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AE_Lecture11_Part1_Audio Oscillators

Module by: Bijay_Kumar Sharma. E-mail the author

Summary: AE_lecture11_Part1 describes a feedback system and the Barkhausen Criteria for sustained oscillation.It describes the circuit diagram and working of Audio Oscillators.

AE_LECTURE 11_Part1.

Section 1. FEEDBACK SYSTEMS STABILITY AND OSCILLATORS

Figure 1
Figure 1 (Picture 1.png)

Figure 1. Block Diagram of feedback system.

Here ‘s’ implies s-domain and s=σ+jω. Just as we have time domain, we have frequency domain, s-domain and z-domain.

In Time-domain we study the response of a system with respect to time.

In Frequency-domain we study the steady state sinusoidal response of a system.

In s-domain we get to see the total response of the system. Whenever a certain input is applied we have a transient response and steady state response. Total response is the sum total of transient plus steady state response.

z-domain is for discrete time systems and s-domain is for continuous time systems.

In the Block Diagram of a feedback system, we have the Basic Amplifier Block A(s) , the feedback network block f(s) and comparison node. f(s) can be frequency independent or frequency selective.

Figure 2
Figure 2 (graphics1.png)
Figure 3
Figure 3 (graphics2.png)
Figure 4
Figure 4 (graphics3.png)
Figure 5
Figure 5 (graphics4.png)
Figure 6
Figure 6 (graphics5.png)
Figure 7
Figure 7 (graphics6.png)

In negative feed back

Figure 8
Figure 8 (graphics7.png)

In positive feed back

Figure 9
Figure 9 (graphics8.png)

Negative feed back system is a degenerative system:

Figure 10
Figure 10 (graphics9.png)

And Aclosed < Aopen Hence a degenerate system.

Positive feedback system is a regenerative system:

Aclosed = Aopen/[1- |L(s)|]

If

Figure 11
Figure 11 (graphics10.png)
then
Figure 12
Figure 12 (graphics11.png)
Hence regenerative system

Figure 13
Figure 13 (graphics12.png)
then
Figure 14
Figure 14 (graphics13.png)
and system becomes unstable and oscillatory.

We can take advantage of instability and realize a pure sine wave oscillator.

When

Figure 15
Figure 15 (graphics14.png)
. This is BARKHAUSEN CRITERIA.

Then we realize a pure sine wave oscillator.

For self starting condition we allow

Figure 16
Figure 16 (graphics15.png)
to be slightly greater than
Figure 17
Figure 17 (graphics16.png)
.This will ensure self starting condition but it will cause a slight distortion.

Section 2. Class of audio oscillators(1 Hz → 100kHz)

Wien Bridge Oscillator, Phase-Shift Oscillator and Quadrature Oscillator.

2.1. Wien Bridge Oscillator.

Here Op.Amp is connected as an Non-inverting amplifier with a gain of 3/_0º. The feedback network is a notch filter providing a dip of exactly 1/3 and phase angle 0º at an angular frequency of

Figure 18
Figure 18 (graphics17.png)
Figure 19
Figure 19 (graphics18.png)
. Thus an exact loop gain of unity with 0º phase shift is achieved. But for selfstarting condition the non-inverting gain is kept slightly larger than 3.

Figure 20
Figure 20 (Picture 2.png)

Figure 2. Circuit Diagram of Wien Bridge Oscillator.

Figure 21
Figure 21 (graphics19.png)
Figure 22
Figure 22 (graphics20.png)
Figure 23
Figure 23 (graphics21.png)
Figure 24
Figure 24 (graphics22.png)

Figure 25
Figure 25 (graphics23.png)

Replacing s by jω,

Figure 26
Figure 26 (graphics24.png)

At

Figure 27
Figure 27 (graphics25.png)

Figure 28
Figure 28 (graphics26.png)
Figure 29
Figure 29 (graphics27.png)

Then Loop Gain=

Figure 30
Figure 30 (graphics28.png)

Oscillation Frequency=

Figure 31
Figure 31 (graphics29.png)

Figure 32
Figure 32 (graphics30.png)
; This gives a non-inverting gain of 3.

2.2. RC PHASE SHIFT OSCILLATOR

Figure 33
Figure 33 (Picture 3.png)

Figure 3. Circuit Diagram of RC phase shift oscillator.

Here Op Amp is connected as an inverting amplifier providing a gain of 29 and phase shift of 180º.

The RC phase shift network gives an attenuation of 1/29 and a phase shift of another 180º.

Thus an exact Loop Gain of 1/_0º is achieved. But for self starting condition the inverting gain is kept slightly larger than 29.

Figure 34
Figure 34 (Picture 4.png)

Figure 4. The feed back network.

The Loop Gain expression is:

L(jω) =

Figure 35
Figure 35 (graphics31.png)
where ∆ =
Figure 36
Figure 36 (graphics32.png)
and
Figure 37
Figure 37 (graphics33.png)
=
Figure 38
Figure 38 (graphics34.png)
;

Re(∆) =

Figure 39
Figure 39 (graphics35.png)

Im(∆)=

Figure 40
Figure 40 (graphics36.png)

To satisfy the Barkhausen Criteria, Re(∆) = 0;

But Re∆=

Figure 41
Figure 41 (graphics37.png)

Therefore by cancelling

Figure 42
Figure 42 (graphics38.png)
through out, we get

Figure 43
Figure 43 (graphics39.png)

Therefore

Figure 44
Figure 44 (graphics40.png)
=
Figure 45
Figure 45 (graphics41.png)
;

Second part of the Barkhausen Criteria says that at oscillatory frequency the phase angle should be zero.

L(ω=

Figure 46
Figure 46 (graphics42.png)
) =
Figure 47
Figure 47 (graphics43.png)
=
Figure 48
Figure 48 (graphics44.png)
=
Figure 49
Figure 49 (graphics45.png)
)

L(ω=

Figure 50
Figure 50 (graphics46.png)
) =
Figure 51
Figure 51 (graphics47.png)
) = 1 ;

Therefore

Figure 52
Figure 52 (graphics48.png)

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