Electric Circuits and Network Theorems

There are certain theorems, which when applied to the solutions of electric networks, wither
simplify the network itself or render their analytical solution very easy. These theorems can also be
applied to an a.c. system, with the only difference that impedances replace the ohmic resistance of
d.c. system. Different electric circuits (according to their properties) are difined below:
I. Circuit. A circuit is a closed conducting path through which an electric current either
flows or is inteneded flow.
2. Parameters. The various elements of an electric circuit are called its parameters like resistance,
inductance and capacitance. These parameters may be lumped or distributed.
3. Liner Circuit. A linear circuit is one whose parameters are constant i.e. they do not change
with voltage or current.
4. Non-linear Circuit. It is that circuit whose parameters change with voltage or current.
5. Bilateral Circuit. A bilateral circuit is one whose properties or characteristics are the same
in either direction. The usual transmission line is bilateral, because it can be made to perform
its function equally well in either direction.
6. Unilateral Circuit. It is that circuit whose properties or characteristics change with the
direction of its operation. A diode rectifier is a unilateral circuit, because it cannot perform
rectification in both directions.
7. Electric Network. A combination of various electric elements, connected in any manner
whatsoever, is called an electric network.
8. Passive Network is one which contains no source of e.m.f. in it.
9. Active Network is one which contains one or more than one source of e.m.f.
10. Node is a junction in a circuit where two or more circuit elements are connected together.
II. Branch is that part of a network which lies between two juncions.
12 Loop. It is a close path in a circuit in which no element or node is epcountered more than
once.
13. Mesh. It is a loop that contains no other loop within it. For example, the circuit of Fig. 2.1
(a) has even branches, six nodes, three loops and two meshes whereas the circuit of Fig.
2.1 (b) has four branches, two nodes, six loops and three meshes.
It should be noted that, unless stated otherwise, an electric network would be assumed passive
in the following treatment.
We will now discuss the various network theorems which are of great help in solving complicated
networks. Incidentally, a network is said to be completely solved or analyzed when all volt-.
ages and all currents in its different elements are determined.
There are two general approaches to network analysis :
(i) Direct Method
Here, the network is left in its original form while determining its different voltages and currents.
Such methods are usually restricted to fairly simple circuits and include Kirchhoff slaws,
Loop analysis, Nodal analysis, superposition theorem, Compensation theorem and Reciprocity theorem
etc.
(ii) Network Reduction Method
Here, the original network is converted into a much simpler equivalent circuit for rapid calculation
of different quantities. This method can be applied to simple as well as complicated networks.
Examples of this method are: Delta/Star and StarlDelta conversions. Thevenin's theorem and
Norton's Theorem etc.
2.2. KirchhoWs Laws *
These laws are more comprehensive than Ohm's law and are used for solving electrical networks
which may not be redily solved by the latter. Kirchhoffs laws, two in number, are particularly
useful (a) in determining the equivalent resistance of a complicated network of conductors and
(b) for calculating the currents flowing in the various conductors. The two-laws are :
1. KirchhoWs Point Law or Current Law (KCL)
It states as follows:
in any electrical network, the algebraic sum of the currents meeting at a point (or junction) is
zero.
Put in another way, it simply means that the total current leaving a juncion is equal to the total
current entering that junction. It is obviously true because there is no accumulation of charge at the
junction of the network. '
Consider the case of a few conductors meeting at a point A as in Fig. 2.2 (a). Some conductors
have currents leading to point A, whereas some have currents leading away from point A. Assuming
the incoming currents to be positive and the outgoing currents negative, we have