equivalent impedance transforms
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An equivalent impedance is an
equivalent circuit In electrical engineering and science, an equivalent circuit refers to a theoretical circuit that retains all of the electrical characteristics of a given circuit. Often, an equivalent circuit is sought that simplifies calculation, and more broadly ...
of an
electrical network An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources ...
of impedance elements which presents the same impedance between all pairs of terminals as did the given network. This article describes mathematical transformations between some
passive Passive may refer to: * Passive voice, a grammatical voice common in many languages, see also Pseudopassive * Passive language, a language from which an interpreter works * Passivity (behavior), the condition of submitting to the influence of o ...
,
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
impedance networks commonly found in electronic circuits. There are a number of very well known and often used equivalent circuits in linear
network analysis Network analysis can refer to: * Network theory, the analysis of relations through mathematical graphs ** Social network analysis, network theory applied to social relations * Network analysis (electrical circuits) See also *Network planning and ...
. These include
resistors in series A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
, resistors in parallel and the extension to series and parallel circuits for
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
s,
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s and general impedances. Also well known are the
Norton Norton may refer to: Places Norton, meaning 'north settlement' in Old English, is a common place name. Places named Norton include: Canada * Rural Municipality of Norton No. 69, Saskatchewan *Norton Parish, New Brunswick **Norton, New Brunswick, a ...
and Thévenin equivalent current generator and voltage generator circuits respectively, as is the
Y-Δ transform The Y-Δ transform, also written wye-delta and also known by many other names, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like t ...
. 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 2.53''n'' and analytically strict bounds are given by a
Farey sequence In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ''n'', arranged in ord ...
of
Fibonacci numbers In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
. This article could never hope to be comprehensive, but there are some generalisations possible.
Wilhelm Cauer Wilhelm Cauer (24 June 1900 – 22 April 1945) was a German mathematician and scientist. He is most noted for his work on the analysis and synthesis of electrical filters and his work marked the beginning of the field of network synthesis. Prio ...
found a transformation that could generate all possible equivalents of a given rational, passive, linear
one-port In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, as a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside ...
, or in other words, any given two-terminal impedance. Transformations of 4-terminal, especially 2-port, networks 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
Sidney Darlington Sidney Darlington (July 18, 1906 – October 31, 1997) was an American electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair. He advanced the state of network theory, developing the insertion-loss synth ...
. According to Darlington, a large number of equivalent circuits were found by
Ronald M. Foster Ronald Martin Foster (3 October 1896 – 2 February 1998), was a Bell Labs mathematician whose work was of significance regarding electronic filters for use on telephone lines. He published an important paper, ''A Reactance Theorem'', (see Fost ...
, 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 and maximum power transfer. They found a number of combinations which might have practical applications and asked the
AT&T AT&T Inc. is an American multinational telecommunications holding company headquartered at Whitacre Tower in Downtown Dallas, Texas. It is the world's largest telecommunications company by revenue and the third largest provider of mobile te ...
patent department to have them patented. The patent department replied that it was pointless just patenting some of the circuits if a competitor could use an equivalent circuit to get around the patent; they should patent all of them or not bother. Foster therefore set to work calculating every last one of them. He arrived at an enormous total of 83,539 equivalents (577,722 if different output ratios are included). This was too many to patent, so instead the information was released into the public domain in order to prevent any of AT&T's competitors from patenting them in the future.Darlington, p.6.


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
one-port In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, as a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside ...
, network. Despite the simple description, there is no limit to the number of meshes, and hence complexity and number of elements, that the impedance network may have. 2-element-kind networks are common in circuit design; filters, for instance, are often LC-kind networks and
printed circuit A printed circuit board (PCB; also printed wiring board or PWB) is a medium used in electrical and electronic engineering to connect electronic components to one another in a controlled manner. It takes the form of a laminated sandwich struct ...
designers favour RC-kind networks because
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s are less easy to manufacture. Transformations are simpler and easier to find than for 3-element-kind networks. One-element-kind networks can be thought of as a special case of two-element-kind. It is possible to use the transformations in this section on a certain few 3-element-kind networks by substituting a network of elements for element ''Z''n. However, this is limited to a maximum of two impedances being substituted; the remainder will not be a free choice. All the transformation equations given in this section are due to
Otto Zobel Otto Julius Zobel (October 20, 1887 – January 1970) was an electrical engineer who worked for the American Telephone & Telegraph Company (AT&T) in the early part of the 20th century. Zobel's work on filter design was revolutionary and led ...
.


3-element networks

One-element networks are trivial and two-element, 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.


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 ''Z''2 being made the same element kind as ''Z''1, 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:


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

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 Kirchhoff's laws. Equivalence is proved between two networks by directly comparing the two sets of equations and equating coefficients. For large networks more powerful techniques are required. A common approach is to start by expressing the network of impedances as a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. This approach is only good for rational networks. Any network that includes
distributed elements : ''This article is an example from the domain of electrical systems, which is a special case of the more general distributed-parameter systems.'' In electrical engineering, the distributed-element model or transmission-line model of electrical E ...
, such as a
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
, cannot be represented by a finite matrix. Generally, an ''n''-mesh network requires an ''n''x''n'' matrix to represent it. For instance the matrix for a 3-mesh network might look like : \mathbf=\begin Z_ & Z_ & Z_ \\ Z_ & Z_ & Z_ \\ Z_ & Z_ & Z_ \end 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 Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the electrical circuit. Planar circuits are circuits that can be drawn on a plane sur ...
): : \mathbf= \mathbf The example diagram in Figure 1, for instance, can be represented as an impedance matrix by : \mathbf=\begin R_1+R_2 & -R_2 \\ -R_2 & R_2+R_3 \end and the associated system of linear equations is : \begin V_1 \\ 0 \end = \begin R_1+R_2 & -R_2 \\ -R_2 & R_2+R_3 \end \begin I_1 \\ I_2 \end In the most general case, each branch ''Z''p of the network may be made up of three elements so that :Z_\mathrm = sL_\mathrm + R_\mathrm + where ''L'', ''R'' and ''C'' represent inductance, resistance, and
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...
respectively and ''s'' is the
complex frequency In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
operator \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 Electrical elastance is the reciprocal of capacitance. The SI unit of elastance is the inverse farad (F−1). The concept is not widely used by electrical and electronic engineers. The value of capacitors is invariably specified in units of c ...
, ''D'', the inverse of capacitance, ''C''. In those terms the general branch impedance can be represented by :sZ_\mathrm = s^2L_\mathrm + sR_\mathrm + D_\mathrm \,\! Likewise, each entry of the impedance matrix can consist of the sum of three elements. Consequently, the matrix can be decomposed into three ''n''x''n'' matrices, one for each of the three element kinds: :s \mathbf= s^2 \mathbf + s \mathbf + \mathbf It is desired that the matrix ''Zrepresent 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 ''Z''11, although it would be perfectly possible to formulate this with connections to any desired nodes. In the following discussion ''Z''(''s'') taken across ''Z''11 is assumed. ''Z''(''s'') may be calculated from ''ZbyE. Cauer ''et al.'', p.4. :Z(s)=\frac where ''z''11 is the
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
of ''Z''11 and , Z, is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of ''Z For the example network above,
:, \mathbf Z, = (R_1+R_2)(R_2+R_3) - ^2 = R_1R_2 + R_1R_3 + R_2R_3 \ , :z_ = Z_ = R_2 + R_3 \ , and, :Z(s) = R_1 + \frac \ .
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 ''Land ''Dcannot be set arbitrarily. For ''Zto be able to realise the impedance ''Z''(''s'') then ''R ''Land ''Dmust all be
positive-definite matrices In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a c ...
. Even then, the realisation of ''Z''(''s'') will, in general, contain ideal transformers within the network. Finding only those transforms that do not require mutual inductances or ideal transformers is a more difficult task. Similarly, if starting from the "other end" and specifying an expression for ''Z''(''s''), this again cannot be done arbitrarily. To be realisable as a rational impedance, ''Z''(''s'') must be positive-real. The positive-real (PR) condition is both necessary and sufficientBelevitch, p.850 but there may be practical reasons for rejecting some
topologies In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
. A general impedance transform for finding equivalent rational one-ports from a given instance of ''Zis due to
Wilhelm Cauer Wilhelm Cauer (24 June 1900 – 22 April 1945) was a German mathematician and scientist. He is most noted for his work on the analysis and synthesis of electrical filters and his work marked the beginning of the field of network synthesis. Prio ...
. The group of real affine transformations :\mathbf = \mathbf^T \mathbf \mathbf :where : \mathbf=\begin 1 & 0 \cdots 0 \\ T_ & T_ \cdots T_ \\ \cdot & \cdots \\ T_ & T_ \cdots T_\end 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.


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 unbalanced topology and is opposed to balanced topology. Balanced topology requires, referring to Figure 3, that the impedance measured between terminals 1 and 3 is equal to the impedance measured between 2 and 4. This is the pairs of terminals ''not'' forming ports: the case where the pairs of terminals forming ports have equal impedance is referred to as
symmetrical Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
. 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 The Y-Δ transform, also written wye-delta and also known by many other names, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like t ...
. 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). 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 Bartlett's bisection theorem is an electrical theorem in network analysis (electrical circuits), network analysis attributed to Albert Charles Bartlett. The theorem shows that any symmetrical two-port network can be transformed into a lattice filt ...
. 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. Reverse transformations from a lattice to an unbalanced topology are not always possible in terms of passive components. For instance, this transform: 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. 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
m-derived filter m-derived filters or m-type filters are a type of electronic filter designed using the image method. They were invented by Otto Zobel in the early 1920s. This filter type was originally intended for use with telephone multiplexing and was a ...
T-section. The circuit is due to
Hendrik Bode Hendrik Wade Bode ( ; ;Van Valkenburg, M. E. University of Illinois at Urbana-Champaign, "In memoriam: Hendrik W. Bode (1905-1982)", IEEE Transactions on Automatic Control, Vol. AC-29, No 3., March 1984, pp. 193–194. Quote: "Something should be ...
who claims that the addition of the bridging resistor of a suitable value will cancel the
parasitic resistance In electrical networks, a parasitic element is a circuit element ( resistance, inductance or capacitance) that is possessed by an electrical component but which it is not desirable for it to have for its intended purpose. For instance, a resis ...
of the shunt inductor. The action of this circuit is clear if it is transformed into T topology - in this form there is a negative resistance in the shunt branch which can be made to be exactly equal to the positive parasitic resistance of the inductor.Bode, Hendrik W., ''Wave Filter'', US patent 2 002 216, filed 7 June 1933, issued 21 May 1935. 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.


Eliminating resistors

A theorem due to
Sidney Darlington Sidney Darlington (July 18, 1906 – October 31, 1997) was an American electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair. He advanced the state of network theory, developing the insertion-loss synth ...
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 ''Zrepresenting 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.E. Cauer et al., pp.6–7.Darlington, p.7.


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. 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. 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.


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. {{DEFAULTSORT:Equivalent Impedance Transforms Circuit theorems Filter theory Analog circuits Electronic design