Af (ωo) =

)()(1 sAs

sAsA

ff

)()()(

1)()()()()( )(

oofo

joofoofo

Aofphase

eAAL o

)()(1 sAs

sAsA

ff

Ch 13 Oscillators 2ECES 352 Winter 2007

Wien Bridge Oscillator

* Based on op amp

* Combination of R’s and C’s in feedback loop so feedback factor βf has a frequency dependence.

* Analysis assumes op amp is ideal. Gain A is very large Input currents are negligibly

small (I+ I_ 0).

Input terminals are virtually shorted (V+ V_ ).

* Analyze like a normal feedback amplifier. Determine input and output

loading. Determine feedback factor. Determine gain with feedback.

* Shunt-shunt configuration.

Vi

V0

ZS

ZP

If

R2

R1

Ch 13 Oscillators 3ECES 352 Winter 2007

Wien Bridge Oscillator

Vi = 0

V0

ZS

ZP

Input Loading Output Loading

sCR

R

sCRZR

ZRZ

sC

sRC

sCRZRZ

CCP

CS

1

111

11

11

Z1 Z2ZP ZP

ZS ZS

2

1

1

1

)1(

1

1

1

11

sCRsCR

sCRR

sCR

sC

R

sCR

ZZZZZ

SPSP

If

Vi

sC

sRCZRZZ CS

12

V0 = 0

R2R1

Define

Ch 13 Oscillators 4ECES 352 Winter 2007

Wien Bridge Oscillator

Z1

V0

Z2IS

IS

R2

R1

IS

Amplifier gain including loading effects

2

1

2

21

1

21

0

1

1

2

1

210

121

11

2121

0

00

)1(

11

)1(

1

1

,0

1

,

sCRsCR

sCRR

R

RA

sosCRsCR

sCRRZwhere

R

RZ

I

V

V

VA

andZI

VISince

R

R

R

RR

V

V

soRRR

VRIVVV

andRR

VIIusewe

V

VgetTo

I

V

V

V

I

VA

r

S

i

ir

S

i

i

oi

o

i

S

i

iSr

Feedback factor

If

V0ZP

ZS

sRC

sC

ZV

I

X

X

So

f

o

ff

1

1

I1 I2

Vi

Ch 13 Oscillators 5ECES 352 Winter 2007

Wien Bridge Oscillator

rf

rrf

rrf

A

AA

sCRsCR

sCR

R

R

sCRsCR

sCRR

R

R

sCR

sC

AsCR

sCA

1

isfeedback Gain with

)1(1

)1(

11

1

1

21

2

21

2

Loop Gain

Oscillation condition

213

11

usingely appropriat

and resistors theselectingby 1get can weThen,

1

frequency n oscillatio at the 0 termimaginary Then

13

11

13

11

311

211

)1(1

Rewriting

1)1(

1only needThen

0. since isalready It .180 toequal of Phase

1

2

1

2

21

1

2

1

2222

1

2

2221

2

21

2

21

2

o

R

Ror

R

R

RRA

RC

CRCRjR

R

sCRsCR

R

R

RCssCR

sCR

R

R

RCssCRsCR

sCR

R

R

sCRsCR

sCR

R

RA

sCRsCR

sCR

R

RA

AA

rf

o

rf

rf

rfrf

Ch 13 Oscillators 6ECES 352 Winter 2007

Wien Bridge Oscillator - Example

Oscillator specifications: o=1x106 rad/s

KKR

K

sradxnFCR

RCnFC

o

o

20)10(2

get weR 2 R sincethen ,10R Choosing

100)/101(10

11

1 fromthen ,10 econveniencfor Selecting

2

121

6

Ch 13 Oscillators 7ECES 352 Winter 2007

Wien Bridge Oscillator

Final note: No input signal is needed. Noise at the desired oscillation frequency will likely be present at the input and when picked up by the oscillator when the DC power is turned on, it will start the oscillator and the output will quickly buildup to an acceptable level.

Ch 13 Oscillators 8ECES 352 Winter 2007

Wien Bridge Oscillator * Once oscillations start, a limiting circuit is needed to prevent

them from growing too large in amplitude

Ch 13 Oscillators 9ECES 352 Winter 2007

Phase Shift Oscillator

* Based on op amp using inverting input

* Combination of R’s and C’s in feedback loop so get additional phase shift. Target 180o to get oscillation.

* Analysis assumes op amp is ideal.

V0

VX

R

IC1

R

IC2IC3

IR1IR2

If Rf

2

23

2

2

223

22

212

112

)(

143

)(

131

12

Finally

)(

131

12

111

11

12

12

12

111

11

sCRsCRsCR

V

sCRsCRsCR

V

sCRsCR

V

sC

IVV

sCRsCRR

V

sCRsCRsCRR

V

sCRR

V

sCRsCRR

VIII

sCRsCRR

V

R

VI

sCRsCR

V

sCsCRR

V

sCR

VZIVV

sCRR

V

R

V

sCRR

VIII

f

o

f

o

f

oCX

f

o

f

o

f

o

f

oCRC

f

oR

f

o

f

o

f

oCC

f

o

f

o

f

oCRC

V1V2

f

o

f

oR

f

oCC

Cf

of

sCRR

V

sCR

V

RR

VI

sCR

VZIVV

IR

VIsoVV

1

0

11

11

1

CC C

Ch 13 Oscillators 10ECES 352 Winter 2007

RR

soR

R

CR

RRCRRC

ωRRC

)L(ω

so)L(ω

RCso

CRCR

CRCRj

RRC

CRCRj

CRj

sCRsCR

sCR

V

VAL

sCRsCRsCR

V

f

fff

of

o

o

ff

f

X

f

oX

12

1123

1

44

get wefor ngsubstituti and14

1 need also wens,oscillatioget To

3

113

so frequency oneat thisachievecan We

zero. togo to termimaginary theneed wens,oscillatioget To

134

)(14

3

1

)(14

3

)()()(

gain loop for theget we

)(

143V

gRearrangin

22

2220

220

0

o

22

2

2

0

2

Phase Shift Oscillator

V0

VX

R

IC1

R

IC2IC3

IR1IR2

If Rf

V1V2

ExampleOscillator specifications: o=1x106 rad/s

KR

sradxnFCR

RC

nFC

f

o

o

67.0)58(12

Then

58)/101(103

1

3

13

1 fromthen

,10 econveniencfor Selecting

6

Note: We get 180o phase shift from op amp since input is to inverting terminal and another 180o from the RC ladder.

CC C

Ch 13 Oscillators 11ECES 352 Winter 2007

Colpitts LC-Tuned Oscillator

* Feedback amplifier with inductor L and capacitors C1 and C2 in feedback network. Feedback is frequency dependent. Aim to adjust components to get

positive feedback and oscillation. Output taken at collector Vo. No input needed, noise at oscillation

frequency o is picked up and amplified.

* RB1 and RB2 are biasing resistors.* RFC is RF Choke (inductor) to allow dc

current flow for transistor biasing, but to block ac current flow to ac ground.

* Simplified circuit shown at midband frequencies where large emitter bypass capacitor CE and base capacitor CB are shorts and transistor capacitances (C and C) are opens.

CB

CE

V0

Vi

V0

Vi

Ch 13 Oscillators 12ECES 352 Winter 2007

Colpitts LC-Tuned Oscillator* Voltage across C2 is just V

* Neglecting input current to transistor (I 0),

* Then, output voltage Vo is

* KCL at output node (C)

* Setting s = j

AC equivalent circuit

VsCZ

VI

CC 2

22

VsCZ

VII

CCL 2

22

22

2 1))(( LCsVsLVsCVZIVV LLo

01

011

01

2122

213

22

12

12

RgCCs

R

LCsCLCs

LCsVsCR

VgVsC

VsCR

VgVsC

m

m

om

01

213

212

2

CLCCCj

R

LC

Rgm

Iπ ≈ 0

sC2V

sC2V

V0

Assuming oscillations have started, then V ≠ 0 and Vo ≠ 0, so

Ch 13 Oscillators 13ECES 352 Winter 2007

Colpitts LC-Tuned Oscillator* To get oscillations, both the real and imaginary

parts of this equation must be set equal to zero.

* From the imaginary part we get the expression for the oscillation frequency

* From the real part, we get the condition on the ratio of C2/C1

01

213

212

2

CLCCCj

R

LC

Rgm

21

2121

21

213

21

1

0

CC

CCL

CLC

CC

CLCCC

o

oo

RgC

C

C

C

CLC

CCLCLCRg

R

LC

Rg

m

om

om

1

2

1

2

21

2122

2

22

11

01

Ch 13 Oscillators 14ECES 352 Winter 2007

Colpitts LC-Tuned Oscillator* Given:

Design oscillator at 150 MHz

Transistor gm = 100 mA/V, R = 0.5 K

* Design:

Select L= 50 nH, then calculate C2, and then C1

sradxxfo /104.91015022 86

50)5.0)(/100(1

2 KVmARgC

Cm

pFpFC

C

pFFxxnHC

C

LC

C

C

LCCLC

CC

o

o

2350

130,1

50

130,11013.1)501()104.9(50

11

1

11

21

928

1

222

1

2

221

21

Example

Ch 13 Oscillators 15ECES 352 Winter 2007

Summary of Oscillator Design* Shown how feedback can be used with

reactive components (capacitors) in the feedback path.

* Can be used to achieve positive feedback. With appropriate choice of the resistor

sizes, can get feedback signal in phase with the input signal.

Resulting circuit can produce large amplitude sinusoidal oscillations.

* Demonstrated three oscillator circuits: Wien Bridge oscillator Phase Shift oscillator Colpitts LC-Tuned oscillator

* Derived equations for calculating resistor and capacitor sizes to produce oscillations at the desired oscillator frequency.

* Key result: Oscillator design depends primarily on components in feedback network, i.e. not on the amplifier’s characteristics.

Wien Bridge Oscillator

Phase Shift Oscillator

Colpitts LC-Tuned Oscillator