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NETWORK THEORY
TERMS:
ELECTRIC NETWORK: An interconnection of various electric elements,
connected in manner whatsoever, is called an electric network.
CIRCUIT: A circuit is a closed conducting path through which an electric
current either flows or intends to flow. If a network contains atleast one closed
path, it is called an electric circuit.
PARAMETERS: Various elements of an electric circuit or circuit elements are
called its parameters. E.g. resistance, inductance, capacitance. Circuit element
consists of two terminals.
ACTIVE ELEMENT: Active elements are those capable of delivering power to
some external device. An active element is defined as one that delivers an average
power greater than zero to some external device over an infinite time interval. E.g.
Ideal energy sources, transistor
PASSIVE ELEMENT: Passive elements are those, which are capable only of
receiving power. A passive element is defined as one that cannot supply an
average power greater than zero over an infinite time interval. E.g. Resistors,
capacitors, inductors (Inductors & Capacitors are capable of storing finite amount
of energy and return it later to an external element)
LINEAR ELEMENT: A circuit element is linear if relation between current and
voltage remains constant or its voltage current characteristic is at all times a
straight line through the origin. It obeys the principle of superposition. E.g.
Resistor
NONLINEAR ELEMENT: If relation between current and voltage is not
constant, a circuit element is nonlinear. E.g. Capacitor, Inductor
BILATERAL ELEMENT: In a bilateral element, the voltage-current relation is
same for current flowing in either direction. E.g. Transmission lines
UNILATERAL ELEMENT: In a unilateral element, the voltage-current relation
is not same for current flowing in both directions. E.g. Vacuum diode, diode
rectifiers etc.
LUMPED ELEMENT: If the element is concentrated at a particular point and is
electrically separable it is known as a lumped element. E.g. Resistors, capacitors,
inductors, transformers
DISTRIBUTED ELEMENT: If the elements are spatially distributed and is not
electrically separable it is known as a distributed element. E.g. Transmission line
NODE: Node is a junction in a circuit where two or more circuit elements are
connected together. Voltage of any node w.r.t a datum or ground is known as
node voltage or nodal voltage. Voltage between any pair of nodes is node-pair
voltage.
BRANCH: Branch is a part of the network, which lies between two junctions.
LOOP: Loop is a closed path in a circuit in which no element or node is
encountered more than once.
MESH: Mesh is a loop that contains no other loop within it.
VECTOR: Vector is a generalized multidimensional quantity having both
magnitude and direction.
PHASOR: Phasor is a two dimensional vector used in electrical technology
which relates to voltage and current.
SOURCES OF ELECTRICAL ENERGY
INDEPENDENT SOURCES
Voltage or current sources, which do not depend on any other quantity in the
circuit, are known as independent sources.
VOLTAGE SOURCE
Ideal voltage source is a 2-terminal element
which maintains a terminal voltage V, whatever is the value of current
through its terminals. At any instant, the value of terminal voltage is a
constant w.r.t current.
In practical voltage source the voltage
across the terminals keeps falling as the current through it increases. Practical
voltage source can be explained as an ideal voltage source in series with a
resistance. Then, the terminal voltage is given by,
v1 = v – i1 r where i1 is the current flowing and r is
internal resistance of a practical voltage
source
Practical voltage source approaches ideal voltage source when r becomes
zero.
CURRENT SOURCE
Ideal current source is a 2-terminal
element, which maintains a current I flowing through its terminals regardless
of the value of voltage. At any instant, the value of current is a constant w.r.t
terminal voltage.
In practical current source the current
through the source decreases as the voltage across it increases. Practical
current source can be explained as an ideal voltage source in series with a
resistance. Then, the terminal voltage is given by,
v1 = v – i1 r where i1 is the current flowing and r is internal resistance of a
practical voltage source
Practical voltage source approaches ideal voltage source when r becomes
zero.
DEPENDENT SOURCES OR CONTROLLED SOURCES
Voltage or current sources, which depend on any other quantity in the circuit (that
may be either current or voltage), are known as dependent sources. They do not
exist as physical entities.
Dependent sources are of 4 types:
(1) Voltage Dependent Voltage Source or Voltage Controlled Voltage Source
In this the voltage source depends upon the voltage across another branch ‘v’. ‘a’
is the constant of proportionality and it has no unit.
(2) Current Dependent Voltage Source or Current Controlled Voltage Source
In this the voltage source depends upon the current in another branch ‘i’. ‘’ is
the constant of proportionality and its unit is .
(3) Voltage Dependent Current Source or Voltage Controlled Current Source
In this the current source depends upon the voltage across another branch ‘v’. ‘g’
is the constant of proportionality and its unit is Siemens.
(4) Current Dependent Current Source or Current Controlled Current Source
In this the current source depends upon the current in another branch ‘i’. ‘’ is
the constant of proportionality and it has no unit.
SOURCE TRANSFORMATION
A real physical source is represented by,
(a) a resistance in series with an ideal voltage source
(b) a resistance parallel with an ideal current source
In fig (a), output voltage at terminal AB,
v1 = v – i1 R (1)
In fig (b) current flowing in the resistance R will be i – i1,
so that voltage at terminal AB,
v1 = i R – i1 R (2)
In order that circuits in (a) and (b) be equal, v = i R
Hence if it is required to convert a voltage source v in series with an internal resistance R
into an equivalent current source, it is done by replacing the voltage source with a current
source of value ‘v / R’, placed in parallel with a resistance R. If a current source i in
parallel with a resistance R is to be converted into a voltage source, it is achieved by
substituting a voltage source ‘i R’ in series with a resistance R.
E.g:
]
CIRCUIT ANALYSIS
Circuit analysis is done in order to simplify complicated networks and to calculate the
solutions easily. Certain laws and methods are applied in order to simplify circuits such
as Kirchoff’s laws, Mesh analysis, Nodal analysis, Superposition theorem, Thevenin’s
theorem etc.
KIRCHOFF’S CURRENT LAW (KCL)
Kirchoff’s current law states as follows, ‘In any electrical network, the algebraic sum of
currents meeting at a point or junction is zero.’ i.e.; current entering a node is equal to the
current leaving the node.
Eg:
Assume the incoming currents as positive and outgoing currents as negative, then,
(- I1 ) + I2 + I3 + (- I4 ) = 0
I2 + I3 = I4 + I1
Incoming Currents = Outgoing Currents
This law is also known as Kirchoff’s point law or Kirchoff’s first law or Kirchoff’s
junction rule (nodal rule).
KIRCHOFF’S VOLTAGE LAW (KVL)
Kirchoff’s voltage law states that, ‘The algebraic sum of the products of currents and
resistance (or impedance) in each of the conductors in any closed path in a network plus
the algebraic sum of e.m.f in the path is zero.’
I R + e.m.f = 0
CONVENTIONS
Following conventions can be used:
(a) Sign of battery e.m.f :
As we move from –‘ve terminal to +’ve terminal of the
battery there is rise in potential, hence this voltage should be given +’ve sign.
As we move from +‘ve terminal to -’ve terminal of the
battery there is fall in potential, hence this voltage should be given -’ve sign.
(b) Sign of ‘IR’ drop:
If we go through a resistor (or impedance) in the same
direction as the current, then there is a fall in potential because current flows from a
higher potential to a lower potential. Hence this voltage fall should be taken as –‘ve.
If we go through a resistor (or impedance) in the opposite
direction as the current, then there is a rise in potential because current flows from a
higher potential to a lower potential. Hence this voltage fall should be taken as +‘ve.
Note: The sign of voltage drop across a resistor (or impedance)
depends on the direction of current through that resistor (or
impedance) but is independent of the polarity of any other
source of e.m.f in the circuit under consideration.
(c) Assumed Direction of Current:
The direction of current flow may be assumed
either clockwise or anticlockwise (preferably clockwise). If the assumed direction of
current is not the actual direction, then on solving the question, this current will be found
to have a minus sign. If the answer is +’ve, then assumed direction is same as the actual
direction.
E.g. Consider the closed path ABCDA. We can write the KVL equation of the loop and
apply the above stated conventions.
Applying KVL to ABDCA,
- I1 R1 – I2 R2 – E2 + I3 R3 + E1 – I4 R4 = 0
E1 – E2 = I1 R1 + I2 R2 – I3 R3 + I4 R4
MAXWELL’S LOOP CURRENT OR MESH METHOD
In this method, we consider loop or mesh currents instead of currents in the various
elements or branch currents. Here currents in different meshes are assigned continuous
paths so that they do not split at a junction into branch currents.
Mesh analysis is applicable only for planar networks. A mesh is defined as a loop
which does not contain any other loops within it .To apply mesh analysis our first step is
to check whether the circuit is planar or not and second is to select the mesh currents.
Finally write kirchoff’s voltage law equations in terms of unknowns and solving them
leads to the final solution.
Figure shows 2 batteries E1 and E2 connected in the network. Let the loop currents
for the 3 meshes be I1, I2 and I3. Assume the direction of all the three currents as
clockwise.
Mesh1:
When Z4 is considered as a part of 1st loop, current through it is (I1 – I2)
Applying KVL,
E1 – (I1 Z1) – Z4 (I1 – I2) = 0
- (I1 Z1) - (I1 Z4) + (I2 Z4) = - E1
(Z1 + Z4) I1 – (Z4) I2 = E1 (1)
Mesh 2:
When Z4 is considered as a part of 2nd loop, current through it is (I2 – I1). When Z5 is
considered as a part of 2nd loop, current through it is (I2 – I3).
Applying KVL,
- (I2 Z2) – (I2 – I3) Z5 – (I2 – I1) Z4 = 0
- (I2 Z2) – (I2 Z5) + (I3 Z5) – (I2 Z4) + (I1 Z4) = 0
- (Z4) I1 +– (Z2 + Z4+ Z5) I2 - (Z5) I3 = 0 (2)
Mesh 3:
When Z5 is considered as a part of 3rd loop, current through it is (I3 – I2).
Applying KVL,
- (I3 – I2) Z5 – (I3 Z3) – E2 = 0
- (I3 Z5) + (I2 Z5) – (I3 Z3) = E2
- (Z5) I2 + (Z5 + Z3) I3 = - E2 (3)
Equations (1), (2) and (3) can be written in matrix form as,
(4)
[Z] [I] = [V] (5)
Equation (5) gives Ohm’s law in matrix form.
From (4),
The value of currents are given by,
MESH ANALYSIS BY INSPECTION
From equation (4),
IMPEDANCE MATRIX
(a), (e) and (j) = Represents the self impedance of mesh (1), (2) and (3) which is sum of
the impedances in mesh (1), (2) and (3) and can be denoted as Z11, Z22 and Z33
respectively.
Rest of the elements in the impedance matrix represents the mutual impedance between
meshes taken two at a time.
(b) = – [Sum of all impedances common to mesh (1) and (2)] ; Denoted as Z12
(c) = – [Sum of all impedances common to mesh (1) and (3)] ; Denoted as Z13
(d) = – [Sum of all impedances common to mesh (2) and (1)] ; Denoted as Z21
(f) = – [Sum of all impedances common to mesh (2) and (3)] ; Denoted as Z23
(g) = – [Sum of all impedances common to mesh (3) and (1)] ; Denoted as Z31
(h) = – [Sum of all impedances common to mesh (3) and (2)] ; Denoted as Z32
Also, Z12 = Z21
Z23 = Z32
Z31 = Z13
CURRENT MATRIX
I1, I2 and I3 represent the current in mesh (1), (2) and (3) respectively.
VOLTAGE MATRIX
E1 – Represent the algebraic sum of voltages in mesh (1)
E2 – Represent the algebraic sum of voltages in mesh (2)
E3 – Represent the algebraic sum of voltages in mesh (3)
The generalized form of the above matrix is given as,
Note: It would be easier if direction of loop current is taken as clockwise because only
then,
(i) all self impedances will always be –‘ve
(ii)all mutual impedances will always be +’ve
SUPER MESH ANALYSIS
Suppose any one of the branches in the network has a current source, then it is difficult to
apply mesh analysis straight forward. An easier method is by using supermesh analysis.
A supermesh is constituted by 2 adjacent loops that have a common current source.
E.g
.
Supermesh AHGFCBA
Applying KVL,
V – (I1Z1) – (I2 – I3) Z3 = 0
V – (I1Z1) – (I2Z3) + (I3Z3) = 0
- (I1Z1) – (I2Z3) + (I3Z3) = -V (1)
Mesh 3
Applying KVL,
- (I3 – I2) Z3 – (I3Z4) = 0
- (I3Z3) + (I2Z3) – (I3Z4) = 0
(I2Z3) – (Z3 + Z4)I3 = 0 (2)
I1 – I2 = I (3)
Solving equations (1), (2) and (3) we can obtain the values of loop currents
NODAL ANALYSIS
Node equation is based directly on KCL. For the application of this method, every
junction in the network where 3 or more branches meet is regarded as a node. One of
these is regarded as the reference node or datum node or zero potential node. Hence the
no. of simultaneous equations to be solved becomes (n – 1) where n is the no. of
independent nodes.
Consider the circuit which has 3 nodes. One of these i.e; node 3 has been taken as the
reference node. Current directions can be chosen arbitrarily. Current direction followed
here is such that all currents leave their respective nodes. Va represents the potential at
node 1 with reference to datum node 3. Similarly, Vb represents the potential at node 2
with reference to datum node 3.
At node (1)I1 + I4 + I2 = 0 (1)
Applying KVL in the 1st branch,Va – E1 – (I1Z1) = 0
I1 = Va – E1 (2) Z1
Applying KVL in the 4th branch,Va – (I4Z4) = 0
I4 = Va (3)Z4
Applying KVL in the 2nd branch,
Va + E3 – (I2Z2) – Vb = 0 I2 = Va + E3 – Vb (4)
Z2
Substituting (2), (3) and (4) in (1),
Va – E1 + Va + Va + E3 – Vb = 0 Z1 Z4 Z2
(1/Z1 + 1/Z2 + 1/Z4) Va – (1/Z2)Vb = E1 – E3 (5) Z1 Z2
At node (2)I5 + I2 + I3 = 0 (6)
Applying KVL in the 5th branch,Vb – (I5Z5) = 0
I5 = Vb (7)
Z5
Applying KVL in the 3rd branch,Vb – (I3Z3) – E3 = 0
I3 = Vb – E2 (8) Z3
Applying KVL in the 2nd branch,
Vb - E3 – (I2Z2) – Va = 0 I2 = Vb - E3 – Va
(9) Z2
Substituting (7), (8) and (9) in (6),
Vb – E2 + Vb + Vb - E3 – Va = 0 Z3 Z5 Z2
– (1/Z2)Va + (1/Z2 + 1/Z3 + 1/Z5) Vb = E2 + E3 (10) Z1 Z2
Solving equations (5) and (10) we can obtain the values of Va and Vb.
NODAL ANALYSIS BY INSPECTION
At node (1), the terms are,
(a) Product of node potential Va and sum of the reciprocal of branch impedances
connected to this node (1/Z1 + 1/Z2 + 1/Z4)
(b) – (Adjacent Potential / Interconnecting Impedance) i.e; Vb/Z2
(c) – (Adjacent Battery Voltage / Interconnecting Impedance) i.e; E1/Z1
(d) + Battery between 2 nodes / Impedance between 2 nodes
Note: Sign of this term depends upon the polarity of the battery
(e) All the above set to zero
Similarly, equation at node (2) can be written.
SUPERNODE ANALYSIS
If there is a voltage source alone between two nodes then it is slightly difficult to apply
nodal analysis. This difficulty is overcome using super node analysis. In this method the
two adjacent nodes that are connected by a voltage source are reduced to a single node
and then equations are formed by applying KCL as usual.
Eg:
In the above circuit,
At node 1,
I = I1 + I2
I = V1 + V1-V2
Z1 Z2
Due to the presence of voltage source Vx in between nodes 2 and 3, combined
equation for node 2 and 3 is given by,
I2 + I3 + I4 + I5 = 0
V2 – V1 + V2 + V3 + V3 – Vy = 0
Z2 Z3 Z5 4
Also, V2 – V3 = Vx
Solving (1), (2) and (3) we obtain the values of node voltages.
(1)
(2)
(3)
COUPLED CIRCUITS
Two circuits are said to be coupled when energy transfer takes place from one circuit to
another when one of the circuits are energized. There are many types of coupling like
conductive coupling, inductive or magnetic coupling .Certain coupled elements are
frequently used in network analysis and synthesis. Transformer, transistors and electronic
pts are some among these circuits.
Eg for Conductive coupling: Potential divider
Eg for Inductive coupling: 2 winding transformer
CONDUCTIVELY COUPLED AND MUTUAL IMPEDANCE
A circuit in which there no magnetic coupling is known as conductively coupled
circuit. It can be represented as a two port network.
The mutual impedance for a conductively coupled circuit can be defined as the voltage
developed at one port per unit current at the other port
SELF INDUCTANCE
Whenever current flows through a coil a magnetic field will be developed across the
coil. When we increase the current the flux will also incase and if decrease the current
correspondingly flux will also decrease. When the flux linking with the coil changes an
emf will be developed across the coil according to Faraday’s laws of electromagnetic
induction. The induced emf can be represented as V = L( di/dt).Its unit is Henry.
MUTUAL INDUCTANCE
It’s a property associated with 2 or more coils or inductors which are in close
proximity and the presence of common magnetic flux which links them.
Consider 2 coils L1 and L2 which are sufficiently close together so that the
flux produced by I1 in coil L1 also links with coil L2
When a voltage V1 is applied across L1 ,a current I1 will start flowing in the coil and
produce a flux Φ. This flux also links coil L2 ,if I1 is changed w.r.t time the flux Φ would
also change w.r.t. time. The time varying flux surrounding the second coil L2 induces an
emf or voltage across the terminals of L2.Thi voltage is proportional to the time rate of
change of current flowing through the first coil L1.
The 2 coils are said to be inductively coupled .Because of this property they are
called coupled elements or coupled circuits and the induced voltage or emf is called the
voltage of mutual induction and is given by
V2(t) = M di1(t)/dt
V2 – voltage induced in coil L2
M –Mutual inductance
If current is made to pass through coil L2 with coil L1 open, would cause voltage
V1 in the coil.
V1(t) = M di2(t)/dt
In general if a pair of linear time variant coupled coils with a non-zero current in
each of the 2 coils, produces a mutual voltage in each coil due to the flow of current in
the other coil. This mutual voltage is present independently of and in addition to the
voltage due to self induction.
Mutual inductance is also measured in Henry and s positive .But the mutually
induced emf may be either positive or negative, depending on the physical construction
of the coil and reference direction.
DOT CONVENTION IN COUPLED CIRCUITS
Dot convention is used to establish the choice of correct sign for the mutually
induced voltages in coupled circuits.
Circular dot marks or special symbols are placed at the one end of the two coils
which are mutually coupled to simplify the diagrammatic representation of the windings
around its core.
When the currents through each of the mutually coupled coils are going away
from the dot or towards the dot the mutual inductance is positive, while when the current
through the coil is leaving the dot from one coil and entering the other, the mutual
inductance is negative.
ELECTRICAL EQUIVALENCE OF MAGNETICALLY
COUPLED CIRCUITS.
From the circuit
V1(t) = L1( di1/dt)+M di2(t)/dt
V2(t) = L2( di2/dt )+M di1(t)/dt. Hence mutually induced voltages can be shown as
controlled voltage sources
.
COEFFICIENT OF COUPLING
The amount of coupling between the inductively coupled coils is expressed in
terms of the coefficient of coupling which is defined as K= M/
Co-efficient f coupling is always less than unity and has a maximum value of 1.This case
for K =1 is called perfect coupling, when the entire flux of one coil links with the other.
ANALYSIS OF MULTI WINDING COUPLED CIRCUITS
Inductively coupled multi winding circuits can be analyzed by using Kirchoff’s
laws by loop current methods.
For such a system inductance can be defined as L =
Where L11, L22, L33 -- self Inductance of coupled circuits
L12, L21, L31 -- mutual Inductance of coupled circuits
Voltage across the coils
[V] =[L] [di/dt] where V and i are the vectors of branch voltages and currents
respectively
SERIES CONNECTION OF COUPLED CIRCUITS
Two inductors of self inductances L1 and L2 and mutual inductance of M
are connected in series .There are 2 kinds of series connection.
1) Series aiding
2) Series opposing
SERIES AIDING
In the case of series aiding connection, the currents in both inductors at any
instant of time are in the same direction. Hence the magnetic fluxes of self induction and
mutual induction linking wit each element add together.
If Φ1 and Φ2 are the flux produced by coils 1 and 2 respectively then the total flux
is
Φ = Φ1 + Φ2
Φ1 = L1i1 + Mi2
Φ2 = L2i2 + Mi1
Φ = Li = L1i1 + Mi2 + L1i1 + Mi2
As series circuit i1 =i2 = i
L eq= L1 + L2 + 2M
SERIES OPPOSING
In the case of series aiding connection, the currents in both inductors at any
instant of time are in the opposite direction.
Φ = Φ1 + Φ2
Φ1 = L1i1 - Mi2
Φ2 = L2i2 - Mi1
Φ = Li = L1i1 - Mi2 + L1i1 - Mi2
As series circuit i1 =i2 = i
L = L1 + L2 - 2M
