## Electrical Circuit Theory | D.C. circuit theory

Kirchhoff’s laws :

(a) Current Law.

At any junction in an electric circuit, the total current flowing towards that junction is equal to the total current flowing away from the junction,

i.e. SI = 0 Thus, referring to Figure 1:

fig.1

(b) Voltage Law.

In any closed loop in a network, the algebraic sum of the voltage drops (i.e. products of current and resistance) taken around the loop is equal to the resultant e.m.f. acting in that loop. Thus, referring to Figure 2:

fig.2

(Note that if current flows away from the positive terminal of a source, that source is considered by convention to be positive. Thus moving anticlockwise around the loop of Figure 2, E1 is positive and E2 is negative.)

The superposition theorem

‘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.’

Thevenin’s theorem

‘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.’ 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 3.

fig.3

Norton’s theorem

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 opencircuit), then determine the resistance r,‘looking-in’ at a break made between A and B

(iv) determine the current I flowing in resistance R from the Norton equivalent network shown in Figure 4,

i.e.

fig.4

Maximum power transfer theorem

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.