2/6/2016
687294932
1/11
A Complex Representation of
Sinusoidal Functions
Q: So, you say (for example) if a linear two-port circuit is driven by a sinusoidal
source with arbitrary frequency o , then the output will be identically
sinusoidal, only with a different magnitude and relative phase.
C


v1t  Vm 1 cos ot  1 
L
R
v2t   Vm 2 cos ot  2 


How do we determine the unknown magnitude Vm 2 and phase 2 of this output?
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687294932
2/11
Eigen values are complex
A: Say the input and output are related by the impulse response g t  :
vv2t =
 L v1t  
t
 g t t  v t dt 
1

We now know that if the input were instead:
v1 t   e j 0t
then:
vv2t =
 L e j 0t   G  0  e j 0t
where:
G  0 

 g t  e
 j 0 t
dt
0
Thus, we simply multiply the input v1 t   e j  t by the complex eigen value G  0 
0
to determine the complex output v2 t  :
vv2t = G  0  e j 0t
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687294932
3/11
Complex voltages and currents
are your friend!
Q: You professors drive me crazy with all this math involving
complex (i.e., real and imaginary) voltage functions. In the lab I can
only generate and measure real-valued voltages and real-valued
voltage functions. Voltage is a real-valued, physical parameter!
A: You are quite correct.
Voltage is a real-valued parameter, expressing electric potential (in Joules) per
unit charge (in Coulombs).
Q: So, all your complex formulations and complex eigen values and complex
eigen functions may all be sound mathematical abstractions, but aren’t they
worthless to us electrical engineers who work in the “real” world (pun
intended)?
A: Absolutely not! Complex analysis actually simplifies our analysis of realvalued voltages and currents in linear circuits (but only for linear circuits!).
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687294932
4/11
Remember Euler
The key relationship comes from Euler’s Identity:
e j t  cos t  j sin t
Meaning:
Re e j t   cos t
Now, consider a complex value C. We of course can write this complex number
in terms of it real and imaginary parts:
C a  j b
 a  Re C 
and
b  Im C 
But, we can also write it in terms of its magnitude C and phase  !
C  C e j
where:
C  C C   a 2  b2
Jim Stiles
  tan 1 b 
The Univ. of Kansas
 a
Dept. of EECS
2/6/2016
687294932
5/11
A complex number has magnitude and phase
Thus, the complex function C e j  t is:
0
C e j t  C e
0
j
e j t
0
 C e j  t 
0
 C cos  0t     j C sin  0t   
Therefore we find:
C cos 0t     Re C e j  t 
0
Now, consider again the real-valued voltage function:
v1t  Vm 1 cos t  1 
This function is of course sinusoidal with a magnitude Vm 1 and phase 1 .
Using what we have learned above, we can likewise express this real function as:

v1t  Vm1 cos t  1   Re V1 e j t
where V1 is the complex number:
Jim Stiles

V1 Vm 1 e j 1
The Univ. of Kansas
Dept. of EECS
2/6/2016
687294932
6/11
But what is the output signal?
Q: I see! A real-valued sinusoid has a magnitude and phase, just like complex
number.
A single complex number (V ) can be used to specify both of the fundamental
(real-valued) parameters of our sinusoid (Vm ,  ).
What I don’t see is how this helps us in our circuit analysis.
After all:
vv2t  =
 G o  Re V1 e j ot 
What then is the real-valued output v 2t  of our two-port network when the
input v1t  is the real-valued sinusoid:
v1t  Vm 1 cos ot  1 
 Re V1 e j ot 
Jim Stiles
???
The Univ. of Kansas
Dept. of EECS
2/6/2016
687294932
7/11
The math will reveal the answer!
A: Let’s go back to our original convolution integral:
vv2t =

t
 g t t  v t dt 
1

If:
v1t  Vm 1 cos ot  1 
 Re V1 e j ot 
then:
vv2t =
t

g t  t   Re V1 e j ot dt 


Now, since the impulse function g t  is real-valued (this is really important!) it
can be shown that:
vv2t  =
t
 g t  t  Re V e
1
j ot 
dt 

t


 Re   g t  t  V1 e j ot dt 
 

Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687294932
8/11
The output signal
Now, applying what we have previously learned;
t


vv2t   Re   g t  t  V1 e j ot dt 
 

 t


 Re V1  g t  t   e j ot dt 
 

 Re V1 G 0  e j ot 
Thus, we finally can conclude the real-valued output v 2t  due to the realvalued input:

v1t  Vm1 cos ot  1   Re V1 e
j ot

is:

vv2t   Re V2 e
j ot
 V
m2
cos ot  2 
where:
V2  G o V1
The really important result here is the last one!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687294932
9/11
The Eigen value of the Linear operator is
its “Frequency Response”
C


v1t  Vm 1 cos ot  1 
L
R
v2t   Re G o V1 e j ot 


The magnitude and phase of the output sinusoid (expressed as complex value V2 )
is related to the magnitude and phase of the input sinusoid (expressed as
complex value V1 ) by the system eigen value G o  :
V2
 G o 
V1
Therefore we find that really often in electrical engineering, we:
1. Use sinusoidal (i.e., eigen function) sources.
2. Express the voltages and currents created by these sources as complex
values (i.e., not as real functions of time)!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
2/6/2016
687294932
10/11
Make sure you know what complex voltages
and currents physically represent!
For example, we might say “ V3  2.0 ”, meaning:
V3  2.0  2.0 e j 0


 v3 t   Re 2.0 e j 0e j ot  2.0 cos ot
Or “ IL  3.0 ”, meaning:
I L  2.0  3.0 e j 

iL t   Re 3.0 e j  e j ot   3.0 cos ot   
Or “Vs  j ”, meaning:
 
Vs  j  1.0 e 2
j 
Jim Stiles

 v s t   Re 1.0 e


 2e j  t  1.0 cos  t  
o
2
j 
o
The Univ. of Kansas

Dept. of EECS
2/6/2016
687294932
11/11
Summarizing
* Remember, if a linear circuit is excited by a sinusoid (e.g., eigen function
exp  j 0t ), then the only unknowns are the magnitude and phase of the
sinusoidal currents and voltages associated with each element of the
circuit.
* These unknowns are completely described by complex values, as complex
values likewise have a magnitude and phase.
* We can always “recover” the real-valued voltage or current function by
multiplying the complex value by exp  j 0t  and then taking the real part,
but typically we don’t—after all, no new or unknown information is revealed
by this operation!

V1

C
L
R

Jim Stiles
V2  G o V1

The Univ. of Kansas
Dept. of EECS