# 11.13: Problems

## 11.13.1: Analysis Problems

1. Using Figure 11.5.1b, determine the loss for a 1 kHz Butterworth secondorder low-pass filter at 500 Hz, 1 kHz, 2 kHz, and 4 kHz.

2. Using Figure 11.6.18, determine the loss for a 2 kHz 3 dB ripple Chebyshev third-order low-pass filter at 500 Hz, 1 kHz, 2 kHz, 4 kHz, and 6 kHz.

3. Using Figure 11.5.1b, determine the loss for a 500 Hz Bessel second-order high-pass filter at 200 Hz, 500 Hz, and 2 kHz.

4. Using Figure 11.6.18, determine the loss one octave above the cutoff frequency for a fourth-order low-pass filter of the following alignments: Butterworth, Bessel, 1 dB-ripple Chebyshev.

5. Repeat Problem 4 for high-pass filters.

6. using Figures 11.5.1b and 11.6.18, determine the loss at 8 kHz for 3 kHz lowpass Butterworth filters of orders 2 through 6.

7. Using Figures 11.5.1b and 11.6.18, determine the loss at 500 Hz for 1.5 kHz high-pass Bessel filters of orders 2 through 6.

8. Repeat Problem 7 using 3 dB-ripple Chebyshevs.

9. Using Figures 11.5.1b and 11.6.18, determine the loss at 500 Hz for 200 Hz high-pass 3 dB-ripple Chebyshev filters of orders 2 through 6.

10. Using Figure 11.6.18, determine the loss one octave below the cutoff frequency for a third-order high-pass filter of the following alignments: Butterworth, Bessel, 1 dB-ripple Chebyshev.

11. Repeat Problem 10 for low-pass filters. 12. An application requires that the stop-band attenuation of a low-pass filter be at least −15 dB at 1.5 times the critical frequency. Determine the minimum order required for the Butterworth and 1 dB and 3 dB-ripple Chebyshev alignments.

13. An application requires that the stop-band attenuation of a high-pass filter be at least −20 dB one octave below the critical frequency. Determine the minimum order required for the Butterworth and 1 dB-ripple and 3 dB-ripple Chebyshev alignments.

14. A band-pass filter has a center frequency of 1020 Hz and a bandwidth of 50 Hz. Determine the filter $$Q$$.

15. A band-pass filter has upper and lower break frequencies of 9.5 kHz and 8 kHz. Determine the center frequency and $$Q$$ of the filter.

16. Design a second-order Butterworth low-pass filter with a critical frequency of 125 Hz. The pass band gain should be unity.

17. Repeat Problem 16 for a high-pass filter.

18. Repeat Problem 16 using a Bessel alignment.

## 11.13.2: Design Problems

19. A particular application requires that all frequencies below 400 Hz should be attenuated. The attenuation should be at least −22 dB at 100 Hz. Design a filter to meet this requirement.

20. Repeat Problem 19 for an attenuation of at least −35 dB at 100 Hz.

21. Audiophile quality stereo systems often use subwoofers to reproduce the lowest possible musical tones. These systems typically use an electronic crossover approach as explained in Example 11.6.4. Design an electronic crossover for this application using third-order Butterworth filters. The crossover frequency should be set at 65 Hz.

22. Explain how the design sequence of Problem 21 is altered if either a new crossover frequency is chosen, or a different alignment is specified.

23. Design a band-pass filter that will only allow frequencies between 150 Hz and 3 kHz. The attenuation slopes should be at least 40 dB per decade. (A filter such as this is useful for “cleaning up” recordings of human speech.)

24. Design a band-pass filter with a center frequency of 2040 Hz and a bandwidth of 400 Hz. The circuit should have unity gain. Also, determine the $$f_{unity}$$ requirement of the op amp(s) used.

25. Repeat Problem 24 for a center frequency of 440 Hz and a bandwidth of 80 Hz.

26. Design a band-pass filter with upper and lower break frequencies of 700 Hz and 680 Hz.

27. Design a notch filter to remove 19 kHz tones. The $$Q$$ of the filter should be 25. (This filter is useful in removing the stereo “pilot” signal from FM radio broadcasts.)

28. Design a second-order low-pass filter with a critical frequency of 30 kHz. Use a state-variable filter. The circuit should have a gain of +6 dB in the pass band.

29. Design a bass/treble equalizer to meet the following specification: maximum cut and boost = 25 dB below 50 Hz and above 10 kHz.

30. Using the MF10, design a fourth-order low-pass Butterworth filter with a critical frequency of 3.5 kHz.

31. Using the MF10, design a low-pass filter that is adjustable from 200 Hz to 10 kHz. Do not ignore the oscillator design.

## 11.13.3: Challenge Problems

32. Design a low-pass second-order filter that may be adjusted by the user from 200 Hz to 2 kHz. Also, make the circuit switchable between Butterworth and Bessel alignments.

33. Design a subsonic filter that will be 3 dB down from the pass band response at 16 Hz. The attenuation at 10 Hz must be at least 40 dB. Although pass band ripple is permissible, the gain should be unity.

34. Design an adjustable band-pass filter with a $$Q$$ range from 10 to 25, and a center frequency range from 1 kHz to 5 kHz.

35. Modify the design of the previous problem so that as the $$Q$$ is varied, the pass-band gain remains constant at unity.

## 11.13.4: Computer Simulation Problems

36. Verify the magnitude response of the circuit designed in Problem 16 by using a simulator. Check both the critical frequency and the roll off rate.

37. Verify the magnitude response of the electronic crossover designed in Problem 21 by using a simulator. Plot both outputs simultaneously on one graph.

38. Verify the magnitude and phase of the filter designed in Problem 24 by using a simulator.

39. Compare the simulations of the circuit designed in Problem 28 using the relatively slow LM741, versus the medium-speed LF411. Is there any noticeable change? What can you conclude from this? Would the results be similar if the break frequency was increased by a factor of 50?

40. It is very common to plot the adjustment range of equalizers on a single graph, as shown in Figure 12-49. Use a simulator to create a plot of the adjustment range of the equalizer designed in Problem 29.

41. Simulate and verify the design of challenge Problem 32.

42. Simulate and verify the design of challenge Problem 33.

43. Verify the design of challenge Problem 34 using a simulator. Include four separate plots, showing maximum and minimum $$Q$$ with maximum and minimum center frequency.

44. Verify the design of challenge Problem 35 using a simulator. Include two simultaneous plots, one showing minimum $$Q$$ with maximum and minimum center frequency, and the other showing maximum $$Q$$ with maximum and minimum center frequency