Linear network analysis  

Elements  
Components  
Series and parallel circuits  
Impedance transforms  
Generator theorems  Network theorems 
Network analysis methods  
Twoport parameters  
An equivalent impedance is an equivalent circuit of an electrical network of impedance elements^{ [note 2] } which presents the same impedance between all pairs of terminals^{ [note 10] } as did the given network. This article describes mathematical transformations between some passive, linear impedance networks commonly found in electronic circuits.
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, that is a simplest form of a more complex circuit in order to aid analysis. In its most common form, an equivalent circuit is made up of linear, passive elements. However, more complex equivalent circuits are used that approximate the nonlinear behavior of the original circuit as well. These more complex circuits often are called macromodels of the original circuit. An example of a macromodel is the Boyle circuit for the 741 operational amplifier.
An electrical network is an interconnection of electrical components or a model of such an interconnection, consisting of electrical elements. An electrical circuit is a network consisting of a closed loop, giving a return path for the current. Linear electrical networks, a special type consisting only of sources, linear lumped elements, and linear distributed elements, have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response.
Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.
There are a number of very well known and often used equivalent circuits in linear network analysis. These include resistors in series, resistors in parallel and the extension to series and parallel circuits for capacitors, inductors and general impedances. Also well known are the Norton and Thévenin equivalent current generator and voltage generator circuits respectively, as is the YΔ transform. None of these are discussed in detail here; the individual linked articles should be consulted.
A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many techniques for calculating these values. However, for the most part, the techniques assume linear components. Except where stated, the methods described in this article are applicable only to linear network analysis.
Components of an electrical circuit or electronic circuit can be connected in series, parallel, or seriesparallel. The two simplest of these are called series and parallel and occur frequently. Components connected in series are connected along a single conductive path, so the same current flows through all of the components but voltage is dropped (lost) across each of the resistances. In a series circuit, the sum of the voltages consumed by each individual resistance is equal to the source voltage. Components connected in parallel are connected along multiple paths so that the current can split up; the same voltage is applied to each component.
A capacitor is a device that stores electrical energy in an electric field. It is a passive electronic component with two terminals.
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 of Fibonacci numbers.^{ [1] } This article could never hope to be comprehensive, but there are some generalisations possible. Wilhelm Cauer found a transformation that could generate all possible equivalents of a given rational,^{ [note 9] } passive, linear oneport,^{ [note 8] } or in other words, any given twoterminal impedance. Transformations of 4terminal, especially 2port, networks are also commonly found and transformations of yet more complex networks are possible.
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 order of increasing size.
Wilhelm Cauer 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. Prior to his work, electronic filter design used techniques which accurately predicted filter behaviour only under unrealistic conditions. This required a certain amount of experience on the part of the designer to choose suitable sections to include in the design. Cauer placed the field on a firm mathematical footing, providing tools that could produce exact solutions to a given specification for the design of an electronic filter.
A twoport network is an electrical network (circuit) or device with two pairs of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the electric current entering one terminal must equal the current emerging from the other terminal on the same port. The ports constitute interfaces where the network connects to other networks, the points where signals are applied or outputs are taken. In a twoport network, often port 1 is considered the input port and port 2 is considered the output port.
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 nondissipative fourports. 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. They found a number of combinations which might have practical applications and asked the AT&T 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.^{ [2] }^{ [3] }
Sidney Darlington was an electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair. He advanced the state of network theory, developing the insertionloss synthesis approach, and invented chirp radar, bombsights, and gun and rocket guidance.
Ronald Martin Foster, 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, which quickly inspired Wilhelm Cauer to begin his program of network synthesis filters which put the design of filters on a firm mathematical footing. He is also known for the Foster census of cubic symmetric graphs and the 90vertex cubic symmetric Foster graph.
George Ashley Campbell was an American engineer. He was pioneer in developing and applying quantitative mathematical methods to the problems of longdistance telegraphy and telephony. His most important contributions were to the theory and implementation of the use of loading coils and the first wave filters designed to what was to become known as the image method. Both these areas of work resulted in important economic advantages for the American Telephone and Telegraph Company (AT&T).
A single impedance has two terminals to connect to the outside world, hence can be described as a 2terminal, or a oneport, network. Despite the simple description, there is no limit to the number of meshes,^{ [note 6] } and hence complexity and number of elements, that the impedance network may have. 2elementkind^{ [note 4] } networks are common in circuit design; filters, for instance, are often LCkind networks and printed circuit designers favour RCkind networks because inductors are less easy to manufacture. Transformations are simpler and easier to find than for 3elementkind networks. Oneelementkind networks can be thought of as a special case of twoelementkind. It is possible to use the transformations in this section on a certain few 3elementkind 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.^{ [4] }
An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.
A printed circuit board (PCB) mechanically supports and electrically connects electronic components or electrical components using conductive tracks, pads and other features etched from one or more sheet layers of copper laminated onto and/or between sheet layers of a nonconductive substrate. Components are generally soldered onto the PCB to both electrically connect and mechanically fasten them to it.
A resistor–capacitor circuit, or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.
Oneelement networks are trivial and twoelement,^{ [note 3] } twoterminal networks are either two elements in series or two elements in parallel, also trivial. The smallest number of elements that is nontrivial is three, and there are two 2elementkind nontrivial transformations possible, one being both the reverse transformation and the topological dual, of the other.^{ [5] }
Dual impedance and dual network are terms used in electronic network analysis. The dual of an impedance is its reciprocal, or algebraic inverse . For this reason the dual impedance is also called the inverse impedance. Another way of stating this is that the dual of is the admittance .
There are four nontrivial 4element transformations for 2elementkind 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 oneelementkind. 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] }


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. This approach is only good for rational^{ [note 9] } networks. Any network that includes distributed elements, such as a transmission line, cannot be represented by a finite matrix. Generally, an nmesh^{ [note 6] } network requires an nxn matrix to represent it. For instance the matrix for a 3mesh network might look like
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):
The example diagram in Figure 1, for instance, can be represented as an impedance matrix by
and the associated system of linear equations is
In the most general case, each branch^{ [note 1] }Z_{p} of the network may be made up of three elements so that
where L, R and C represent inductance, resistance, and capacitance respectively and s is the complex frequency operator .
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
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:
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 Z_{11}, 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 Z_{11} is assumed. Z(s) may be calculated from [Z] by^{ [7] }
where z_{11} is the complement of Z_{11} 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 positivedefinite matrices. Even then, the realisation of Z(s) will, in general, contain ideal transformers^{ [note 5] } 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 positivereal. The positivereal (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 oneports from a given instance of [Z] is due to Wilhelm Cauer. The group of real affine transformations
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] }
When discussing 4terminal networks, network analysis often proceeds in terms of 2port networks, which covers a vast array of practically useful circuits. "2port", 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 2port. This idea is demonstrated in Figure 2.
A 3terminal network can also be used as a 2port. 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 4terminal 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. Strictly speaking, any network that does not meet the balance condition is unbalanced, but the term is most often referring to the 3terminal topology described above and in Figure 3. Transforming an unbalanced 2port network into a balanced network is usually quite straightforward: all seriesconnected 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 3terminal network transform that is not restricted to 2ports is the YΔ 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 YΔ transform, its inverse the ΔY transform, and the nterminal analogues of these two transforms (starpolygon transforms) represent the minimal additional transforms required to solve the general case. Series and parallel are, in fact, the 2terminal 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 2ports only.
Symmetric 2port 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.
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 bridgedT topology is a modification of a midseries mderived filter Tsection. 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. 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.^{ [14] }
Description  Network  Transformed network 

Transform 3.5 Transform of a bridgedT lowpass filter section to a Tsection.^{ [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] }
A theorem due to Sidney Darlington states that any PR function Z(s) can be realised as a lossless twoport 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 LCkind 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 3elementkind 2port networks to 2elementkind (LC) 2port networks provided the output port is terminated in a resistance of the required value.^{ [8] }^{ [16] }^{ [17] }
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 samekind 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 stepdown transformer.  
Transform 5.2 Elimination of a stepup transformer.  
Example 3. Example of transform 5.1. 
Example 3 shows the result is a Πnetwork rather than an Lnetwork. 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] }
Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electrical elements interconnected by wires. Where the elements roughly correspond to real components the representation can be in the form of a schematic diagram or circuit diagram. This is called a lumpedelement circuit model. In other cases infinitesimal elements are used to model the network, in a distributedelement model.
A gyrator is a passive, linear, lossless, twoport electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is nonreciprocal. Gyrators permit network realizations of two(ormore)port devices which cannot be realized with just the conventional four elements. In particular, gyrators make possible network realizations of isolators and circulators. Gyrators do not however change the range of oneport devices that can be realized. Although the gyrator was conceived as a fifth linear element, its adoption makes both the ideal transformer and either the capacitor or inductor redundant. Thus the number of necessary linear elements is in fact reduced to three. Circuits that function as gyrators can be built with transistors and opamps using feedback.
Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributedelement filters. That is, using components and interconnections that, in analysis, can be considered to exist at a single point. These components can be in discrete packages or part of an integrated circuit.
Zobel networks are a type of filter section based on the imageimpedance design principle. They are named after Otto Zobel of Bell Labs, who published a muchreferenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the input impedance is fixed in the design independently of the transfer function. This characteristic is achieved at the expense of a much higher component count compared to other types of filter sections. The impedance would normally be specified to be constant and purely resistive. For this reason, Zobel networks are also known as constant resistance networks. However, any impedance achievable with discrete components is possible.
A lattice phase equaliser or lattice filter is an example of an allpass filter. That is, the attenuation of the filter is constant at all frequencies but the relative phase between input and output varies with frequency. The lattice filter topology has the particular property of being a constantresistance network and for this reason is often used in combination with other constant resistance filters such as bridgeT equalisers. The topology of a lattice filter, also called an Xsection is identical to bridge topology. The lattice phase equaliser was invented by Otto Zobel. using a filter topology proposed by George Campbell.
Network synthesis is a method of designing signal processing filters. It has produced several important classes of filter including the Butterworth filter, the Chebyshev filter and the Elliptic filter. It was originally intended to be applied to the design of passive linear analogue filters but its results can also be applied to implementations in active filters and digital filters. The essence of the method is to obtain the component values of the filter from a given rational function representing the desired transfer function.
The topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their positions on a circuit diagram, similarly to the mathematic concept of topology it is only concerned with what connections exist between the components. There may be numerous physical layouts and circuit diagrams that all amount to the same topology.
Bartlett's bisection theorem is an electrical theorem in network analysis attributed to Albert Charles Bartlett. The theorem shows that any symmetrical twoport network can be transformed into a lattice network. The theorem often appears in filter theory where the lattice network is sometimes known as a filter Xsection following the common filter theory practice of naming sections after alphabetic letters to which they bear a resemblance.
Analogue filters are a basic building block of signal processing much used in electronics. Amongst their many applications are the separation of an audio signal before application to bass, midrange, and tweeter loudspeakers; the combining and later separation of multiple telephone conversations onto a single channel; the selection of a chosen radio station in a radio receiver and rejection of others.
An antimetric electrical network is an electrical network that exhibits antisymmetrical electrical properties. The term is often encountered in filter theory, but it applies to general electrical network analysis. Antimetric is the diametrical opposite of symmetric; it does not merely mean "asymmetric". It is possible for networks to be symmetric or antimetric in their electrical properties without being physically or topologically symmetric or antimetric.
Commensurate line circuits are electrical circuits composed of transmission lines that are all the same length; commonly oneeighth of a wavelength. Lumped element circuits can be directly converted to distributedelement circuits of this form by the use of Richards' transformation. This transformation has a particularly simple result; inductors are replaced with transmission lines terminated in shortcircuits and capacitors are replaced with lines terminated in opencircuits. Commensurate line theory is particularly useful for designing distributedelement filters for use at microwave frequencies.
The impedance analogy is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially in the field of filters. By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in electromechanical systems: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole.
The starmesh transform, or starpolygon transform, is a mathematical circuit analysis technique to transform a resistive network into an equivalent network with one less node. The equivalence follows from the Schur complement identity applied to the Kirchhoff matrix of the network.
In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside circuit, that meets the port condition; the currents flowing into the two nodes must be equal and opposite.
A loss free resistor (LFR) is a resistor that does not lose energy. The first implementation is due to Singer and it has been implemented in various settings.
The mobility analogy, also called admittance analogy or Firestone analogy, is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially in the field of filters. By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in electromechanical systems: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole.
In control system theory, and various branches of engineering, a transfer function matrix, or just transfer matrix is a generalisation of the transfer functions of singleinput singleoutput (SISO) systems to multipleinput and multipleoutput (MIMO) systems. The matrix relates the outputs of the system to its inputs. It is a particularly useful construction for linear timeinvariant (LTI) systems because it can be expressed in terms of the splane.