Positive Feedback

 When input and feedback signal both are in same phase, It is

called a positive feedback.

 Positive feedback is used in analog and digital systems.

 A primary use of +ve feedback is in the production of oscillators.

+ Vi
Vs Σ A(f) Vo
+

Vf SelectiveNetwork
β(f)
Vo A
Vo = AVi = A(Vs + V f ) and V f = βVo
=
Vs 1 − Aβ

Barkhausen Criterion: for oscillator βA=1 and +ve feedback

2
 Oscillator Circuit
 Oscillator is an electronic circuit which converts dc signal into ac
signal.

 Oscillator is basically a positive feedback amplifier with unity loop

gain.

 For an inverting amplifier- feedback network provides a phase shift

of 180°° while for non-inverting amplifier- feedback network provides
a phase shift of 0°° to get positive feedback .
Vo A If βA=1 then Vo = ∞ ;
= Very high output with zero input.
Vs 1 − Aβ
Use positive feedback through frequency-selective feedback network to ensure sustained oscillation at ω0

Use of Oscillator Circuits

Clock input for CPU, DSP chips …

Local oscillator for radio receivers, mobile receivers, etc

As a signal generators in the lab

Clock input for analog-digital and digital-analog converters 3
Oscillators

 If the feedback signal is not

positive and gain is less than
unity, oscillations dampen out.

 If the gain is higher than unity

then oscillation saturates.

Type of Oscillators
Oscillators can be categorized according to the types of feedback
network used:

 RC Oscillators: Phase shift and Wien Bridge Oscillators

 LC Oscillators: Colpitt and Hartley Oscillators

 Crystal Oscillators 4
RC Oscillators

R −1 XC
Vo = ( )Vin and φ = tan ( )
R − jX c R

 Φ =0o if Xc=0 and Φ =90o if R=0

 However adjusting R to zero is impractical because it would lead

to no voltage across R, thus in a RC circuit, phase shift is always
≤ 90o and it is a function of frequency.

 Hence to get 180o phase shift from the feedback network, we

need 3 RC circuits.

 RC oscillators build by using inverting amplifier and 3 RC

circuits is known as phase shift oscillator.
5
 RC Oscillators: Phase shift Oscillator
 Use of an inverting Rf
amplifier.
R1
 The additional 180o − C C C
phase shift is provided
by an RC ladder network. +
R R R
 It can be used for very
low frequencies and
provides good frequency
stability.

A phase shift of 180°° is obtained at a frequency f, given by

1
f =
2πCR 6
vo 1
=−
At this frequency the gain of the network is vi 29
6
 RC Oscillators: Phase shift Oscillator
At node V2
I3
V2 = Vo + I 3 X c = Vo +
jwC
V
But I 3 = o
R
1
V2 = Vo (1 + )
jω CR

V V V 1 Vo 1
I 2 = I 3 + 2 = o + o (1 + ) I 2 = (2 + )
R R R jωCR R jωCR

At node V1 I2 3 1
V1 = V2 + = Vo (1 + − 2 2 2)
jω C jωcR ω C R
7
RC Oscillators: Phase shift Oscillator

I2 3 1
V1 = V2 + = Vo (1 + − 2 2 2)
jω C jωcR ω C R
V1 V0 4 1
I1 = I 2 + = ( 3 + − 2 2 2)
R R jωcR ω C R

I1 6 5 1
Vi = V1 + = Vo (1 + − 2 2 2− 3 3 3
)
jω C jωcR ω C R jω C R

Output voltage should be real hence imaginary part equal to zero.

6 1 1
− 3 3 3
=0 6ω 2C 2 R 2 = 1 ω=
jωcR jω C R RC 6

At this frequency: Vi =-29 Vo therefore, Av = -29 8

RC Oscillators: Wien Bridge Oscillator
Rf
Z1
R1
− R1 C1 Z2
+
Vi C2 R2 Vo
C R
Vo
R Z1
C Z2
Feedback Network

 Feedback network is a lead-lag circuit where R1, C1 form the lag

portion and R2, C2 form the lead portion. Thus feedback network
provides 0o phase shift.

9
 RC Oscillators: Wien Bridge Oscillator
1 1 Z1
Let X C1 = X C2 = and
ωC1 ωC2 R1 C1 Z2
Z1 = R1 − jX C1
1
−1 Vi C2 R2 Vo
1  − jR2 X C 2
Z2 =  +  =
 R2 − jX C 2  R2 − jX C 2
Therefore, the feedback factor,
Vo Z2 (− jR2 X C 2 / R2 − jX C 2 )
β= = =
Vi Z1 + Z 2 ( R1 − jX C1 ) + (− jR2 X C 2 / R2 − jX C 2 )

− jR2 X C 2
β=
( R1 − jX C1 )( R2 − jX C 2 ) − jR2 X C 2

R2 X C 2
β=
R1 X C 2 + R2 X C1 + R2 X C 2 + j ( R1 R2 − X C1 X C 2 )
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RC Oscillators: Wien Bridge Oscillator
For Barkhausen Criterion, imaginary part = 0, R1 R2 − X C1 X C 2 = 0
1 1
R1R2 = ω = 1 / R R C C 0.34
ωC1 ωC2 1 2 1 2
0.32

Feedback factor β
0.3
Supposing, R1=R2=R and XC1= XC2=XC, 0.28
β=1/3
0.26
RX C
β= 0.24
3RX C + j ( R 2 − X C2 ) 0.22
0.2
At this frequency: β = 1 / 3 and phase shift = 0o f(R=Xc)
1

Due to Barkhausen Criterion, gain Avβ=1

0.5 Phase=0
Where Av : Gain of the amplifier
Phase
Rf Rf 0

Av β = 1 ⇒ Av = 3 = 1 + =2
R1 R1 -0.5

-1
Frequency 11
Stabilization method for Wien Bridge Oscillator

Rf R C
R1 R3
− +
Vo
+ R C −
C R
R2

R
C Blub

When dc power is first applied, both zener diode is off.

R f + R3 R3 Because
Rf
Av = 1 + = 3+ =2
R1 R1 R1
Initially, a small +ve feedback signal develops from noise or turn-on transients. This
feedback signal is amplified and continually reinforced , resulting in a buildup of the output
voltage. When the output voltage reaches the zener breakdown voltage , zener diode
conducts and effective short out R3 and thus lowers the close loop voltage gain to 3.
12