Download Linear Circuit Elements

5/13/2017
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Eigen Values of the
Laplace Transform
Well, I fibbed a little when I stated that the
Eigen function of linear, time-invariant systems
(circuits) is:
L e jω t   G ω  e jω t
Instead, the more general Eigen function is:
L e st   G s  e st
Where s is a complex (i.e., real and imaginary) frequency of
the form:
s σ  j ω
such that:
e
st
e
σ  j ω  t
σt
e e
jωt
Note then, if σ  0 , the Eigen function e
jω t
previously described Eigen function e !
st
becomes the
Q: Yikes! I understand e st even less than I understood e
What does this function mean?
jω t
!
st
A: Remember, the function e is a complex function—it is
actually an expression of two real-value functions.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
5/13/2017
582762100
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These two real-valued functions could be its real and
imaginary components:
e
st
σt
e e
e
σt
e
σt
j ω t
 cos ωt  j sin ωt 
cos ωt  j e
1.0
σt
sin ωt
σ  0.2
ω  3.2
0.5
2
4
6
8
10
0.5
1.0
Or, the two real-valued functions could alternatively be the
complex values magnitude and phase:
σ  0.2
ω  3.2
3
2
1
2
4
6
8
10
1
2
3
st
j ω t
If σ  0 , then e  e
, and we’re back to the timeharmonic Eigen function:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
5/13/2017
582762100
3/6
1.0
σ  0.0
ω  3.2
0.5
2
4
6
1
v2 t  
2
0.5
8
10

j t
V

e
d


2


1.0
Q: What about basis functions? Can we use these Eigen
function to expand a signal?
A: Sure! Instead of the Fourier Transform, the result of
st
expanding a signal with basis function e is the Laplace
Transform.
For example, again consider the following linear circuit:
i1t 
i2t 
C

v1t 

Jim Stiles

L
R


  g t t  v t dt 
v2t   L v1t  

1

The Univ. of Kansas
Dept. of EECS
5/13/2017
582762100
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Using the Laplace transform, we can determine the output
voltage v 2t  by:
1. Expand the input signal v1t  using the basis function
e
st
:
V1  s  

 v t  e
0
1
s t
(or use a look-up table!)
dt
2. Determine the Eigen value of the linear operator
relating v1t  to v 2t  :

  g t t  v t dt 
v2t   L v1t  


1

V2  s   G  s V1 s 
where:
G s  


g t  e s t dt

3. Determine v 2t  from the inverse Laplace transform
of V2  s  (definitely use a look-up table!).
Q: But how do we determine G  s  ?
A: It’s just pretty darn simple!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
5/13/2017
582762100
5/6
Again, we determine the Eigen value of each linear operator
of our three linear circuit elements—only this time we use the
st
Eigen function e !
iR t   L vR t   
R
L e

R
Y
st
st

VR  s 
R
iC t   L vC t    C
L e

st
st
 C d e

dt
d vC t 
 sC e
dt
st
iC t 

vC t 
C

IC  s   sC VC  s 
Jim Stiles
R
vR t 
C
Y
C
Y

e

R
IR  s  
iR t 
vR t 
R
Y
The Univ. of Kansas
Dept. of EECS
5/13/2017
582762100
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iL t 
t
1
L v L t    iL t    v L t   dt 
L 
L
Y
L e

L
Y
st

t
  1 e s t dt   1 e s t
 L 
sL

IL s  
L
v L t 

VL  s 
sL
As a result we can determine the Eigen value G  s  of a linear
circuit by applying our circuit theory:
I 1 s 
1
I 2 s 
sC


V1 s 
R
sL


V2  s 
V1  s 
Jim Stiles
V2 s 
 G21 s  
sL R
1
 sL R
sC
The Univ. of Kansas
Dept. of EECS