Introduction to Electric Circuits

234 Duals and analogues

Solution

The energy stored in the capacitor is given by W- (CV2)/2. The dual of

capacitance is inductance and the dual of voltage is current. Replacing the

component parts of the above equation by their duals we get, for the required

dual expression, that the energy W stored in an inductor of inductance L

through which is flowing a current I is given by W - (LI2)//2.

10.2 DUAL CIRCUITS

The circuits described by Equations (10.1) and (10.2) are shown in Fig. 10.1(a)

and (b), respectively. In Fig. 10.1(a) the voltage is the source or stimulus, and

the current through the impedance is the circuit response, whereas in the circuit

of Fig. 10.1(b) the current is the source or stimulus, and the voltage across the

admittance is the circuit response. Notice that the two parts of the previous

sentence are dual statements.

Figure 10.1

vC

I z

~ L_~

)

(a) (b)

m

v

Series and parallel circuits

If we have a series circuit such as that shown in Fig. 10.2(a) for which

V- I(Z~ + Z2 § Z3 + Z4), the dual equation is obtained by replacing the

elements of the equation by their duals so that we get

I Zl Z2 Z3 Z4

H H t-[ J---

vC)

Figure 10.2

(a)

Y2~

(b)

Y3

T