Antimetric (electrical networks)
An antimetric electrical network is one that exhibits anti-symmetrical electrical properties. The term is often encountered in filter theory, but it applies to general electrical network analysis. Antimetric is the diametrical opposite of symmetric; it does not merely mean "asymmtetric" (i.e., "lacking symmetry").
Definition
References to symmetry and antimetry of a network usually refer to the input impedances of a two-port network when correctly terminated. A symmetric network will have the two equal 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]
- <math>\frac {Z_{i1}}{R_0} = \frac {R_0}{Z_{i2}}</math>
which is well-defined because R0 ≠ 0 and Zi2 ≠ 0. Hence,
- <math>Z_{i1} Z_{i2} = {R_0}^2.</math>
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]
Other network parameters may also be referred to as antimetric. For instance, for a two-port network described by scattering parameters (S-parameters),
- <math>S_{11} = S_{22}</math>
if the network is symmetric, and
- <math>S_{11} = -S_{22}</math>
if the network is antimetric.[2]
Physical and electrical antimetry
Symmetric and antimetric networks are often also topologically symmetric and antimetric, respectively. That is, the physical arrangement of their components and values are symmetric or antimetric as in the ladder example above. However, it is not a necessary condition for electrical antimetry. For example, if the example networks from the preceding section have an additional T-section added to the left-hand side, then the networks remain topologically symmetric and antimetric. However, the network resulting from the application of Bartlett's bisection theorem[3] applied to the first T-section in each network are neither physically symmetric nor antimetric but retain their electrical symmetric (in the first case) and antimetric (in the second case) properties.[4]
Mechanics
Antimetry appears in mechanics as a property of forces, motions, and oscillations. Symmetric forces produce translational motion and normal stress, and antimetric forces produce rotational motion and shear stress. Any asymmetric pair of forces can be expressed as a linear combination of a symmetric and an antimetric pair.[5]
References
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<references />, or <references group="..." />- ↑ 1.0 1.1 Matthaei, Young, Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, pp. 70–72, McGraw-Hill, 1964.
- ↑ Carlin, HJ, Civalleri, PP, Wideband circuit design, pp. 299–304, CRC Press, 1998. ISBN 0849378974.
- ↑ 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.
- ↑ Ray, SS. Structural steelwork: analysis and design, pp. 44–46, Wiley-Blackwell, 1998. ISBN 0632038578.
