Introduction to Electric Circuits

10.3 Ana/ogues 239

Solution

The magnetic field analogues of D and E are, respectively, B and H so that the
analogous energy equation is W - BH//2 joules per cubic metre.

Electric and magnetic circuits

It is often convenient to use the analogies between electric and magnetic
circuits when analysing the latter. For example, the series-parallel magnetic
circuit shown in Fig. 10.6(a) is the analogue of the series-parallel electric circuit
shown in Fig. 10.6(b).

Figure 10.6

R2

12 X I1

R

E

(a) (b)

R1

The resistance of an electrical conductor (copper, for example) is given by
R - l/o'A where l is the length of the conductor, A is its cross-sectional area,
and cr is the conductivity of the copper. The reluctance of a magnetic
'conductor' (iron, for example) is given by S = l/~A where l is the length of the
iron path, A is its cross-sectional area and/x is the permeability of the iron.
Resistance is a measure of how difficult it is for current to flow in an electric
circuit and reluctance is a measure of how difficult it is for magnetic flux to
'flow' in a magnetic circuit. Resistance and reluctance are analogues of one
another.
In the circuits of Fig. 10.6, the resistance R~ of the electric circuit is analogous
to the reluctance S~ of the right-hand limb of the magnetic circuit. Similarly,
resistors R2 and R are analogous to $2 and S (respectively, the reluctances of the
left-hand limb and the centre limb of the magnetic circuit). The electromotive
force (emf) E in the central branch of the electric circuit is analogous to the
magnetomotive force (mmf) F (-NI) in the centre limb of the magnetic circuit.
Finally, the currents I~, 12 and I are the electric circuit analogues of the fluxes (I)1,
q~2 and q~, respectively, in the magnetic circuit.
Applying KVL to the left-hand mesh gives

E = IR + I2R2 (10.8)

Applying KVL to the right-hand mesh gives

E = IR + I1R1 (10.9)