Association of electrical terms into pairs based on interchanging voltage and current
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:
History
The use of duality in circuit theory is due to Alexander Russell who published his ideas in 1904.[1][2]
Examples
Constitutive relations
- Resistor and conductor (Ohm's law)
![{\displaystyle v=iR\iff i=vG\,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- Capacitor and inductor – differential form
![{\displaystyle i_{C}=C{\frac {d}{dt}}v_{C}\iff v_{L}=L{\frac {d}{dt}}i_{L}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- 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 }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Impedance and admittance
- Resistor and conductor
![{\displaystyle Z_{G}={1 \over G}\iff Y_{R}={1 \over R}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- Capacitor and inductor
![{\displaystyle Z_{L}=Ls\iff Y_{c}=Cs}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
See also
References
- ^ Belevitch, V, "Summary of the history of circuit theory", Proceedings of the IRE, vol 50, Iss 5, pp. 848–855, May 1962 doi:10.1109/JRPROC.1962.288301.
- ^ Alexander Russell, A Treatise on the Theory of Alternating Currents, volume 1, chapter XVII, Cambridge: University Press 1904 OCLC 264936988.
- Turner, Rufus P, Transistors Theory and Practice, Gernsback Library, Inc, New York, 1954, Chapter 6.