10 Duals and analogues
10.1 DUALS OF CIRCUIT ELEMENTS
We have seen that in linear circuit theory there is an intimate relationship
between voltage and current. Their relationship is expressed in terms of
impedance or admittance by the following equations:
V = IZ (10.1)
I = vr (10.2)
These equations ultimately give the same information and the operations
involved in solving for V or I are the same. Each equation is said to be the dual
of the other. The elements of the equations are similarly dual pairs so that
voltage is the dual of current and impedance is the dual of admittance. The
component parts of impedance are resistance and inductive (or capacitive)
reactance whose duals are, respectively, conductance and capacitive (or induc-
tive) susceptance. The dual of (R + jXL) is thus (G- jBc) and the dual of
(R -jXc) is (G + jBL).
Given an equation, therefore, its dual can immediately be written down by
replacing each one of its component parts by its dual. Table 10.1 shows the
duals of the circuit elements.Table 10.1
Quantity ,----, Dual
Voltage Current
Impedance Admittance
Resistance Conductance
Inductance Capacitance
Example 10.1
Obtain the dual of the expression for the energy stored in a capacitor of
capacitance C across which is maintained a voltage V.