Equivalent impedance transforms

This article describes mathematical transformations between some passive, linear impedance networks commonly found in electronic circuits.

The vast scale of the topic of equivalent circuits is underscored in a story told by Sidney Darlington.

According to Darlington, a large number of equivalent circuits were found by Ronald M. Foster, following his and George Campbell's 1920 paper on non-dissipative four-ports.

In the course of this work they looked at the ways four ports could be interconnected with ideal transformers[note 5] and maximum power transfer.

[2][3] A single impedance has two terminals to connect to the outside world, hence can be described as a 2-terminal, or a one-port, network.

Further transformations are possible in the special case of Z2 being made the same element kind as Z1, that is, when the network is reduced to one-element-kind.

Any network that includes distributed elements, such as a transmission line, cannot be represented by a finite matrix.

Generally, an n-mesh[note 6] network requires an nxn matrix to represent it.

For the example network above, This result is easily verified to be correct by the more direct method of resistors in series and parallel.

However, such methods rapidly become tedious and cumbersome with the growth of the size and complexity of the network under analysis.

The positive-real (PR) condition is both necessary and sufficient[8] but there may be practical reasons for rejecting some topologies.

[7] A general impedance transform for finding equivalent rational one-ports from a given instance of [Z] is due to Wilhelm Cauer.

It is possible to take exactly the same network and connect it to external circuitry in such a way that it is no longer behaving as a 2-port.

Strictly speaking, any network that does not meet the balance condition is unbalanced, but the term is most often referring to the 3-terminal topology described above and in Figure 3.

Its importance arises from the fact that the total impedance between two terminals cannot be determined solely by calculating series and parallel combinations except for a certain restricted class of network.

The method is limited to symmetric networks but this includes many topologies commonly found in filters, attenuators and equalisers.

Reverse transformations from a lattice to an unbalanced topology are not always possible in terms of passive components.

The following circuit in bridged-T topology is a modification of a mid-series m-derived filter T-section.

The circuit is due to Hendrik Bode who claims that the addition of the bridging resistor of a suitable value will cancel the parasitic resistance of the shunt inductor.

[15] A theorem due to Sidney Darlington states that any PR function Z(s) can be realised as a lossless two-port terminated in a positive resistor R. That is, regardless of how many resistors feature in the matrix [Z] representing the impedance network, a transform can be found that will realise the network entirely as an LC-kind network with just one resistor across the output port (which would normally represent the load).

This technique requires that the transformer is next to (or capable of being moved next to) an "L" network of same-kind impedances.

The transform in all variants results in the "L" network facing the opposite way, that is, topologically mirrored.

Fig. 1. Simple example of a network of impedances using resistors only for clarity. However, analysis of networks with other impedance elements proceed by the same principles. Two meshes are shown, with numbers in circles. The sum of impedances around each mesh, p, will form the diagonal of the entries of the matrix, Z pp . The impedance of branches shared by two meshes, p and q, will form the entries - Z pq . Z pq , p≠q, will always have a minus sign provided that the convention of loop currents are defined in the same (conventionally counter-clockwise) direction and the mesh contains no ideal transformers or mutual inductors.
Fig. 2. A 4-terminal network connected by ports (top) has equal and opposite currents in each pair of terminals. The bottom network does not meet the port condition and cannot be treated as a 2-port. It could, however, be treated as an unbalanced 3-port by splitting one of the terminals into three common terminals shared between the ports.
Equivalent unbalanced and balanced networks. The impedance of the series elements in the balanced version is half the corresponding impedance of the unbalanced version.
Fig. 3. To be balanced, a network must have the same impedance in each "leg" of the circuit.