Introduction to Electric Circuits

10.2 Dual circuits 235

I = V(Y1 + Yz + Y3 + Y4) and the circuit described by this equation is shown in

Fig. 10.2(b), which is thus the dual of that in Fig. 10.2(a).

It follows that:

9 a parallel circuit is the dual of a series circuit;

9 'impedances in series are added' and 'admittances in parallel are added' are

dual statements; and

9 'when elements are in series, voltages are added' and 'when elements are in

parallel currents are added' are dual statements.

Kirchhofrs current law and Kirchhoff's voltage law

From the circuits of Fig. 10.2 we see that the voltage V is the sum of the voltages

across the impedances Z~, Z2, Z3 and Z4 (which is KVL), while the current I is

the sum of the currents in }11, Y2, Y3 and Y4 (KCL). Thus Kirchhoff's current law

is the dual of his voltage law.

Nodal voltage and mesh current

It follows from the previous paragraph that the dual of a closed path (a loop or

mesh) is a node and that mesh current analysis and nodal voltage analysis are

dual procedures.

Thevenin's theorem and Norton's theorem

t 1

Eo()

Figure 10.3

"Z o

I I A

lZL ,so

B

Ysc YL

(a) (b)

I

A

lvL

IB

The Thevenin equivalent circuit of Fig. 10.3(a) consists of an open circuit
voltage Eo in series with an impedance Zo. The current through the load
impedance ZL connected across the output terminals A and B is then calculated
from the equation
i, = Eo/(Zo + z,) (10.3)
The dual of a voltage source is a current source and the dual of a series
impedance is a parallel admittance. The dual of the circuit of Fig. 10.3(a) is thus
that of Fig. 10.3(b), which is the Norton equivalent circuit. This circuit consists