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An '''antimetric electrical network''' is an [[electrical network]] that exhibits anti-[[symmetry|symmetrical]] electrical properties. The term is often encountered in [[filter theory]], but it applies to general electrical [[network analysis (electrical circuits)|network analysis]]. Antimetric is the diametrical opposite of symmetric; it does not merely mean "asymmetric" (i.e., "lacking symmetry").  It is possible for networks to be symmetric or antimetric in their electrical properties without being physically or [[topology (electronics)|topologically]] symmetric or antimetric.


==Definition==
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[[File:Antimetric physical example.svg|thumb|300px|'''Figure 1.'''  Examples of symmetry and antimetry: both networks are [[low-pass filter]]s but one is symmetric (left) and the other is antimetric (right). For a symmetric ladder the 1st element is equal to the ''n''th, the 2nd equal to the (''n''-1)th and so on.  For an antimetric ladder, the 1st element is the dual of the ''n''th and so on.]]
References to symmetry and antimetry of a network usually refer to the input impedances<ref group=note>'''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.</ref> of a [[two-port network]] when correctly terminated.<ref group=note>'''"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 line]]s.  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]].</ref> A symmetric network will have the two equal input impedances, ''Z''<sub>i1</sub> and ''Z''<sub>i2</sub>.  For an antimetric network, the two impedances must be the [[dual impedance|dual]] of each other with respect to some nominal impedance ''R''<sub>0</sub>.  That is,<ref name=Matt>Matthaei, Young, Jones, ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'', pp.&nbsp;70&ndash;72, McGraw-Hill, 1964.</ref>
 
:<math>\frac {Z_{i1}}{R_0} = \frac {R_0}{Z_{i2}}</math>
 
which is well-defined because ''R''<sub>0</sub> &ne; 0 and ''Z''<sub>''i''2</sub> &ne; 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 ''R''<sub>0</sub>.  Hence, they are both symmetric and antimetric at the same time.<ref name=Matt/>
 
==Physical and electrical antimetry==
[[File:Antimetric intermediate example.svg|thumb|300px|'''Figure 2.'''  Adding an additional T-section to the ladders of figure 1]]
[[File:Antimetric electrical example.svg|thumb|300px|'''Figure 3.'''  Examples of symmetric (top) and antimetric (bottom) networks which do not exhibit topological symmetry nor antimetry.]]
Symmetric and antimetric networks are often also [[Topology (electronics)|topologically]] symmetric and antimetric, respectively. 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 of figure 1 have an additional identical [[Topology (electronics)#Simple filter topologies|T-section]] added to the left-hand side as shown in figure 2, then the networks remain topologically symmetric and antimetric. However, the network resulting from the application of [[Bartlett's bisection theorem]]<ref>Bartlett, AC, "An extension of a property of artificial lines", ''Phil.&nbsp;Mag.'', '''vol 4''', p.&nbsp;902, November 1927.</ref> applied to the first T-section in each network, as shown in figure 3, are neither physically symmetric nor antimetric but retain their electrical symmetric (in the first case) and antimetric (in the second case) properties.<ref>[[Vitold Belevitch|Belevitch, V]], "Summary of the History of Circuit Theory", ''Proceedings of the IRE'', '''vol 50''', p.&nbsp;850, May 1962.</ref>
 
==Two-port parameters==
The conditions for symmetry and antimetry can be stated in terms of [[two-port parameters]]. For a two-port network described by [[impedance parameters]] (''z''-parameters),
 
:<math>z_{11} = z_{22}</math>
 
if the network is symmetric, and
 
:<math>z_{11} = -z_{22}</math>
 
if the network is antimetric.  Passive networks of the kind illustrated in this article are also [[Reciprocity (electromagnetism)#Reciprocity for electrical networks|reciprocal]], which requires that
 
:<math>z_{12} = z_{21}</math>
 
and results in a ''z''-parameter matrix of,
 
:<math>\left [ \mathbf z \right ] = \begin{bmatrix} z_{11} & z_{12} \\ z_{12} & z_{11} \end{bmatrix} </math>
 
for symmetric networks and
 
:<math>\left [ \mathbf z \right ] = \begin{bmatrix} z_{11} & z_{12} \\ z_{12} & -z_{11} \end{bmatrix} </math>
 
for antimetric networks.<ref>Dietmar Findeisen, ''System Dynamics and Mechanical Vibrations'', p. 32, Springer, 2000 ISBN 3540671447.</ref>
 
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.<ref>Carlin, HJ, Civalleri, PP, ''Wideband circuit design'', pp.&nbsp;299&ndash;304, CRC Press, 1998. ISBN 0-8493-7897-4.</ref>  The condition for reciprocity is,
 
:<math>z_{12} = z_{21}</math>
 
resulting in an ''S''-parameter matrix of,
 
:<math>\left [ \mathbf S \right ] = \begin{bmatrix} S_{11} & S_{12} \\ S_{12} & S_{11} \end{bmatrix} </math>
 
for symmetric networks and
 
:<math>\left [ \mathbf S \right ] = \begin{bmatrix} S_{11} & S_{12} \\ S_{12} & -S_{11} \end{bmatrix} </math>
 
for antimetric networks.<ref>Leo Maloratsky, ''Passive RF & Microwave Integrated Circuits'', pp. 86-87, Elsevier, 2003 ISBN 0080492053.</ref>
 
==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 [[Band-pass filter|bandpass]] Butterworth with an even number of resonators will be antimetric, as will a Butterworth [[mechanical filter]] with an even number of mechanical resonators.<ref>Robert A. Johnson, ''Mechanical Filters in Electronics'', p. 145, John Wiley & Sons Australia, Limited, 1983 ISBN 0471089192.</ref>
 
==Glossary notes==
{{reflist|group=note}}
 
==References==
{{reflist}}
 
[[Category:Linear filters]]
[[Category:Filter theory]]
[[Category:Analog circuits]]
[[Category:Electronics terminology]]
[[Category:Electronic design]]

Latest revision as of 23:54, 8 November 2014

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