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...

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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 co
ect 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 co
espond 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 a
itrary constants.
??(??) =
??
(??2 + ??1?? + ??0)

??(??) =
????
(??2 + ??1?? + ??0)

??(??) =
????2
(??2 + ??1?? + ??0)

a. low-pass, high-pass, band-pass
. 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 + ?? XXXXXXXXXX)


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
elow represents one stage of our multistage filter, find the value of C2 (in F).
(a XXXXXXXXXX
(b XXXXXXXXXX
(c XXXXXXXXXX
(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
????
=
??
??
XXXXXXXXXX
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
. 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 XXXXXXXXXXΩ
???? =
????
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
) 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
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