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
Passive may refer to:
* Passive voice, a grammatical voice common in many languages, see also Pseudopassive
* Passive language, a language from which an interpreter works
* Passivity (behavior), the condition of submitting to the influence of o ...
networks. The reciprocity theorem is a feature of a more general principle of
reciprocity in electromagnetism.
Description
If a
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
,
, injected into
port
A port is a maritime facility comprising one or more wharves or loading areas, where ships load and discharge cargo and passengers. Although usually situated on a sea coast or estuary, ports can also be found far inland, such as H ...
A produces a
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
,
, 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; applying voltage,
, at port A producing current
at port B and
at port B producing current
at port A. In general,
passive
Passive may refer to:
* Passive voice, a grammatical voice common in many languages, see also Pseudopassive
* Passive language, a language from which an interpreter works
* Passivity (behavior), the condition of submitting to the influence of o ...
networks are reciprocal. Any network that consists entirely of ideal
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...
s,
inductances (including
mutual inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of th ...
s), and
resistances, that is, elements that are
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and
bilateral
Bilateral may refer to any concept including two sides, in particular:
*Bilateria, bilateral animals
*Bilateralism, the political and cultural relations between two states
*Bilateral, occurring on both sides of an organism ( Anatomical terms of l ...
, 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
circulator
A circulator is a passive, non-reciprocal three- or four-port device that only allows a microwave or radio-frequency signal to exit through the port directly after the one it entered. Optical circulators have similar behavior. Ports are where an ...
s and
isolators.
The
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
of a reciprocal network has the property that it is symmetrical about the
main diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
if expressed in terms of a
z-parameter,
y-parameter
Admittance parameters or Y-parameters (the elements of an admittance matrix or Y-matrix) are properties used in many areas of electrical engineering, such as power, electronics, and telecommunications. These parameters are used to describe the el ...
, or
s-parameter
Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals.
The parameters are useful f ...
matrix. A non-symmetrical matrix implies a non-reciprocal network. A
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
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
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
and
ABCD-parameters
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
, but they all have some other condition for reciprocity that can be calculated from the parameters. For h-parameters the condition is
and for the ABCD parameters it is
. These representations mix voltages and currents in the same
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
and therefore do not even have matching units in
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
d 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 fairly self-evidently reciprocal.
Injecting six
amp #REDIRECT Amp
{{Redirect category shell, {{R from other capitalisation{{R from ambiguous page ...
s into port 1 of this network produces 24
volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defin ...
s at port 2.
Injecting six amps 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
In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time- harmonic electric current densities (sources) and the resulting electromagnetic fields in Maxwell's equations for time-invarian ...
, 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
In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittan ...
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
In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between " nodes" (points where elements or branches connect) in an electrical circuit ...
. 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,
:
where
:
is the current injected into node ''k'' by a generator (which amounts to zero if no current source is connected to node ''k'')
:
is the voltage at node ''k'' with respect to the reference node (one could also say, it is the electric potential at node ''k'')
:
(''j'' ≠ ''k'') is the negative of the admittance directly connecting nodes ''j'' and ''k'' (if any)
:
is the sum of the admittances connected to node ''k'' (regardless of the other node the admittance is connected to).
This representation corresponds to the one obtained by
nodal analysis
In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between " nodes" (points where elements or branches connect) in an electrical circuit ...
. 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''. The matrix is therefore symmetrical. For the case where
the matrix reduces to,
:
.
From which it can be seen that,
:
and
:
But since
then,
:
which is synonymous with the condition for reciprocity. In words, the ratio of the current at one port to the voltage at another is the same ratio if the ports being driven and measured are interchanged. Thus reciprocity is proven for the case of
.
For the case of a matrix of arbitrary size, the order of the matrix can be reduced through
node elimination. After eliminating the ''s''th node, the new admittance matrix will have the form,
:
It can be seen that this new matrix is also symmetrical. Nodes can continue to be eliminated in this way until only a 2×2 symmetrical matrix remains involving the two nodes of interest. 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
Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the electrical circuit. Planar circuits are circuits that can be drawn on a plane sur ...
demonstrates reciprocity where one node is driven by a current and voltage is measured at another.
[Guillemin, pp. 149–150]
References
Bibliography
* Bakshi, U.A.; Bakshi, A.V., ''Electrical Networks'', Technical Publications, 2008 .
* Guillemin, Ernst A., ''Introductory Circuit Theory'', New York: John Wiley & Sons, 1953
* Kumar, K. S. Suresh, ''Electric Circuits and Networks'', Pearson Education India, 2008 .
* Harris, Vincent G., "Microwave ferrites and applications", ch. 14 in, Mailadil T. Sebastian, Rick Ubic, Heli Jantunen, ''Microwave Materials and Applications'', John Wiley & Sons, 2017 .
* Zhang, Kequian; Li, Dejie, ''Electromagnetic Theory for Microwaves and Optoelectronics'', Springer Science & Business Media, 2013 {{ISBN, 3662035537.
Circuit theorems
Linear electronic circuits