Matching law

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 — serial (circuits)
• resistance — conductance
• impedance — admittance
• capacitance — inductance
• reactance — susceptance
• short circuit — open circuit
• Kirchhoff's current law — Kirchhoff's voltage law.
• 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

References

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