MODULE 1
Common Terms used in Circuit Theory
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A circuit is a closed conducting path through which an electrical current either flows or
is intended to flow. A circuit consists of active and passive elements.
Parameters are the various elements of an electrical circuit (for example, resistance,
capacitance, and inductance).
Linear circuit – a circuit in which the parameters are constant with time, do not change
with voltage or current, and obey Ohm’s law. In a non-linear circuit the parameters
change with voltage and current.
A passive network is a one which contains no source of EMF.
An active network is a one which contains one or more sources of EMF.
A bilateral circuit is one whose properties or characteristics are same in either direction
of current. Example: the usual transmission line is bilateral.
A unilateral circuit is that circuit in which properties or characteristics change with the
direction of operation. Example: a diode rectifier can rectify only in one direction.
A Node is a point in a circuit where two or more circuit elements are connected
together.
Branch is a part of a network which lies between two nodes.
Loop is a closed path in a circuit in which no element or node is encountered more than
once.
Mesh is a loop that contains no other loop within it.
Kirchoff's Law
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Kirchoff's Voltage Law (KVL) states that the algebraic sum of the voltages across
any set of branches in a closed loop is zero.
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Kirchoff's Current Law(KCL) states that the sum of the currents entering any node
(i.e., any junction of wires) equals the sum of the currents leaving that node.
Mesh Analysis
Where:
[ V ] gives the total battery voltage for loop 1 and then loop 2.
[ I ] states the names of the loop currents which we are trying to find.
[ R ] is called the resistance matrix.
1. Label all the internal loops with circulating currents. (I1, I2, …IL etc)
2. Write the [ L x 1 ] column matrix [ V ] giving the sum of all voltage sources in each
loop.
3. Write the [ L x L ] matrix, [ R ] for all the resistances in the circuit as follows;
R11 = the total resistance in the first loop.
Rnn = the total resistance in the Nth loop.
RJK = the resistance which directly joins loop J to Loop K.
4. Write the matrix or vector equation [V] = [R] x [I] where [I] is the list of currents to
be found.
Nodal Analysis
Nodal Voltage Analysis uses the “Nodal” equations of Kirchoff’s first law to find the
voltage potentials around the circuit.
The basic procedure for solving Nodal Analysis equations is as follows:
1. Write down the current vectors, assuming currents into a node are positive. ie, a (N
x 1) matrices for “N” independent nodes.
2. Write the admittance matrix [Y] of the network where:
Y11 = the total admittance of the first node.
Y22 = the total admittance of the second node.
RJK = the total admittance joining node J to node K.
3. For a network with “N” independent nodes, [Y] will be an (N x N) matrix and that
Ynn will be positive and Yjk will be negative or zero value.
4. The voltage vector will be (N x L) and will list the “N” voltages to be found.
Superposition theorem
Superposition theorem: It states that “if a network of linear impedances contains
more than one generator, the current which flows at any point is the vector sum of
all currents which would flow at that point if each generator was considered
separately and all other generators are replaced at that time by impedance equal
to their internal impedances”
Let the currents due to V1 alone be I1’ and I2’ and currents due to V2 alone be I1’’
and I2’’ and the currents due to V1 and V2 acting together be I1 and I2.
I1 = I1 ’ + I1 ”
I2 = I2 ’ + I2 ”
Thevenin’s Theorem
Thevenin’s Theorem states that “Any linear circuit containing several voltages and
resistances can be replaced by just one single voltage in series with a single resistance
connected across the load“.
RL is the load resistor
Rs is the source resistance value looking back into the circuit and Vs is the
open circuit voltage at the terminals.
Vs is one single equivalent voltage
The basic procedure for solving a circuit using Thevenin’s Theorem is as follows:
1. Remove the load resistor RL or component concerned.
2. Find RS by shorting all voltage sources or by open circuiting all the current sources.
3. Find VS by the usual circuit analysis methods.
4. Find the current flowing through the load resistor RL.
Nortons Theorem
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Nortons Theorem states that “Any linear circuit containing several energy sources
and resistances can be replaced by a single Constant Current generator in parallel
with a Single Resistor“.
RL is the load resistance, concerned this single resistance,
RS is the value of the resistance looking back into the network with all the current
sources open circuited
IS is the short circuit current at the output terminals as shown below.
The basic procedure for solving a circuit using Nortons Theorem is as follows:
1. Remove the load resistor RL or component concerned.
2. Find RS by shorting all voltage sources or by open circuiting all the current sources.
3. Find IS by placing a shorting link on the output terminals A and B.
4. Find the current flowing through the load resistor RL.
Maximum Power Transfer Theorem
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The Maximum Power Transfer Theorem states that the maximum amount of
power will be dissipated by a load resistance if it is equal to the Thevenin or
Norton resistance of the network supplying power.
For the load resistance to absorb the maximum power possible it has to be
“Matched” to the impedance of the power source and this forms the basis of
Maximum Power Transfer.
Millman’s theorem
Millman’s theorem states that if n voltage sources V1, V2, ---,Vn having internal
impedances Z1, Z2, ---, Zn respectively are connected in parallel, then these sources
may be replaced by a single voltage source Vm having internal series impedance Zm
where Vm and Zm are given by the equations
where Y1, Y2, ---, Yn are the admittances corresponding to Z1, Z2, ---, Zn
COMPENSATION THEOREM
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COMPENSATION THEOREM: The theorem may be stated as “Any impedance linear
or nonlinear, may be replaced by a voltage source of zero internal impedance and
voltage source equal to the instantaneous potential difference produced across
the replaced impedance by the current flowing through it”.
Consider a network of impedance and voltage source, together with the particular
impedance Z1, that is to be replaced, considered as the load.
By Kirchhoff’s law to fig (b),
SI1Z1+Z I = SV1.
Where the summation extends over a number of unspecified impedances in the mesh
shown in fig(b).
By Kirchhoff’s law to fig 5.ii.(a), the equation is the same as for fig (b) except the
equation for the right hand side mesh.
SZ1I1 = -IZ1 + SV1.
All the equations are identical for the two networks, and so are the currents and
voltages through out the two networks, that is networks are equivalent.