Antimetric electrical network
An antimetric electrical network is an
Definition
References to symmetry and antimetry of a network usually refer to the input impedances[note 1] of a two-port network when correctly terminated.[note 2] A symmetric network will have two equal input impedances, Zi1 and Zi2. For an antimetric network, the two impedances must be the dual of each other with respect to some nominal impedance R0. That is,[1]
or equivalently
It is necessary for antimetry that the terminating impedances are also the dual of each other, but in many practical cases the two terminating impedances are resistors and are both equal to the nominal impedance R0. Hence, they are both symmetric and antimetric at the same time.[1]
Physical and electrical antimetry
Symmetric and antimetric networks are often also
Two-port parameters
The conditions for symmetry and antimetry can be stated in terms of
if the network is symmetric, and
if the network is antimetric. Passive networks of the kind illustrated in this article are also reciprocal, which requires that
and results in a normalized z-parameter matrix of,
for symmetric networks and
for antimetric networks.[4]
For a two-port network described by scattering parameters (S-parameters),
if the network is symmetric, and
if the network is antimetric.[5] The condition for reciprocity is,
resulting in an S-parameter matrix of,
for symmetric networks and
for antimetric networks.[6]
Applications
Some circuit designs naturally output antimetric networks. For instance, a low-pass Butterworth filter implemented as a ladder network with an even number of elements will be antimetric. Similarly, a bandpass Butterworth with an even number of resonators will be antimetric, as will a Butterworth mechanical filter with an even number of mechanical resonators.[7]
Glossary notes
- ^ input impedance. The input impedance of a port is the impedance measured across that network port with nothing connected to it externally and all other ports terminated with a defined impedance.[citation needed]
- ^ "correctly terminated". This will most usually mean termination with the system nominal impedance which, in turn, is usually chosen to equal the nominal characteristic impedance of the system transmission lines. This is the impedance the circuit is expected to be connected to in operation and impedance matching is of some importance in telecommunications. In some design contexts a more theoretical impedance is considered such as image impedance.[citation needed]
References
- ^ a b Matthaei, Young, Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, pp. 70–72, McGraw-Hill, 1964.
- ^ Bartlett, AC, "An extension of a property of artificial lines", Phil. Mag., vol 4, p. 902, November 1927.
- ^ Belevitch, V, "Summary of the History of Circuit Theory", Proceedings of the IRE, vol 50, p. 850, May 1962.
- ^ Baher, H.,Synthesis of electrical networks, John Wiley & Sons, 1984, §5.7.3.
- ISBN 0-8493-7897-4.
- ISBN 0080492053.
- ISBN 0471089192.