Download Exponential Function

AC Circuit Analysis
Represent sinusoidal signals by Ve jt 
where V cos(t   )  Re Ve jt   and
j t 


V sin(t   )  Im Ve
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Exponential Function
We focus on the exponential function est where s  j
d t
Note:
e  et and  et dt  et
dt
d st
But,
e  s  est and  est dt  1s est
dt
Properties of exponential function:
(1) Essentially all waveforms encountered in practice
can be expressed as a sum of exponential functions
(2) The response of a “Linear Time-Invariant” (LTI)
system to an exponential function e st is also an
exponential function, H(s)e st
So we only need to know H(s) – the transfer function.
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The Complex Plane
Remember, s    j , where  is radian frequency.
LHP
j
RHP

-
What about negative frequencies?
sin t  
e

2j
cos t  
e
2
1
1
jt
jt
e
 jt
 e  jt

exponentially
decreasing
signals
exponentially
-j increasing
signals

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Imaginary axis projection with phase t
t
unit circle
A phasor can represent either voltage or current.
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Rotating Phasor at radian frequency 
Im axis
Projection onto
imaginary axis
Phasor
Re axis
Sine Function
Cosine Function
Projection onto
real axis
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Voltage and current
are in phase (0 )
Current leads voltage
by 90 (= /2)
Voltage leads current
by 90
?
Resistor
Capacitor
Inductor
Voltage in blue
Current in red
Phasors shown projected onto real axis.
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Resistor : v(t )  i(t )R and i(t )  e st
So v(t )  Re st
v(t )
i( t )
1
Z(s ) 
 R ; Y (s) 
G
i( t )
v(t )
R
Inductor: v(t )  L
di(t )
and i(t )  e st
dt
v(t )  sLe st
v(t )
i( t ) 1
Z(s ) 
 sL ; Y (s ) 

i( t )
v(t ) sL
Capacitor: i(t )  C
dv(t )
and v(t )  e st
dt
i(t )  sCe st
i( t )
v(t ) 1
Y (s ) 
 sC ; Z(s ) 

v(t )
i(t ) sC
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 A phasor is a complex number whose magnitude is the magnitude
of a corresponding sinusoid, and whose phase is the phase of that
corresponding sinusoid.
 A phasor is complex, and does not exist. Voltages and currents are
real, and do exist.
 A voltage is not equal to its phasor. A current is not equal to its
phasor.
 A phasor is a function of frequency, . A sinusoidal voltage or
current is a function of time, t. The variable t does not appear in the
phasor domain. The square root of –1, or j, does not appear in the
time domain.
 Phasor variables are often given as upper-case boldface variables,
with lowercase subscripts. For hand-drawn letters, a bar are typically
placed over the variable to indicate that it is a phasor.
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Example: R-L Circuit
di(t )
Ri(t )  L
 v(t ); Let i(t )  Ie st and v(t )  Ve st
dt
Therefore ,
( Re st  sLe st )I  Ve st
i( t ) I
1
H (s) 
 
v(t ) V R  sL
R
So , i(t )  H (s )Ve st
v(t)
i(t)
L
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General Form of H(s)
Transfer Function H (s )

s 
s  
s 
1
 1 
   1 




bn 
z 1 
z2 
zn 

H (s) 

am 
s 
s  
s 
1
  1 
 1 






p 1  p 2   pm 

We must factor polynomials in variable s
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A representative pole-zero diagram
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Plot of H(s)
3 poles
2 zeros
j

http://lpsa.swarthmore.edu/Representations/SysRepZPK.html
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