596 Active Filter Circuits 15.7 a) Using only three components from Appendix H, design a low-pass filter with a cutoff frequency and passband gain as close as possible to the specifications in Problem 15.6(a). Draw the cir- cuit diagram and label all component values. b) Calculate the percent error in this new filter's cutoff frequency and passband gain when com- pared to the values specified in Problem 15.6(a). 15.8 Design an op amp-based high-pass filter with a cut- off frequency of 4 kHz and a passband gain of 8 using a 250 nF capacitor. a) Draw your circuit, labeling the component val- ues and the output voltage. b) If the value of the feedback resistor in the filter is changed but the value of the resistor in the forward path is unchanged, what characteristic of the filter is changed? 15.9 The input to the high-pass filter designed in Problem 15.8 is 250 cos ot mV. a) Suppose the power supplies are ±V cc . What is the smallest value of V cc that will still cause the op amp to operate in its linear region? b) Find the output voltage when a) = (o c . c) Find the output voltage when u> = 0.2&><). d) Find the output voltage when w = 5ft> 0 . 15.10 a) Use the circuit in Fig. 15.4 to design a high-pass filter with a cutoff frequency of 8 kHz and a passband gain of 14 dB. Use a 3.9 nF capacitor in the design. b) Draw the circuit diagram of the filter and label all the components. 15.11 Using only three components from Appendix H, design a high-pass filter with a cutoff frequency and passband gain as close as possible to the specifica- tions in Problem 15.10. a) Draw the circuit diagram and label all compo- nent values. b) Calculate the percent error in this new filter's cut- off frequency and passband gain when compared to the values specified in Problem 15.10(a). Section 15.2 15.12 The voltage transfer function for either high-pass prototype filter shown in Fig. P15.12 is Figure P15.12 C = IF If R = l a v, (a) R = \a L = 1 H v (b) 15.13 The voltage transfer function of either low-pass prototype filter shown in Fig. P15.13 is H(s) 1 s + 1 Show that if either circuit is scaled in both magni- tude and frequency, the scaled transfer function is H'(s) 1 (s/k f ) + 1 + C = 1 F v„ (a) L= 1H R = 1 fl v (b) H(s) = s + 1 15.14 The voltage transfer function of the prototype bandpass filter shown in Fig. P15.14 is Show that if either circuit is scaled in both magni- tude and frequency, the scaled transfer function is H'(s) = {s/kf) (s/k f ) + 1 H(s) s 2 + Q s + 1 Problems 597 Show that if the circuit is scaled in both magnitude and frequency, the scaled transfer function is !)(± H>(s) Figure P15.14 <h + 1 C=1F L = 1H R =^ v 0 15.15 a) Specify the component values for the prototype passive bandpass filter described in Problem 15.14 if the quality factor of the filter is 20. b) Specify the component values for the scaled bandpass filter described in Problem 15.14 if the quality factor is 20; the center, or resonant, frequency is 40 krad/s; and the impedance at resonance is 5 kft. c) Draw a circuit diagram of the scaled filter and label all the components. 15.16 An alternative to the prototype bandpass filter illustrated in Fig. P15.14 is to make m 0 = 1 rad/s, R = 1 ft, and L = Q henrys. a) What is the value of C in the prototype filter circuit? b) What is the transfer function of the prototype filter? c) Use the alternative prototype circuit just described to design a passive bandpass filter that has a qual- ity factor of 16, a center frequency of 25 krad/s, and an impedance of 10 kft at resonance. d) Draw a diagram of the scaled filter and label all the components. e) Use the results obtained in Problem 15.14 to write the transfer function of the scaled circuit. DESIGN PROBLEM 15.17 The passive bandpass filter illustrated in Fig. 14.22 has two prototype circuits. In the first prototype circuit, co 0 = 1 rad/s, C = 1 F, L = 1 H, and R = Q ohms. In the second prototype circuit, co 0 = 1 rad/s, R = 1 ft, C = Q farads, and L = (1/(2) henrys. a) Use one of these prototype circuits (your choice) to design a passive bandpass filter that has a quality factor of 25 and a center frequency of 50 krad/s. The resistor R is 40 kft. b) Draw a circuit diagram of the scaled filter and label all components. 15.18 The transfer function for the bandreject filter shown in Fig. 14.28(a) is s 2 + H(s) 1 LC '.I l$ LC Show that if the circuit is scaled in both magnitude and frequency, the transfer function of the scaled circuit is equal to the transfer function of the unsealed circuit with s replaced by (s/kf), where kf is the frequency scale factor. 15.19 Show that the observation made in Problem 15.18 with respect to the transfer function for the circuit in Fig. 14.28(a) also applies to the bandreject filter circuit (lower one) in Fig. 14.31. 15.20 The passive bandreject filter illustrated in Fig. 14.28(a) has the two prototype circuits shown in Fig. P15.20. a) Show that for both circuits, the transfer function is H(s) = s 2 + l s 2 + ( - )s + 1 b) Write the transfer function for a bandreject fil- ter that has a center frequency of 50 krad/s and a quality factor of 5. Figure PI5.20 -pvQ; (b) 15.21 The two prototype versions of the passive band- reject filter shown in Fig. 14.31 (lower circuit) are shown in Fig. P15.21(a) and (b). Show that the transfer function for either ver- sion is H(s) s 2 + 1 s 2 + s + 1 598 Active Filter Circuits Figure P15.21 1H { Q> (a) (b) 15.22 The circuit in Fig. P13.22 is scaled so that the 1 Q resistors are replaced by 1 kll resistors and the 1 F capacitor is replaced by a 200 nF capacitor. a) What is the scaled value of L? b) What is the expression for i (> in the scaled circuit? 15.23 Scale the circuit in Problem 13.31 so that the 50 0. resistor is increased to 5 kH and the frequency of the voltage response is increased by a factor of 5000. Find/„(0- 15.24 Scale the bandpass filter in Problem 14.22 so that the center frequency is 200 kHz and the quality fac- tor is still 8, using a 2.5 nF capacitor. Determine the values of the resistor, the inductor, and the two cut- off frequencies of the scaled filter. 15.25 Scale the bandreject filter in Problem 14.35 to get a center frequency of 50 krad/s, using a 200 /u,H inductor. Determine the values of the resistor, the capacitor, and the bandwidth of the scaled filter. 15.26 a) Show that if the low-pass filter circuit illustrated in Fig. 15.1 is scaled in both magnitude and fre- quency, the transfer function of the scaled circuit is the same as Eq. 15.1 with s replaced by s/kf, where kf is the frequency scale factor. b) In the prototype version of the low-pass filter circuit in Fig. 15.1, to c = 1 rad/s, C = 1 F, R 2 = 1 O, and R x = \/K ohms. What is the transfer function of the prototype circuit? c) Using the result obtained in (a), derive the transfer function of the scaled filter. 15.27 a) Show that if the high-pass filter illustrated in Fig. 15.4 is scaled in both magnitude and fre- quency, the transfer function is the same as Eq. 15.4 with s replaced by s/kf, where kf is the frequency scale factor. b) In the prototype version of the high-pass filter circuit in Fig. 15.4, o) c = 1 rad/s, R\ = 1 il, C = 1 F, and R 2 = K ohms. What is the transfer function of the prototype circuit? c) Using the result in (a), derive the transfer func- tion of the scaled filter. Section 15.3 15.28 a) Using 0.1 fx¥ capacitors, design an active broad- OESIGN band first-order bandpass filter that has a lower PROBLEM r PSPICE cutoff frequency of 1000 Hz, an upper cutoff fre- MULTISIM quency of 5000 Hz, and a passband gain of 0 dB. Use prototype versions of the low-pass and high-pass filters in the design process (see Problems 15.26 and 15.27). b) Write the transfer function for the scaled filter. c) Use the transfer function derived in part (b) to find H(ja) v ), where o) () is the center frequency of the filter. d) What is the passband gain (in decibels) of the fil- ter at o) a ? e) Using a computer program of your choice, make a Bode magnitude plot of the filter. 15.29 a) Using 10 nF capacitors, design an active broad- DESIGN band first-order bandreject filter with a lower PROBLEM J PSPICE cutoff frequency of 400 Hz, an upper cutoff MULTISIM frequency of 4000 Hz, and a passband gain of 0 dB. Use the prototype filter circuits intro- duced in Problems 15.26 and 15.27 in the design process. b) Draw the circuit diagram of the filter and label all the components. c) What is the transfer function of the scaled filter? d) Evaluate the transfer function derived in (c) at the center frequency of the filter. e) What is the gain (in decibels) at the center frequency? f) Using a computer program of your choice, make a Bode magnitude plot of the filter trans- fer function. 15.30 Design a unity-gain bandpass filter, using a cascade connection, to give a center frequency of 200 Hz and a bandwidth of 1000 Hz. Use 5 fx¥ capacitors. Specify f ch f cZ , R L , and R H . 15.31 Design a parallel bandreject filter with a center fre- quency of 1000 rad/s, a bandwidth of 4000 rad/s, and a passband gain of 6. Use 0.2 JJL¥ capacitors, and specify all resistor values. 15.32 Show that the circuit in Fig. P15.32 behaves as a bandpass filter. {Hint—find the transfer function for this circuit and show that it has the same form as Problems 599 the transfer function for a bandpass filter. Use the result from Problem 15.1.) a) Find the center frequency, bandwidth and gain for this bandpass filter. b) Find the cutoff frequencies and the quality for this bandpass filter. Section 15.4 15.34 The circuit in Fig. 15.21 has the transfer function given by Eq. 15.34. Show that if the circuit in Fig. 15.21 is scaled in both magnitude and fre- quency, the transfer function of the scaled circuit is 1 Figure P15.32 400 O + »i • 50 (xF \( i \\ * 1 10 JJLF \( \\ 5kO r P^ , ^^ 1 . + % o H'(s) = JR 2 C,C l<-2 RCMf) + R 2 CtC 2 *f 15.33 For circuits consisting of resistors, capacitors, induc- tors, and op amps, \H(jco) I 2 involves only even pow- ers of a). To illustrate this, compute \H(ja))\ 2 for the three circuits in Fig. PI5.33 when Figure P15.33 ™-v,- R dB/dec. + • w < < > 1 ~sC i —• + V, —• (a) Ri -AA/V- R 2 -vw- 1 sC sL (b) 15.35 The purpose of this problem is to illustrate the advantage of an «th-order low-pass Butterworth fil- ter over the cascade of n identical low-pass sections by calculating the slope (in decibels per decade) of each magnitude plot at the corner frequency o) c . To facilitate the calculation, let y represent the magni- tude of the plot (in decibels), and let x = log 10 &>. Then calculate dy/dx at a> c for each plot. a) Show that at the corner frequency (w c = 1 rad/s) of an wth-order low-pass proto- type Butterworth filter, dy -T = -10« dB/dec. dx b) Show that for a cascade of n identical low-pass prototype sections, the slope at (o c is di _ -20n(2 ]/ " - 1) dx ~ 2 V " c) Compute dy/dx for each type of filter for n = 1,2, 3, 4, and oo. d) Discuss the significance of the results obtained in part (c). 15.36 a) Determine the order of a low-pass Butterworth filter that has a cutoff frequency of 2000 Hz and a gain of no more than -30 dB at 7000 Hz. b) What is the actual gain, in decibels, at 7000 Hz? 15.37 a) Write the transfer function for the prototype low-pass Butterworth filter obtained in Problem 15.36(a). b) Write the transfer function for the scaled filter in (a) (see Problem 15.34). c) Check the expression derived in part (b) by using it to calculate the gain (in decibels) at 7000 Hz. Compare your result with that found in Problem 15.36(b). *—•V, DESIGN PROBLEM (c) 15.38 a) Using 1 kO resistors and ideal op amps, design a circuit that will implement the low-pass Butterworth filter specified in Problem 15.36. The gain in the passband is one. b) Construct the circuit diagram and label all com- ponent values. 600 Active Filter Circuits 15.39 a) Using 10 nF capacitors and ideal op amps, PROBLEM design a high-pass unity-gain Butterworth filter with a cutoff frequency of 2 kHz and a gain of no more than -48 dB at 500 Hz. b) Draw a circuit diagram of the filter and label all component values. 15.40 Verify the entries in Table 15.1 for n = 5 and n = 6. 15.45 Show that if co ( , = 1 rad/s and C = 1 F in the cir- cuit in Fig. 15.26, the prototype values of R h R 2 , and R 3 are R, = R, = R, Q Q 2Q 2 2Q. K 15.41 The circuit in Fig. 15.25 has the transfer function given by Eq. 15.47. Show that if the circuit is scaled in both magnitude and frequency, the transfer func- tion of the scaled circuit is H'(s) = ft + •\kfl H " D D^2 R { R 2 a DESIGN PROBLEM Hence the transfer function of a scaled circuit is obtained from the transfer function of an unsealed circuit by simply replacing s in the unsealed trans- fer function by s/kf, where kf is the frequency scal- ing factor. 15.42 a) Using 1 kf! resistors and ideal op amps, design a low-pass unity-gain Butterworth filter that has a cutoff frequency of 8 kHz and is down at least 48 dB at 32 kHz. b) Draw a circuit diagram of the filter and label all the components. 15.43 The high-pass filter designed in Problem 15.39 is cascaded with the low-pass filter designed in Problem 15.42. a) Describe the type of filter formed by this interconnection. b) Specify the cutoff frequencies, the mid- frequency, and the quality factor of the filter. c) Use the results of Problems 15.36 and 15.40 to derive the scaled transfer function of the filter. d) Check the derivation of (c) by using it to calculate H(Ja) 0 ), where 0) o is the midfrequency of the filter. 15.44 a) Use 20 nF capacitors in the circuit in Fig. 15.26 to design a bandpass filter with a quality factor of 16, a center frequency of 6.4 kHz, and a pass- band gain of 20 dB. b) Draw the circuit diagram of the filter and label all the components. DESIGN PROBLEM 15.46 DESIGN PROBLEM 15.47 15.48 15.49 15.50 DESIGN PROBLEM a) Design a broadband Butterworth bandpass fil- ter with a lower cutoff frequency of 500 Hz and an upper cutoff frequency of 4500 Hz. The pass- band gain of the filter is 20 dB. The gain should be down at least 20 dB at 200 Hz and 11.25 kHz. Use 15 nF capacitors in the high-pass circuit and 10 kH resistors in the low-pass circuit. b) Draw a circuit diagram of the filter and label all the components. a) Derive the expression for the scaled transfer function for the filter designed in Problem 15.46. b) Using the expression derived in (a), find the gain (in decibels) at 200 Hz and 1500 Hz. c) Do the values obtained in part (b) satisfy the fil- tering specifications given in Problem 15.46? Derive the prototype transfer function for a sixth- order high-pass Butterworth filter by first writing the transfer function for a sixth-order prototype low-pass Butterworth filter and then replacing s by \/s in the low-pass expression. The sixth-order Butterworth filter in Problem 15.48 is used in a system where the cutoff frequency is 25 krad/s. a) What is the scaled transfer function for the filter? b) Test your expression by finding the gain (in deci- bels) at the cutoff frequency. The purpose of this problem is to guide you through the analysis necessary to establish a design procedure for determining the circuit components in a filter circuit. The circuit to be analyzed is shown in Fig. P15.50. a) Analyze the circuit qualitatively and convince yourself that the circuit is a low-pass filter with a passband gain of Rj/Rh b) Support your qualitative analysis by deriving the transfer function V 0 fV- v {Hint: In deriving the transfer function, represent the resistors with their equivalent conductances, that is, Gx = l/i?i, and so forth.) To make the transfer function useful in terms of the entries in Table 15.1, put it in the form H(s) = -Kb, s 2 + bis + b 0 Problems 601 c) Now observe that we have five circuit compo- nents—/?!, R 2 , i?3, C b and C 2 —and three trans- fer function constraints—#, b h and b 0 . At first glance, it appears we have two free choices among the five components. However, when we investigate the relationships between the circuit components and the transfer function constraints, we see that if C 2 is chosen, there is an upper limit on C\ in order for R 2 {G 2 ) to be realizable. With this in mind, show that if C 2 = 1 F, the three con- ductances are given by the expressions G\ — KG 2 \ [ G 2 ]( G 2 = h } ± Vbj - 4b () (l + K)C { 2(1 + K) For G 2 to be realizable, C, 46,,(1 +K)' d) Based on the results obtained in (c), outline the design procedure for selecting the circuit com- ponents once K, b w and b\ are known. Figure P15.50 + /?j »i | 1 f&2 * i "C! «3 -Q i ' + v„ DESIGN PROBLEM 15.51 Assume the circuit analyzed in Problem 15.50 is part of a third-order low-pass Butter worth filter having a passband gain of 4. (Hint: implement the gain of 4 in the second-order section of the filter.) a) If C 2 = 1 F in the prototype second-order sec- tion, what is the upper limit on Ci? b) If the limiting value of C] is chosen, what are the prototype values of R\, R 2 , and /? 3 ? c) If the corner frequency of the filter is 2.5 kHz and C 2 is chosen to be 10 nF, calculate the scaled values of C b R h R 2 , and /? 3 . d) Specify the scaled values of the resistors and the capacitor in the first-order section of the filter. e) Construct a circuit diagram of the filter and label all the component values on the diagram. DESIGN PROBLEM 15.52 Interchange the Rs and Cs in the circuit in Fig. P15.50; that is, replace R x with C h R 2 with C 2 , i?3 with C 3 , C\ with R\, and C 2 with R 2 . a) Describe the type of filter implemented as a result of the interchange. b) Confirm the filter type described in (a) by deriv- ing the transfer function V a /Vj. Write the trans- fer function in a form that makes it compatible with Table 15.1. c) Set C 2 = C 3 = 1 F and derive the expressions for Q, /?i, and R 2 in terms of K, b h and b 0 . (See Problem 15.50 for the definition of b\ and b (r ) d) Assume the filter described in (a) is used in the same type of third-order Butterworth filter that has a passband gain of 8. With C 2 = C 3 = 1 F, calculate the prototype values of C h R h and R 2 in the second-order section of the filter. DESIGN PROBLEM 15.53 a) Use the circuits analyzed in Problems 15.50 and 15.52 to implement a broadband bandreject fil- ter having a passband gain of 0 dB, a lower cor- ner frequency of 400 Hz, an upper corner frequency of 6400 Hz, and an attenuation of at least 30 dB at both 1000 Hz and 2560 kHz. Use 10 nF capacitors whenever possible. b) Draw a circuit diagram of the filter and label all the components. 15.54 a) Derive the transfer function for the bandreject filter described in Problem 15.53. b) Use the transfer function derived in part (a) to find the attenuation (in decibels) at the center frequency of the filter. DESIGN PROBLEM 15.55 The purpose of this problem is to develop the design equations for the circuit in Fig. PI5.55. (See Problem 15.50 for suggestions on the development of design equations.) a) Based on a qualitative analysis, describe the type of filter implemented by the circuit. b) Verify the conclusion reached in (a) by deriving the transfer function V 0 /Vi. Write the transfer function in a form that makes it compatible with the entries in Table 15.1. c) How many free choices are there in the selec- tion of the circuit components? d) Derive the expressions for the conductances G\ = l/Ri and G 2 = l/R 2 in terms of C h C 2 , and the coefficients b 0 and b%. (See Problem 15.50 for the definition of b 0 and b\.) e) Are there any restrictions on C\ or C 2 ? f) Assume the circuit in Fig. P15.55 is used to design a fourth-order low-pass unity-gain Butterworth filter. Specify the prototype values of R x and R 2 in each second-order section if 1 F capacitors are used in the prototype circuit. 602 Active Filter Circuits Figure PI5.55 Section 15.5 15.56 The fourth-order low-pass unity-gain Buttcrworth PROBLEM ^ ter * n Problem 15.55 is used in a system where the cutoff frequency is 3 kHz. The filter has 4.7 nF capacitors. a) Specify the numerical values of R { and R 2 in each section of the filter. b) Draw a circuit diagram of the filter and label all the components. 15.57 Interchange the Rs and Cs in the circuit in DESIGN fig. P15.55, that is, replace i?, with Ci, Ri with C 2 , PROBLEM ° * r ' l • - and vice versa. a) Analyze the circuit qualitatively and predict the type of filter implemented by the circuit. b) Verify the conclusion reached in (a) by deriving the transfer function V Q /Vf. Write the transfer function in a form that makes it compatible with the entries in Table 15.1. c) How many free choices are there in the selec- tion of the circuit components? d) Find R\ and R 2 as functions of b m b\, C h and C 2 . e) Are there any restrictions on C } and C 2 1 f) Assume the circuit is used in a third-order Butterworth filter of the type found in (a). Specify the prototype values of R\ and R 2 in the second- order section of the filter if C\ = C 2 = 1 F. DESIGN PROBLEM 15.58 a) The circuit in Problem 15.57 is used in a third- order high-pass unity-gain Butterworth filter that has a cutoff frequency of 5 kHz. Specify the values of R l and R 2 if 75 nF capacitors are avail- able to construct the filter. b) Specify the values of resistance and capacitance in the first-order section of the filter. c) Draw the circuit diagram and label all the components. d) Give the numerical expression for the scaled transfer function of the filter. e) Use the scaled transfer function derived in (d) to find the gain in dB at the cutoff frequency. 15.59 a) Show that the transfer function for a prototype narrow band bandreject filter is H{s) s l + 1 s 2 + (1/Q)s + 1 DESIGN PROBLEM b) Use the result found in (a) to find the transfer function of the filter designed in Example 15.13. 15.60 a) Using the circuit shown in Fig. 15.29, design a narrow-band bandreject filter having a center frequency of 1 kHz and a quality factor of 20. Base the design on C = 15 nF. b) Draw the circuit diagram of the filter and label all component values on the diagram. c) What is the scaled transfer function of the filter? Sections 15.1-15.5 15.61 Using the circuit in Fig. 15.32(a) design a volume J£2S™ control circuit to give a maximum gain of 20 dB and a gain of 17 dB at a frequency of 40 Hz. Use an 11.1 kfi resistor and a 100 kfl potentiometer. Test your design by calculating the maximum gain at o) = 0 and the gain at &> = X/R^Cy using the selected values of R [} R 7 , and C v PERSPECTIVE DESIGN PROBLEM 15.62 Use the circuit in Fig. 15.32(a) to design a bass vol- PERSPECTIVE ume control circuit that has a maximum gain of DESIGN 13.98 dB that drops off 3 dB at 50 Hz. PROBLEM 15.63 Plot the maximum gain in decibels versus a when <w = 0 for the circuit designed in Probh a vary from 0 to 1 in increments of 0.1. PRACTICAL w = 0 for the circuit designed in Problem 15.61. Let PERSPECTIVE PRACTICAL PERSPECTIVE 15.64 a) Show that the circuits in Fig. PI 5.64(a) and (b) are equivalent. b) Show that the points labeled x and y in Fig. P15.64(b) are always at the same potential. c) Using the information in (a) and (b), show that the circuit in Fig. 15.33 can be drawn as shown in Fig. P15.64(c). d) Show that the circuit in Fig. PI5.64(c) is in the form of the circuit in Fig. 15.2, where /?! + (! - a)R 2 + RiR 2 C x s Zi z / = 1 + R 2 C]S R } + aR 2 + R^C^ 1 + R 2 C t s Problems 603 Figure P15.64 lfsCi l-a a HM-K (l-a)R 2 otR 2 (a) (l-a)R 2 y cxR 2 (b) R 4 + 2/? 3 (c) 15.65 An engineering project manager has received a PRACTICAL proposal from a subordinate who claims the circuit PERSPECTIVE r r shown in Fig. PI 5.65 could be used as a treble vol- ume control circuit if R 4 » R^ + R$ + 2R 2 , The subordinate further claims that the voltage transfer function for the circuit is *«-£ -{(2J? 3 + R 4 ) + [(1 - /3)i? 4 + Rg](fiR4 + R 3 )C 2 s} {(27? 3 + Hi) + [(1 - Aft* + ^1(/^4 + #e>)C2*} where i? ( , = /? t + 7? 3 + 2R 2 . Fortunately the project engineer has an electrical engineering undergraduate student as an intern and therefore asks the student to check the subordinate's claim. The student is asked to check the behavior of the transfer function as co—>0; as w—»oo; and the behavior when co = oo and /3 varies between 0 and 1. Based on your testing of the transfer function do you think the circuit could be used as a treble volume control? Explain. Figure P15.65 V s • VQ 15.66 In the circuit of Fig. P15.65 the component values PRACTICAL are R x = R 2 = 20 kll, R^ = 5.9 kO, R 4 = 500 kft, PERSPECTIVE L > O * and C 2 = 2.7 nF. a) Calculate the maximum boost in decibels. b) Calculate the maximum cut in decibels. c) Is R 4 significantly greater than R a 7 d) When p = 1, what is the boost in decibels when co = 1/R 3 C 2 7 e) When /3 = 0, what is the cut in decibels when co = l/i? 3 C 2 ? f) Based on the results obtained in (d) and (e), what is the significance of the frequency \/R 3 C 2 when R 4 » i? 0 ? 15.67 Using the component values given in PRACTICAL Problem 15.66, plot the maximum gain in decibels PERSPECTIVE ' r ° versus /3 when co is inifinite. Let /3 vary from 0 to 1 in increments of 0.1. CHAPTER » v M Fourier Series CHAPTER CONTENTS 16.1 Fourier Series Analysis: An Overview p. 607 16.2 The Fourier Coefficients p. 608 16.3 The Effect of Symmetry on the Fourier Coefficients p. 611 16.4 An Alternative Trigonometric Form of the Fourier Series p. 617 16.5 An Application p. 529 16.6 Average-Power Calculations with Periodic Functions p. 623 16.7 The rms Value of a Periodic Function p. 6£6" 16.8 The Exponential Form of the Fourier Series p. 627 16.9 Amplitude and Phase Spectra p. 530 1 Be able to calculate the trigonometric form of the Fourier coefficients for a periodic waveform using the definition of the coefficients and the simplifications possible if the waveform exhibits one or more types of symmetry. 2 Know how to analyze a circuit's response to a periodic waveform using Fourier coefficients and superposition. 3 Be able to estimate the average power delivered to a resistor using a small number of Fourier coefficients. 4 Be able to calculate the exponential form of the Fourier coefficients for a periodic waveform and use them to generate magnitude and phase spectrum plots for that waveform. 604 In the preceding chapters, we devoted a considerable amount of discussion to steady-state sinusoidal analysis. One reason for this interest in the sinusoidal excitation function is that it allows us to find the steady-state response to nonsinusoidal, but peri- odic, excitations. A periodic function is a function that repeats itself every T seconds. For example, the triangular wave illus- trated in Fig. 16.1 on page 606 is a nonsinusoidal, but periodic, break waveform. A periodic function is one that satisfies the relationship fit) = f(t ± nT), (16.1) where n is an integer (1,2, 3, ) and T is the period. The func- tion shown in Fig. 16.1 is periodic because f(to) = f(to ~T) = f(t Q + T) = f(t 0 + 2T) = for any arbitrarily chosen value of t 0 . Note that T is the smallest time interval that a periodic function may be shifted (in either direction) to produce a function that is identical to itself. Why the interest in periodic functions? One reason is that many electrical sources of practical value generate periodic waveforms. For example, nonfiltered electronic rectifiers driven from a sinusoidal source produces rectified sine waves that are nonsinusoidal, but periodic. Figures 16.2(a) and (b) on page 606 show the waveforms of the full-wave and half-wave sinusoidal rectifiers, respectively. The sweep generator used to control the electron beam of a cathode-ray oscilloscope produces a periodic triangular wave like the one shown in Fig. 16.3 on page 606. Electronic oscillators, which are useful in laboratory testing of equipment, are designed to produce nonsinusoidal periodic waveforms. Function generators, which are capable of producing square-wave, triangular-wave, and rectangular-pulse waveforms, are found in most testing laboratories. Figure 16.4 on page 606 illustrates typical waveforms. m Practical Perspective Active High-Q Filters In Chapters 14 and 15, we discovered that an important char- acteristic of bandpass and bandreject filters is the quality fac- tor, Q. The quality factor provides a measure of how selective the filter is at its center frequency. For example, a bandpass fil- ter with a large value of Q will amplify signals at or near its center frequency and will attentuate signals at all other fre- quencies. On the other hand, a bandreject filter with a small value of Q will not effectively distinguish between signals at the center frequency and signals at frequencies quite different from the center frequency. In this chapter, we learn that any periodic signal can be represented as a sum of sinusoids, where the frequencies of the sinusoids in the sum are comprised of the frequency of the periodic signal and integer multiples of that frequency. We can use a periodic signal like a square wave to test the quality fac- tor of a bandpass or bandreject filter. To do this, we choose a square wave whose frequency is the same as the center fre- quency of a bandpass filter, for example. If the bandpass filter has a high quality factor, its output will be nearly sinusoidal, thereby transforming the input square wave into an output sinusoid. If the filter has a low quality factor, its output will still look like a square wave, as the filter is not able to select from among the sinusoids that make up the input square wave. We present an example at the end of this chapter. h A fi AAA VIMy IAAAAA/ 1 uvuVuu High-(? Bandpass Filter A A \J V 605 . in Fig. 14.28(a) has the two prototype circuits shown in Fig. P15.20. a) Show that for both circuits, the transfer function is H(s) = s 2 + l s 2 + ( - )s + 1 b) Write the transfer function. ^1(/^4 + #e>)C2*} where i? ( , = /? t + 7? 3 + 2R 2 . Fortunately the project engineer has an electrical engineering undergraduate student as an intern and therefore asks the student to check. identical to itself. Why the interest in periodic functions? One reason is that many electrical sources of practical value generate periodic waveforms. For example, nonfiltered