EECE 290 Homework, Spring 2008

14.5 A resistor denoted as Ry, is connected in parallel with the capacitor in the circuit in Fig. 14.7. The loaded low-pass filter circuit is shown in Fig. P14.5. a) Derive the expression for the voltage trans- fer function V,/V;. b) At what frequency will the magnitude of A (j@) be maximum? c) What is the maximum value of the magni- tude of H(jw)? d) At what frequency will the magnitude of H (jw) equal its maximum value divided by 2? e) Assume a resistance of 10 kQ is added in parallel with the 100 nF capacitor in the cir- cuit in Fig. P14.4. Find w,, H(j0), H(j@c), H(j0.1a,), and H(j10w<). Figure P14.5 R eww? ® + + Vj ANC Ry Vp e 6 ° R,(1/sC) Rt Rr +1/sC RiCs+1 Z Z+R Rr ~ RR,Cst+R+Rz (1/RC) = n (FR) °T RRC (1/RC) (wu? + [(R + Rz)/RRLCP P14.5° [a] LetZ= Then H(s) = b] |H(jw)| |H(jw)| is maximum at w = 0 fel LU) mas = (a) IHGedl = sy a ess ee sae FETE = 5 (1 + (R/RL)) [e] wy, = [1 + (10?/10*)] = 10,000(1 + 0.1) = 11,000 rad/s (10°)(10-") 10,000 (0) = 7 G60 = 0.9091/0° 10,000 iw.) = ! = 0.6428/— 45° H(j~e) = T6004 jiL.000 f= 48 10,000 0.1) = = 0.9046/—5.71° H(j0.140) 11,000 +1100 ~ 2-90462=5.71° 10,000 }10u-) = = 0.0905/— 84.29° A(j10s.) 11.000 + j110,000 5/= 84.29" 14.6 Usea0.5F capacitor to design a low-pass passive ate filter with a cutoff frequency of 50 krad/s. P a) b) c) Specify the cutoff frequency in hertz. Specify the value of the filter resistor. Assume the cutoff frequency cannot in- crease by more than 5%. What is the small- est value of load resistance that can be con- nected across the output terminals of the filter? If the resistor found in (c) is connected across the output terminals, what is the magnitude of H(jw) when w = 0? P 14.6 _ we _ 50,000 50 ; _ 3 sore = [a] fe = on =e x 10° = 7957.75 Hz 1 icf 3 [b] Re = 30 x 10 1 R= — = 40 (50 x 103)(0.5 x 10-8) RC R, R — = 0.05 -. Ry, =20R = 8002 Ry 800 [d] H(j0) = Rp 0.9524 R+Rr, 840 9 Usinga100nF capacitor, designa filter with a cutoff frequency of 300 Hz. Specify the value of R in high-pass passive kilohms. A 47 kQ resistor is connected across the output terminals of the fi cutoff frequency, in hertz, ter? ter. What is the of the loaded fil- 1 P14.9 [a] w= Ro = 2n(300) = 600n rad/s 1 1 - = 5305.16 we (6007)(100 x 10-9) R= [b] R. = 5305.16]/47,000 = 4767.08. 1 1 We= a5 = = 2097.7 rad/ “e™ RoC ~ (a767.08)(100 x 10-5) Cf 7aCls (14,14 Use a5 nF capacitor to design a series RLC band- of pass filter, as shown at the top of Fig. 14.27. The Pp center frequency of the filter is 8 kHz, and the qual- : | ity factor is 2. a) Specify the values of R and L. b) What is the lower cutoff frequency in kilo- hertz? d) What is the upper cutoff frequency in kilo- hertz? What is the bandwidth of the filter in kilo- hertz?