Duality (electrical circuits)

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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

Video Duality (electrical circuits)

History

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

Maps Duality (electrical circuits)

Examples

Constitutive relations

• Resistor and conductor (Ohm's law)

${\displaystyle v=iR\iff i=vG\,}$

• Capacitor and inductor - differential form

${\displaystyle i_{C}=C{\frac {d}{dt}}v_{C}\iff v_{L}=L{\frac {d}{dt}}i_{L}}$

• Capacitor and inductor - integral form

${\displaystyle v_{C}(t)=V_{0}+{1 \over C}\int _{0}^{t}i_{C}(\tau )\,d\tau \iff i_{L}(t)=I_{0}+{1 \over L}\int _{0}^{t}v_{L}(\tau )\,d\tau }$

Voltage division -- current division

${\displaystyle v_{R_{1}}=v{\frac {R_{1}}{R_{1}+R_{2}}}\iff i_{G_{1}}=i{\frac {G_{1}}{G_{1}+G_{2}}}}$

Impedance and admittance

• Resistor and conductor

${\displaystyle Z_{R}=R\iff Y_{G}=G}$

${\displaystyle Z_{G}={1 \over G}\iff Y_{R}={1 \over R}}$

• Capacitor and inductor

${\displaystyle Z_{C}={1 \over Cs}\iff Y_{L}={1 \over Ls}}$

${\displaystyle Z_{L}=Ls\iff Y_{c}=Cs}$

src: farhek.com

See also

• Duality (electricity and magnetism)
• Duality (mechanical engineering)
• Dual impedance
• Dual graph

References

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

Source of article : Wikipedia