EECS 40 Spring 2003 Lecture 6
Copyright, Regents University of California
S. Ross and W. G. Oldham
Lecture 6
FINDING VOLTAGES IN A CIRCUIT
So far, we have been applying KVL and KCL “haphazardly” to
find voltages and currents in a circuit.
Good for developing intuition, finding things quickly…
…but what if the circuit is complicated? What if you get stuck?
Systematic way to find all node voltages in a circuit:
Nodal Analysis
EECS 40 Spring 2003 Lecture 6
S. Ross and W. G. Oldham
Copyright, Regents University of California
NOTATION: NODE VOLTAGES
The voltage drop from node X to a reference node
(ground) is called the node voltage Vx.
Example:
a
+
Va
_
b
+
+
_
Vb
_
ground
EECS 40 Spring 2003 Lecture 6
Copyright, Regents University of California
S. Ross and W. G. Oldham
FORMAL CIRCUIT ANALYSIS USING KCL:
NODAL ANALYSIS
(Memorize these steps and apply them rigorously!)
1 Choose a reference node (ground, node 0)
(look for the one with the most connections!)
2 Define unknown node voltages (those not fixed by
voltage sources)
3 Write KCL at each unknown node, expressing current
in terms of the node voltages (using the I-V
relationships of branch elements*)
4 Solve the set of equations (N equations for N unknown
node voltages)
* with floating voltages we will use a modified Step 3
EECS 40 Spring 2003 Lecture 6
S. Ross and W. G. Oldham
Copyright, Regents University of California
EXAMPLE OF NODE ANALYSIS
node voltage set
R1

+
-
V1
Va
R
3
R2
Vb
What if we used
different ref node?
R4
IS
 reference node
1. Choose a reference node.
2. Define the node voltages (except reference node and
the one set by the voltage source).
3. Apply KCL at the nodes with unknown voltage.
Va  V1 Va Va  Vb


0
R1
R2
R3
Vb  Va Vb

 IS
R3
R4
4. Solve for Va and Vb in terms of circuit parameters.
EECS 40 Spring 2003 Lecture 6
S. Ross and W. G. Oldham
Copyright, Regents University of California
EXAMPLE OF NODE ANALYSIS
R1
Va
R3
V
1
R2
I1
R4
R5
V2
Va   V1 Va Va  V2


 I1
R1
R4
R5
EECS 40 Spring 2003 Lecture 6
S. Ross and W. G. Oldham
Copyright, Regents University of California
NODAL ANALYSIS WITH “FLOATING” VOLTAGE SOURCES
A “floating” voltage source is a voltage source for which
neither side is connected to the reference node. VLL in
the circuit below is an example.
VLL
Va
Vb
- +
I1
R2
R4
I2
Problem: We cannot write KCL at node a or b
because there is no way to express the current
through the voltage source in terms of Va  Vb.
Solution: Define a “supernode” – that chunk of the circuit
containing nodes a and b. Express KCL at this supernode.
EECS 40 Spring 2003 Lecture 6
S. Ross and W. G. Oldham
Copyright, Regents University of California
FLOATING VOLTAGE SOURCES (cont.)
Use a Gaussian surface to enclose the floating voltage source;
supernode
write KCL for that surface.
VLL
Va
-
I1
Vb
+
R2
I2
R4
Two unknowns: Va and Vb.
Get one equation from KCL at supernode: Va
Vb

 I1  I2
R2 R 4
Get a second equation from the property of the voltage source:
VLL  Vb  Va
EECS 40 Spring 2003 Lecture 6
S. Ross and W. G. Oldham
Copyright, Regents University of California
ANOTHER EXAMPLE
V2
R1
Va
Vb
| +
V1
+

R2
R3
R4
Can we choose ground to avoid a floating voltage source?
Not in this circuit.
Va  V1 Va Vb


0
R1
R2 R 4
Vb  Va  V2