1. Introduction to Frequency Selective Circuits
CHAPTER 14
Introduction to Frequency
Selective Circuits

2. CHAPTER CONTENTS 14.1 Some Preliminaries 14.2 Low-Pass Filters
14.3 High-Pass Filters
14.4 Bandpass Filters
14.5 Bandreject Filters

3. CHAPTER OBJECTIVES
Know the RL and RC circuit configurations that act as low-pass filters and be able to design RL and RC circuit component values to meet a specified cutoff frequency.
Know the RL and RC circuit configurations that act as high-pass filters and be able to design RL and RC circuit component values to meet a specified cutoff frequency.

4. CHAPTER OBJECTIVES
Know the RLC circuit configurations that act as bandpass filters, understand the definition of and relationship among the center frequency, cutoff frequencies, bandwidth, and quality factor of a bandpass filter, and be able to design RLC circuit component values to meet design specifications.
Know the RLC circuit configurations that act as bandreject filters, understand the definition of and relationship among the center frequency, cutoff frequencies, bandwidth, and quality factor of a bandreject filter, and be able to design RLC circuit component values to meet design specification.

5. Frequency-selective circuits
Varying source frequency on circuit voltages and currents. The result of this analysis is the frequency response of a circuit.
Frequency-selective circuits are also called filters, such as telephones, radios, televisions, and satellites, employ frequency-selective circuits.
Figure The action of a filter on an input signal results in an output signal.

6. 14.1 Some Preliminaries
The signals passed from the input to the output fall within a band of frequencies called the passband. Input voltages outside this band have their magnitudes attenuated by the circuit and are thus effectively prevented from reaching the output terminals of the circuit. Frequencies not in a circuit’s passband are in its stopband. Frequency-selective circuits are categorized by the location of the passband.
Figure A circuit with voltage input and output.

7. One way of identifying the type of frequency-selective circuit is to examine a frequency response plot. A frequency response plot shows how a circuit’s transfer function (both amplitude and phase) changes as the source frequency changes.
A frequency response plot has two parts. One is a graph of |H(jω)|versus frequency ω. This part of the plot is called the magnitude plot. The other part is a graph of θ(jω) versus frequency ω. This part is called the phase angle plot.

8. One passband and one stopband, which are defined by the cutoff frequency.
Low-pass filter, which passes signals at frequencies lower than the cutoff frequency from the input to the output.
High-pass filter, which passes signals at frequencies higher than the cutoff frequency.
Bandpass filter, which passes a source voltage to the output only when the source frequency is within the band defined by the two cutoff frequencies.
Bandreject filter, which passes a source voltage to the output only when the source frequency is outside the band defined by the two cutoff frequencies.

9. Figure Ideal frequency response plots of the four types of filter circuits. (a) An ideal low-pass filter. (b) An ideal high-pass filter. (c) An ideal bandpass filter. (d) An ideal bandreject filter.

10. Passive filters
Passive filters, their filtering capabilities depend only on the elements: resistors, capacitors, and inductors.

11. 14.2 Low-Pass Filters The Series RL Circuit— Qualitative Analysis
Figure (a) A series RL low-pass filter. (b) The equivalent circuit at ω = 0 and (c) The equivalent circuit at ω = ∞.

12. Figure 14.5 The frequency response plot for the series RL circuit in Fig. 14.4(a).

13. Defining the Cutoff Frequency
The definition for cutoff frequency widely used by electrical engineers is the frequency for which the transfer function magnitude is decreased by the factor from its maximum value: where Hmax is the maximum magnitude of the transfer function.

14. The Power Delivered

15. At the cutoff frequency the average power delivered by the circuit is one half the maximum average power. Thus ωc is also called the half-power frequency.

16. The Series RL Circuit—Quantitative Analysis
Figure The s-domain equivalent for the circuit in Fig. 14.4(a).

17. Cutoff frequency for RL filters

18. Example 14.1
Electrocardiology is the study of the electric signals produced by the heart. These signals maintain the heart’s rhythmic beat, and they are measured by an instrument called an electrocardiograph. This instrument must be capable of detecting periodic signals whose frequency is about 1 Hz (the normal heart rate is 72 beats per minute). The instrument must operate in the presence of sinusoidal noise consisting of signals from the surrounding electrical environment, whose fundamental frequency is 60 Hz—the frequency at which electric power is supplied.

19. Example 14.1
Choose values for R and L in the circuit of Fig. 14.4(a) such that the resulting circuit could be used in an electrocardiograph to filter out any noise above 10 Hz and pass the electric signals from the heart Vo at or near 1 Hz. Then compute the magnitude of at 1 Hz, 10 Hz, and 60 Hz to see how well the filter performs.

20. Example 14.1

21. Example 14.1

22. Example 14.1

23. A Series RC Circuit as a low-pass filter
Zero frequency (ω = 0): The impedance of the capacitor is infinite, and the capacitor acts as an open circuit. The input and output voltages are thus the same.
Frequencies increasing from zero: The impedance of the capacitor decreases relative to the impedance of the resistor, and the source voltage divides between the resistive impedance and the capacitive impedance. The output voltage is thus smaller than the source voltage.
Infinite frequency (ω = ∞): The impedance of the capacitor is zero, and the capacitor acts as a short circuit. The output voltage is thus zero.
Figure A series RC low-pass filter.

24. Example 14.2 For the series RC circuit in Fig. 14.7:
a) Find the transfer function between the source voltage and the output voltage.
b) Determine an equation for the cutoff frequency in the series RC circuit.
c) Choose values for R and C that will yield a lowpass filter with a cutoff frequency of 3 kHz.
Figure A series RC low-pass filter.

25. Example 14.2
Figure The s-domain equivalent for the circuit in Fig

26. Example 14.2

27. Example 14.2

28. Transfer function for a low-pass filter
Figure Two low-pass filters, the series RL and the series RC, together with their transfer functions and cutoff frequencies.
Transfer function for a low-pass filter

29. Relating the Frequency Domain to the Time Domain
This result is a direct consequence of the relationship between the time response of a circuit and its frequency response.

31. 14.3 High-Pass Filters The Series RC Circuit— Qualitative Analysis
Figure (a) A series RC high-pass filter; (b) the equivalent circuit at ω = 0 and (c) the equivalent circuit at ω = ∞.

32. Figure 14.11 The frequency response plot for the series RC circuit in Fig. 14.10(a).

33. The Series RC Circuit—Quantitative Analysis
Figure The s-domain equivalent of the circuit in Fig (a).

35. Example 14.3
Show that the series RL circuit in Fig also acts like a high-pass filter:
a) Derive an expression for the circuit’s transfer function.
b) Use the result from (a) to determine an equation for the cutoff frequency in the series RL circuit.
c) Choose values for R and L that will yield a highpass filter with a cutoff frequency of 15 kHz.

36. Example 14.3
Figure The circuit for Example 14.3.

37. Example 14.3

38. Example 14.3

39. Example 14.3
Figure The s-domain equivalent of the circuit in Fig

40. Example 14.4
Examine the effect of placing a load resistor in parallel with the inductor in the RL high-pass filter shown in Fig :
a) Determine the transfer function for the circuit in Fig
b) Sketch the magnitude plot for the loaded RL high-pass filter, using the values for R and L from the circuit in Example 14.3(c) and letting. On the same graph, sketch the magnitude plot for the unloaded RL high-pass filter of Example 14.3(c).

41. Example 14.4 Figure 14.15 The circuit for Example 14.4.
Figure The s-domain equivalent of the circuit in Fig

42. Example 14.4

43. Example 14.4

44. Example 14.4
Figure The magnitude plots for the unloaded RL high-pass filter of Fig and the loaded RL high-pass filter of Fig

45. Transfer function for a high-pass filter
Figure Two high-pass filters, the series RC and the series RL, together with their transfer functions and cutoff frequencies.

47. 14.4 Bandpass Filters
Ideal bandpass filters have two cutoff frequencies which identify the passband.
For realistic bandpass filters, cutoff frequencies are defined as the frequencies for which the magnitude of the transfer function equals

48. Center Frequency, Bandwidth, and Quality Factor
Center frequency ωo, defined as the frequency for which a circuit’s transfer function is purely real.
Resonant frequency, is the same name given to the frequency that characterizes the natural response of the second-order circuits.
When a circuit is driven at the resonant frequency, we say that the circuit is in resonance.
Bandwidth, b, is the width of the passband.
Quality factor, is the ratio of the center frequency to the bandwidth.

49. The Series RLC Circuit—Qualitative Analysis
Figure (a) A series RLC bandpass filter; (b) the equivalent circuit for ω = 0; and (c) the equivalent circuit for ω = ∞.

50. The frequency response plot for the series RLC bandpass filter circuit
Figure The frequency response plot for the series RLC bandpass filter circuit in Fig

51. The Series RLC Circuit—Quantitative Analysis
Figure The s-domain equivalent for the circuit in Fig (a).

52. Center frequency

54. Cutoff frequencies, series RLC filters

55. Relationship between center frequency and cutoff frequencies

56. Relationship between bandwidth and cutoff frequencies

57. Quality factor
The ratio of center frequency to bandwidth.

58. Cutoff frequencies

59. Example 14.5
A graphic equalizer is an audio amplifier that allows you to select different levels of amplification within different frequency regions. Using the series RLC circuit in Fig (a), choose values for R, L, and C that yield a bandpass circuit able to select inputs within the 1–10 kHz frequency band. Such a circuit might be used in a graphic equalizer to select this frequency band from the larger audio band (generally 0–20 kHz) prior to amplification.

60. Example 14.5

61. Example 14.5

62. Example 14.5

63. Example 14.5

64. Example 14.6
a) Show that the RLC circuit in Fig is also a bandpass filter by deriving an expression for the transfer function b) Compute the center frequency, c) Calculate the cutoff frequencies, and the bandwidth, and the quality factor, Q. d) Compute values for R and L to yield a bandpass filter with a center frequency of 5 kHz and a bandwidth of 200 Hz, using a capacitor.

65. Example 14.6
Figure The circuit for Example 14.6.

66. Example 14.6
Figure The s-domain equivalent of the circuit in Fig

67. Example 14.6

68. Example 14.6

69. Example 14.6

70. Example 14.6

71. Example 14.6

72. Example 14.7
For each of the bandpass filters we have constructed, we have always assumed an ideal voltage source, that is, a voltage source with no series resistance. Even though this assumption is often valid, sometimes it is not, as in the case where the filter design can be achieved only with values of R, L, and C whose equivalent impedance has a magnitude close to the actual impedance of the voltage source. Examine the effect of assuming a nonzero source resistance, Ri, on the characteristics of a series RLC bandpass filter.

73. Example 14.7
a) Determine the transfer function for the circuit in Fig b) Sketch the magnitude plot for the circuit in Fig , using the values for R, L, and C from Example 14.5 and setting Ri = R. On the same graph, sketch the magnitude plot for the circuit in Example 14.5, where Ri = 0.
Figure The circuit for Example 14.7.

74. Example 14.7
Figure The s-domain equivalent of the circuit in Fig

75. Example 14.7

76. Example 14.7

77. Example 14.7

78. Example 14.7

79. Example 14.7
Figure The magnitude plots for a series RLC bandpass filter with a zero source resistance and a nonzero source resistance.

80. Figure Two RLC bandpass filters, together with equations for the transfer function, center frequency, and bandwidth of each.

81. Transfer function for RLC bandpass filter

82. Relating the Frequency Domain to the Time Domain

84. 14.5 Bandreject Filters
Bandreject filters are characterized by the same parameters as bandpass filters: the two cutoff frequencies, the center frequency, the bandwidth, and the quality factor.

85. The Series RLC Circuit—Qualitative Analysis
Figure (a) A series RLC bandreject filter. (b) The equivalent circuit for ω = 0 (c) The equivalent circuit for ω = ∞.

86. Figure The frequency response plot for the series RLC bandreject filter circuit in Fig (a).

87. The Series RLC Circuit—Quantitative Analysis

88. Figure 14.30 The s-domain equivalent of the circuit in Fig. 14.28(a).

90. Example 14.8
Using the series RLC circuit in Fig (a), compute the component values that yield a bandreject filter with a bandwidth of 250 Hz and a center frequency of 750 Hz. Use a 100 nF capacitor. Compute values for R, L, wc1, wc2 and Q.
Figure (a) A series RLC bandreject filter.

91. Example 14.8

92. Example 14.8

93. Transfer function for RLC bandreject filter
Figure Two RLC bandreject filters, together with equations for the transfer function, center frequency, and bandwidth of each.

95. Summary
A frequency selective circuit, or filter, enables signals at certain frequencies to reach the output, and it attenuates signals at other frequencies to prevent them from reaching the output. The passband contains the frequencies of those signals that are passed; the stopband contains the frequencies of those signals that are attenuated.
The cutoff frequency, wc, identifies the location on the frequency axis that separates the stopband from the passband. At the cutoff frequency, the magnitude of the transfer function equals

96. Summary
A low-pass filter passes voltages at frequencies below and attenuates frequencies above wc. Any circuit with the transfer function functions as a low-pass filter.
A high-pass filter passes voltages at frequencies above and attenuates voltages at frequencies below Any circuit with the transfer function functions as a high-pass filter.

97. Summary
Bandpass filters and bandreject filters each have two cutoff frequencies, wc1 and wc2.These filters are further characterized by their center frequency (wo), bandwidth (b), and quality factor (Q). These quantities are defined as

98. Summary
A bandpass filter passes voltages at frequencies within the passband, which is between wc1 and wc2. It attenuates frequencies outside of the passband.Any circuit with the transfer function functions as a bandpass filter.

99. Summary
A bandreject filter attenuates voltages at frequencies within the stopband, which is between wc1 and wc2. It passes frequencies outside of the stopband. Any circuit with the transfer function functions as a bandreject filter.

100. Summary
Adding a load to the output of a passive filter changes its filtering properties by altering the location and magnitude of the passband. Replacing an ideal voltage source with one whose source resistance is nonzero also changes the filtering properties of the rest of the circuit, again by altering the location and magnitude of the passband.