Duality (electrical circuits)

In electrical engineering, electrical terms are associated into pairs called duals. A dual of a relationship is formed by interchanging voltage and current in an expression. The dual expression thus produced is of the same form, and the reason that the dual is always a valid statement can be traced to the duality of electricity and magnetism.

Here is a partial list of electrical dualities:

• voltage – current
• parallel – series (circuits)
• resistance – conductance
• voltage division – current division
• impedance – admittance
• capacitance – inductance
• reactance – susceptance
• short circuit – open circuit
• Kirchhoff's current law – Kirchhoff's voltage law. KVL and KCL
• Thévenin's theorem – Norton's theorem

History

The use of duality in circuit theory is due to Alexander Russell who published his ideas in 1904.

Examples

Constitutive relations

• Resistor and conductor (Ohm's law) $$v=iR⟺i=vG$$
• Capacitor and inductor – differential form $$i_C=C\frac{d}{dt}{v}_C⟺v_L=L\frac{d}{dt}{i}_L$$
• Capacitor and inductor – integral form $$v_C(t)=V_0+\frac{1}{C}\underset{0}{\overset{t}{\int }}{i}_C(\tau )d\tau ⟺i_L(t)=I_0+\frac{1}{L}\underset{0}{\overset{t}{\int }}{v}_L(\tau )d\tau$$

Voltage division — current division

$$v_{R_1}=v\frac{R_1}{R_1+R_2}⟺i_{G_1}=i\frac{G_1}{G_1+G_2}$$

Impedance and admittance

• Resistor and conductor $$Z_R=R⟺Y_G=G$$ $$Z_G=\frac{1}{G}⟺Y_R=\frac{1}{R}$$
• Capacitor and inductor $$Z_C=\frac{1}{Cs}⟺Y_L=\frac{1}{Ls}$$ $$Z_L=Ls⟺Y_c=Cs$$

See also

• Duality (electricity and magnetism)
• Duality (mechanical engineering)
• Dual impedance
• Dual graph
• Mechanical–electrical analogies
• List of dualities

References

• Turner, Rufus P, Transistors Theory and Practice, Gernsback Library, Inc, New York, 1954, Chapter 6.