EEL 205 Control Engineering
Practice Problem Set 5
1. For the circuit shown in Fig. P1, obtain the response vo(t),
following the closure of switch at t = 0, if the initial voltage
on the capacitor is zero. Consider two cases of the source
voltage as
i. 2e−tii. t + e−2t
[2te−t;−1+t+2e−t−e−2t]
2. Fig. P2 shows the initial state of a certain
resistive-capacitive circuit. Obtain the response
v1(t) and v2(t) if the switch is closed at t = 0.
1
2[1+e−2t];1
2[1−e−2t]
3. A chemical reactor has a heating coil mounted at one end, and a temperature sensor at the other. Because of
reactants in the tank between the two, the sensor follows the temperature of the heating coil with a delay, and it
has been found that between them there exists an effective transfer function of first order with time constant of
2s. Starting with an initial room temperature of 30oC, if the heating coil has its temperature increasing at the
steady rate of 1oC/s, obtain the error between the reactor temperature and the sensor reading.
[e(t)=−1+e−t/2]
4. In Fig. P4, L = 0.01H, C = 1µF, R = 100Ω, initial value of v(t) is 10V,
and initial rate of change of v(t) is zero. Write the system differential
equation in a single variable and find
ζ
and
ω
n. Find v(t) for time
greater than zero.
[vC(t)=11.547 $e−5000tcos(8660t−30o)]
5. In Fig. P5, the position x(t) of mass M is used as a servo
measure for the difference in speed (v1(t) and v2(t)) of two
reverse going belts.
i. Find the response of x(t) for a step change in
v1(t)−v2(t).
ii. What condition would determine if the response is
overdamped, underdamped, or critically damped ?
iii. How could you make the dynamics to step change
settle faster ?
iv. How could you increase the sensitivity ∆x/∆(v1−v2) ?
x(t)= Dv
K1−1
1−D2/(KM)e−Dt/Msin K
M−D2
M2t+tan−1KM
D2−1
6. The roots of a second order system have
σ
= −400rad/s and
ω
= !300rad/s.
i. Determine the natural frequency and damping ratio. [500rad/s, 0.8]
ii. If the damping ratio is reduced by 25%, show the effect on the roots of the system.
[
σ
= −300rad/s and
ω
= !400rad/s]
EEL 205/PS5
v t
( )
10 Ω
6
1µ F
P1
v t
( ) o
v
2
(0) = 0V
1Ω
1F
P2
v
1
(0) = 1V 1F
t
( )L R C v
P4
k
M
D
x
v1
v2
P5
