Reciprocity (electrical networks)

Reciprocity in electrical networks is a property of a circuit that relates voltages and currents at two points.

The reciprocity theorem is valid for almost all passive networks.

Equivalently, reciprocity can be defined by the dual situation; applying voltage,

Any network that consists entirely of ideal capacitances, inductances (including mutual inductances), and resistances, that is, elements that are linear and bilateral, will be reciprocal.

Examples of passive components deliberately designed to be non-reciprocal include circulators and isolators.

[3] The transfer function of a reciprocal network has the property that it is symmetrical about the main diagonal if expressed in terms of a z-parameter, y-parameter, or s-parameter matrix.

Common examples are h-parameters and ABCD-parameters, but they all have some other condition for reciprocity that can be calculated from the parameters.

These representations mix voltages and currents in the same column vector and therefore do not even have matching units in transposed elements.

[5] An example of reciprocity can be demonstrated using an asymmetrical resistive attenuator.

In this example, the port that is not injecting current is left open circuit.

If, on the other hand, one wished to apply voltages and measure the resulting current, then the port to which the voltage is not applied would be made short circuit.

This is because a voltage generator applying zero volts is a short circuit.

This proof shows reciprocity for a two-node network in terms of its admittance matrix, and then shows reciprocity for a network with an arbitrary number of nodes by an induction argument.

If we further require that network is made up of passive, bilateral elements, then since the admittance connected between nodes j and k is the same element as the admittance connected between nodes k and j.

Since this matrix is symmetrical it is proved that reciprocity applies to a matrix of arbitrary size when one node is driven by a voltage and current measured at another.

A similar process using the impedance matrix from mesh analysis demonstrates reciprocity where one node is driven by a current and voltage is measured at another.