Modeling of Electrical Systems
Voltage, Current, Charge Relationship for Capacitor,
Resistor, and Inductor.
Component Symbol V-I Relation I-V Relation
v R (t )
Resistor v R (t ) i R (t )R i R (t )
R
1 dvc (t )
Capacitor vc (t ) ic (t )dt ic (t ) C
C dt
diL (t ) 1
Inductor v L (t ) L iL (t ) v L (t )dt
dt L
Voltage, Current, Charge Relationship for Capacitor,
Resistor, and Inductor.
Transform Impedance (Resistor, Inductor & Capacitor)
iR(t) IR(s)
+ +
Transformation
vR(t) ZR = R VR(s)
- -
IL(s)
iL(t) +
+
vL(t) ZL=Ls VL(s)
- -
ic(t) Ic(s)
+ +
vc(t) ZC(s)=1/Cs Vc(s)
- -
Equivalent Transform Impedance (Series)
• Consider following arrangement, find out equivalent transform
impedance.
ZT Z R Z L Z C
C
1 R
ZT R Ls
Cs
Equivalent Transform Impedance (Parallel)
L
1 1 1 1
ZT Z R Z L ZC
C
1 1 1 1
ZT R Ls 1 R
Cs
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.
vi( t) i(t) C vo(t)
Vi(s) I(s) 1/Cs Vo(s)
1
I ( s ) Vi ( s ) /( R )
Cs
1
Vo ( s ) I ( s ).
Cs
Vo ( s ) 1
Vo ( s ) 1
Vi ( s ) 1
Cs( R ) Vi ( s ) 1 RCs
Cs
• The system has one pole at
1
1 RCs 0 s
RC
Example: 2
The two-port network shown in the following figure has ei(t) as the
input voltage and eo(t) as the output voltage. Find the transfer
function Eo(s)/Ei(s) of the network.
Solution: The transfer function, Eo(s)/Ei(s), can be obtain by applying the voltage-divider
rule, hence
I 2 ( s)
Example-3: Find the transfer function
V ( s)
Example-3 continue
( R1 Ls ) I1 ( s ) Ls I 2 ( s ) V ( s ) (1)
1
Ls I1 ( s ) Ls R2 I 2 ( s ) 0 (2)
Cs
Example-3 continue
( R1 Ls ) I1 ( s ) Ls I 2 ( s ) V ( s ) (1)
1
Ls I1 ( s ) Ls R2 I 2 ( s ) 0 (2)
Cs
VC ( s )
Example-4: Find the transfer function
V (s)
Example-4 continue
Example-4 continue
VC ( s )
V (s)
Vo ( s )
Example-5: Find the transfer function
Vi ( s )
by mesh and nodal method
Vo ( s )
Example-5: Mesh method
Vi ( s )
(2s 1) I1 ( s ) I 2 ( s ) Vi ( s ) (1)
2
I1 ( s ) 3s 1 I 2 ( s ) 0 (2)
s
Example-5: Mesh method Continue
(2s 1) I1 ( s ) I 2 ( s ) Vi ( s ) (1)
2
I1 ( s ) 3s 1 I 2 ( s ) 0 (2)
s
Vo ( s ) I 2 ( s ).3s
Vo ( s ) 3s 2
3
Vi ( s ) 6s 5s 2 4s 2
Example-5: Nodal method
Vo ( S )
Example-6: Find the transfer function
Vi ( S )
Solution:
Vo ( S )
Example-7: Find the transfer function
Vi ( S )
Example-7: Continue
Example-7: Continue
Example-3: Consider the signal flow graph below and
identify the following:
a) Determine the loop gains of the feedback loops.
b) Determine the path gains of the forward paths.
c) Determine the non-touching loop gains.
Example-3: Answers
• There are four loop gains;
• There are two forward path gains;
• Nontouching loop gains;
Mason’s Rule
• The TRANSFER FUNCTION, C(s)/R(s), of a system represented by a
signal-flow graph is;
n
Pi i
C( s ) i 1
R( s )
Where
n = number of forward paths.
Pi = the i th forward-path gain.
∆ = Determinant of the system
∆i = Determinant of the ith forward path
• ∆ is called the signal flow graph determinant or characteristic function.
Since ∆=0 is the system characteristic equation.
Transfer function from signal flow graph can be calculated using
MASON’S GAIN formula
n
Pi i
C( s ) i 1
R( s )
∆ = 1- (sum of all individual loop gains) + (sum of the products of the gains
of all possible two loops that do not touch each other) – (sum of the
products of the gains of all possible three loops that do not touch each
other) + … and so forth with sums of higher number of non-touching loop
gains
∆i = value of Δ for the part of the signal flow graph that does not
touch the i-th forward path.
Example-4: Construct the signal flow graph of the block diagram of the
canonical feedback control system and find the control ratio C/R.
The characteristic function
Since the loop touch the forward path
Example-5: Determine the control ratio C/R and the canonical
block diagram of the feedback control system.
Example-5:Continue. (finding the control ratio C/R )
Example-5:Continue. (finding the canonical block diagram)
Example-6: Construct the signal flow graph for the
following set of simultaneous equations.
• There are four variables in the equations (i.e., x1,x2,x3,and x4) therefore four nodes are required
to construct the signal flow graph.
• Arrange these four nodes from left to right and connect them with the associated branches.
Example-7: Determine the transfer function C/R for the block diagram below
by signal flow graph techniques.
Example-7: Continue
• There are two forward paths. The path gains are
• The three feedback loop gains are
• No loops are non-touching, hence
• Because the loops touch the nodes of P1, hence
• Since no loops touch the nodes of P2, therefore
Example-8a: Determine C/R of each of the following
system using the signal flow graph techniques.
Example-8b: Determine C/R of each of the following system using
the signal flow graph techniques.
Example-8c: Determine C/R of each of the following system using
the signal flow graph techniques.
Example-9: Find the transfer function C/R for the system in which k is constant.
Example-9: Solution
Example-10: Find the control ratio C/R for the system given below.
Example-10: Solution
• The two forward path gains are • There are no non-touching loops, hence
• The five feedback loop gains are
• All feedback loops touches the two forward paths, hence
Example-11: Determine C/R for the system given below using signal flow graph
techniques and then put G3 = G1G2H2.
Example-12: Obtain the transfer function C(s)/R(s), for the signal-flow graph.
No.of forward paths = 1
C ( S ) P11
Forward path gain
R( S )
Δ formula terms
Individual loop gains
Two nontouching loop combinations and their gains
Three nontouching loop combinations and their gains
C ( S ) P11
R( S )
Example-13: Obtain the transfer function C(s)/R(s), for the signal-flow graph.
C ( S ) P11
R( S )
P1 G1G2G3G4 ; 1 1
Example-14: Obtain the transfer function C(s)/R(s), for the signal-flow graph.
Example-15: Obtain the transfer function C(s)/R(s), for the signal-flow graph.
C ( S ) P11 P2 2 P3 3
No.of forward paths = 3
R( S )
