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PowerPoint Presentation Final Exam solutions The circuit shown is a Butterworth lowpass filter with the following specifications: Amax = 2 dB at ωp = 2,000 rad/s Amin = 50 dB at ωs = 20,000 rad/s Find the maximum cutoff frequency, ωC, max, for this filter (in rad/s). (a) 1,419 (b) 1,818 (c) 2,000 (d) 2,208 (e) 2,493 (f) 2,935 (g) 3,172 (h) 3,261 n=3 ???? 100.1???????? − 1 1 2?? = 20,000 105 − 1 1/6 = ??,?????? 1) Convert ?? = 1 ?????? = 2 20,000 ,?? = 1 ?????? = 2 10,000 , 2) Magnitude scale by 50 ?? = 50 ?????? = 25 2 10,000 = ??.?? ??,???? ?? = 1 50?????? = 2 500,000 = ?? ?? ???? A Butterworth highpass filter is desired with the following specifications: Amax = 3 dB at ωp = 10,000 rad/s Amin = 70 dB at ωs = 100 rad/s Rin = 50 Ω The associated normalized Butterworth LPF is shown below. Find the correct combination of inductance (in mH) and capacitance (in µF) for the Butterworth HPF. (a) L = 2.5√2 mH C = 2√2 μF (b) L = 2√2 mH C = 2.5√2 μF (c) L = 25√2 mH C = 2√2 μF (d) L = 2√2 mH C = 25√2 μF (e) L = 100√2 mH C = 20√2 μF (f) L = 25√2 mH C = 100√2 μF (g) L = 125√2 mH C = 100√2 μF 1) Evaluate at zero ?? 0 = ?? (??0) ?? 0 = 0 (??0) = 0 ?? 0 = 0 (??0) = 0 2) Take limit as s approaches infinity lim ??→∞ ??/??2 (1 + ??1/?? + ??0/??2) = 0 lim ??→∞ ??/?? (1 + ??1/?? + ??0/??2) = 0 lim ??→∞ ?? (1 + ??1/?? + ??0/??2) = ?? 3) lowpass, bandpass, high pass The following transfer functions correspond to types of filters. Find the proper identification of filter type for the first, second, and third transfer functions listed below. Note: K, α1, and α2 are arbitrary constants. ??(??) = ?? (??2 + ??1?? + ??0) ??(??) = ???? (??2 + ??1?? + ??0) ??(??) = ????2 (??2 + ??1?? + ??0) a. low-pass, high-pass, band-pass b. low-pass, band-pass, high-pass c. high-pass, low-pass, band-pass d. high-pass, band-pass, low-pass e. band-pass, low-pass, high-pass f. band-pass, high-pass, low-pass g. low-pass, low-pass, high-pass h. low-pass, high-pass, high-pass A multistage filter is characterized by the transfer function below. ??(??) = 6(?? + 4) (?? + 3)(??2 + ??10 + 100) Recall that if complex poles are embedded in a transfer function, they may be represented by the circuit below, where R1 and R2 have been set to 1 ohm. represents an active filter with a complex pole. Knowing that the circuit below represents one stage of our multistage filter, find the value of C2 (in F). (a) 0.01 (b) 0.02 (c) 0.05 (d) 0.1 (e) 0.2 (f) 0.5 (g) 1 (h) 2 From the complex poles ??2 + ???0 + 100 1 (??2| ??1 ??1 = 2 ??1 = 10 ⇒ ??1 = 1 5 1 ??1??2??1??2 = 5 ??2 = 100 ⇒ ???? = ??.???? ?? = ?????????????? + ???? ??/?? ??2 + 1???? + ?? ?? ?? ?? ?? ⇒ ?? ?? + ??0?? ?????? + 1 ??0???????? 1 ???? = ??02 ⇒ 1 ?? = ??02?? ?? ?? + ??0 ???? + ??02?? ??0?????? = ?? ?? + ??0 ???? + ??0 ???? = ?? ?? + 50 + 25 75 radians ?? = ?? ?? = ?? ?? ???? = 1 ???? ⇒ ?? = 1 ?? ?? ??0 = ?? ?? = 1 ???? = 1 5 × 10−2 = ???? ?????? = ??0 ???? + ????0tan(??) ??0 + ??????tan(??) = ???? ∴ ???????? ????????? ∠ ?????? ???? = ?? Consider a transmission line with characteristic impedance 50 Ω, terminating in a load of 100 Ω. At the operating frequency, the line is half a wavelength long. How much (in radians) does the transmission line shift the phase at the load? a. 0 b. π/6 c. π /4 d. π /2 e. π f. 3 π /2 g. 3 π /4 h. 3 π /8 5 V ?? For the circuit assume: ???? = 1 mA, ???? = 2 ???? ?? , ????? = 1 ??, and ?? = 0. Find ??. (a) 1000 Ω (b) 1032 Ω (c) 2000 Ω (d) 1999 Ω (e) 3000 Ω (f) 3970 Ω (g) 4000 Ω (h) 5000 Ω ???? = ???? 2 ?????? − ????? 2 ⇒ 1 ??2 = ?????? − ????? 2 ⇒ ?????? = ???? = ???? = 2 ?? ?? = ??−???? ?????? = ?? ???? Cut-off ???? = 0 Triode ???? = ???? ?????? − ????? ?????? − 1 2 ??????????2 Saturation ???? = ???? 2 ?????? − ????? 2 0 V For the single stage amplifier circuit. The DC equivalent model is generated by replacing the small signal voltage source by _________, and capacitors by _______. The small signal equivalent model is generated by replacing DC voltage sources by _________, and capacitors by ________. a) a short;an open circuit; a short; a short b) a short;an open circuit; a short; an open circuit c) a short;a short;a short; a short d) a short; a short;a short; ; an open circuit e) an open circuit;an open circuit; an open circuit; a short f) an open circuit;an open circuit; an open circuit; ; an open circuit g) an open circuit; a short; an open circuit; a short h) an open circuit; a short; an open circuit; ; an open circuit Final Exam solutions Slide Number 2 Slide Number 3 Slide Number 4 Slide Number 5 Slide Number 6 Slide Number 7 Slide Number 8 Slide Number 9 Slide Number 10