2/6/2016
687274019
1/11
The Eigen Values
of Linear Circuits
Recall the linear operators that define a capacitor:
LCY vC t   iC t   C
1
L iC t   vC t  
C
C
Z
d vC t 
dt
t
 i t  dt 
C

We now know that the eigen function of these linear, timeinvariant operators—like all linear, time-invariant
operartors—is exp  j  t  .
The question now is, what is the eigen value of each of these
operators? It is this value that defines the physical behavior
of a given capacitor!
For vC t   exp  j t  , we find:
iC t   LCY vC t 
d e j t
C
dt
  j C  e j t
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687274019
2/11
Just as we expected, the eigen function exp  j t  “survives”
the linear operation unscathed—the current function i t  has
precisely the same form as the voltage function
v t   exp  j t  .
The only difference between the current and voltage is the
multiplication of the eigen value, denoted as GYC   .
i t   LCY v t   e j t   GYC   e j t
Since we just determined that for this case:
i t    j C  e j t
it is evident that the eigen value of the linear operation:
i t   LCY v t   C
d v t 
dt
is:
GYC    j  C   C e
j 2
!!!
So for example, if:
v t   Vm cos ot   

 Re Vm e
j
e 
j ot
we will find that:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687274019
3/11
LCY Vm e j   e j ot   GYC o  Vm e j   e j ot

 C e
j

 C Vm e
Therefore:

iC t   Re C Vme

j  
 V e
2
2
m

j

e
j ot
j  2 
 e j t
e j  t

o

 C Vm cos ot    
 C Vm sin ot   
2
o

Hopefully, this example again emphasizes that these realvalued sinusoidal functions can be completely expressed in
terms of complex values. For example, the complex value:
VC Vme j 
means that the magnitude of the sinusoidal voltage is VC Vm ,
and its relative phase is VC   .
The complex value:
IC  GYC  VC

 C e
j 2
V
C
likewise means that the magnitude of the sinusoidal current
is:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687274019
4/11
IC  GYC  VC
 GYC   VC
 C Vm
And the relative phase of the sinusoidal current is:
IC  GYC    VC
 2
We can thus summarize the behavior of a capacitor with the
simple complex equation:
IC   j C VC

IC   j C VC

 C e
j2
V
C
VC
C

Now let’s return to the second of the two linear operators
that describe a capacitor:
1
vC t   L iC t  
C
C
Z
t
 i t  dt 
C

Now, if the capacitor current is the eigen function
iC t   exp  j t , we find:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687274019
L e
C
Z
j t
1
 
C
t
e
5/11
j t 
dt 

 1  j t

e
j

C


where we assume i t     0 .
Thus, we can conclude that:
 1  j t
LCZ e j t   GZC   e j t  
e
j

C


Hopefully, it is evident that the eigen value of this linear
operator is:
GZC   
1
j C

j
1 j 3 2

e
C C
And so:
 1 
VC  
 IC
j

C


Q: Wait a second! Isn’t this essentially the same result as
the one derived for operator LCY ??
A: It’s precisely the same! For both operators we find:
VC
1

IC
j C
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687274019
6/11
This should not be surprising, as both operators LCY and LCZ
relate the current through and voltage across the same
device (a capacitor).
The ratio of complex voltage to complex current is of course
referred to as the complex device impedance Z.
Z
V
I
An impedance can be determined for any linear, time-invariant
one-port network—but only for linear, time-invariant one-port
networks!
Generally speaking, impedance is a function of frequency. In
fact, the impedance of a one-port network is simply the eigen
value GZ   of the linear operator LZ :
 I
V Z I
V
LZ i t   v t 
Z
Z  GZ  

Note that impedance is a complex value that provides us with
two things:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687274019
7/11
1. The ratio of the magnitudes of the sinusoidal voltage and
current:
V
Z 
I
2. The difference in phase between the sinusoidal voltage
and current:
Z  V  I
Q: What about the linear operator:
LY v t   i t  ??
A: Hopefully it is now evident to you that:
GY   
1
GZ  

1
Z
The inverse of impedance is admittance Y:
Y
1 I

Z V
Now, returning to the other two linear circuit elements, we
find (and you can verify) that for resistors:
Jim Stiles
LRY vR t   iR t 
 GYR    1 R
LRZ iR t   vR t 
 GZR    R
The Univ. of Kansas
Dept. of EECS
2/6/2016
687274019
8/11
and for inductors:
1
LLY v L t   iL t 
 GYL   
LZL iL t   v L t 
 GZL    j  L
j L
meaning:
ZR 
1
 R  R ej0
YR
ZL 
and
j  
1
 j L  L e 2
YL
Now, note that the relationship
Z 
V
I
forms a complex “Ohm’s Law” with regard to complex
currents and voltages.
Additionally, ICBST (It Can Be Shown That) Kirchoff’s Laws
are likewise valid for complex currents and voltages:
I
n
n
0
V
n
n
0
where of course the summation represents complex addition.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687274019
9/11
As a result, the impedance (i.e., the eigen value) of any oneport device can be determined by simply applying a basic
knowledge of linear circuit analysis!
Returning to the example:
I
C

Z 
V
I
L
V
R

And thus using out basic circuits knowledge, we find:
Z  ZC  Z R Z L 
1
j C
 R j L
Thus, the eigen value of the linear operator:
LZ i t   v t 
For this one-port network is:
GZ   
1
j C
 R j L
Look what we did! We were able to determine GZ   without
explicitly determining impulse response gZ t  , or having to
perform any integrations!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687274019
10/11
Now, if we actually need to determine the voltage function
v t  created by some arbitrary current function i t  , we
integrate:
1
v t  
2

1

2

G
Z
   I   e j  t d 


1
 R j  L  I   e j  t d 
j C

where:
I   

 j t
i
t
e
dt




Otherwise, if our current function is time-harmonic (i.e.,
sinusoidal with frequency  ), we can simply relate complex
current I and complex voltage V with the equation:
V Z I
  1 j C  R j  L  I
Similarly, for our two-port example:
C


V1
L
V2


Jim Stiles
R
The Univ. of Kansas
Dept. of EECS
2/6/2016
687274019
11/11
we can likewise determine from basic circuit theory the eigen
value of the linear operator:
L21 v1 t   v2 t 
is:
G21  
so that:
Z L ZR
ZC  Z L Z R

j L R
1
 j L R
j C
V2  G21 V1
or more generally:
1
v2t  
2

j t
G

V

e
d




21
1


where:
V1  

 v t  e
1
 j t
dt

Jim Stiles
The Univ. of Kansas
Dept. of EECS