Reciprocity (engineering)


Reciprocity in linear systems is the principle that a response Rab, measured at a location a, when the system has an excitation signal applied at a location b, is exactly equal to Rba which is the response at location b, when that same excitation is applied at a. This applies for all frequencies of the excitation signal. If Hab is the transfer function between a and b then Hab = Hba, if the system is linear.
In the special case of a modal analysis this is known as Maxwell's reciprocity theorem. In electromagnetism the concept is known as Lorentz reciprocity.
The reciprocity principle is also used in the analysis of structures. When combined with superposition, symmetry and anti-symmetry, it can be used to resolve complex load conditions.

Electrical networks

If a current,, injected into port A produces a voltage,, at port B and injected into port B produces at port A, then the network is said to be reciprocal. Equivalently, reciprocity can be defined by the dual situation; applyng voltage,, at port A producing current at port B and at port B producing current at port A. In general, passive networks are reciprocal. Any network that comprises entirely of resistances will be reciprocal. However, [passive components">ideal capacitances, inductances, and resistances will be reciprocal. However, [passive components that are non-reciprocal do exist. Any component containing ferromagnetic material is likely to be non-reciprocal. Examples of passive components deliberately designed to be non-reciprocal include circulators and isolators.
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. A non-symmetrical matrix implies a non-reciprocal network. A symmetric matrix does not imply a symmetric network. In some parametisaations of networks, the representative matrix is not symmetrical for reciprocal networks. 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.