Antimetric electrical network

Figure 1. Examples of symmetry and antimetry: both networks are low-pass filters but one is symmetric (left) and the other is antimetric (right). For a symmetric ladder the 1st element is equal to the nth, the 2nd equal to the (n-1)th and so on. For an antimetric ladder, the 1st element is the dual of the nth and so on.

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, Z_i1 and Z_i2. For an antimetric network, the two impedances must be the dual of each other with respect to some nominal impedance R₀. That is,[1]

${\displaystyle {\frac {Z_{i1}}{R_{0}}}={\frac {R_{0}}{Z_{i2}}}}$

or equivalently

${\displaystyle Z_{i1}Z_{i2}={R_{0}}^{2}.}$

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]

Figure 2. Adding another T-section to the ladders of figure 1

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 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 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[2] 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.[3]

The conditions for symmetry and antimetry can be stated in terms of two-port parameters. For a two-port network described by normalized impedance parameters (z-parameters),

${\displaystyle z_{11}=z_{22}}$

if the network is symmetric, and

${\displaystyle z_{11}z_{22}-z_{12}z_{21}=1}$

if the network is antimetric. Passive networks of the kind illustrated in this article are also reciprocal, which requires that

${\displaystyle z_{12}=z_{21}}$

and results in a normalized z-parameter matrix of,

${\displaystyle \left[\mathbf {z} \right]={\begin{bmatrix}z_{11}&z_{12}\\z_{12}&z_{11}\end{bmatrix}}}$

for symmetric networks and

${\displaystyle \left[\mathbf {z} \right]={\begin{bmatrix}z_{11}&z_{12}\\z_{12}&(z_{12}^{2}+1)/z_{11}\end{bmatrix}}}$

for antimetric networks.[4]

For a two-port network described by scattering parameters (S-parameters),

${\displaystyle S_{11}=S_{22}}$

if the network is symmetric, and

${\displaystyle S_{11}=-S_{22}}$

if the network is antimetric.[5] The condition for reciprocity is,

${\displaystyle S_{12}=S_{21}}$

resulting in an S-parameter matrix of,

${\displaystyle \left[\mathbf {S} \right]={\begin{bmatrix}S_{11}&S_{12}\\S_{12}&S_{11}\end{bmatrix}}}$

for symmetric networks and

${\displaystyle \left[\mathbf {S} \right]={\begin{bmatrix}S_{11}&S_{12}\\S_{12}&-S_{11}\end{bmatrix}}}$

for antimetric networks.[6]

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]