Electrical Circuit Analysis (3130906) Darshan Unit-1,5

38 Pages·2020·2.237 MB·English

by  Darshan Institute of Engineering & Technology

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
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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
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sources. A voltage source has a driving role of emf whereas the current source has a
driving job of current.
ecruoS eg ecruoS eg ecruoS tn ecruoS tn
atloV atloV erruC erruC
Figure 1. 1 Ideal voltage Figure 1. 2 Practical voltage Figure 1. 3 Ideal current Figure 1. 4 Practical current
source source source 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) I i(t)
-

Figure 1. 5 Independent voltage source 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.
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1. Network Theorems

(r) Dependent energy sources
+ + + Icd +
a c a c
Vab μ Vab+- Vcd Vab gmVab Vcd
b- -d b- -d
Figure 1. 7 Voltage controlled voltage source (VCVS) Figure 1. 8 Voltage controlled current source (VCCS)

i i
+ i + + ab cd +
a ab c a c
r iab +- Vcd β iab
b- -d b- -d

Figure 1. 9 Current controlled voltage source (CCVS) 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 V depends upon the
cd
controlled voltage V and constant μ.
ab
Voltage controlled current source (VCCS) is the four terminal network components
that establishes a current i in the branch of circuit. Value of i depends on the
cd cd
controlled voltage V and constant g .
ab m
Current controlled voltage source (CCVS) is the four terminal network components
that establishes a voltage V between two-point c and d. Value of V depends upon the
cd cd
controlled current i and constant r.
ab
Current controlled current source (CCCS) is the four terminal network components
that establishes a current i in the branch of circuit. Value of i depends upon the
cd cd
controlled current i and constant β.
ab
(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.

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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,
dq
vRiR
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
tv(t)
 v(t)dt
R
0
1T
 V2dt
R
0
V2
 T
R
2
If v(0)0,v(t)V sin(t) and energy dissipated for time period T
m 
v(t) V sin(t)
i(t)  m
R R
t
e(t)p(t)dt
0
t
i(t)v(t)dt
0
tV sin(t)
 m V sin(t) dt
R m
0
V2 t
 msin2(t)dt
R
0
V2T(1cos(2t))
 m dt
R 2
0
V2
 mT
2R
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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,
d
v
dt
 When there is an initial charge of ψ is stored on inductor and it is increasing linearly with
o
time, then charge on inductor at any instant of time is,
 kt
o
d
 k
dt
 Hence, it can be observed that voltage in the inductive system is independent of initial
charge.
t
 vdt

0 t
 vdtvdt
 0
t
 vdt
o
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
d d(Li) di(t)
v(t)  L
dt dt dt
1 t
i(t) v(t)dt
L

t
e(t)p(t)dt
0
t
i(t)v(t)dt
0
t di(t)
i(t)L dt
dt
0
I
Lidi
0
1.4. 1LI2
2
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2
If i(0) 0,i(t) I sin(t) and energy dissipated for time period T
m 
 
di(t) d I sin(t)
v(t)L L m LI cos(t)
dt dt m
t
e(t)p(t)dt
0
t
i(t)v(t)dt
0
t
I sin(t)LI cos(t) dt
m m
0
LI2t
 m 2cos(t)sin(t)dt
2
0
LI2T
 m sin(2t)dt
2
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,
dq
i
dt
 When there is an initial charge of q is stored on capacitor and it is increasing linearly
o
with time, then charge on capacitor at any instant of time is,
qq kt
o
dq
 k
dt
 Hence, it can be observed that current in the capacitive system is independent of initial
charge.
t
q idt

0 t
 idtidt
 0
t
q idt
o
0
 Given voltage v (t) across and current i (t) through a capacitor C and then associated
energy e (t) is,
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If v(0)0 and v(t)V
 
dq(t) dCv(t) dv(t)
i(t)  C
dt dt dt
1 t
v(t) i(t)dt
C

t
e(t)p(t)dt
0
t
i(t)v(t)dt
0
t dv(t)
C v(t)dt
dt
0
V
Cvdv
0
1
 Cv2
2
2
If v(0)0,v(t)V sin(t) and energy dissipated for time period T
m 
 
dv(t) dV sin(t)
i(t)C C m CVcos(t)
dt dt m
t
e(t)p(t)dt
0
t
i(t)v(t)dt
0
t
CVcos(t)V sin(t) dt
m m
0
CV2t
 m 2cos(t)sin(t)dt
2
0
CV2T
 m sin(2t)dt
2
0
0

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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.
R1 R2 Current through resistanceR,I I' I"
1 1 1 1
I1 I I2 Current through resistanceR,I I' I"
3 2 2 2 2
Current through resistanceR,I I' I"
V R3 V 3 3 3 3
1 2

 Consider voltage source V only
1
R R RR
1 2
Equivalent resistance,R R  2 3
eq 1 R R
I I 2 3
1 I 2
3 V
Current through resistanceR,I'  1
1 1 R
eq

V1 R3 S.C.  R 
Current through resistanceR,I'  3 I'
2 2 R R  1
 
2 3
 R 
Current through resistanceR,I'  2 I'
3 3 R R  1
 
2 3
 Consider voltage source V only
2
R R RR
1 2
Equivalent resistance,R R  1 3
eq 2 R R
1 3
I I
1 I 2
3 V
Current through resistanceR,I'  2
2 2 R
eq

S.C. R3 V  R 
2 Current through resistanceR,I'  3 I'
1 1 R R  2
 
1 3
 R 
Current through resistanceR,I'  1 I'
3 3 R R  2
1 3
 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.
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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 Z and current I .
x x
 Impedance Z can be replaced by a compensating voltage source having magnitude V = I
x x x
Z or can be replaced by current source having magnitude I = V / Z .
x x x x
A A A
I
x
Vx Zx = Ix OR Vx
B 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 = V at node B and keep node A and node C at same
x
potential.
A A A=C A=C
C
Vx Zx = E Zx E Zx E
B B B 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 =
I is connected between node A and node C such that addition of current cause the current
x
in short circuit branch zero.
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