EE101
Handout #7
Prof. A. El Gamal
January 30, 2003
Homework Assignment #4
Due:
Thursday 2/6
Reminder:
Quiz 2 will be next Thursday 2/6 during the Frst 15 minutes of the lecture. It
will be on material covered by homework sets 3 and 4.
In addition to solving the following problems you may want to review the Matrix Primer
posted on the class webpage. It has more than you need for EE 101.
1. The following is the reduced incidence matrix for a circuit.
A
=
1
1
0
0
0
1

1
0
1
1
0
0
0
0
0

1

1

1
(a) ±ind the incidence matrix
˜
A
.
(b) Draw a schematic of the circuit using boxes to symbolize elements, label the
branches and nodes, and show the reference directions for each branch.
2. In this problem we show how the KCL and KVL matrix formulation discussed in the
lecture can be generalized to a circuit with disconnected subcircuits. The formulation
assumes that we are only interested in Fnding the branch voltages and currents and
not the absolute node potentials (since each subcircuit may have di²erent reference
ground node potential).
The procedure to writing the KCL and KVL for such a circuit is as follows:
(a) Choose one reference ground node per subcircuit;
(b) Connect (
i.e.
, short) all “ground” nodes together. This should not change branch
voltages and currents, since no current will ³ow among these nodes. Remove all
but one of the ground node labels, as we don’t need them any more;
(c) Write the KCL and KVL matrix equations for the modiFed circuit.
Apply this procedure to the circuit in the following Fgure, and write down the KCL
and KVL matrix equations.
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3.
A generalization of power conservation.
Consider a circuit with
b
branches and
n
nodes, with associated reference directions assigned.
As in the notes,
v
denotes the
vector of branch voltages,
i
denotes the vector of branch currents, and
e
denotes the
vector of node potentials (excluding the reference node). We saw in the notes that no
matter what the branch elements are, we always have
v
T
i
= 0,
i.e.
, the total net power
dissipated in the circuit is always zero.
Now suppose we change one or more of the elements in this circuit, but we do not change
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 Fall '08
 PingLi
 Graph Theory, Voltage source, Norton's theorem, node voltage equations

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