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EECS 16A
Fall 2016
Designing Information Devices and Systems I
Babak Ayazifar, Vladimir Stojanovic
Discussion 5A
v2
−
+
u1
+
u4
R1
i1
+
v1
i2
R4
−
i4
u2 −
v4
R2
i3
i5
R5
R3
+
u3
−
u5
−
+
v3
I
1. Circuits
Last week we saw how we could view circuits as traffic flows. Today we will go over how to incorporate
node potentials into our model, so we can solve for the voltage across every element and the current through
every element.
(a) Translate the above circuit into a directed graph, this time follow convention and label currents as
going in through the + terminal and out through the - terminal. Write the incidence matrix for the
graph. Ignore the current source for now.
(b) Let ~v be the vector of node potentials. How many elements are in ~v? What is the product F~v? What
does this product represent?
(c) We now have a way to represent the voltage difference across elements. Last discussion we had
defined a vector of currents through elements. What equation can be used to relate the voltage across
and element and the current through it? Write down such an equation for each of the resistors in the
above graph. Ignore the current source for now.
(d) Let R be the diagonal matrix of branch resistances. Write ~i in terms of R, F, and ~v.
(e) The equation in d) represents the branch currents in terms of the node voltages. From last discussion
we have another relation for the branch currrents, namely, F T~i = 0. Now we will model the current
source as a steady current flowing into one node and a steady current flowing out of another. Let
~f represent this vector of independent currents. Restate KCL to include this vector of independent
sources ~f .
(f) At this point we have two equations that independently describe the current through the branches ( ~i ).
Use these equations to come up with one equation we could use to solve for the node potentials ~v.
(g) Given we now have ~v, how could we find ~i?
EECS 16A, Fall 2016, Discussion 5A
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