EECE 290 Homework 4, Spring 2015

5 H 100  vO + _ – + vI + _ Figure P15.3.1 20 k 10 k 1 H 1 k Homework 4 P15.1.6 (a) Derive the transfer function of the circuit of Figure P15.1.6; (b) determine L and C so as to have a second-order, lowpass, normalized Butterworth response, with R1 = 2  and R2 = 1 ; (c) scale the parameters so as to have R2 = 10 k and a 3-dB cutoff frequency f0 = 10 kHz. Solution: (a) The impedance of C in parallel with L and R2 is:= . It follows that: . (b) Substituting R1 = 2  and R2 = 1 , the denominator becomes: . For a normalized Butterworth response, and . This gives = 1.6730 F and = 0.8966 H, or = 0.4483 F and = 3.3460 H, with 0 = 1 rad/s. (c) To scale, RL = R2 to 10 k, km  104; to scale f0  10 kHz, kf  ; hence  = 5.3254 nF, or 1.4269 nF, and = 0.1427, or 0.5325 H. P15.3.1 Specify the response VO/VI of the filter circuit of Figure P15.3.1, and determine the passband gain. Solution: As   , the inductors behave as open circuits, and the source and op amp are isolated, with the noninverting input connected to ground. Hence, VO = 0. When   0, the inductors behave as short circuits, with the op amp in the noninverting configuration having a gain of (1 + 2/1) = 3. It follows that the filter is a second-order, lowpass filter of passband gain of 3. + – VO ( j) R1 = 2  – + VSRC ( j) C j 1 jL R2 =1  Figure P15.1.6 P15.4.4 Design a broadband bandpass filter having 3-dB cutoff frequencies of 100 Hz and 10 kHz and a passband gain of 2 using 0.2 F capacitors. The filter is to have a very high input impedance. Solution: For the lowpass filter, Rfl = 1/(c2Cfl) = 1/(210,000210-7) 79.58  80 . The gain of -2 can be achieved by making Rrl = 40 . For the high-pass filter, Rrh = 1/(c1Crh) = 1/(2100210-7) = 7958  8 k = Rfh, so that the gain of this stage is -1. To achieve a very high input impedance, a unity-gain amplifier may be added at the input. P15.4.7 It is required to design a third- order highpass Butterworth filter using a first-order highpass filter cascaded with a second-order highpass, noninverting filter of the type shown in Figure 15.4.1, reproduced in Figure P15.4.7 using a unity-gain amplifier. The filter should have a gain of 20 dB and a 3-dB cutoff frequency of 10 krad/s, using 0.1 µF capacitors. Determine the required values of resistances. Solution: From Table 15.2.2, the second order filter of a normalized third-order Butterworth filter should have Q = 1. From Equation 15.4.2, R2 = 2  and R1 = 0.5 . To move the cutoff frequency from 1 rad/s to 10 krad/s requires a frequency scale factor kf = 104. To use R1 C2 C1 R2 vO + _ – + Figure P15.4.7 vI + – 0.2 F 40  0.2 F vO vI + – + – – + – + 8 k 8 k 80  – + capacitors of 0.1 F requires a magnitude scale factor of km = 103. Resistances are multiplied by km, so that R1 = 1030.5 = 500  and R2 = 2 k. To have a first-order highpass filter, an active RC filter having Rr = 1/chC = 1/(10410-7) = 1 k and Rf = 10 k.