Reciprocity in electrical networks is a property of a circuit that relates voltages and currents at two points. The reciprocity theorem states that the current at one point in a circuit due to a voltage at a second point is the same as the current at the second point due to the same voltage at the first. The reciprocity theorem is valid for almost all passive networks. The reciprocity theorem is a feature of a more general principle of reciprocity in electromagnetism.
Description If a current, I A {\displaystyle I_{\text{A}}} , injected into port A produces a voltage, V B {\displaystyle V_{\text{B}}} , at port B and I A {\displaystyle I_{\text{A}}} injected into port B produces V B {\displaystyle V_{\text{B}}} at port A, then the network is said to be reciprocal. Equivalently, reciprocity can be defined by the dual situation; applying voltage, V A {\displaystyle V_{\text{A}}} , at port A producing current I B {\displaystyle I_{\text{B}}} at port B and V A {\displaystyle V_{\text{A}}} at port B producing current I B {\displaystyle I_{\text{B}}} at port A. In general, passive networks are reciprocal. 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. 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-, y-, or s-parameter matrix. A non-symmetrical matrix implies a non-reciprocal network. A symmetric matrix does not imply a symmetric network. In some parametisations 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. For h-parameters the condition is h 12 = − h 21 {\displaystyle h_{12}=-h_{21}} and for the ABCD parameters it is A D − B C = 1 {\displaystyle AD-BC=1} . These representations mix voltages and currents in the same column vector and therefore do not even have matching units in transposed elements.
Example An example of reciprocity can be demonstrated using an asymmetrical resistive attenuator. An asymmetrical network is chosen as the example because a symmetrical network is self-evidently reciprocal.
Injecting 6 amperes into port 1 of this network produces 24 volts at port 2.
Injecting 6 amperes into port 2 produces 24 volts at port 1.
Hence, the network is reciprocal. In this example, the port that is not injecting current is left open circuit. This is because a current generator applying zero current is an 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.
Proof Reciprocity of electrical networks is a special case of Lorentz reciprocity, but it can also be proven more directly from network theorems. 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. A linear network can be represented as a set of linear equations through nodal analysis. For a network consisting of n+1 nodes (one being a reference node) where, in general, an admittance is connected between each pair of nodes and where a current is injected in each node (provided by an ideal current source connected between the node and the reference node), these equations can be expressed in the form of an admittance matrix,
… excerpt ends here. Continue reading the full article.
