# 7.9: Problems

## 7.9.1: Review Questions

1. What are the advantages of active rectifiers versus passive rectifiers?

2. What are the disadvantages of active rectifiers versus passive rectifiers?

3. What is a peak detector?

4. What is a limiter?

5. What is the function of a clamper?

6. What are the differences between active and passive clampers?

7. What is a transfer function generator circuit, and what is its use?

8. Explain how a function circuit might be used to linearize an input transducer.

9. What is a Schmitt trigger?

10. What is the advantage of a Schmitt-type comparator versus an ordinary open-loop op amp comparator?

11. What are the advantages of dedicated comparators such as the LM311 versus ordinary op amp comparators?

12. How are log and anti-log amplifiers formed?

13. What is the effect of passing a signal through a log or anti-log amplifier?

14. What is a four-quadrant multiplier?

15. How might a multiplier be used?

16. What is the difference between a multiplier and a VCA?

## 7.9.2: Problems

### Analysis Problems

1. Sketch the output of Figure 7.2.2 if the input is a 100 Hz 2 V peak triangle wave.

2. Repeat Problem 1 for a square wave input.

3. Sketch the output of Figure 7.2.8 if $$R_f$$ = 20 k$$\Omega$$ and $$R_i$$ = 5 k$$\Omega$$. Assume that $$V_{in}$$ is a 0.5 V peak triangle wave.

4. Repeat Problem 3, but with the diode reversed.

5. Sketch the output of Figure 7.2.9 if $$C$$ = 50 nF and $$R$$ = 5 M$$\Omega$$. Assume that the input is a 1 V peak pulse with 10% duty cycle. The input frequency is 1 kHz.

6. Repeat Problem 5 with the diode reversed

7. Sketch the output of Figure 7.2.14 if $$R$$ = 25 k$$\Omega$$ and $$V_{in} = 2 sin 2 \pi 500 t. 8. Repeat Problem 7 with \(V_{in}$$ equal to a 3 V peak square wave.

9. Sketch the output of Figure 7.2.15 if the diodes are reversed. Assume that $$V_{in}$$ is a 100 mV peak-to-peak sine wave at 220 Hz.

10. In Figure 7.3.5, assume that $$R$$ = 10 k$$\Omega$$, $$R_l$$ = 1 M$$\Omega$$, $$C$$ = 100 nF and $$V_{offset}$$ = 2 V. Sketch the output for a 10 kHz 1 V peak sine wave input.

11. Sketch the output of Figure 7.3.9 if $$R_i$$ = 10 k$$\Omega$$, $$R_f$$ = 40 k$$\Omega$$ and the Zener potential is 3.9 V. The input signal is a 2 V peak sine wave.

12. Sketch the transfer curve for the circuit of Problem 11.

13. In Figure 7.4.1, sketch the transfer curve if $$R_i$$ = 5 k$$\Omega$$, $$R_f$$ = 33 k$$\Omega$$, $$R_a$$ = 20 k$$\Omega$$ and $$V_{Zener}$$ = 5.7 V.

14. Sketch the output of the circuit in Problem 13 for an input signal equal to a 2 V peak triangle wave.

15. Repeat Problem 14 with a square wave input.

16. If $$R_i$$ = 4 k$$\Omega$$, $$R_a$$ = 5 k$$\Omega$$, $$R_f$$ = 10 k$$\Omega$$, and $$V_z$$ = 2.2 V in Figure 7.4.3, determine the minimum and maximum input impedance.

17. Draw the transfer curve for the circuit of Problem 16.

18. Sketch the output voltage for the circuit of Problem 16 if the input signal is a 3 V peak triangle wave.

19. Sketch the transfer curve for the circuit of Figure 7.4.4 if $$R_i$$ = 1 k$$\Omega$$, $$R_f$$ = 10 k$$\Omega$$, $$R_a$$ = 20 k$$\Omega$$, $$R_b$$ = 18 k$$\Omega$$, $$V_{Zener-a}$$ = 3.9 V, and $$V_{Zener-b}$$ = 5.7 V.

20. Sketch the output of the circuit in Problem 19 if the input is a 1 V peak triangle wave.

21. If $$R_1$$ = 10 k$$\Omega$$ and $$R_2$$ = 33 k$$\Omega$$ in Figure 7.5.3, determine the upper and lower thresholds if the power supplies are $$\pm$$15 V.

22. Determine the upper and lower thresholds for Figure 7.5.4 if $$R_1$$ = 4.7 k$$\Omega$$ and $$R_2$$ = 2.2 k$$\Omega$$, with $$\pm$$12 V power supplies.

23. Sketch the output of Figure 7.5.9 if $$V_{in} = 2 \sin 2 \pi 660 t$$, $$V_{ref} = 1 V DC$$, and $$V_{strobe} = 5 V DC. 24. Repeat Problem 23 for \(V_{strobe}$$ = 0 V DC.

25. Determine the output of Figure 7.6.1 if $$V_{in} = 0.1 V$$, $$R_i = 100 k\Omega$$, and $$I_s = 60 nA$$.

26. Determine the output of Figure 7.6.3 if $$V_{in} = 300 mV$$, $$R_f = 20 k\Omega$$ and $$I_s = 40 nA$$.

27. Sketch the output signal of Figure 7.6.6 if $$V_{in} = 2 \sin 2 \pi 1000 t$$, $$K$$ = 0.1, and $$V_{control} = 1 V$$ DC, -2 V DC, and 5 V DC.

28. Sketch the output of Figure 7.6.7 if $$V_{in} = 1 \sin 2 \pi 500 t$$, and $$K$$ = 0.1.

29. Sketch the output of Figure 7.6.8 if $$V_{in} = 5 \sin 2 \pi 2000 t$$, $$K$$ = 0.1, $$V_x$$ = 4 V DC, and $$R_1$$ = $$R_2$$ = 10 k$$\Omega$$.

### Design Problems

30. Determine values of $$R$$ and $$C$$ in Figure 7.2.9 so that stage 1 slewing is at least 1V/μs, along with a time constant of 1 ms. Assume that op amp 1 can produce at least 20 mA of current.

31. In Figure 7.3.3, assume that $$C$$ = 100 nF. Determine an appropriate value for Rl if the input signal is at least 2 kHz. Use a time constant factor of 100 for your calculations.

32. Determine new values for $$R_a$$ and $$R_f$$ in Problem 13, so that the slopes are -5 and -3.

33. Determine new resistor values for the circuit of Figure 7.4.3 such that the slopes are -10 and -20. The input impedance should be at least 3 k$$\Omega$$.

34. Determine the resistor values required in Figure 7.4.7 to produce slopes of -5, -8, and -12, if the input impedance must be at least 1 k$$\Omega$$.

35. Determine the resistor values required in Figure 7.4.12 for $$S_0$$ = -1, $$S_1$$ = $$S_{-1}$$ = -5, $$S_2$$ = $$S_{-2}$$ = -12. Also $$V_1$$ = |$$V_{-1}$$| = 3 V, and $$V_2$$ = |$$V_{-2}$$| = 4 V. Use $$R_f$$ = 20 k$$\Omega$$.

36. Sketch the transfer curve for Problem 35.

37. Determine a value for $$R_i$$ in Figure 7.6.1 so that a 3 V input will produce an output of 0.5 V. Assume $$I_s$$ = 60 nA.

38. Determine a value for $$R_f$$ in Figure 7.6.3 such that a 0.5 V input will produce a 3 V output. Assume $$I_s$$ = 40 nA.

39. If the multiplier of Figure 7.6.8 can produce a maximum current of 10 mA, what should the minimum sizes of $$R_1$$ and $$R_2$$ be? (Assume $$\pm$$15 V supplies).

### Challenge Problems

40. Assuming that 5% resistors are used in Figure 7.2.14, determine the worstcase mismatch between the two halves of the rectified signal.

41. Design a circuit that will light an LED if the input signal is beyond $$\pm$$5 V peak. Make sure that you include some form of pulse-stretching element so that the LED remains visible for short duration peaks.

42. A pressure transducer produces an output of 500 mV per atmosphere up to 5 atmospheres. From 5 to 10 atmospheres, the output is 450 mV per atmosphere. Above 10 atmospheres, the output falls off to 400 mV per atmosphere. Design a circuit to linearize this response using the Zener form

43. Repeat the Problem above using the biased-diode form.

44. Design a circuit to produce the transfer characteristic shown in Figure $$\PageIndex{1}$$

45. Sketch the output waveform of Figure 7.5.4 if $$R_1$$ = 22 k$$\Omega$$, $$R_2$$ = 4.7 k$$\Omega$$, and $$V_{in} = 2 + 8 \sin 2 \pi 120 t$$. Assume that the power supplies are $$\pm$$15 V.

Figure $$\PageIndex{1}$$

46. Determine the output of Figure 7.6.5 if $$K = 0.1$$, $$V_x = 1 \sin 2 \pi 440 t$$, and $$V_y = 3 \sin 2 \pi 200000 t$$.

## Computer Simulation Problems

47. In preceding work it was noted that the use of an inappropriate device model can produce computer simulations that are way off mark. An easy way to see this is to simulate the circuit of Figure 7.2.2 using an accurate model (such as the 741 simulation presented in the chapter) and a simple model (such as the controlled-voltage source version presented in Chapter 2). Run two simulations of this circuit for each of these models. One simulation should use a lower input frequency, such as 1 kHz, and the second simulation should use a higher frequency where the differences in the models is very apparent, such as 500 kHz.

48. Simulate the circuit designed in Problem 30 using a square wave input. Perform the simulation for several different input frequencies between 10 Hz and 10 kHz. Do the resulting waveforms exhibit the proper shapes?

49. Perform simulations for the circuit of Figure 7.2.14. Use $$R = 12$$ k$$\Omega$$. For the input waveform, use both sine and square waves, each being 1 V peak at 200 Hz.

50. Component tolerances can directly affect the rectification accuracy of the full-wave circuit shown in Figure 7.2.14. This effect is easiest to see using a square wave input. If the positive and negative portions of the input signal see identical gains, the output of the circuit will be a DC level. Any inaccuracy or mismatch will produce a small square wave riding on this DC level. This effect can be simulated using the Monte Carlo analysis option. If Monte Carlo analysis is not available on your system, you can still see the effect by manually altering the resistor values within a preset tolerance band for each of a series of simulation runs.

51. Use a simulator to verify the results of the limiter examined in Example 7.3.2.

52. Use a simulator to verify the response of the circuit shown in Figure 7.4.7. For the stimulus, use a low frequency triangle wave of 4 V peak amplitude. Does the resulting waveform exhibit the same hard “corners” shown on the transfer curve of Figure 7.4.7?