546 Introduction to Frequency Selective Circuits R ,6 <\> C (c) Figure 14.28 • (a) A series RLC bandreject filter. (b) The equivalent circuit for 00 = 0. (c) The equivalent circuit for co = 00. \ff(h*)\ 90° - Figure 14.29 • The frequency response plot for the series RLC bandreject filter circuit in Fig. 14.28(a). before they reach the output at frequencies between the two cutoff fre- quencies (the stopband). Bandpass filters and bandreject filters thus per- form complementary functions in the frequency domain. Bandreject filters are characterized by the same parameters as band- pass filters: the two cutoff frequencies, the center frequency, the band- width, and the quality factor. Again, only two of these five parameters can be specified independently. In the next sections, we examine two circuits that function as band- reject filters and then compute equations that relate the circuit compo- nent values to the characteristic parameters for each circuit. The Series RLC Circuit—Qualitative Analysis Figure 14.28(a) shows a series RLC circuit. Although the circuit components and connections are identical to those in the series RLC bandpass filter in Fig. 14.19(a), the circuit in Fig. 14.28(a) has an important difference: the out- put voltage is now defined across the inductor-capacitor pair. As we saw in the case of low- and high-pass filters, the same circuit may perform two dif- ferent filtering functions, depending on the definition of the output voltage. We have already noted that at to = 0, the inductor behaves like a short circuit and the capacitor behaves like an open circuit, but at a) = 00, these roles switch. Figure 14.28(b) presents the equivalent cir- cuit for a) = 0; Fig. 14.28(c) presents the equivalent circuit for to = 00. In both equivalent circuits, the output voltage is defined over an effective open circuit, and thus the output and input voltages have the same mag- nitude. This series RLC bandreject filter circuit then has two pass- bands—one below a lower cutoff frequency, and the other above an upper cutoff frequency. Between these two passbands, both the inductor and the capacitor have finite impedances of opposite signs. As the frequency is increased from zero, the impedance of the inductor increases and that of the capac- itor decreases. Therefore the phase shift between the input and the out- put approaches —90° as toL approaches 1/OJC. AS soon as coL exceeds 1/wC, the phase shift jumps to +90° and then approaches zero as to con- tinues to increase. At some frequency between the two passbands, the impedances of the inductor and capacitor are equal but of opposite sign. At this frequency, the series combination of the inductor and capacitor is that of a short cir- cuit, so the magnitude of the output voltage must be zero. This is the cen- ter frequency of this series RLC bandreject filter. Figure 14.29 presents a sketch of the frequency response of the series RLC bandreject filter from Fig. 14.28(a). Note that the magnitude plot is overlaid with that of the ideal bandreject filter from Fig. 14.3(d). Our qual- itative analysis has confirmed the shape of the magnitude and phase angle plots. We now turn to a quantitative analysis of the circuit to confirm this frequency response and to compute values for the parameters that charac- terize this response. ^(•v) Figure 14.30 • The s-domain equivalent of the circuit in Fig. 14.28(a). The Series RLC Circuit—Quantitative Analysis After transforming to the s-domain, as shown in Fig. 14.30, we use voltage division to construct an equation for the transfer function: H(s) = ^Tc R + sL + — sC s 2 + 1 LC •> R l (14.43) Substitute jco for 5 in Eq. 14.43 and generate equations for the transfer function magnitude and the phase angle: [H(jm)\ = LC LC + ?)" (14.44) 6(ja>) = — tan -1 wR L (14.45) LC Note that Eqs. 14.44 and 14.45 confirm the frequency response shape pictured in Fig. 14.29, which we developed based on the qualita- tive analysis. Wc use the circuit in Fig. 14.30 to calculate the center frequency. For the bandreject filter, the center frequency is still defined as the frequency for which the sum of the impedances of the capacitor and inductor is zero. In the bandpass filter, the magnitude at the center frequency was a maxi- mum, but in the bandreject filter, this magnitude is a minimum. This is because in the bandreject filter, the center frequency is not in the pass- band; rather, it is in the stopband. It is easy to show that the center fre- quency is given by V LC (14.46) Substituting Eq. 14.46 into Eq. 14.44 shows that \H(jco 0 )\ = 0. The cutoff frequencies, the bandwidth, and the quality factor are defined for the bandreject filter in exactly the way they were for the bandpass filters. Compute the cutoff frequencies by substituting the constant (l/V2)// max for the left-hand side of Eq. 14.44 and then solv- ing for co cl and w c2 - Note that for the bandreject filter, #max = 1^(/0)1 = \H(j oo)\, and for the series RLC bandreject filter in Fig. 14.28(a), H BUX = l.Thus, 0>c\ = . -A _, . fJLY 2L + 2LJ LC (14.47) R ^- = 2L + R_ 2L + LC (14.48) Use the cutoff frequencies to generate an expression for the band- width, jS: 0 = R/L. (14.49) Finally, the center frequency and the bandwidth produce an equation for the quality factor, Q: Q = R 2 C (14.50) 548 Introduction to Frequency Selective Circuits Again, we can represent the expressions for the two cutoff frequencies in terms of the bandwidth and center frequency, as we did for the band- pass filter: cl 2 (14.51) to* + + cof (14.52) Alternative forms for these equations express the cutoff frequencies in terms of the quality factor and the center frequency: (0 ct w c2 = t0 o ' ¾ + V 1 + (¾) k + V 1 + Qa)\ i (14.53) (14.54) Example 14.8 presents the design of a series RLC bandreject filter. Example 14.8 Designing a Series RLC Bandreject Filter Using the series RLC circuit in Fig. 14.28(a), com- pute the component values that yield a bandreject filter with a bandwidth of 250 Hz and a center fre- quency of 750 Hz. Use a 100 nF capacitor. Compute values for R, L, a) ch a> c . 2 , and Q. Solution We begin by using the definition of quality factor to compute its value for this filter: Q = <oJ(3 = 3. Use Eq. 14.46 to compute L, remembering to con- vert io a to radians per second: colC [2TT(750)] 2 (100 X 10" 9 ) = 450 mH. Use Eq. 14.49 to calculate R: R = (3L = 277(250)(450 x 10" 3 ) = 707 a The values for the center frequency and band- width can be used in Eqs. 14.51 and 14.52 to com- pute the two cutoff frequencies: co c] = 0 + f 1 3992.0 rad/s, 0) c2 P + PV + uf = 5562.8 rad/s. The cutoff frequencies are at 635.3 Hz and 885.3 Hz. Their difference is 885.3 - 635.3 = 250 Hz, con- firming the specified bandwidth. The geometric mean is V(635.3)(885.3) = 750 Hz, confirming the specified center frequency. 14.5 Bandreject Filters 549 As you might suspect by now, another configuration that produces a bandreject filter is a parallel RLC circuit. Whereas the analysis details of the parallel RLC circuit are left to Problem 14.34, the results are summa- rized in Fig. 14.31, along with the series RLC bandreject filter. As we did for other categories of filters, we can state a general form for the transfer functions of bandreject filters, replacing the constant terms with /3 and OJ ( ;. //(5) s 2 + d s z + jSs + <a% 2' (14.55) A Transfer function for RLC bandreject filter Equation 14.55 is useful in filter design, because any circuit with a transfer function in this form can be used as a bandreject filter. H(s)=~2 s 2 + l/LC s 1 + (R/L)s + l/LC > 0 = VT/LC P = R/L "•© sL sC + H(s) = s 2 + l/LC s 2 + s/RC + l/LC co 0 = VljLC p = l/RC Figure 14.31 • Two RLC bandreject filters, together with equations for the transfer function, center frequency, and bandwidth of each. /ASSESSMENT PROBLEMS Objective 4—Know the RLC circuit configurations that act as bandreject filters 14.10 Design the component values for the series RLC bandreject filter shown in Fig. 14.28(a) so that the center frequency is 4 kHz and the quality factor is 5. Use a 500 nF capacitor. Answer: L = 3.17 mH, R = 15.92 H. NOTE: Also try Chapter Problems 14.35 and 14.36. 14.11 Recompute the component values for Assessment Problem 14.10 to achieve a band- reject filter with a center frequency of 20 kHz. The filter has a 100 fi resistor. The quality fac- tor remains at 5. Answer: L = 3.98 mH, C = 15.92 nF. 550 Introduction to Frequency Selective Circuits Practical Perspective 69? ** 770 # % 8521» 1 941 H* O ' Figure 14.32 • Tones generated by the rows and columns of telephone pushbuttons. Pushbutton Telephone Circuits In the Practical Perspective at the start of this chapter, we described the dual-tone-multiple-frequency (DTMF) system used to signal that a button has been pushed on a pushbutton telephone. A key element of the DTMF system is the DTMF receiver—a circuit that decodes the tones produced by pushing a button and determines which button was pushed. In order to design a DTMF reciever, we need a better understanding of the DTMF system. As you can see from Fig. 14.32, the buttons on the tele- phone are organized into rows and columns. The pair of tones generated by pushing a button depends on the button's row and column. The button's row determines its low-frequency tone, and the button's column determines its high-frequency tone. 1 For example, pressing the "6" button produces sinu- soidal tones with the frequencies 770 Hz and 1477 Hz. At the telephone switching facility, bandpass filters in the DTMF receiver first detect whether tones from both the low-frequency and high-frequency groups are simultaneously present. This test rejects many extraneous audio signals that are not DTMF. If tones are present in both bands, other filters are used to select among the possible tones in each band so that the frequencies can be decoded into a unique button signal. Additional tests are performed to prevent false button detection. For example, only one tone per frequency band is allowed; the high- and low-band frequencies must start and stop within a few milliseconds of one another to be considered valid; and the highl- and low-band signal amplitudes must be sufficiently close to each other. You may wonder why bandpass filters are used instead of a high-pass filter for the high-frequency group of DTMF tones and a low-pass filter for the low-frequency group of DTMF tones. The reason is that the telephone system uses frequencies outside of the 300-3 kHz band for other signal- ing purposes, such as ringing the phone's bell. Bandpass filters prevent the DTMF receiver from erroneously detecting these other signals. NOTE: Assess your understanding of this Practical Perspective by trying Chapter Problems 14.46-14.48. 1 A fourth high-frequency tone is reserved at 1633 Hz. This tone is used infrequently and is not produced by a standard 12-button telephone. Summary A frequency selective circuit, or filter, enables signals at certain frequencies to reach the output, and it attenu- ates signals at other frequencies to prevent them from reaching the output. The passband contains the fre- quencies of those signals that are passed; the stopband contains the frequencies of those signals that are atten- uated. (See page 524.) The cutoff frequency, co c , 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 (1/V2)// raax . (See page 527.) A low-pass filter passes voltages at frequencies below (a c and attenuates frequencies above co c . Any circuit with the transfer function /.> H{s) S + (o c functions as a low-pass filter. (See page 531.) Problems 551 A high-pass filter passes voltages at frequencies above and attenuates voltages at frequencies below Any cir- cuit with the transfer function H{s) = - functions as a high-pass filter. (See page 536.) Bandpass filters and bandreject filters each have two cut- off frequencies, a) c] and a; c2 . These filters are further char- acterized by their center frequency («„), bandwidth (/3), and quality factor (Q). These quantities are defined as w 0 = Vco t i • (o c2 , (3 = (o c2 - o) d , Q = "o/P. (See pages 539-540.) A bandpass filter passes voltages at frequencies within the passband, which is between a> cl and u) c2 - It attenuates frequencies outside of the passband. Any circuit with the transfer function His) Ps r + Pi functions as a bandpass filter. (See page 544.) A bandreject filter attenuates voltages at frequencies within the stopband, which is between co c] and w c2 . It passes frequencies outside of the stopband. Any circuit with the transfer function His) JT + (t)n S 2 + p s + <4 functions as a bandreject filter. (See page 549.) Adding a load to the output of a passive filter changes its filtering properties by altering the location and mag- nitude 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. (See page 542.) Problems Section 14.2 14.1 a) Find the cutoff frequency in hertz for the RL fil- ter shown in Fig. P14.1. b) Calculate H(Jm) at G> C , 0.2W C ., and 5w c c) If Vj = lOcosatf V, write the steady-state expression for v a when co = w ( ., co = 0.2w r , and a) = 5oo c . Figure P14.1 10 mH m rv^r*r\ « m 127 a 14.2 Use a 1 mH inductor to design a low-pass, RL, pas- OESIGN s ive filter with a cutoff frequency of 5 kHz. PROBLEM " J PSPICE a ) Specify the value of the resistor. MULTISIM b) A load having a resistance of 68 fi is connected across the output terminals of the filter. What is the corner, or cutoff, frequency of the loaded fil- ter in hertz? c) If you must use a single resistor from Appendix H for part (a), what resistor should you use? What is the resulting cutoff frequency of the filter? 14.3 A resistor, denoted as R/, is added in series with the inductor in the circuit in Fig. 14.4(a). The new low- pass filter circuit is shown in Fig. P14.3. a) Derive the expression for His) where His) = VJV b b) At what frequency will the magnitude of H{J<D) be maximum? c) What is the maximum value of the magnitude of »(/»)? d) At what frequency will the magnitude of Hija)) equal its maximum value divided by V2? e) Assume a resistance of 75 O is added in series with the 10 mH inductor in the circuit in Fig. P14.1. Find ta e , #(/0), ff(/» e ) f //(/0.30, and Hij3a) c ). Figure P14.3 Ri +• • L 4 :R » + • 14.4 a) Find the cutoff frequency (in hertz) of the low- pass filter shown in Fig. PI4.4. b) Calculate H{joo) at w t ., 0.1w t ., and 10w r . 552 Introduction to Frequency Selective Circuits DESIGN PROBLEM PSPICE MULT1SIM c) If Vj = 200 cos col mV, write the steady-state expression for v a when to = co c , 0.1co c , and 10co c . Figure P14.4 o VW 100 nF 14.5 Use a 500 nF capacitor to design a low-pass passive filter with a cutoff frequency of 50 krad/s. a) Specify the cutoff frequency in hertz. b) Specify the value of the filter resistor. c) Assume the cutoff frequency cannot increase by more than 5%. What is the smallest value of load resistance that can be connected across the output terminals of the filter? d) If the resistor found in (c) is connected across the output terminals, what is the magnitude of H(joi) when co = 0? 14.6 Design a passive RC low pass filter (see Fig. 14.7) with a cutoff frequency of 100 Hz using a 4.7/AF capacitor. a) What is the cutoff frequency in rad/s? b) What is the value of the resistor? c) Draw your circuit, labeling the component val- ues and output voltage. d) What is the transfer function of the filter in part (c)? e) If the filter in part (c) is loaded with a resistor whose value is the same as the resistor part (b), what is the transfer function of this loaded filter? f) What is the cutoff frequency of the loaded filter from part (e)? g) What is the gain in the pass band of the loaded filter from part (e)? 14.7 A resistor denoted as R L is connected in parallel with the capacitor in the circuit in Fig. 14.7. The loaded low-pass filter circuit is shown in Fig. P14.7. a) Derive the expression for the voltage transfer function V 0 /V r b) At what frequency will the magnitude of H(joo) be maximum? c) What is the maximum value of the magnitude d) At what frequency will the magnitude of H(joo) equal its maximum value divided by V2? e) Assume a resistance of 10 kH is added in paral- lel with the 100 nF capacitor in the circuit in Fig. P14.4. Find a> c , H(jO), H(Jm c ), H(j0.1w c ), and H(jl0co c ). Figure P14.7 + m— R O 4 :R L + • 14.8 Study the circuit shown in Fig. PI 4.8 (without the load resistor). a) As co —> 0, the inductor behaves like what circuit component? What value will the output voltage v 0 have? b) As co —>• oo, the inductor behaves like what cir- cuit component? What value will the output voltage v 0 have? c) Based on parts (a) and (b), what type of filtering does this circuit exhibit? d) What is the transfer function of the unloaded filter? e) If R = 330 O and L = 10 mH, what is the cutoff frequency of the filter in rad/s? Figure P14.8 <P R\v,, R, His) VM V,<s) 14.9 Suppose we wish to add a load resistor in parallel with the resistor in the circuit shown in Fig. PI 4.8. a) What is the transfer function of the loaded filter? b) Compare the transfer function of the unloaded filter (part (d) of Problem 14.8) and the trans- fer function of the loaded filter (part (a) of Problem 14.9). Are the cutoff frequencies differ- ent? Are the passband gains different? c) What is the smallest value of load resistance that can be used with the filter from Problem 14.8(e) such that the cutoff frequency of the resulting filter is no more than 5% different from the unloaded filter? Section 14.3 14.10 a) Find the cutoff frequency (in hertz) for the high- pass filter shown in Fig. P14.10. b) Find H(joo) at oo c , 0.2co c , and 5co c . Problems 553 c) If Vj = 500 cos cot mV, write the steady-state expression for v a when co = co c , to = 0.2co ct and &) = 5co r . Figure P14.10 5nF ? 1(- 50 kn 14.11 A resistor, denoted as R c , is connected in series with the capacitor in the circuit in Fig. 14.10(a). The new high-pass filter circuit is shown in Fig. P14.ll. a) Derive the expression for H(s) where H(s) = VJV, b) At what frequency will the magnitude of H(joo) be maximum? c) What is the maximum value of the magnitude of H(joo)? d) At what frequency will the magnitude of H(jco) equal its maximum value divided by V2? e) Assume a resistance of 12.5 kfl is connected in series with the 5 nF capacitor in the circuit in Fig. P14.10. Calculate w c , H(jto c ), H(jQ.2a> c ), and H(j5(i) c ). Figure P14.ll Re • -vw- c If A' 14.12 Design a passive RC high pass filter (see Fig. 14.10[a]) with a cutoff frequency of 500 Hz using a 220 pF capacitor. a) What is the cutoff frequency in rad/s? b) What is the value of the resistor? c) Draw your circuit, labeling the component val- ues and output voltage. d) What is the transfer function of the filter in part (c)? e) If the filter in part (c) is loaded with a resistor whose value is the same as the resistor in (b), what is the transfer function of this loaded filter? f) What is the cutoff frequency of the loaded filter from part (e)? g) What is the gain in the pass band of the loaded filter from part (e)? 14.13 Using a 100 nF capacitor, design a high-pass passive filter with a cutoff frequency of 300 Hz. a) Specify the value of R in kilohms. b) A 47 kfl resistor is connected across the output terminals of the filter. What is the cutoff fre- quency, in hertz, of the loaded filter? DFSIGN PROBLEM PSPICE MULTISIM DESIGN PROBLEM PSPICE MULTISIM 14.14 Using a 100 juH inductor, design a high-pass, RL, passive filter with a cutoff frequency of 1500 krad/s. a) Specify the value of the resistance, selecting from the components in Appendix H. b) Assume the filter is connected to a pure resistive load. The cutoff frequency is not to drop below 1200 krad/s. What is the smallest load resistor from Appendix H that can be connected across the output terminals of the filter? 14.15 Consider the circuit shown in Fig. P14.15. a) With the input and output voltages shown in the figure, this circuit behaves like what type of filter? b) What is the transfer function, H(s) = 1/,,(^)/^:(^), of this filter? c) What is the cutoff frequency of this filter? d) What is the magnitude of the filter's transfer function at s — jcojl Figure P14.15 150 a -AW- <b + lOmHH'o 14.16 Suppose a 150 ft load resistor is attached to the fil- ter in Fig. P14.15. a) What is the transfer function, H(s) = V ( ,(s)/Vi(s), of this filter? b) What is the cutoff frequency of this filter? c) How does the cutoff frequency of the loaded fil- ter compare with the cutoff frequency of the unloaded filter in Fig. P14.15? d) What else is different for these two filters? Section 14.4 14.17 Show that the alternative forms for the cutoff fre- quencies of a bandpass filter, given in Eqs. 14.36 and 14.37, can be derived from Eqs. 14.34 and 14.35. 14.18 Calculate the center frequency, the bandwidth, and the quality factor of a bandpass filter that has an upper cutoff frequency of 121 krad/s and a lower cutoff frequency of 100 krad/s. 554 Introduction to Frequency Selective Circuits 14.19 A bandpass filter has a center, or resonant, frequency of 50 krad/s and a quality factor of 4. Find the band- width, the upper cutoff frequency, and the lower cut- off frequency. Express all answers in kilohertz. 14.20 Use a 5 nF capacitor to design a series RLC band- PMBLEM P ass filter ' as snown at tne t0 P of Fig- 14 .27. Th e cen " PSPFCE ter frequency of the filter is 8 kHz, and the quality MumsiM factor is 2. a) Specify the values of R and L. b) What is the lower cutoff frequency in kilohertz? c) What is the upper cutoff frequency in kilohertz? d) What is the bandwidth of the filter in kilohertz? 14.21 Design a series RLC bandpass filter using only three components from Appendix Ff that comes closest to meeting the filter specifications in Problem 14.20. a) Draw your filter, labeling all component values and the input and output voltages. b) Calculate the percent error in this new filter's center frequency and quality factor when com- pared to the values specified in Problem 14.20. 14.22 For the bandpass filter shown in Fig. P14.22, find "SPICE (a) co a , (b) f m (c) Q, (d) » At (e) f ch (f) co c2 , (g) / c2 , MULT,5IM and(h)iS. Figure P14.22 + •— 8kfi T~ llOmH ^ -A i ~10nF i + • DESIGN PROBLEM PSPICE MULTISIM 14.23 Using a 50 nF capacitor in the bandpass circuit shown in Fig. 14.22, design a filter with a quality fac- tor of 5 and a center frequency of 20 krad/s. a) Specify the numerical values of R and L. b) Calculate the upper and lower cutoff frequen- cies in kilohertz. c) Calculate the bandwidth in hertz. 14.24 Design a series RLC bandpass filter using only three components from Appendix H that comes closest to meeting the filter specifications in Problem 14.23. a) Draw your filter, labeling all component values and the input and output voltages. b) Calculate the percent error in this new filter's center frequency and quality factor when com- pared to the values specified in Problem 14.23. 14.25 For the bandpass filter shown in Fig. P14.25, calculate -SPICE the following: (a) f 0 ; (b) Q; (c) / cl ; (d) f c2 ; and (e) 0. MULTISIM Figure P14.25 20 H 40 mH 40 . " F -AW ^nrv> 1 ^ 180 n v» 14.26 The input voltage in the circuit in Fig. PI 4.25 is 500 cos cot mV. Calculate the output voltage when (a) co = co 0 ; (b) co = co c i; and (c) co = co c2 . 14.27 Design a series RLC bandpass filter (see Fig. 14.19[aJ) with a quality of 8 and a center frequency of 50 krad/s, using a 0.01 /xF capacitor. a) Draw your circuit, labeling the component val- ues and output voltage. b) For the filter in part (a), calculate the bandwidth and the values of the two cutoff frequencies. 14.28 The input to the series RLC bandpass filter designed in Problem 14.27 is 50costttf mV. Find the voltage drop across the resistor when (a) co = eo () ; (b) eo= (o cl ; (c) co = o) c2 \ (d) co = 0.2w o ; (e) co = 5co () . 14,29 The input to the series RLC bandpass filter designed in Problem 14.27 is 50coswt mV. Find the voltage drop across the series combination of the inductor and capacitor when (a) eo = co a ; (b) to = o> <;1 ; (c) co = co c2 \ (d) co = 0.2oo o ; (e) oo = 5co () . 14.30 A block diagram of a system consisting of a sinu- soidal voltage source, an RLC series bandpass fil- ter, and a load is shown in Fig. P14.30. The internal impedance of the sinusoidal source is 80 + ;0 fl, and the impedance of the load is 480 + /0 11. The RLC series bandpass filter has a 20 nF capacitor, a center frequency of 50 krad/s, and a quality factor of 6.25. a) Draw a circuit diagram of the system. b) Specify the numerical values of L and R for the filter section of the system. c) What is the quality factor of the interconnected system? d) What is the bandwidth (in hertz) of the inter- connected system? Problems 555 Figure P14.30 Sourc *> Filt er Load Figure P14.32 lOOkft -AAV 400 kH 14.31 The purpose of this problem is to investigate how a resistive load connected across the output termi- nals of the bandpass filter shown in Fig. 14.19 affects the quality factor and hence the bandwidth of the filtering system. The loaded filter circuit is shown in Fig. PI 4.31. a) Calculate the transfer function VJV-, for the cir- cuit shown in Fig. P14.31. b) What is the expression for the bandwidth of the system? c) What is the expression for the loaded band- width (/3{J as a function of the unloaded band- width (ft/)? d) What is the expression for the quality factor of the system? e) What is the expression for the loaded quality factor (Qi) as a function of the unloaded quality factor (0j)? f) What are the expressions for the cutoff frequen- cies a) cl and ft> c2 ? Figure PI4.31 14.33 The parameters in the circuit in Fig. P14.31 are R = 2.4 kO, C = 50 pF, and L = 2 fiH. The quality factor of the circuit is not to drop below 7.5. What is the smallest permissible value of the load resistor /? L ? Section 14.5 14.34 a) Show (via a qualitative analysis) that the circuit in Fig. P14.34 is a bandreject filter. b) Support the qualitative analysis of (a) by finding the voltage transfer function of the filter. c) Derive the expression for the center frequency of the filter. d) Derive the expressions for the cutoff frequen- cies avi an d o> C 2- e) What is the expression for the bandwidth of the filter? f) What is the expression for the quality factor of the circuit? Figure P14.34 C It- R 14.32 Consider the circuit shown in Fig. P14.32. a) Find co (> . PSPICE MULTISIM b) Find (3. c) FmdQ. d) Find the steady-state expression for v„ when Vi = 250 cos a) 0 t mV. e) Show that if R L is expressed in kilohms the Q of the circuit in Fig. PI4.32 is Q = 20 1 + 100//? L f) Plot Q versus R L for 20 kfl </? L <2MH. 14.35 For the bandreject filter in Fig. PI4.35, calculate PSPICE (a) (o a ; (b) f a ; (c) Q; (d) m cl ; (e) / cl ; (f) a> c2 ; (g) /, 2 ; HULTC,M and (h) j8 in kilohertz. Figure P14.35 50 fiH 20 nF 750 a . for a) = 0; Fig. 14.28(c) presents the equivalent circuit for to = 00. In both equivalent circuits, the output voltage is defined over an effective open circuit, and thus the output and