modeling of system electrical, Basic Elements Modeling-R,L,C Solved Examples with RLC circuit L, C Modeling with Non-Zero Initial condition

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Waqas Afzal

Basic Elements Modeling-R,L,C Solved Examples with RLC circuit L, C Modeling with Non-Zero Initial condition

Modeling of Electrical System
• The time domain expression relating voltage and current for the
resistor is given by Ohm’s law i-e
R
t
i
t
v R
R )
(
)
( 
• The Laplace transform of the above equation is
R
s
I
s
V R
R )
(
)
( 
R
s
I
s
V R
R 
)
(
/
)
(

Basic Elements of Electrical Systems
• The time domain expression relating voltage and current for the
Capacitor is given as:
dt
t
i
C
t
v c
c 
 )
(
)
(
1
• The Laplace transform of the above equation (assuming there is no
charge stored in the capacitor) is
)
(
)
( s
I
Cs
s
V c
c
1

Cs
s
I
s
V c
c
1
)
(
/
)
( 

Basic Elements of Electrical Systems
• The time domain expression relating voltage and current for the
inductor is given as:
dt
t
di
L
t
v L
L
)
(
)
( 
• The Laplace transform of the above equation (assuming there is no
energy stored in inductor) is
)
(
)
( s
LsI
s
V L
L 
Ls
s
I
s
V L
L 
)
(
/
)
(

V-I and I-V relations
5
Component Laplace V-I Relation I-V Relation
Resistor R
Capacitor
Inductor
dt
t
di
L
t
v L
L
)
(
)
( 
dt
t
i
C
t
v c
c 
 )
(
)
(
1
R
t
i
t
v R
R )
(
)
( 
R
t
v
t
i R
R
)
(
)
( 
dt
t
dv
C
t
i c
c
)
(
)
( 
dt
t
v
L
t
i L
L 
 )
(
)
(
1
Ls
Cs
1

Example-1
• The two-port network shown in the following figure has vi(t) as
the input voltage and vo(t) as the output voltage. Find the
transfer function Vo(s)/Vi(s) of the network.
6
C
i(t)
vi( t) vo(t)


 dt
t
i
C
R
t
i
t
vi )
(
)
(
)
(
1

 dt
t
i
C
t
vo )
(
)
(
1

Example-1
• Taking Laplace transform of both equations, considering initial
conditions to zero.
• Re-arrange both equations as:
7


 dt
t
i
C
R
t
i
t
vi )
(
)
(
)
(
1

 dt
t
i
C
t
vo )
(
)
(
1
)
(
)
(
)
( s
I
Cs
R
s
I
s
Vi
1

 )
(
)
( s
I
Cs
s
Vo
1

Cs
s
I
s
V
s
I
s
CsV
o
o
/
)
(
)
(
)
(
)
(


)
)(
(
)
(
Cs
R
s
I
s
Vi
1



Example-1
• .
8
Cs
s
I
s
Vo /
)
(
)
( 
)
)(
(
)
(
Cs
R
s
I
s
Vi
1


)
(
/
)
( s
V
s
V
nction
TransferFu i
o

)
1
)(
(
/
)
(
)
(
)
(
Cs
R
s
I
Cs
s
I
s
V
s
V
i
o


RCs
s
V
s
V
i
o


1
1
)
(
)
(
)
1
(
1
)
(
)
(
Cs
R
Cs
s
V
s
V
i
o



Mathematical modeling of electrical systems
9
Example-2

Mathematical modeling of electrical systems
10

Mathematical modeling of electrical systems
Where:-
11
Example-4

Mathematical modeling of electrical systems
12

Mathematical modeling of electrical systems
13

Mathematical modeling of electrical systems
14

Example-4
Use Laplace transforms to find v(t) for t > 0.
+
_
 

t = 0 6 k 
3 k 
100 F
+
_
v(t)
12 V
volts
v 4
)
0
( 

Circuit theory problem:
+
_
vc(t) i(t)
3 k 
100 F
6 k 
 
 
  0
5
)
(
0
6
^
10
*
100
*
3
^
10
*
2
)
(
0
)
(
0
)
(
)
(
)
(
)
(
)
(
)
(
)
arg
(
)
(
)
(














t
v
dt
t
dv
t
v
dt
t
dv
RC
t
v
dt
t
dv
t
v
dt
t
dv
RC
dt
t
dv
RC
t
v
R
t
i
t
v
ing
disch
dt
t
dv
C
t
i
c
c
c
c
c
c
c
c
c
c
c
c
c
c
Take the Laplace transform
of this equations including
the initial conditions on vc(t)

Circuit theory problem:
)
(
4
)
(
5
4
)
(
0
)
(
5
4
)
(
0
)
(
5
)
(
5
t
u
e
t
v
s
s
V
s
V
s
sV
t
v
dt
t
dv
t
c
c
c
c
c
c










An inductor in the s domain
 iv-relation in the time domain
v(t)  L
d
i(t).
dt
 By operational Laplace transform:
Lv(t) LLi(t) L Li(t),
V(s)  LsI(s)  I0  sL I(s)  LI0.
initial current
1

Equivalent circuit of an inductor
 Series equivalent:  Parallel equivalent:
Thévenin 
Norton
1

A capacitor in the s domain
 iv-relation in the time domain
i(t)  C
d
v(t).
dt
 By operational Laplace transform:
Li(t) LCv(t) C  Lv(t),
 I(s)  C sV(s) V0  sC V (s)  CV0.
initial voltage
2

Equivalent circuit of a capacitor
 Parallel equivalent:  Series equivalent:
Norton 
Thévenin
2

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The document discusses the Z-transform, which is the discrete-time equivalent of the Laplace transform. Some key points covered include: - The Z-transform is used to analyze discrete-time signals and discrete-time systems. It allows representation in the frequency domain through pole-zero analysis. - Causal, anti-causal, and two-sided sequences have different regions of convergence for their Z-transforms. Stability depends on poles lying inside or outside the unit circle. - The Z-transform has various properties that allow computations and transformations. The inverse Z-transform can be obtained through techniques like partial fraction expansion or long division.

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Frequency modulation (FM) is a type of angle modulation where the instantaneous frequency of the carrier signal varies linearly with the modulating signal. There are two types of FM: narrowband FM (NBFM) where the modulation index is less than 1, and wideband FM (WBFM) where the modulation index is greater than 1. The bandwidth of an FM signal can be estimated using Carson's rule, which states that nearly all the signal power lies within a bandwidth equal to twice the maximum frequency deviation plus the maximum modulating frequency. FM signals have constant amplitude but varying frequency, so their average power does not depend on the modulating signal and remains constant.

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