Transcribed Text
1. Consider a C.T. system below. (Assume all initial conditions are zero).
(a) Find transfer function, H(s), for the system. (include all detailed steps, no
points for just the answer).
dt2 + 4 dt
(b) State Response, y2s(1) when the system when input just signal the answer). is
x(t) Find = Zero e 5tu(t). (include all detailed steps, no points for
x(1) = 8(1)
X(s) =1
1
x(1) =1
X(s) =
s
dx(t)
1
sX(s)~x(0")
x(t) = ed
X(s)=
dt
s+a
5
d²x(t)
52x(s).  sx(07107
x(1) = cos(bt)
x(s)==167 =
dt²
b
1
x(t) = sin(bt)
X(s)= 3+6
x(1)dr
X(s)
s
Laplace Transform Properties
Basic Laplace Transform Pair
1
2. Consider a C.T. system implemented by using opamps. Complete steps below.
(a) Find Transfer Function, A(s) = Y(s) / X(s), for a C.T. system with opamps below.
Assume all initial conditions are zero.
(Partial credits will be given, be sure to include ALL detailed steps)
(b) Find Transfer Function, B(s) = Y(s) / X(s), for a C.T. system with opamps below.
Assume all initial conditions are zero.
(Partial credits will be given, be sure to include ALL detailed steps)
M
Cn
Rn
Ro
W
If
M
Cp
R11
Rn

+
+
+
N(s) Ca
Y(s)
X(s)
Y(s)


A
Figure 2(a) For A(s) = Y(s)/X(s),
Figure 2(b) For B(s) = Y(s) / X(s),
an mostom with below
+
 0
+
X(s) Cn
+
Y(s)
T(t)


Figure 2(a) For A(s) = Y(s) / X(s),
Figure 2(b) For B(s) = Y(s)/X(s),
(c) Use results from (a) and (b) to realize (build) a C.T. system with below
description. Find values for R/1, Ril, R/2, Ri2, and Cp when C=0.01 [uF] and Cp
= 0.01 [uF]. (Assume all initial conditions are zero.). Use an additional inverting
gain opamp if needed. (Partial credits will be given, be sure to include ALL detailed
steps)
d²y(t) dt2 + 35 dy(t) dt + 5 + 50x(t)
3. Consider a periodic signal x(t) shown below, where Amplitude = 4 [V], To=2 (ms)
(period of signal), wo (fundamental frequency), and n fo (harmonic frequencies).
5
4
3
2
1
1
3
2
1
1
2
3
4
4
10°3
(a) Find Co, C1, C3, and C5 using the definition of Compact Trigonometric
Fourier Series Coefficients. (Partial credits will be given, be sure to include ALL detailed
steps)
(Note)
(b) Draw frequencydomain representation of signal (C6/v5(n)6), n: integer). = 0).
2D graph, similar to spectrum analyzer traces. (note, G=G=C=
mol
Co=ao
Cn= an+b2
b,
x(1)=af+)
an
If
n=
nol
On
= tan"¹ bey) an
1
af=  To S x(t)dt
T3
2 dt (m = 0)
Toir
X.=7 1
2 dt (m
# 0)
To is
a) Find Fourier Co, C1, C3, and C5 using the definition of Compact
steps) Series Coefficients. (Partial credits will be given, be sure Trigonometric to include ALL detailed
(Note) an=2n f==2x/To
(b) Draw frequencydomain representation of signal n: integer).
2D graph, similar to spectrum analyzer traces. (note, C=G=C6 =0).
x
x(1)=C C, Il
n=1
Co =ao Cr= an+b2
00
80
11 nost
n=1
br
1
On =tan~¹
=
To
x
2
x(I) =
anT!!
(m # 0)
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