Network Theorems

Superposition Theorem

The superposition theorem states:

‘In any network made up of linear resistances and containing more than one source of e.m.f., the resultant current flowing in any branch is the algebraic sum of the currents that would flow in that branch if each source was considered separately, all other sources being replaced at that time by their respective internal resistances.’
Or
For a circuit consisting of linear elements and sources, the response (voltage or current) in any element in the circuit is the algebraic sum of the responses in this element obtained by applying one independent source at a time. When one independent source is applied, all other independent sources are deactivated.

It may be noted that a deactivated voltage source behaves as a short circuit, whereas a deactivated current source behaves as an open circuit. 

Thevenin’s theorem

Thevenin’s theorem states:

‘The current in any branch of a network is that which would result if an e.m.f. equal to the p.d. across a break made in the branch, were introduced into the branch, all other e.m.f.’s being removed and represented by the internal resistances of the sources.’
or
A network consisting of linear resistors and dependent and independent sources with a pair of accessible terminals can be represented by an equivalent circuit with a voltage source and a series resistance as shown in Figure 1. 

VTH is equal to the open circuit voltage across the two terminals A and B, and RTH is the resistance measured across nodes A and B.
 The RTH can also be determined as RTH = Voc / Isc , where Voc is the open circuit voltage across terminals A and B and where Isc is the short circuit current that will flow from A to B through an external zero resistance connection (short circuit) if one is made.

The procedure adopted when using Thevenin’s theorem is summarized below. To determine the current in any branch of an active network (i.e. one containing a source of e.m.f.):

(i) remove the resistance R from that branch,

(ii) determine the open-circuit voltage, E, across the break,

(iii) remove each source of e.m.f. and replace them by their internal resistances and then determine the resistance, r, ‘looking-in’ at the break,

(iv) determine the value of the current from the equivalent circuit shown in Figure 1,

i.e. I = ­­__E___

             R + r

Norton’s theorem 

Norton’s theorem states: 

‘The current that flows in any branch of a network is the same as that which would flow in the branch if it were connected across a source of electrical energy, the short-circuit current of which is equal to the current that would flow in a short-circuit across the branch, and the internal resistance of which is equal to the resistance which appears across the open-circuited branch terminals.’

The procedure adopted when using Norton's theorem is summarized below.

To determine the current flowing in a resistance R of a branch AB of an active network:

(i) short-circuit branch AB

(ii) determine the short-circuit current Isc flowing in the branch

(iii) remove all sources of e.m.f. and replace them by their internal resistance (or, if a current source exists, replace with an open-circuit), then determine the resistance r/looking-in' at a break made between A and B

(iv) determine the current 1 flowing in resistance R from the Norton equivalent network shown in Figure, i.e.

Maximum power transfer theorem

The maximum power transfer theorem states:

 ‘The power transferred from a supply source to a load is at its maximum when the resistance of the load is equal to the internal resistance of the source.’

Hence, in Figure , when R = r the power transferred from the source to the load is a maximum.