Download linalg paper 1

Zack Christensen
Elementary Applications of Linear Algebra in Circuit Analysis
For the beginning student of electrical engineering, or those interested in learning
the basic working of a circuit, the task of analyzing even the simplest of circuits can seem
daunting. However, once it is known that you can organize the information into a system
of equations that can them be solved by linear algebra the task becomes far less
intimidating.
The resistor network. Beginning with the simplest collection of components we
will construct a working model through which the process can be explained.
Consider this circuit where V is some constant voltage source; R1, R2 and R3 are
resistors of some known value; and the labeled points a nodes with “ref.” being the
reference node. The object is to find the voltage at every node except the reference node.
To do this first we apply Ohm’s law and Kirchoff’s Current Law [1] to each node. This
gives us the equations:
v1  V
v 2  v1 v 2  v3

0
R1
R2
v3  v 2 v3

0
R2
R3
This can then be arranged in matrix for:
0
0   v1  V 
 1
 R 2 R1  R 2
 R1  v2    0  .

 0
 R3
R 2  R3 v3   0 
And then these equations are solved giving the values v1  V ,
R 2( R 2  R3) * V
R 2 * R3 * V
, and v3 
. This can be
v2 
( R1  R 2)( R 2  R3)  R1 * R3
( R1  R 2)( R 2  R3)  R1 * R3
applied to any resistive network so long as the equations are written correctly.
Time dependent networks. With the introduction of circuit elements like inductor coils
and capacitors voltage in a circuit no longer remains constant with a direct current source,
it will vary over time. Let us look at our example circuit from before but this time we
will replace R2 with an inductor with value L and R3 with a capacitor of value C.
Zack Christensen
Elementary Applications of Linear Algebra in Circuit Analysis
Now we have a simple RLC circuit; with R standing for the resistor, L for the inductor,
and C for the capacitor; where we will apply Ohm’s Law; the current equations for an
dv
1
inductor and a capacitor, il   vl dt and ic  C c ; and Kirchoff’s Current Law [1].
L
dt
Applying these we get the equations:
v1  V
v 2  v1 1
  v 2  v3 dt  0
R
L
dv
1
v3  v 2 dt  C 3  0

L
dt
These equations can be written with indefinite intervals because in the next step they are
going to be differentiated and written in matrix form so that a known time interval is not
needed.
Now rewrite the equations so that they involve only derivatives and put them in
matrix form.


 1
 v
0
0
 1  V 
 d R d
R    

 v2  0 .


L    
 dt L dt

1
1
d 2  v3   0 
0


LC
LC dt 2 

As it stands this set of equations cannot be solved because it will produce
differential equations and there have been no initial conditions set forth. The method,
though, is the same as it was for a resistor network only now incorporating differential
equations to produce a more powerful tool for the analysis of simple electronic circuits.
References
1. David Cunningham and John Stuller, Circuit Analysis, Houghton Mifflin, 1995.