Equivalent impedance transforms

Linear network analysis
Elements
Components
Series and parallel circuits
Impedance transforms
Generator theorems	Network theorems
Network analysis methods
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An equivalent impedance is an

impedance networks commonly found in electronic circuits.

There are a number of very well known and often used equivalent circuits in linear

. None of these are discussed in detail here; the individual linked articles should be consulted.

The number of equivalent circuits that a linear network can be transformed into is unbounded. Even in the most trivial cases this can be seen to be true, for instance, by asking how many different combinations of resistors in parallel are equivalent to a given combined resistor. The number of series and parallel combinations that can be formed grows exponentially with the number of resistors, n. For large n the size of the set has been found by numerical techniques to be approximately 2.53ⁿ and analytically strict bounds are given by a

are also commonly found and transformations of yet more complex networks are possible.

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

2-terminal, 2-element-kind networks

A single impedance has two terminals to connect to the outside world, hence can be described as a 2-terminal, or a

3-element networks

One-element networks are trivial and two-element,[note 3] two-terminal networks are either two elements in series or two elements in parallel, also trivial. The smallest number of elements that is non-trivial is three, and there are two 2-element-kind non-trivial transformations possible, one being both the reverse transformation and the topological dual, of the other.^[5]

Description	Network	Transform equations	Transformed network
Transform 1.1 Transform 1.2 is the reverse of this transform.		${\displaystyle p_{1}=1+m_{1}\ ,}$  ${\displaystyle p_{2}=m_{1}(1+m_{1})\ ,}$  ${\displaystyle p_{3}=(1+m_{1})^{2}\ .}$
Transform 1.2 The reverse transform, and topological dual, of Transform 1.1.		${\displaystyle p_{1}={\frac {{m_{1}}^{2}}{1+m_{1}}}\ ,}$  ${\displaystyle p_{2}={\frac {m_{1}}{1+m_{1}}}\ ,}$  ${\displaystyle p_{3}=\left({\frac {m_{1}}{1+m_{1}}}\right)^{2}\ .}$
Example 1. An example of Transform 1.2. The reduced size of the inductor has practical advantages.		${\displaystyle m_{1}=0.5\ ,}$  ${\displaystyle p_{1}=\textstyle {\frac {1}{6}}\ ,}$ ${\displaystyle p_{2}=\textstyle {\frac {1}{3}}\ ,}$ ${\displaystyle p_{3}=\textstyle {\frac {1}{9}}\ .}$

4-element networks

There are four non-trivial 4-element transformations for 2-element-kind networks. Two of these are the reverse transformations of the other two and two are the dual of a different two. 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. The number of possible networks continues to grow as the number of elements is increased. For all entries in the following table it is defined:^[6]

${\displaystyle q_{1}:=1+m_{1}+m_{2}\,\!}$ , ${\displaystyle q_{2}:={\sqrt {{q_{1}}^{2}-4m_{1}m_{2}}}\,\!}$ , ${\displaystyle q_{3}:={\frac {(1+m_{1})(1+m_{2})}{(m_{1}-m_{2})^{2}}}}$ ,	${\displaystyle q_{4}:={\frac {q_{2}-q_{1}+2m_{2}}{2q_{2}}}}$ , ${\displaystyle q_{5}:={\frac {q_{2}+q_{1}-2m_{2}}{2q_{2}}}}$ .

Description	Network	Transform equations	Transformed network
Transform 2.1 Transform 2.2 is the reverse of this transform. Transform 2.3 is the topological dual of this transform.		${\displaystyle p_{1}={\frac {q_{1}+q_{2}}{2q_{5}}}\ ,}$ ${\displaystyle p_{2}={\frac {q_{1}-q_{2}}{2q_{4}}}\ ,}$ ${\displaystyle p_{3}={\frac {m_{2}}{q_{5}}}\ ,}$ ${\displaystyle p_{4}={\frac {m_{2}}{q_{4}}}\ .}$
Transform 2.2 Transform 2.1 is the reverse of this transform. Transform 2.4 is the topological dual of this transform.		${\displaystyle p_{1}={\frac {1}{q_{3}(1+m_{2})}}\ ,}$ ${\displaystyle p_{2}={\frac {m_{1}}{1+m_{1}}}\ ,}$ ${\displaystyle p_{3}={\frac {1}{q_{3}(1+m_{1})}}\ ,}$ ${\displaystyle p_{4}={\frac {m_{2}}{1+m_{2}}}\ .}$
Transform 2.3 Transform 2.4 is the reverse of this transform. Transform 2.1 is the topological dual of this transform.		${\displaystyle p_{1}={\frac {q_{4}(q_{1}+q_{2})}{2m_{2}}}\ ,}$ ${\displaystyle p_{2}={\frac {q_{5}(q_{1}-q_{2})}{2m_{2}}}\ ,}$ ${\displaystyle p_{3}=q_{4}\ ,}$ ${\displaystyle p_{4}=q_{5}\ .}$
Transform 2.4 Transform 2.3 is the reverse of this transform. Transform 2.2 is the topological dual of this transform.		${\displaystyle p_{1}=1+m_{1}\ ,}$ ${\displaystyle p_{2}=m_{1}q_{3}(1+m_{1})\ ,}$ ${\displaystyle p_{3}=1+m_{2}\ ,}$ ${\displaystyle p_{4}=m_{1}q_{3}(1+m_{2})\ .}$
Example 2. An example of Transform 2.2.		${\displaystyle m_{1}=3\ ,}$ ${\displaystyle m_{2}=1\ ,}$ ${\displaystyle q_{3}=2\ ,}$ ${\displaystyle p_{1}=\textstyle {\frac {1}{4}}\ ,}$ ${\displaystyle p_{2}=\textstyle {\frac {3}{4}}\ ,}$ ${\displaystyle p_{3}=\textstyle {\frac {1}{8}}\ ,}$ ${\displaystyle p_{4}=\textstyle {\frac {1}{2}}\ .}$

2-terminal, n-element, 3-element-kind networks

loop currents

are defined in the same (conventionally counter-clockwise) direction and the mesh contains no ideal transformers or mutual inductors.

Simple networks with just a few elements can be dealt with by formulating the network equations "by hand" with the application of simple network theorems such as

network requires an nxn matrix to represent it. For instance the matrix for a 3-mesh network might look like

${\displaystyle \mathbf {[Z]} ={\begin{bmatrix}Z_{11}&Z_{12}&Z_{13}\\Z_{21}&Z_{22}&Z_{23}\\Z_{31}&Z_{32}&Z_{33}\end{bmatrix}}}$

The entries of the matrix are chosen so that the matrix forms a system of linear equations in the mesh voltages and currents (as defined for mesh analysis):

${\displaystyle \mathbf {[V]} =\mathbf {[Z][I]} }$

The example diagram in Figure 1, for instance, can be represented as an impedance matrix by

${\displaystyle \mathbf {[Z]} ={\begin{bmatrix}R_{1}+R_{2}&-R_{2}\\-R_{2}&R_{2}+R_{3}\end{bmatrix}}}$

and the associated system of linear equations is

${\displaystyle {\begin{bmatrix}V_{1}\\0\end{bmatrix}}={\begin{bmatrix}R_{1}+R_{2}&-R_{2}\\-R_{2}&R_{2}+R_{3}\end{bmatrix}}{\begin{bmatrix}I_{1}\\I_{2}\end{bmatrix}}}$

In the most general case, each branch^{[note 1]} Z_p of the network may be made up of three elements so that

${\displaystyle Z_{\mathrm {p} }=sL_{\mathrm {p} }+R_{\mathrm {p} }+{1 \over sC_{\mathrm {p} }}}$

where L, R and C represent

operator ${\displaystyle \scriptstyle s=\sigma +i\omega }$.

This is the conventional way of representing a general impedance but for the purposes of this article it is mathematically more convenient to deal with elastance, D, the inverse of capacitance, C. In those terms the general branch impedance can be represented by

${\displaystyle sZ_{\mathrm {p} }=s^{2}L_{\mathrm {p} }+sR_{\mathrm {p} }+D_{\mathrm {p} }\,\!}$

Likewise, each entry of the impedance matrix can consist of the sum of three elements. Consequently, the matrix can be decomposed into three nxn matrices, one for each of the three element kinds:

${\displaystyle s\mathbf {[Z]} =s^{2}\mathbf {[L]} +s\mathbf {[R]} +\mathbf {[D]} }$

It is desired that the matrix [Z] represent an impedance, Z(s). For this purpose, the loop of one of the meshes is cut and Z(s) is the impedance measured between the points so cut. It is conventional to assume the external connection port is in mesh 1, and is therefore connected across matrix entry Z₁₁, although it would be perfectly possible to formulate this with connections to any desired nodes.^{[note 7]} In the following discussion Z(s) taken across Z₁₁ is assumed. Z(s) may be calculated from [Z] by^[7]

${\displaystyle Z(s)={\frac {|\mathbf {Z} |}{z_{11}}}}$

where z₁₁ is the complement of Z₁₁ and |Z| is the determinant of [Z].

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 entries of [R], [L] and [D] cannot be set arbitrarily. For [Z] to be able to realise the impedance Z(s) then [R],[L] and [D] must all be

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

${\displaystyle \mathbf {[Z']} =\mathbf {[T]} ^{T}\mathbf {[Z]} \mathbf {[T]} }$

where

${\displaystyle \mathbf {[T]} ={\begin{bmatrix}1&0\cdots 0\\T_{21}&T_{22}\cdots T_{2n}\\\cdot &\cdots \\T_{n1}&T_{n2}\cdots T_{nn}\end{bmatrix}}}$

is invariant in Z(s). That is, all the transformed networks are equivalents according to the definition given here. If the Z(s) for the initial given matrix is realisable, that is, it meets the PR condition, then all the transformed networks produced by this transformation will also meet the PR condition.^[7]

3 and 4-terminal networks

When discussing 4-terminal networks, network analysis often proceeds in terms of 2-port networks, which covers a vast array of practically useful circuits. "2-port", in essence, refers to the way the network has been connected to the outside world: that the terminals have been connected in pairs to a source or load. 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. This idea is demonstrated in Figure 2.

A 3-terminal network can also be used as a 2-port. To achieve this, one of the terminals is connected in common to one terminal of both ports. In other words, one terminal has been split into two terminals and the network has effectively been converted to a 4-terminal network. This topology is known as

. 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. Transforming an unbalanced 2-port network into a balanced network is usually quite straightforward: all series-connected elements are divided in half with one half being relocated in what was the common branch. Transforming from balanced to unbalanced topology will often be possible with the reverse transformation but there are certain cases of certain topologies which cannot be transformed in this way. For example, see the discussion of lattice transforms below.

An example of a 3-terminal network transform that is not restricted to 2-ports is the Y-Δ transform. This is a particularly important transform for finding equivalent impedances. 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. In the general case additional transformations are required. The Y-Δ transform, its inverse the Δ-Y transform, and the n-terminal analogues of these two transforms (star-polygon transforms) represent the minimal additional transforms required to solve the general case. Series and parallel are, in fact, the 2-terminal versions of star and polygon topology. A common simple topology that cannot be solved by series and parallel combinations is the input impedance to a bridge network (except in the special case when the bridge is in balance).^[9] The rest of the transforms in this section are all restricted to use with 2-ports only.

Lattice transforms

Symmetric 2-port networks can be transformed into lattice networks using Bartlett's bisection theorem. The method is limited to symmetric networks but this includes many topologies commonly found in filters, attenuators and equalisers. The lattice topology is intrinsically balanced, there is no unbalanced counterpart to the lattice and it will usually require more components than the transformed network.

Some common networks transformed to lattices (X-networks)
Description	Network	Transform equations	Transformed network
Transform 3.1 Transform of T network to lattice network.[10]		${\displaystyle Z_{\mathrm {A} }=Z_{1}\ ,\!}$ ${\displaystyle Z_{\mathrm {B} }=Z_{1}+2Z_{2}\ .\!}$
Transform 3.2 Transform of Π network to lattice network.[10]		${\displaystyle Z_{\mathrm {A} }={\frac {Z_{1}Z_{2}}{Z_{1}+2Z_{2}}}\ ,\!}$ ${\displaystyle Z_{\mathrm {B} }=Z_{2}\ .}$
Transform 3.3 Transform of Bridged-T network to lattice network.[11]		${\displaystyle Z_{\mathrm {A} }={\frac {Z_{1}Z_{0}}{Z_{1}+2Z_{0}}}\ ,\!}$ ${\displaystyle Z_{\mathrm {B} }=Z_{0}+2Z_{2}\ .\!}$

Reverse transformations from a lattice to an unbalanced topology are not always possible in terms of passive components. For instance, this transform:

Description	Network	Transformed network
Transform 3.4 Transform of a lattice phase equaliser to a T network.[12]

cannot be realised with passive components because of the negative values arising in the transformed circuit. It can however be realised if mutual inductances and ideal transformers are permitted, for instance, in this circuit. Another possibility is to permit the use of active components which would enable negative impedances to be directly realised as circuit components.^[13]

It can sometimes be useful to make such a transformation, not for the purposes of actually building the transformed circuit, but rather, for the purposes of aiding understanding of how the original circuit is working. The following circuit in bridged-T topology is a modification of a mid-series

Description	Network	Transformed network
Transform 3.5 Transform of a bridged-T low-pass filter section to a T-section.[14]

Any symmetrical network can be transformed into any other symmetrical network by the same method, that is, by first transforming into the intermediate lattice form (omitted for clarity from the above example transform) and from the lattice form into the required target form. As with the example, this will generally result in negative elements except in special cases.[15]

Eliminating resistors

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). No resistors within the network are necessary in order to realise the specified response. Consequently, it is always possible to reduce 3-element-kind 2-port networks to 2-element-kind (LC) 2-port networks provided the output port is terminated in a resistance of the required value.^[8][16][17]

Eliminating ideal transformers

An elementary transformation that can be done with ideal transformers and some other impedance element is to shift the impedance to the other side of the transformer. In all the following transforms, r is the turns ratio of the transformer.

Description	Network	Transformed network
Transform 4.1 Series impedance through a step-down transformer.
Transform 4.2 Shunt impedance through a step-down transformer.
Transform 4.3 Shunt and series impedance network through a step-up transformer.

These transforms do not just apply to single elements; entire networks can be passed through the transformer. In this manner, the transformer can be shifted around the network to a more convenient location.

Darlington gives an equivalent transform that can eliminate an ideal transformer altogether. 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.^[2]

Description	Network	Transformed network
Transform 5.1 Elimination of a step-down transformer.
Transform 5.2 Elimination of a step-up transformer.
Example 3. Example of transform 5.1.

Example 3 shows the result is a Π-network rather than an L-network. The reason for this is that the shunt element has more capacitance than is required by the transform so some is still left over after applying the transform. If the excess were instead, in the element nearest the transformer, this could be dealt with by first shifting the excess to the other side of the transformer before carrying out the transform.^[2]

Terminology

References

Bibliography

• Bartlett, A. C., "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, Iss 5, pp. 848–855, May 1962.
• E. Cauer, W. Mathis, and R. Pauli, "Life and Work of Wilhelm Cauer (1900 – 1945)", Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems, Perpignan, June, 2000.
• Foster, Ronald M.; Campbell, George A., "Maximum output networks for telephone substation and repeater circuits"
  , Transactions of the American Institute of Electrical Engineers, vol.39, iss.1, pp. 230–290, January 1920.
• Darlington, S., "A history of network synthesis and filter theory for circuits composed of resistors, inductors, and capacitors", IEEE Trans. Circuits and Systems, vol 31, pp. 3–13, 1984.
• Farago, P. S., An Introduction to Linear Network Analysis, The English Universities Press Ltd, 1961.
• Khan, Sameen Ahmed, "Farey sequences and resistor networks", Proceedings of the Indian Academy of Sciences (Mathematical Sciences), vol.122, iss.2, pp. 153–162, May 2012.
• Zobel, O. J.,Theory and Design of Uniform and Composite Electric Wave Filters, Bell System Technical Journal, Vol. 2 (1923), pp. 1–46.