Oscillator

•  Introduction of Oscillator •  Linear Oscillator

– Wien Bridge Oscillator – RC Phase-Shift Oscillator – LC Oscillator

•  Stability

Oscillators

Oscillation: an effect that repeatedly and regularly fluctuates about the mean value

Oscillator: circuit that produces oscillation Characteristics: wave-shape, frequency,

amplitude, distortion, stability

Application of Oscillators

•  Oscillators are used to generate signals, e.g. – Used as a local oscillator to transform the RF

signals to IF signals in a receiver; – Used to generate RF carrier in a transmitter – Used to generate clocks in digital systems; – Used as sweep circuits in TV sets and CRO.

Linear Oscillators

1.  Wien Bridge Oscillators 2.  RC Phase-Shift Oscillators 3.  LC Oscillators 4.  Stability

Integrant of Linear Oscillators

For sinusoidal input is connected “Linear” because the output is approximately sinusoidal A linear oscillator contains: - a frequency selection feedback network - an amplifier to maintain the loop gain at unity

Σ

+

+ Amplifier (A)

Frequency-SelectiveFeedback Network (β)

Vf

Vs VoVε

PositiveFeedback

Basic Linear Oscillator Σ

+

+

SelectiveNetwork β(f)

Vf

Vs VoVεA(f)

)( fso VVAAVV +== ε and of VV β=

βAA

VV

s

o

−=⇒1

If Vs = 0, the only way that Vo can be nonzero is that loop gain Aβ=1 which implies that

01||=∠

=

β

β

AA (Barkhausen Criterion)

Wien Bridge Oscillator Frequency Selection Network Let

11

1C

XC ω= and

111 CjXRZ −=2

21C

XC ω=

22

22

1

222

11

C

C

C jXRXjR

jXRZ

−

−=⎥

⎦

⎤⎢⎣

⎡

−+=

−

Therefore, the feedback factor,

)/()()/(

222211

2222

21

2

CCC

CC

i

o

jXRXjRjXRjXRXjR

ZZZ

VV

−−+−

−−=

+==β

222211

22

))(( CCC

C

XjRjXRjXRXjR

−−−

−=β

Vi Vo

R1 C1

R2C2

Z1Z2

β can be rewritten as:

)( 2121221221

22

CCCCC

C

XXRRjXRXRXRXR

−+++=β

For Barkhausen Criterion, imaginary part = 0, i.e.,

02121 =− CC XXRR

Supposing, R1=R2=R and XC1= XC2=XC,

2121

2121

/1

11or

CCRR

CCRR

=⇒

=

ω

ωω

)(3 22CC

C

XRjRXRX

−+=β

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

Feed

back

fact

or β

-1

-0.5

0

0.5

1

Phas

e

Frequency

β=1/3

Phase=0

f(R=Xc)

Example Rf

+

−

R

R

C

CZ1

Z2

R1

Vo

By setting , we get Imaginary part = 0 and

RC1

=ω

31

=β

Due to Barkhausen Criterion, Loop gain Avβ=1 where Av : Gain of the amplifier

1

131RR

AA fvv +==⇒=β

21

=RRfTherefore, Wien Bridge Oscillator

RC Phase-Shift Oscillator

+

−

Rf

R1

R R R

C C C

§  Using an inverting amplifier §  The additional 180o phase shift is provided by an RC

phase-shift network

Applying KVL to the phase-shift network, we have

R R R

C C CV1 Vo

I1 I2 I3Solve for I3, we get

)2(0

)2( 0)(

32

321

211

C

C

C

jXRIRIRIjXRIRI

RIjXRIV

−+−=

−−+−=

−−=

C

C

C

C

C

jXRRRjXRR

RjXRRjXRR

VRjXR

I

−−

−−−

−−−

−−

−−

=

202

00002

3

1

)2(])2)[(( 222

21

3CCC jXRRRjXRjXR

RVI−−−−−

=Or

The output voltage,

)2(])2)[(( 222

31

3CCC

o jXRRRjXRjXRRV

RIV−−−−−

==

Hence the transfer function of the phase-shift network is given by,

)6()5( 2323

3

1 CCC

o

XRXjRXRR

VV

−+−==β

For 180o phase shift, the imaginary part = 0, i.e.,

RC

RXXRX CCC

61

6X

(Rejected) 0or 0622

C

23

=

=⇒

==−

ω

and,

291

−=β

Note: The –ve sign mean the phase inversion from the voltage

LC Oscillators

+

−

~Av Ro

Z1 Z2

Z3

12

Zp

§  The frequency selection network (Z1, Z2 and Z3) provides a phase shift of 180o

§  The amplifier provides an addition shift of 180o

Two well-known Oscillators: •  Colpitts Oscillator •  Harley Oscillator

+~Av Ro

Z1 Z2

Z3Zp

Vf Vo

321

312

312

)(

)//(

ZZZZZZZZZZ p

++

+=

+=

For the equivalent circuit from the output

po

pv

i

o

p

o

po

iv

ZRZA

VV

ZV

ZRVA

+

−==

+

−or

Therefore, the amplifier gain is obtained,

)()()(

312321

312

ZZZZZZRZZZA

VV

Ao

v

i

o

++++

+−==

oof VZZ

ZVV

31

1

+== β

Zp−AvVi

Ro

+−

−

+Vo

Io

The loop gain,

)()( 312321

21

ZZZZZZRZZA

Ao

v

++++

−=β

If the impedance are all pure reactances, i.e., 332211 and , jXZjXZjXZ ===

The loop gain becomes, )()( 312321

21

XXXXXXjRXXA

Ao

v

+−++=β

The imaginary part = 0 only when X1+ X2+ X3=0 §  It indicates that at least one reactance must be –ve (capacitor) §  X1 and X2 must be of same type and X3 must be of opposite type

2

1

31

1

XXA

XXXAA vv =

+

−=βWith imaginary part = 0,

For Unit Gain & 180o Phase-shift, 1

2 1XXAA v =⇒=β

R L1

L2C

RC1

C2L

Hartley Oscillator Colpitts Oscillator

To LC

1=ω

1

2

RCCgm =

21

21

CCCCCT +

=CLLo )(1

21 +=ω

2

1

RLLgm =

Colpitts Oscillator

RC1

C2L

Equivalent circuit

In the equivalent circuit, it is assumed that: §  Linear small signal model of transistor is used §  The transistor capacitances are neglected §  Input resistance of the transistor is large enough

+−Vπ gmVπ

R C1C2

L

)(11 LjiVV ωπ +=

At node 1,

where,

πω VCji 21 =

)1( 22

1 LCVV ωπ −=⇒

Apply KCL at node 1, we have

0111

2 =+++ VCjRVVgVCj m ωω ππ

01)1( 122

2 =⎟⎠

⎞⎜⎝

⎛ +−++ CjR

LCVVgVCj m ωωω πππ

[ ] 0)(121

321

22

=−++⎟⎟⎠

⎞⎜⎜⎝

⎛−+ CLCCCj

RLC

Rgm ωω

ω

For Oscillator Vπ must not be zero, therefore it enforces,

+−Vπ

gmVπR C1C2

Lnode 1

I1

I2I3

I4

V1

Imaginary part = 0, we have

[ ] 0)(121

321

22

=−++⎟⎟⎠

⎞⎜⎜⎝

⎛−+ CLCCCj

RLC

Rgm ωω

ω

To LC

1=ω

1

2

RCCgm =

21

21

CCCCCT +

=

Real part = 0, yields

Frequency Stability •  The frequency stability of an oscillator is

defined as •  Use high stability capacitors, e.g. silver

mica, polystyrene, or teflon capacitors and low temperature coefficient inductors for high stable oscillators.

Cppm/ T

o

oo dd

ωωω

ω =⎟⎠

⎞⎜⎝

⎛⋅1

Amplitude Stability •  In order to start the oscillation, the loop gain

is usually slightly greater than unity. •  LC oscillators in general do not require

amplitude stabilization circuits because of the selectivity of the LC circuits.

•  In RC oscillators, some non-linear devices, e.g. NTC/PTC resistors, FET or zener diodes can be used to stabilized the amplitude

Wien-bridge oscillator with bulb stabilization

Vrms

irms

Operating point

+

−

R

R

C

C

R2

Blub

Wien-bridge oscillator with diode stabilization

Rf

+

−

R

R

C

C

R1

Vo

Twin-T Oscillator

+

−

low pass filter

high pass filter

low pass region high pass region

fr f

Filter output

Bistable Circuit

+

−vo

v1

v+Vth

+Vcc

-Vcc

vo

v1

-Vth

+Vcc

-Vcc

vo

v1 Vth

+Vcc

-Vcc

vo

v1-Vth

A Square-wave Oscillator

+

−

vo

vc

vf

vc

vo

+vf¡Ðvf+vmax¡Ðvmax