1. Network Theorems
1.1. Define following terms
(a)
Charge
Electrons in the outer orbit of an atom can easily evacuated by application of some
external force. Electrons that are forced out of their orbits can result in a lack of electrons,
hence called positive charge i.e. more protons than electrons. Electrons where they come
to rest can result in excess of electrons, hence called negative charge i.e. more electrons
than protons. A positive or negative charge is an effect of absence or excess of electrons.
The number of protons remains constant. Charge is measured in coulombs.
(b)
Potential
Work done against the force of repulsion to bring a charge closer to the one another is
called potential. Potential is measured in volt.
(c)
Potential difference
The potential difference between two points is “One volt” when one joule of work is done
to displace a unit charge of one coulomb from the point of lower potential to point of
higher potential. Potential difference is measured in volt
(d)
Electro motive force (EMF)
Emf is the potential difference that moves the electrons to flow in any conductor. Emf is
measured in volt.
(e)
Current
An amount of charge passing through the conductor in unit time is called current. It is
measured in ampere.
(f)
Current density
It is the amount of current flowing per unit cross section area of a conductor. Current
density is measured in A/mm2.
(g)
Power
Rate of change of energy with respect to time is called power. It is measured in watt.
(h)
Electrical energy
Electrical power consumed in unit time is called electric energy. It is measured in Kwh.
(i)
Linear element and Nonlinear element
An elements such as resistor, inductor and capacitor whose voltage vs current
characteristics is linear and their resistance, inductance and capacitance do not vary with
the change in applied voltage or circuit current are called linear elements.
An elements such as semiconductor devices whose voltage vs current characteristics is
nonlinear and their resistance, inductance and capacitance may vary with the change in
applied voltage or circuit current are called nonlinear elements.
(j)
Active element and Passive element
An element such as vacuum tube, transistor, Opams with the capacity of boosting the
energy level of signal passing through it are called active elements.
An element such as resistor, inductor, capacitor, thermistor that do not have capacity of
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
1
1. Network Theorems
boosting the energy level of signal passing through it are called passive elements.
(k)
Unilateral element and Bilateral element
When the amount of current passing through element is affected by the change in polarity
of applied voltage then it is called unilateral element. This element offers varying
impedance with the variation in current. Diode, transistors etc. are the examples of
unilateral elements.
When the amount of current passing through element is not affected by the change in
polarity of applied voltage then it is called bilateral element. This element offers same
impedance irrespective of variation in current. Resistance, inductance and capacitance
are the examples of bilateral elements.
(l)
Lumped network and Distributed network
A network in which circuit elements like resistance, inductance and capacitance are
physically separable for analysis purposes, is called lumped network. Most of the electric
networks are lumped in nature.
A network in which circuit elements like resistance, inductance and capacitance cannot
be physically separated for analysis purposes, is called distributed network. A
transmission line where resistance, inductance and capacitance of a transmission line are
distributed all along its length and cannot separated anywhere in the circuit.
(m)
Linear network and Non-linear network
A network whose parameters remain constant irrespective of the change in time, voltage,
temperature etc. is known as linear circuit. Ohm’s law is applicable to such network. This
type of circuit can be solved using super position law.
A network whose parameters change their values with the change in time, voltage,
temperature etc. is known as non-linear circuit. Ohm’s law is not applicable to such
network. This type of circuit does not follow super position law.
(n)
Unilateral network and Bilateral network
A network whose characteristic dependents on the direction of current i.e. characteristics
changes if direction of current is changed. Network with diode, transistors etc. that has
diverse characteristics in different direction of current.
A network whose characteristic is independent of the direction of current i.e.
characteristics remains same if direction of current is changed. Network with only
resistance has similar characteristics in different direction of current.
(o)
Active network and Passive network
A network that contains one or more energy source such as voltage or current is called
active network.
A network that does not contain any energy source such as voltage or current is called
passive network.
(p)
Ideal energy source and Particle energy source
Energy sources are the devices that converts any source of energy into electrical energy.
Types of sources available in the electrical network are voltage source and current
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
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1. Network Theorems
Figure 1. 1 Ideal voltage
source

Figure 1. 2 Practical voltage
source
Current Source
Current Source
Voltage Source
Voltage Source
sources. A voltage source has a driving role of emf whereas the current source has a
driving job of current.
Figure 1. 3 Ideal current
source
Figure 1. 4 Practical current
source
Voltage source
Ideal voltage source is a two-terminal device whose voltage at any instant of time is
constant and is independent of the current drawn from it. Internal resistance of ideal
voltage source is zero, but practically an ideal voltage source cannot be achieved.
Practical voltage source is a two-terminal device whose voltage at any instant of time
changes with the current drawn from it. Due to internal resistance of voltage source,
when current flows voltage drop takes place and it causes terminal voltage to fall down.

Current source
Ideal current source is a two-terminal device that provides constant current to any load
from zero to infinity. Internal resistance of ideal current source is infinite, but practically
an ideal current source cannot be achieved.
Practical current source is a two-terminal device whose current at any instant of time
changes. Amount off current depends upon the load.
(q)
Independent energy sources
V
+
-
v(t)
Figure 1. 5 Independent voltage source
I
i(t)
Figure 1. 6 Independent current source
Independent voltage source is the two terminal element that provides a specific voltage
across its terminal. The value of this voltage at any instant is independent of value or
direction of the current that flow through it.
Independent current source is the two-terminal elements that provides a specific
current across its terminal. The value and direction of this current at any instant is
independent of value or direction of the voltage that appears across the terminal of
source.
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
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1. Network Theorems
(r)
Dependent energy sources
a
+
+
Vab
b
μ Vab
+
-
c
Vcd
-
-
d
a
Vab
b
Figure 1. 7 Voltage controlled voltage source (VCVS)
a
+ i ab
+
r iab
b
-
+
-
Figure 1. 9 Current controlled voltage source (CCVS)
+
c
Vcd
g m Vab
-
-
d
Figure 1. 8 Voltage controlled current source (VCCS)
c
a
+
d
icd
i ab
+
c
β iab
Vcd
-
Icd
+
b
-
-
d
Figure 1. 10 Current controlled current source (CCCS)
Voltage controlled voltage source (VCVS) is the four terminal network components
that establishes a voltage between two-point c and d. Value of Vcd depends upon the
controlled voltage Vab and constant μ.
Voltage controlled current source (VCCS) is the four terminal network components
that establishes a current icd in the branch of circuit. Value of icd depends on the
controlled voltage Vab and constant gm.
Current controlled voltage source (CCVS) is the four terminal network components
that establishes a voltage Vcd between two-point c and d. Value of Vcd depends upon the
controlled current iab and constant r.
Current controlled current source (CCCS) is the four terminal network components
that establishes a current icd in the branch of circuit. Value of icd depends upon the
controlled current iab and constant β.
(s)
Single port network
An active or passive network with two terminals is treated as single port network.
(t)
Two port network
An active or passive network with two pairs of terminals is treated as two port network.
Where one pair of terminal is designated as input and other pair of terminal is designated
as output.
(u)
Multi-port network
An active or passive network with n- number of pairs of terminals is treated as multi-port
network. Where some pair of terminals are designated as input and some pair of
terminals are designated as output.
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
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1. Network Theorems
1.2. Relation between energy (E) and power (P) for two terminal resistor
element

Relation between voltage and current in resistor element in terms of charge is,
v  Ri  R

dq
dt
Given voltage v (t) across and current i (t) through a resistor L and then associated energy
e (t) is,
If v (0)  0 and v (t ) V
v (t )  Ri (t )
v (t )
i (t ) 
R
t
e (t )   p (t )dt
0
t
  i (t )v (t )dt
0
t
v (t )
v (t )dt
R
0


1
T
V
R
2
dt
0

V2
T
R
If v (0)  0,v (t ) V m sin(t ) and energy dissipated for time period T 
i (t ) 
2

v (t ) V m sin(t )

R
R
t
e (t )   p (t )dt
0
t
  i (t )v (t )dt
0
V m sin(t )
V m sin(t ) dt
R
0
t

V m2 t 2

sin (t )dt
R 0
V m2 T (1  cos(2t ))

dt
R 0
2
V m2
 T
2R
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
5
1. Network Theorems
1.3. Relation between energy (E) and power (P) for two terminal inductor
element

Relation between voltage and current in inductor element in terms of charge is,
v

d
dt
When there is an initial charge of ψo is stored on inductor and it is increasing linearly with
time, then charge on inductor at any instant of time is,
   o  kt
d

k
dt

Hence, it can be observed that voltage in the inductive system is independent of initial
charge.
t
   vdt

t
0
  vdt  vdt

0
t
  o  vdt
0

Given voltage v (t) across and current i (t) through a inductor L and then associated
energy e (t) is,
If i (0)  0 and i (t )  I
v (t ) 
i (t ) 
d d (Li )
di (t )

L
dt
dt
dt
t
1
 v (t )dt
L 
t
e (t )   p (t )dt
0
t
  i (t )v (t )dt
0
t
  i (t )L
0
di (t )
dt
dt
I
 L  idi
0
1.4.
1
 LI 2
2
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
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1. Network Theorems
If i (0)  0, i (t )  I m sin(t ) and energy dissipated for time period T 
v (t )  L
2

d  I m sin(t ) 
di (t )
L
 LI m cos(t )
dt
dt
t
e (t )   p (t )dt
0
t
  i (t )v (t )dt
0
t
  I m sin(t )LI m cos(t ) dt
0


LI m2  t
2
 2cos(t )sin(t )dt
0
LI m2  T
2
 sin(2t )dt
0
0
1.5. Relation between energy (E) and power (P) for two terminal capacitor
element

Relation between voltage and current in capacitor element in terms of charge is,
i

dq
dt
When there is an initial charge of qo is stored on capacitor and it is increasing linearly
with time, then charge on capacitor at any instant of time is,
q  qo  kt
dq

k
dt

Hence, it can be observed that current in the capacitive system is independent of initial
charge.
t
q   idt

t
0

 idt   idt

0
t
 qo   idt
0

Given voltage v (t) across and current i (t) through a capacitor C and then associated
energy e (t) is,
Shital Patel, EE Department
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1. Network Theorems
If v (0)  0 and v (t ) V
i (t ) 
v (t ) 
dq (t ) d Cv (t )
dv (t )

C
dt
dt
dt
t
1
C
 i (t )dt

t
e (t )   p (t )dt
0
t
  i (t )v (t )dt
0
t
 C
0
dv (t )
v (t )dt
dt
V
 C vdv
0
1
 Cv 2
2
If v (0)  0,v (t ) V m sin(t ) and energy dissipated for time period T 
i (t )  C
2

d V m sin(t ) 
dv (t )
C
 CV m cos(t )
dt
dt
t
e (t )   p (t )dt
0
t
  i (t )v (t )dt
0
t
 CV m cos(t )V m sin(t ) dt
0

CV m2 t

CV m
2
2
2
 2cos(t )sin(t )dt
0
T
 sin(2t )dt
0
0
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
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1. Network Theorems
1.6. Superposition theorem

Statement: In a linear circuit having several independent sources, the current or voltage
of a circuit element equals the algebraic sum of the component voltages or currents
produced by the independent sources acting alone.

To reflect the effect of each sources alone, a voltage source that makes no contribution is
replaced by a short circuit. Whereas a current source that makes no contribution is
replaced by an open-circuit. The internal resistance of the source is kept as it is.

For better understanding consider below circuit with two voltage sources.
R2
R1
I1

I2
I3
R3
V1
I1
Current through resistanceR3 , I 3  I 3'  I 3"
V2
R2

Equivalent resistance, Req  R1 
I2
I3
R2R3
R 2  R3
Current through resistanceR1 , I 1' 
R3
V1
S.C.
V1
Req
 R3  '
Current through resistanceR2 , I 2'  
 I 1
 R 2  R3 
 R2  '
Current through resistanceR3 , I 3'  
 I 1
 R 2  R3 
Consider voltage source V2 only
R2
R1
1
S.C.

Current through resistanceR 2 , I 2  I 2'  I 2"
Consider voltage source V1 only
R1
I
Current through resistanceR1 , I 1  I 1'  I 1"
I
3
R3
Equivalent resistance, Req  R 2 
I
R1R3
R1  R3
2
Current through resistanceR2 , I 2' 
V2
V2
Req
 R3  '
Current through resistanceR1 , I 1'  
 I 2
 R1  R3 
 R1  '
Current through resistanceR3 , I 3'  
 I 2
 R1  R3 
Superposition theorem is applicable to linear networks i.e. time varying or time invariant
with independent sources, linear dependent sources, linear passive elements such as
resistors, inductors, capacitors and linear transformers.
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
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1. Network Theorems
1.7. Substitution theorem

Statement: In any circuit if, current through branch or voltage across that branch is
known then this branch can be replace by combination of same set of terminal voltage
and current without disturbing voltages and currents in entire circuit.

For better understanding consider below circuit with a branch x between node A and B
having impedance Zx and current Ix.

Impedance Zx can be replaced by a compensating voltage source having magnitude Vx = Ix
Zx or can be replaced by current source having magnitude Ix = Vx / Zx.
A
A
A
Ix
Vx
Zx
=
Ix
B
Vx
OR
B
B

While applying substitution theorem, branch k should not be connected to other element
i.e. neither the part of magnetically coupled circuit nor part of controlled source.

This theorem is generally used for the circuits that contains single non-liner or time
varying elements.

Connect voltage source of magnitude E = Vx at node B and keep node A and node C at same
potential.
A
A
A=C
A=C
C
Vx
Zx
B
=
E
Zx
B
E
Zx
B
E
B

As branch x, is in parallel with voltage source and hence it can be removed without
affecting the other part of circuit i.e. branch x is replaced by independent voltage source.

Similarly branch x can replaced by a current source. Let, current source of magnitude I =
Ix is connected between node A and node C such that addition of current cause the current
in short circuit branch zero.
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
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1. Network Theorems
Ix
Vx
Zx
I = Ix
I
Ix
=
B

A
A
A
Zx
Zx
B
B
As branch x, is in series with current source and hence it can be removed without affecting
the other part of circuit i.e. branch x is replaced by independent current source.
1.8. Compensation theorem

Statement: In any linear time invariant network when the resistance of R of an uncoupled
branch, carrying a current I is changed by ΔR, then currents in all the branches will
change. The change in current ΔI is obtained by assuming that an ideal voltage source VC
= I (ΔR) is connected in series with (R+ΔR) when all other sources in the network are
replaced by their internal resistances.

As it is known that voltage drop across element is replaced by ideal voltage source and
current through element is replaced by ideal current source without affecting rest of
circuit.

But, if impedance of an element is changed then redistribution of current and voltage in
entire circuit takes places.

This theorem is useful to determine current and voltage change in a circuit element when
value of its impedance is changed.

Let suppose, circuit is supplied by Thevenin’s voltage and resistance of circuit is changed
to RL+ΔR such that current changes from IL to I’L.
Rth
Rth
Rth
D IL
IL
IL
RL
Vth
Vth
RL
RL
DR
DR
VC=(IL)DR
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
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1. Network Theorems

Thus, change in branch current
DI  I L'  I L

Vth
Vth

Rth  RL  DR Rth  R L
 R  R  R  R  DR 
L
th
L

Vth  th
  Rth  R L  DR  Rth  R L  


  DR Vth

Rth  RL  DR Rth  RL 
 Vth 
DR
  


 Rth  R L  Rth  R L  DR


DR
 I L 

 Rth  R L  DR 
I L  DR

Rth  RL  DR





Vc
Rth  RL  DR
The voltage source Vc = (IL) DR is called compensation voltage source.
1.9. Thevenin's theorem

Statement: Any linear bilateral network with circuit element and active source
connected to the load can be replaced by single two terminal networks consisting of a
single voltage source (Vth) in series with impedance (Zth).

Single voltage source (Vth) is the voltage across load terminal when load ZL is removed i.e.
open circuit voltage across load terminal.

Series impedance Zth is the equivalent impedance of passive network viewed from load
terminal when ZL is removed. Passive network means effect of sources are considered
zero i.e. voltage sources are short circuited and current sources are open circuit.

Thevenin's theorem is used to find current through any branch of the circuit.

For better understanding consider below circuit with two voltage sources.
Rth
RL=R2
R1
IL
V1
Shital Patel, EE Department
R3
V2
IL
Vth
RL
IL 
Vth
Rth  R L
Electrical Circuits Analysis (3130906)
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1. Network Theorems

Determination of Vth
R1
Thevenin's voltage,Vth  IR3 V 2
Vth
 V1 
 
 R3 -V 2
R

R
3 
 1
I
R3
V1

V2
Determination of Rth
R1
Thevenin's resistance, Rth  R1 R3
Rth

R3
S.C.
R1R3
R1  R3
S.C.
1.10. Norton's theorem

Statement: Any linear bilateral network with circuit element and active source
connected to the load can be replaced by single two terminal networks consisting of a
single current source (In) in parallel with impedance (Zn).

Single current source (In) is the current through load terminal when load ZL is removed
and terminals are short circuited i.e. short circuit current across load terminal.

Parallel impedance Zn is the equivalent impedance of passive network viewed from load
terminal when ZL is removed. Passive network means effect of sources are considered
zero i.e. voltage sources are short circuited and current sources are open circuit.

Norton's theorem is used to find current through any branch of the circuit.

For better understanding consider below circuit with two voltage sources.
RL=R2
R1
IL
IL

V1
R3
Shital Patel, EE Department
V2
In
Rn
RL

 I n
 R n  RL 
I L  
Rn
Electrical Circuits Analysis (3130906)
13
1. Network Theorems

Determination of In
R1
I1
Norton's current, I n  I 1  I 2
Vx
I2
R3
V1

In
V V  V 
  x 1    x 
 R1   R3 
V V   V 
  2 1    2 
 R1   R3 
V2
Determination of Rn
R1
Norton's resistance, R n  R1 R3
Rn

R3
S.C.
R1R3
R1  R3
S.C.
1.11. Reciprocity theorem

Statement: In any linear, bilateral network, the current due to a single source of voltage
in the network is equal to the current through that branch in which the source was
originally placed when the source is again put in the branch in which the current was
originally obtained.

Limitations of reciprocity theorem are
o Applicable to the network with only one source of excitation
o Network is initially relaxed i.e. all initial condition are zero
o Network must be linear and bilateral
o Impedance matrix of a network must be symmetric matrix
o Network with dependent or controlled sources are excluded even if it is linear

For better understanding consider below circuit.
R2
R1
I1
V1
I3
R3
Shital Patel, EE Department
I2
 RR 
Equivalent resistance, Req  R1   2 3 
 R2  R3 
V
Current through resistance R1 , I1  1
Req
 R3 
Current through resistance R2 , I2  
 I1
 R2  R3 
 R2 
Current through resistance R3 , I3  
 I1
 R2  R3 
Electrical Circuits Analysis (3130906)
14
1. Network Theorems
R2
R1
I1
I3
R3
 RR 
Equivalent resistance, Req  R2   1 3 
 R1  R3 
V
Current through resistance R2 , I2  1
Req
I2
 R3 
Current through resistance R1 , I1  
 I2
 R1  R3 
 R1 
Current through resistance R3 , I3  
 I2
 R1  R3 
V1
1.12. Maximum power transfer theorem
(a)

Maximum power transfer theorem helps to determine value of load impedance that
allows maximum power to be transferred from source to load.

This theorem is generally used for telecommunication circuit i.e. small amount of power
handling capacity and aim is to transfer maximum power from source to load.

It is never used for power system i.e. large amount of power handling capacity and aim is
to achieve maximum power transfer efficiency.

For better understanding consider DC circuit and AC circuit separately.
DC circuit
Rth
Load current, I L 
IL
Vth
RL
Vth
Rth  RL
 Vth
Power transferred to load, P  I L R L  
 Rth  R L
2
For power to be maximum,
2

 R L

dP
0
dRL
 R R 2 1  R 2 R R 
 th L     L   th L  
dP

Vth2 
4


dRL
Rth  RL 


 R R 2 1  R 2 R R 
 th L     L   th L  
0 Vth 
4


Rth  RL 


2
0   Rth  R L  1   R L  2 Rth  R L 
2
0  Rth2  2Rth R L  R L2  2Rth R L  2R L2
0  Rth2  R L2
RL2  Rth2
RL  Rth
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
15
1. Network Theorems
 This shows that, in DC circuit maximum power can be transferred when load resistance
is equal to the internal resistance of network.
Maximum power, Pmax
 Vth
 I L RL  
 Rth  RL
2
 RL
Voltage across load, V L  
 Rth  RL
(b)
2

 Vth
 RL  
 RL  RL


 RL
Vth  
 RL  RL

2

Vth2
R

 L
4R L


Vth
Vth 
2

AC circuit with variable resistive load
Rth
Xth
IL
Vth
RL
Load current, I L 
Vth

Z th  RL
Vth
R
th
 R L   X th2
2

Vth2
Power transferred to load, P  I L R L  
2

R

R
 X th2


L
 th
2
For power to be maximum,
dP
0
dRL

R
 L



RL
dP
2


V
dRL th   R  R 2  X 2 
 th
L
th 





2
2
  Rth  R L   X th 1   R L  2Rth  R L  

0 Vth2 
2
2


Rth  RL   X th2






0   Rth  R L   X th2 1   R L  2 Rth  R L 
2
0  Rth2  2Rth R L  R L2  X th2  2Rth R L  2R L2
0  Rth2  X th2  R L2
RL2  Rth2  X th2
RL  Rth2  X th2
RL  Z th
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
16
1. Network Theorems
(c)
AC circuit with variable resistive and inductive load
Rth
Xth
Load current, I L 
IL
Vth
Z th  Z L
Vth

R
 RL    X th  X L 
2
th
RL
2


Vth2

R
Power transferred to load, P  I L R L 
2
2  L


  Rth  RL    X th  X L  
2
Vth
XL
For power to be maximum, X L  X th
For power to be maximum,
dP
0
dRL



RL
dP

Vth2 
 R R 2  X  X 2 
dRL
 th L  
  th
L




2
2
  Rth  RL    X th  X L  1   RL  2Rth  R L  

0 Vth2 
2
2
2


Rth  RL    X th  X L 


2
2
0    Rth  RL   X th   X th   1  RL  2Rth  R L 






0  Rth2  2Rth RL  RL2  2Rth RL  2RL2
0  Rth2  RL2
RL  Rth
So, maximum power transferred to the load when
RL  jX L  Rth  jX th
Z L  Z th*
(d)
AC circuit with fixed resistive and variable inductive load
Rth
Xth
IL
Load current, I L 
Vth
Z th  Z L

Vth
R
th
 RL    X th  X L 
2
2
RL


Vth2
R
Power transferred to load, P  I L2RL  
2
2  L


  Rth  RL    X th  X L  
XL
For power to be maximum, X L  X th
Vth
So, maximum power transferred to the load when
RL  jX L  Rth  jX th
Z L  Z th*
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
17
1. Network Theorems
(e)
AC circuit with variable resistive and fixed inductive load
Rth
Xth
IL
RL
Vth
Load current, I L 
Vth

Z th  Z L
Vth
R
 RL    X th  X L 
2
th
2


Vth2
R
Power transferred to load, P  I L2R L  
2
2  L


  Rth  RL    X th  X L  
XL
For power to be maximum,

dP
0
dRL


RL
dP

Vth2 
 R R 2  X  X 2 
dRL
 th L  
  th
L




2
2
  Rth  RL    X th  X L  1   R L  2 Rth  R L  

0 Vth2 
2
2
2


Rth  RL    X th  X L 




0   Rth  RL    X th  X L 
2
2

 1    R  2 R
L
th
 RL 
0  Rth2  2Rth RL  RL2   X th  X L   2Rth RL  2RL2
2
0  Rth2   X th  X L   RL2
2
RL  Rth2   X th  X L 
2
RL   Rth  jX th   jX L
RL  Z th  jX L
1.13. Millman's theorem

Statement: Number of voltage sources with their internal resistance are connected in
parallel can be replaced by single equivalent voltage source with equivalent internal
resistance connected in series.

It is applicable to the circuit that contains only parallel branches with only one resistance
and source in a branch.

It is easier to apply theorem to a circuit if all the branches contains same type of source
either voltage or current.

It is not applicable to the complex mesh of parallel/series network or to the circuit where
resistance elements are connected between the sources.
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
18
1. Network Theorems
I1

Z1
I2
I3
I
In
Z1
Z2
Z3
Zn
Z
E1
E2
E3
En
E
Using source transformation technique each branch voltage source and its internal
resistance is replaced with equivalent current source in parallel with internal resistance.
I1
Z2
Z3
I2
Zn
I3
Total current, I  I 1  I 2  I 3 

Z
In
I
In
E1 E2 E3
  
Z1 Z2 Z3

En
Zn
 E 1Y1  E 2Y2  E 3Y3 
 E nYn
n
  E iYi
i 1
Total impedance,
1
Z

1
Z1

1
Z2

1
Z3
Y  Y1 Y2 Y3 


1
Zn
 Yn
n
 Yi
i 1
Equivalent source, E=IZ
1
=I
Y
n

E iYi

i
1
n
Yi

i
1
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
19
1. Network Theorems
1.14. Duality of a network

Sometimes statement of Kirchhoff’s current law for one network is almost similar to the
statement of Kirchhoff’s voltage law of another network i.e. voltage replaced with current
or mesh analysis replaced with nodal analysis.
R1
v2(t)
L1
v1(t)
v 1 (t )  L1
G2
i2(t)
C1
i1(t)
di 1
1
 R1i 1   i 1dt
dt
C1
i 2(t )  C 2
L2
C2
dv 2
1
 G 2v 2  v 2dt
dt
L2

These two equation are identical mathematical operations, only the part of voltage and
current is interchanged. Solution of first equation is the solution of second equation. The
similarity between two networks is termed as duality.

Two networks are said to be dual if node equation of one have the same mathematical
form as mesh equations of other. The voltage and current variables are not same.
Duality of network elements
R and G
 vdt and idt
L and C
L
i and v
di
dv
and C
dt
dt
1
1
vdt and  idt

L
C
q and ψ
Steps to draw dual network

Place a node in each individual mesh and one reference node outside the network i.e. 1,
2, 3 node number in each mesh and 0 node number outside.
C2
R1
v(t)
C2
L
C1
3
R1
R2
v(t)
1
C1
L
2
R2
0
Shital Patel, EE Department
Electrical Circuits Analysis (3130906)
20
1. Network Theorems

Join two nodes through each elements at a time. Stay with the same procedure until all
possible number of path through each element is considered. Replace each element by its
dual element between two connected nodes.
L2
C2
G1
3
R1
1
G1
L
L1
v(t)
L2
C
1
2
C1
R2
i(t)
0
i(t)
Shital Patel, EE Department
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L1
C
2
G2
G2
0
Electrical Circuits Analysis (3130906)
21