1. TUTORIAL QUESTIONS – OSCILLATOR
With the aid of a suitable diagram, explain briefly what are meant by 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
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.

19. TUTORIAL QUESTIONS – OSCILLATOR
Solution

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

21. 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. Q5

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

23. TUTORIAL QUESTIONS – ACTIVE FILTERS
A single-pole low-pass filter as shown in Fig.5-1 has a RC network with the resistance R of 2 kΩ and the capacitance C of 0.04μF. If the resistances of resistor R1 and R2 are 20 and 4 Ω respectively, determine:
cutoff frequency, fC
pass band voltage gain or the gain of non-inverting amplifier, Acl
expression of output voltage, Vo a
Figure 5-1. 23

24. TUTORIAL QUESTIONS – ACTIVE FILTERS
Sol1
(a)
The cutoff frequency for a single-pole low-pass filter is
(b)
The pass band voltage gain or the gain of non-inverting amplifier is calculated as follows: 24

25. TUTORIAL QUESTIONS – ACTIVE FILTERS
Sol1_Cont’d
Applying the voltage divider rule, the input voltage, which is the voltage across the capacitor C, is determined as follows:
Multiplying the numerator and the denominator by j and substituting for XC results in the following: 25

26. TUTORIAL QUESTIONS – ACTIVE FILTERS
Sol1_Cont’d
At the cutoff frequency fc, the magnitude of the capacitive reactance XC equals the resistance of the resistor R.
Substituting for 2πRC in Eq. results in the following equation for va:
Where f is the operating frequency and fc is the cutoff frequency. 26

27. TUTORIAL QUESTIONS – ACTIVE FILTERS
Sol1_Cont’d
The output vo is the amplified version of vi.
where Acl(NI) is the pass-band voltage gain or the gain of the non-inverting amplifier. 27

28. TUTORIAL QUESTIONS – ACTIVE FILTERS
Determine the critical frequency, pass-band voltage gain, and damping factor for the second-order low-pass active filter in Fig. 5-2 with the following circuit components:
RA = 5 kΩ, RB = 8 kΩ, CA = 0.02 μF, CB = 0.05 μF, R1 = 10 kΩ, R2 = 20 kΩ
Figure 5-2. 28

29. TUTORIAL QUESTIONS – ACTIVE FILTERS
Sol2
The critical frequency for the second-order low-pass active filter is 29

30. TUTORIAL QUESTIONS – ACTIVE FILTERS
Sol2_Cont’d
The pass-band voltage gain set by the values of R1 and R2 is 30

31. Q3
Design a multiple-feedback band-pass filter with the maximum gain, Ao = 8, quality factor, Q = 25 and center frequency, fo =10 kHz. Assume that C1 = C2 = 0.01µF.
Draw the circuit design of the active band-pass

32. Sol3

33. Sol3_Con’t
The drawing of the circuit diagram :

34. Q4
Determine the center frequency, fc, quality factor, Q and bandwidth, BW for the band-pass output of the state-variable filter in following figure.
Given that R1 = R2 = R3 = 25 kΩ, R4 = R7 =2.5 kΩ, R5 = 150 kΩ,
R6 = 2.0 kΩ, C1 = C2 = 0.003µF

35. Sol4_Con’t
i) Center frequency, fc
ii) Quality factor, Q

36. Sol4_Con’t iii) Bandwidth, BW
The critical frequency, fo of the integrators usually made equal
to the critical frequency, fc