 Electronic filter topology

Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected.
Filter design characterises filter circuits primarily by their transfer function rather than their topology. Transfer functions may be linear or nonlinear. Common types of linear filter transfer function are; highpass, lowpass, bandpass, bandreject or notch and allpass. Once the transfer function for a filter is chosen, the particular topology to implement such a prototype filter can be selected so that, for example, one might choose to design a Butterworth filter using the Sallen–Key topology.
Filter topologies may be divided into passive and active types. Passive topologies do not contain a generator of energy, either in reality or, due to nonlinearity, in their equivalent circuit, but only capacitors, inductors and also, in some topologies, resistors: particularly, variable resistors are often included in order to control the depth of filtering. Active topologies include components that require power. Further, topologies may be implemented either in unbalanced form or else in balanced form when employed in balanced circuits. Implementations such as electronic mixers and stereo sound may require arrays of identical circuits.
Contents
Passive topologies
Passive filters have been long in development and use. Most are built from simple twoport networks called "sections". There is no formal definition of a section except that it must have at least one series component and one shunt component. Sections are invariably connected in a "cascade" or "daisychain" topology, consisting of either repeats of the same section or of completely different sections. Impedance would combine two sections consisting only of series components or shunt components into a single section.
Some passive filters, consisting of only one or two filter sections, are given special names including the Lsection, Tsection and Πsection, which are unbalanced filters, and the Csection, Hsection and boxsection, which are balanced. All are built upon a very simple "ladder" topology (see below). The chart at the bottom of the page shows these various topologies in terms of general constant k filters.
Filters designed using network synthesis usually repeat the simplest form of Lsection topology though component values may change in each section. Image designed filters, on the other hand, keep the same basic component values from section to section though the topology may vary and tend to make use of more complex sections.
Lsections are never symmetrical but two Lsections backtoback form a symmetrical topology and many other sections are symmetrical in form.
Ladder topologies
Ladder topology, often called Cauer topology after Wilhelm Cauer (inventor of the Elliptical filter), was in fact first used by George Campbell (inventor of the Constant k filter). Campbell published in 1922 but had clearly been using the topology for some time before this. Cauer first picked up on ladders (published 1926) inspired by the work of Foster (1924). There are two forms of basic ladder topologies; unbalanced and balanced. Cauer topology is usually thought of as an unbalanced ladder topology.
A ladder network consists of cascaded asymmetrical Lsections (unbalanced) or Csections (balanced). In low pass form the topology would consist of series inductors and shunt capacitors. Other bandforms would have an equally simple topology transformed from the lowpass topology. The transformed network will have shunt admittances that are dual networks of the series impedances if they were duals in the starting network  which is the case with series inductors and shunt capacitors.
Image filter sections Unbalanced L Half section T Section Π Section Ladder network Balanced C Halfsection H Section Box Section Ladder network X Section (midTDerived) X Section (midΠDerived) N.B. Textbooks and design drawings usually show the unbalanced implementations, but in telecoms it is often required to convert the design to the balanced implementation when used with balanced lines. edit Modified ladder topologies
Image filter design commonly uses modifications of the basic ladder topology. These topologies, invented by Otto Zobel,^{[1]} have the same passbands as the ladder on which they are based but their transfer functions are modified to improve some parameter such as impedance matching, stopband rejection or passbandtostopband transition steepness. Usually the design applies some transform to a simple ladder topology: the resulting topology is ladderlike but no longer obeys the rule that shunt admittances are the dual network of series impedances: it invariably becomes more complex with higher component count. Such topologies include;
The mtype (mderived) filter is by far the most commonly used modified image ladder topology. There are two mtype topologies for each of the basic ladder topologies; the seriesderived and shuntderived topologies. These have identical transfer functions to each other but different image impedances. Where a filter is being designed with more than one passband, the mtype topology will result in a filter where each passband has an analogous frequencydomain response. It is possible to generalise the mtype topology for filters with more than one passband using parameters m_{1}, m_{2}, m_{3} etc., which are not equal to each other resulting in general m_{n}type^{[2]} filters which have bandforms that can differ in different parts of the frequency spectrum.
The mm'type topology can be thought of as a double mtype design. Like the mtype it has the same bandform but offers further improved transfer characteristics. It is, however, a rarely used design due to increased component count and complexity as well as its normally requiring basic ladder and mtype sections in the same filter for impedance matching reasons. It is normally only found in a composite filter.
BridgedT topologies
Zobel constant resistance filters^{[3]} use a topology that is somewhat different from other filter types, distinguished by having a constant input resistance at all frequencies and in that they use resistive components in the design of their sections. The higher component and section count of these designs usually limits their use to equalisation applications. Topologies usually associated with constant resistance filters are the bridgedT and its variants, all described in the Zobel network article;
 BridgedT topology
 Balanced bridgedT topology
 Opencircuit Lsection topology
 Shortcircuit Lsection topology
 Balanced opencircuit Csection topology
 Balanced shortcircuit Csection topology
The bridgedT topology is also used in sections intended to produce a signal delay but in this case no resistive components are used in the design.
Lattice topology
Both the Tsection (from ladder topology) and the bridgeT (from Zobel topology) can be transformed into a lattice topology filter section but in both cases this results in high component count and complexity. The most common application of lattice filters (Xsections) is in allpass filters used for phase equalisation.^{[4]}
Although T and bridgedT sections can always be transformed into Xsections the reverse is not always possible because of the possibility of negative values of inductance and capacitance arising in the transform.
Lattice topology is identical to the more familiar bridge topology, the difference being merely the drawn representation on the page rather than any real difference in topology, cicuitry or function.
Active topologies
Multiple feedback topology
Multiple feedback topology is an electronic filter topology which is used to implement an electronic filter by adding two poles to the transfer function. A diagram of the circuit topology for a second order low pass filter is shown in the figure on the right.
The transfer function of the multiple feedback topology circuit, like all secondorder linear filters, is:
 .
In an MF filter,
Biquad filter
For the digital implementation of a biquad filter, check digital biquad filter.
A biquad filter is a type of linear filter that implements a transfer function that is the ratio of two quadratic functions. The name biquad is short for biquadratic.
Biquad filters are typically active and implemented with a singleamplifier biquad (SAB) or twointegratorloop topology.
 The SAB topology uses feedback to generate complex poles and possibly complex zeros. In particular, the feedback moves the real poles of an RC circuit in order to generate the proper filter characteristics.
 The twointegratorloop topology is derived from rearranging a biquadratic transfer function. The rearrangement will equate one signal with the sum of another signal, its integral, and the integral's integral. In other words, the rearrangement reveals a state variable filter structure. By using different states as outputs, any kind of secondorder filter can be implemented.
The SAB topology is sensitive to component choice and can be more difficult to adjust. Hence, usually the term biquad refers to the twointegratorloop state variable filter topology.
TowThomas Biquad Example
For example, the basic configuration in Figure 1 can be used as either a lowpass or bandpass filter depending on where the output signal is taken from.
The secondorder lowpass transfer function is given by
where lowpass gain G_{lpf} = R_{2} / R_{1}. The secondorder bandpass transfer function is given by
. with bandpass gain G_{bpf} = − R_{4} / R_{2}. In both cases, the
 Natural frequency is .
 Quality factor is .
The bandwidth is approximated by B = ω_{0} / Q, and Q is sometimes expressed as a damping constant ζ = 1 / 2Q. If a noninverting lowpass filter is required, the output can be taken at the output of the second operational amplifier. If a noninverting bandpass filter is required, the order of the second integrator and the inverter can be switched, and the output taken at the output of the inverter's operational amplifier.
Sallen–Key topology
Main article: SallenKey topologySee also
Notes
 ^ Zobel, 1923
 ^ There is no universallyrecognised name for this kind of filter: Zobel (1923, p.11) used the title General Wavefilters having any Preassigned Transmitting and Attenuating Bands and Propagation Constants Adjustable Without Changing one Midpoint Characteristic Impedance. Since Zobel refers to the parameters as m_{1}, m_{2} etc., the shorthand general m_{n}type seems reasonable terminology to use here.
 ^ Zobel, 1928
 ^ Zobel, 1931
References

 Campbell, G A, "Physical Theory of the Electric WaveFilter", Bell Systems Technical Journal, November 1922, vol. 1, no. 2, pp. 132.
 Zobel, O J, "Theory and Design of Uniform and Composite Electric Wave Filters", Bell Systems Technical Journal, Vol. 2 (1923).
 Foster, R M, "A reactance theorem", Bell Systems Technical Journal, Vol. 3, pp. 259–267, 1924.
 Cauer, W, "Die Verwirklichung der Wechselstromwiderst ande vorgeschriebener Frequenzabh angigkeit", Archiv f¨ur Elektrotechnik, 17, pp. 355–388, 1926.
 Zobel, O J, "Distortion correction in electrical networks with constant resistance recurrent networks", Bell Systems Technical Journal, Vol. 7 (1928), p. 438.
 Zobel, O J, Phaseshifting network, US patent 1 792 523, filed 12 March 1927, issued 17 Feb 1931.
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