I
Network (A)
R
2
R
1
L
i(t)
i(0) = i o
Network (B)
V
+
-
R
1
R
2
C
+
v(t)
-
v(0) = v o
Problem 7.4: This problem examines the relation between transient responses of linear systems.
The network shown below is first driven by a step at t = 0, then driven by a ramp at t = 0, and finally driven by a stepped ramp at t = 0. In the first two cases, the capacitor has zero initial voltage.
(A) Find the capacitor voltage v(t) in response to the step shown below. Assume that v(0) = 0.
(B) Find the capacitor voltage v(t) in response to the ramp shown below. Again assume that
v(0) = 0. Hint: since the step input can be constructed from the ramp input according to v
Step
(t) = --d dt v
Ramp
(t), their respective ZSR responses are related in a similar manner.
(C) Finally, find the capacitor voltage v(t) in response to the stepped-ramp shown below assuming that the capacitor has the known initial voltage v(0) = v o
. Hint: think superposition.
V(t) = Ramp V(t) = Stepped Ramp V(t) = Step
V o
0 t
0
V
0
t t
V o
0
V
0
1 +
t
t
V(t)
+
-
R
C
+
v(t)
-
Part (A): v(0) = 0
Part (B): v(0) = 0
Part (C): v(0) = v o

(B) Next, at t = T, v
IN
turns the MOSFET off. Determine both i
R
(t) and v
DS
(t) for t ≥ 0. Hint: i
R
(t) is continuous at t = T.
(C) Sketch and clearly label graphs of both i
R
(t) and v
DS
(t) for t
≥
0 assuming that T
≈
5L
R
/R
R
and
R
X
= R
R
.
(D) The relay control circuit would be less expensive without the external resistor, which may be
‘‘removed’’ from the circuit by considering the limit R
X
→ ∞
. Why might such a cost reduction be unwise?
V
S
Relay
L
R i
R
R
R
+
v
DS v
IN
+
-
R x
Problem 7.3: This problem illustrates the superposition of a zero-input response (ZIR) and a zero-state response (ZSR) as a means of determining the total response of a network.
(A) Solve the differential equation τ dt
+ x = S for t ≥ 0 given x(0). This is equivalent to finding the step response of a general linear first-order time-invariant system having a nonzero initial condition.
(B) Use the result from Part (A) to show that the step response of a linear time-invariant first-order system takes the form x(t) = x(0)e
-t/
τ
+ x(
∞
)(1 - e
-t/
τ
). Explain why the two terms in this response are the ZIR and ZSR of the system, respectively.
(C) For each network shown below, find the network state at t = ∞ and the network time constant; note that I and V are constants. Hint: see Exercise 7.3. Next, use the results of Part (B) to find the network state for t
≥
0. You should consider whether you find the superposition of a ZIR and
ZSR to be a simple and intuitive method of determining the response of a linear system.

Problem 7.1: At t = 0
-
, the networks shown below have zero initial state. That is, the capacitor voltage v(t) and the inductor current i(t) are both zero at t = 0
-
. At t = 0, the voltage source produces an impulse of area Λ , and the current source produces an impulse of area Q.
(A) Derive the differential equation which relates v(t) to V(t) and i(t) to I(t). Hint: consider using
Thevenin or Norton equivalent networks to simplify the work.
(B) Find the capacitor voltage v(t) and the inductor current i(t) at both t = 0
+
and t =
∞
. One way to find the states at t = 0
+
is to integrate the corresponding differential equations from t = 0
to t = 0
+
under the assumption that each state remains finite during that time; you should justify this assumption. Then, substitute the initial conditions at t = 0
-
into the results to determine the states at t = 0
+
. Try to determine the states at t =
∞
through physical, rather than mathematical, reasoning.
(C) Next, find the time constant by which each state goes from its initial value at t = 0
+ value at t = ∞ . Hint: see Exercise 7.3.
to its final
(D) Using the previous results, and without necessarily solving the differential equations directly, construct v(t) and i(t) for t ≥ 0.
(E) Verify that the solutions to Part (D) are correct by substituting them into the differential equation found in Part (A).
V(t)
+
-
V(t)
R
1
R
2 C
+ v(t)
-
0 t
I(t)
0
I(t)
Q
R
1
R
2
L i(t) t
Problem 7.2:
R
X
In the circuit shown below, a MOSFET and an external resistor having resistance
are used to control the current i
R in the winding of a relay. Here, the relay is modeled as a series inductor and resistor having inductance L
R
and resistance R
R
, respectively. The MOSFET may be modeled as an ideal switch.
(A) At t = 0, v
IN
turns the MOSFET on so that v
DS
= 0. Determine i
R
(t) for t ≥ 0 given that i
R
(t = 0) = 0.

Exercise 7.1: v
OUT
Consider an amplifier with an input-output relation that takes the form
= V
A
(v
IN
/V
B
)
3
, where V
A
and V
B
are voltage constants. Determine its output bias voltage V
OUT and its small-signal gain v out
/v in
for a given input bias voltage V
IN
.
Exercise 7.2: Find the capacitance of the all-capacitor network, and the inductance of the allinductor network, shown below.
C
2
C
1
C
3
L
1
L
2
L
3
Exercise 7.3: Each network shown below has a non-zero initial state at t = 0, as indicated. Find the network state for t ≥ 0. Hint: what equivalent resistance is in parallel with each capacitor or inductor, and what decay time results from this combination?
+ v(t) -
R
1
C
R
2
v(0) = V
R
1
C
+
v(t) R
-
2
v(0) = V
i(t) L
R
1
R
2
i(0) = I
R
1
L
i(t)
i(0) = I
R
2