1. TUTORIAL QUESTIONS – OSCILLATOR
With the aid of a suitable diagram, explain briefly the following terms;
a) open-loop gain;
b) Loop gain
c) closed-loop gain

2. TUTORIAL QUESTIONS – OSCILLATOR
Solution a)
The open-loop gain is the gain of the operational amplifier without feedback. Referring to the figure, the open-loop gain is A and is expressed as;

3. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d) b)
The loop gain is the amplification (or attenuation) experienced by the signal as it travels from the input to the operational amplifier to the output of the feedback network. From the figure, the loop gain is A and is expressed as;

4. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d) c)
The closed-loop gain is the amplification of the input signal by the amplifier with feedback. From the figure, the closed-loop gain is Af and is expressed as;

5. TUTORIAL QUESTIONS – OSCILLATOR
With the aid of suitable figures, describe the term Barkhausen criterion as applied to sinusoidal oscillators

6. TUTORIAL QUESTIONS – OSCILLATOR
Solution
In the figure, it can be shown that the closed-loop gain, Af, is given by the expression;

7. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
When the magnitude of the loop gain is unity and the phase is zero i.e. when;
the system will produce an output for zero input.

8. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
Barkhausen criterion states that in order to start and sustain an oscillation, the loop gain must be unity and the phase shift through the loop must be 0

9. TUTORIAL QUESTIONS – OSCILLATOR
The following figure shows a Wien-Bridge oscillator employing an ideal operational amplifier A. Derive an expression for the frequency of oscillation o in terms of R and C.

10. TUTORIAL QUESTIONS – OSCILLATOR
Solution

11. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)

12. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)

13. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)

14. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)

15. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
and
The loop gain;

16. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
Substituting for s;
Since A must be real at the oscillation frequency o, it follows that;

17. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
or;

18. TUTORIAL QUESTIONS – OSCILLATOR
The transfer function, (s) of the phase-shift network in the following figure may be expressed as;
Show that the frequency of oscillation, fo is given the expression;

19. TUTORIAL QUESTIONS – OSCILLATOR
Q4 (cont’d)

20. TUTORIAL QUESTIONS – OSCILLATOR
Solution
Substituting j for s;

21. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
At the frequency of oscillation o;
At this frequency the transfer function must be real i.e. the imaginary part must be zero;

22. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
or;

23. TUTORIAL QUESTIONS – OSCILLATOR
Describe the operation of the relaxation oscillator shown in the following figure: Q5

24. TUTORIAL QUESTIONS – OSCILLATOR
Solution
Assume that the initial capacitor voltage, vC is zero. When the supply voltage vBB is first applied, the UJT is in the OFF state, IE is zero and C charges exponentially towards VBB through R1. This is the charging phase (See the following figure)

25. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)

26. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
When the voltage vC = VP, the UJT fires and C discharges exponentially towards zero (ground) through the internal resistance RB1 of the UJT and R2. This is the discharging phase (See the following figure).
When vC reduces to the valley potential VV, the UJT is OFF and C reenters the charging phase again but for the second cycle onwards, the initial value of vC is equal to VV.

27. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)

28. 07/10/08

29. TUTORIAL QUESTIONS – OSCILLATOR
For the Wien-Bridge oscillator shown in the following figure, show that, to guarantee oscillation, the ratio:

30. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
It can be shown that, for the oscillator in the figure;
where o the oscillation angular frequency.

31. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
It can be also shown that the loop gain is;

32. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
Substituting for o in the expression for A gives us;
Since |A| must be unity, it follows that;

33. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
Therefore, in order to guarantee oscillation;

34. TUTORIAL QUESTIONS – OSCILLATOR
Show that for the phase-shift oscillator shown in the following figure,in order to sustain oscillation, the ratio;

35. TUTORIAL QUESTIONS – OSCILLATOR
Solution
It can be shown that, for the given oscillator circuit, the frequency of oscillation o is given by the expression;
and at this frequency the feedback factor  is;

36. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
Substituting for o;
The signal, in going through the phase-shift network has experienced attenuation by a factor of 29 and a phase shift of 180. The amplifier gain, A must be such that the loop gain will be unity i.e.;

37. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
The signal, in going through the phase-shift network has experiences attenuation by a factor of 29 and a phase shift of 180. The amplifier gain, A must be such that the loop gain will be unity i.e.;
The amplifier shall also introduce an additional phase shift of 180 (which implies the use of inverting configuration)

38. TUTORIAL QUESTIONS – OSCILLATOR
For the relaxation oscillator shown in the figure, sketch and label the waveforms of vC and vR2 and indicate in your sketch, the relevant mathematical equations describing various sections of the waveform.

39. TUTORIAL QUESTIONS – OSCILLATOR
Solution

40. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)

41. TUTORIAL QUESTIONS – OSCILLATOR
Calculate the frequency of oscillation fo in the following figure if R = 10 k and C = 796 pF.

42. TUTORIAL QUESTIONS – OSCILLATOR
Solution
Substituting values;

43. TUTORIAL QUESTIONS – OSCILLATOR
Design a phase shift oscillator in the following figure, to obtain a sinusoidal wave of 3 kHz. Use C = 22 nF and R1 = 50 k.
Q10

44. TUTORIAL QUESTIONS – OSCILLATOR
Solution
From the expression;
we obtain;
Substituting values;

45. TUTORIAL QUESTIONS – OSCILLATOR
The following figure shows a basic relaxation oscillator employing a unijunction transistor (UJT). The UJT has the following properties:
Q11
and
during the discharge phase
of the capacitor C.

46. TUTORIAL QUESTIONS – OSCILLATOR
Q11 (cont’d)
If:
determine;
a) the value of VP to switch-on the UJT;
b) the range of R1 to switch-on and switch-off the UJT;
c) the frequency of oscillation.
Assume Vpn = 0.7 V.

47. TUTORIAL QUESTIONS – OSCILLATOR
Q11 (cont’d)

48. TUTORIAL QUESTIONS – OSCILLATOR
Solution a)
Substituting values;

49. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
To switch-on the UJT;
Substituting values;

50. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d) b)
To ensure switching-on and switching-off of the UJT, R1 must be in the following range;

51. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
Substituting values;

52. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d) c)
Substituting values;

53. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)
Substituting values;

54. TUTORIAL QUESTIONS – OSCILLATOR
Solution (cont’d)