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Electronic filtertopology defineselectronic 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 theirtransfer function rather than theirtopology. Transfer functions may belinear ornonlinear. Common types of linear filter transfer function are;high-pass,low-pass,bandpass,band-reject or notch andall-pass. Once the transfer function for a filter is chosen, the particular topology to implement such aprototype filter can be selected so that, for example, one might choose to design aButterworth filter using theSallen–Key topology.

Filter topologies may be divided intopassive andactive types. Passive topologies are composed exclusively ofpassive components: resistors, capacitors, and inductors. Active topologies also includeactive components (such as transistors, op amps, and other integrated circuits) that require power. Further, topologies may be implemented either inunbalanced form or else inbalanced form when employed inbalanced circuits. Implementations such aselectronic mixers andstereo sound may require arrays of identical circuits.

Passive filters have beenlong in development and use. Most are built from simpletwo-port 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"daisy-chain" topology, consisting of additional copies of the same section or of completely different sections. The rules of series and parallelimpedance 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 L-section, T-section and Π-section, which are unbalanced filters, and the C-section, H-section and box-section, 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 generalconstant k filters.

Filters designed usingnetwork synthesis usually repeat the simplest form of L-section 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.

L-sections are never symmetrical but two L-sections back-to-back form a symmetrical topology and many other sections are symmetrical in form.

Ladder topology, often calledCauer topology afterWilhelm Cauer (inventor of theelliptic filter), was in fact first used byGeorge Campbell (inventor of theconstant 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 L-sections (unbalanced) or C-sections (balanced). Inlow pass form the topology would consist of series inductors and shunt capacitors. Other bandforms would have an equally simple topologytransformed from the lowpass topology. The transformed network will have shunt admittances that aredual 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 Half-section H Section Box Section Ladder network X Section (mid-T-Derived) X Section (mid-Π-Derived)
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 withbalanced lines. edit

Image filter design commonly uses modifications of the basic ladder topology. These topologies, invented byOtto Zobel,^[1] have the samepassbands as the ladder on which they are based but their transfer functions are modified to improve some parameter such asimpedance matching,stopband rejection or passband-to-stopband transition steepness. Usually the design applies some transform to a simple ladder topology: the resulting topology is ladder-like 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;

• m-derived filter
• mm'-type filter
• General m_n-type filter

The m-type (m-derived) filter is by far the most commonly used modified image ladder topology. There are two m-type topologies for each of the basic ladder topologies; the series-derived and shunt-derived 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 m-type topology will result in a filter where each passband has an analogous frequency-domain response. It is possible to generalise the m-type topology for filters with more than one passband using parameters m₁, m₂, m₃ 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 m-type design. Like the m-type 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 m-type sections in the same filter for impedance matching reasons. It is normally only found in acomposite filter.

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 bridged-T and its variants, all described in theZobel network article;

• Bridged-T topology
• Balanced bridged-T topology
• Open-circuit L-section topology
• Short-circuit L-section topology
• Balanced open-circuit C-section topology
• Balanced short-circuit C-section topology

The bridged-T topology is also used in sections intended to produce a signal delay but in this case no resistive components are used in the design.

Both the T-section (from ladder topology) and the bridge-T (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 (X-sections) is inall-pass filters used forphase equalisation.^[4]

Although T and bridged-T sections can always be transformed into X-sections 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 familiarbridge topology, the difference being merely the drawn representation on the page rather than any real difference in topology, circuitry or function.

Theelementary feedback topology is based on the simpleinverting amplifier configuration. The transfer function is:

${\displaystyle H(s)=\frac{V_o}{V_i}=-\frac{Z_2}{Z_1}}$

Multiple feedback topology is an electronic filter topology which is used to implement anelectronic filter by adding two poles to thetransfer 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 second-orderlinear filters, is:

${\displaystyle H(s)=\frac{V_o}{V_i}=-\frac{1}{A{s}^2+Bs+C}=\frac{K{{\omega }_0}^2}{s^2+\frac{{\omega }_0}{Q}s+{{\omega }_0}^2}}$.

In an MF filter,

${\displaystyle A=(R_1{R}_3{C}_2{C}_5)}$

${\displaystyle B=R_3{C}_5+R_1{C}_5+R_1{R}_3{C}_5/R_4}$

${\displaystyle C=R_1/R_4}$

${\displaystyle Q=\frac{\sqrt{R_3{R}_4{C}_2{C}_5}}{(R_4+R_3+|K|R_3)C_5}}$ is thequality factor.

${\displaystyle K=-R_4/R_1}$ is the DC voltagegain

${\displaystyle {\omega }_0=2\pi f_0=1/\sqrt{R_3{R}_4{C}_2{C}_5}}$ is the corner frequency

For finding suitable component values to achieve the desired filter properties, a similar approach can be followed as in theDesign choices section of the alternative Sallen–Key topology.

For the digital implementation of a biquad filter, seeDigital biquad filter.

Abiquad filter is a type oflinear filter that implements atransfer function that is the ratio of twoquadratic functions. The namebiquad is short forbiquadratic. Any second-order filter topology can be referred to as abiquad, such as the MFB or Sallen-Key.^[5][6] However, there is also a specific "biquad" topology. It is also sometimes called the 'ring of 3' circuit.^{[citation needed]}

Biquad filters are typicallyactive and implemented with asingle-amplifier biquad (SAB) ortwo-integrator-loop topology.

• The SAB topology uses feedback to generatecomplexpoles and possibly complexzeros. In particular, the feedback moves thereal poles of anRC circuit in order to generate the proper filter characteristics.
• The two-integrator-loop 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 astate variable filter structure. By using different states as outputs, any kind of second-order filter can be implemented.

The SAB topology is sensitive to component choice and can be more difficult to adjust. Hence, usually the termbiquad refers to the two-integrator-loop state variable filter topology.

For example, the basic configuration in Figure 1 can be used as either alow-pass orbandpass filter depending on where the output signal is taken from.

The second-order low-pass transfer function is given by

${\displaystyle H(s)=\frac{G_{\mathrm{l}\mathrm{p}\mathrm{f}}{{\omega }_0}^2}{s^2+\frac{{\omega }_0}{Q}s+{{\omega }_0}^2}}$

where low-pass gain${\displaystyle G_{\mathrm{l}\mathrm{p}\mathrm{f}}=-R_2/R_1}$. The second-order bandpass transfer function is given by

${\displaystyle H(s)=\frac{G_{\mathrm{b}\mathrm{p}\mathrm{f}}\frac{{\omega }_0}{Q}s}{s^2+\frac{{\omega }_0}{Q}s+{{\omega }_0}^2}}$.

with bandpass gain${\displaystyle G_{\mathrm{b}\mathrm{p}\mathrm{f}}=-R_3/R_1}$. In both cases, the

• Natural frequency is${\displaystyle {\omega }_0=1/\sqrt{R_2{R}_4{C}_1{C}_2}}$.
• Quality factor is${\displaystyle Q=\sqrt{\frac{{R_3}^2{C}_1}{R_2{R}_4{C}_2}}}$.

The bandwidth is approximated by${\displaystyle B={\omega }_0/Q}$, and Q is sometimes expressed as adamping constant${\displaystyle \zeta =1/2Q}$. If a noninverting low-pass filter is required, the output can be taken at the output of the secondoperational amplifier, after the order of the second integrator and the inverter has been switched. 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.

Figure 2 shows a variant of the Tow-Thomas topology, known asAkerberg-Mossberg topology, that uses an actively compensated Miller integrator, which improves filter performance.

The Sallen-Key design is a non-inverting second-order filter with the option of high Q and passband gain.

• Prototype filter
• Topology (electronics)
• Linear filter
• State variable filter

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