Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.002 – Circuits & Electronics
Spring 2006
Problem Set #9
Issued 4/12/06 – Due 4/19/06
Exercise 9.1:
Determine i(t) and v(t) for t ≥ 0+ in the network shown below. Note that
the inductor and capacitor both have nonzero states at t = 0, and that both sources are impulse
sources. Hint: use superposition to establish initial conditions at t = 0+ .
L
Qδ(t)
i(t)
+
v(t)
−
C
v(0) = V
Λδ(t)
i(0) = I
Exercise 9.2:
Shown below is a series resistor-inductor-capacitor network. Also shown below
is a graph of the resistor voltage v(t) as a function of time; the voltage is the homogeneous response
of the network to some set of initial conditions. From the voltage graph, determine the inductance
of the inductor and the capacitance of the capacitor given that the resistance of the resistor is 10 Ω.
Note: the next to last page of this problem set contains a larger graph of the common voltage. It
can be turned in with your problem set solutions.
Homogeneous Response: Resistor Voltage
1
0.8
0.6
Resistor Voltage [V]
0.4
L
R
0.2
0
−0.2
C
−0.4
−0.6
−0.8
0
0.2
0.4
0.6
0.8
1
Time [ms]
1.2
1.4
1.6
1.8
2
Problem 9.1:
The network shown below is driven in steady state by the sinusoidal input
voltage vI (t) = VI cos(ωt). The output of the network is the voltage vO (t), which takes the form
vO (t) = VO cos(ωt + φ). Find VO and φ as functions of ω as follows.
R1
vI (t)
C
R2
+
vO (t)
−
(A) Using the Taylor Series expansions for ex , cos(x)
and sin(x), show that ejx = cos(x) + j sin(x).
jx
Following this, recognize that cos(x) = e .
√
(B) Show
that A + Bj = A2 + B 2 ej arctan(B/A) . Thus, the magnitude and phase of A + Bj are
√
A2 + B 2 and arctan(B/A), respectively.
(C) Find a differential equation that can be solved for vO (t) given vI (t).
(D) Following Part A, let vI (t) = VI ejωt . Also, let vO (t) = V̂O ejωt where V̂O is a complex function
of the circuit parameters, ω and VI . With these substitutions, use the differential equation to
find V̂O .
(E) Following Parts A and B, first express vO from Part (D) in the form vO (t) = |V̂O |ej(ωt+ V̂O ) ,
and determine |V̂O | and V̂O as functions of the circuit parameters, ω and VI . Then, find VO
and φ for the original cosine input, again both as functions of the circuit parameters, ω and
VI .
(F) Sketch and clearly label VO /VI and φ as functions of ω. Identify the low-frequency and highfrequency asymptotes on the sketch.
Problem 9.2:
This problem concerns the sinusoidal-steady-state behavior of the networks
shown below, both of which have two ports.
+
Port #1
−
R1
R2
C
L
+
Port #1
−
L
+
Port #2
−
R2
+
Port #2
−
C
R1
(A) Determine the impedance of each network as viewed into Port #1 under the assumption that
Port #2 is open.
(B) Assume that Port #1 of each network is driven in sinusoidal steady state by the voltage
V1 cos(ωt), and that Port #2 is open. Determine the current into the positive terminal of each
network at Port #1. Express the current in the form I1 cos(ωt + φ1 ) where I1 is an amplitude
and φ1 is a phase angle.
(C) Assume that Port #1 of each network is again driven in sinusoidal steady state by the voltage
V1 cos(ωt), and that Port #2 is again open. Determine the voltage that appears at Port #2.
Express the voltage in the form V2 cos(ωt + φ2 ) where V2 is an amplitude and φ2 is a phase
angle.
Note that the results of this problem are useful when completing the pre-lab exercises to Lab #3.
Problem 9.3:
When the two-port network shown below is driven with the input voltage
vIN (t) = VIN cos(ωt), its output voltage takes the form vOUT (t) = VIN A(ω) cos(ωt + φ(ω)). The
normalized output voltage magnitude A(ω) is also shown below for a specific combination of R1 ,
R2 , R3 and C. Note: the last page of this problem set contains a larger graph of the normalized
output voltage magnitude. It can be turned in with your problem set solutions.
(A) Assuming that C = 1 µF, and that the smallest resistor has a resistance of 1 kΩ, determine R1 ,
R2 and R3 . Hint: consider first the low-frequency and high-frequency behavior of the network.
(B) There is a second but different network topology that also contains three resistors and one
capacitor, and that also provides the same input-output voltage relation as does the network
shown below. Determine the second network topology. You need not determine component
values for that network.
Normalized Output Voltage Magnitude A(ω)
0
10
R2
+
vIN
R3
C
vOUT
−
A(ω) [1]
R1
−1
10
−2
10
0
10
1
10
2
10
3
4
10
10
Frequency ω [rad/s]
5
10
6
10
Problem 9.4:
The input-output voltage relation of the network shown in Problem 9.3 can also
be produced by a network containing three resistors and one inductor. As in Problem 9.3, there are
actually two such network topologies. Determine both topologies; you need not determine resistor
and inductor values to match the graphical relation given in Problem 9.3.
Homogeneous Response: Resistor Voltage
1
0.8
0.6
Resistor Voltage [V]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
0.2
0.4
0.6
0.8
1
Time [ms]
1.2
1.4
1.6
1.8
2
Normalized Output Voltage Magnitude A(ω)
0
A(ω) [1]
10
−1
10
−2
10
0
10
1
10
2
10
3
4
10
10
Frequency ω [rad/s]
5
10
6
10