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Reciprocity Theorem

 

Reciprocity Theorem (Voltage and Current) formulas

  • This formula states that if a voltage source  at one point in a network produces a current  at another point, then a voltage source  at the second point will produce a current  at the first point, where the ratio of voltage to current is equal in both cases.
\(  \dfrac{ V_1 }{ I_2 }  \;=\;  \dfrac{ V_2 }{ I_1 }  \)
Symbol English Metric
\( V_1 \) = Voltage Source 1 \(V\) \(V\)
\( I_2 \) = Current Produced at One Point \(A\) \(A\)
\( V_2 \) = Voltage Source 2 \(V\) \(V\)
\( I_1 \) = Current Produced at One Point \(A\) \(A\)

Reciprocity theorem is a principle in electrical engineering and circuit theory that applies to linear, bilateral, and time-invariant networks.  It states that in such a network, the ratio of a response (such as voltage or current) at one point to an excitation (such as a voltage or current source) at another point remains the same if the positions of the excitation and response are interchanged, provided the network contains only linear and bilateral components. 

In simpler terms, if a voltage source at one location produces a specific current at another location, then placing the same voltage source at the second location will produce the same current at the first location.  This theorem is rooted in the symmetry of Maxwell’s equations and is widely used in analyzing electrical circuits, antennas, and other systems to simplify calculations and understand mutual interactions. 

Reciprocity Theorem (Impedance) formulas

  • In a two-port network, the transfer impedance from port 2 to port 1 () is equal to the transfer impedance from port 1 to port 2 ().  This means the effect of a current at one port on the voltage at the other is symmetrical.
\(  Z_{12}  \;=\;  Z_{21}  \)
Symbol English Metric
\( Z_{12} \) = Impedance from Port 1 \(V\) \(V\)
\( Z_{21}  \) = Impedance from Port 2 \(A\) \(A\)

The reciprocity theorem in electrical circuit theory doesn't have a single, specific formula but is expressed as a principle about the interchangeability of excitation and response in a linear, bilateral, time-invariant network. 

For the Reciprocity Theorem to be Valid, the Network Must be:
Linear  -  The components (like resistors, inductors, and capacitors) have a linear relationship between voltage and current.
Bilateral  -  The components allow current to flow equally well in both direction.
Passive  -  The network should not contain any independent sources (e.g., batteries or amplifiers) other than the one being used for the test.

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