Electronic filter topology

An electronic filter topology is an electronic analog filter circuit in which the values of the components remain undefined. A particular topology is then characterized entirely by the manner in which the components are connected, and not by their values.

There are many different types of electronic filters and they are characterized by their transfer function, but not by any particular topology. Once the transfer function for a filter is chosen, it remains to select the particular topology to implement that filter. For example, one might choose to design a Butterworth filter using the Sallen–Key topology.

Passive topologies

A passive topology is one which uses only passive components in its implementation. By passive components it is meant only components that do not, either in reality or in their equivalent circuit due to non-linearity, contain a generator of energy. In electronics terms, this means that all the components are capacitors and inductors and also, in some topologies, resistors.

In virtually all cases, passive electronic filters are built up with simple two-port networks called sections. These are invariably connected in a cascade topology and may be multiple repeats of the same section or completely different sections. There is no formal definition of what constitutes a section, but as a minimum it must have one series component and one shunt component. Two sections consisting of just series components could be combined into a single element by impedances in series and an analogous argument applies to shunt components.

Typically, filters designed using network synthesis methods will repeat the topology from section to section but the component values will change at each section. Also, the section used is invariably the simplest form of L-section. Image designed filters on the other hand, keep the same basic component values from section to section but the topology can vary along the filter. Also, image design tends to make use of the 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 topologies

Ladder topology is often called Cauer topology after Wilhelm Cauer (inventor of the Elliptical filter), but the topology was, in fact, first used by George Campbell (inventor of the Constant k filter).^[1] Cauer topology is usually thought of as being an unbalanced ladder topology. However, there are two forms of basic ladder topologies;

• Unbalanced ladder topology
• Balanced ladder topology

A ladder network is a topology built up of cascaded asymmetrical L-sections (unbalanced) or C-sections (balanced). In its lowpass bandform, the ladder topology would consist of series inductors and shunt capacitors. Other bandforms would have an equally simple topology transformed from the lowpass topology. Or to put it another way, the shunt admittance is always a dual network of the series impedance.

Filters built with ladder topology that consist of only one or two filter sections are given special names. These include the L-section, T-section and Π-section (unbalanced filters) and the C-section, H-section and box-section (balanced filters). The chart below shows these various topologies in terms of general constant k filters. They can of course, be used to implement any kind of filter.

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 with balanced lines. edit

Modified ladder topologies

series m-derived topology

In image filter design it is common to use topologies that are a modification of the basic ladder topology. These topologies, invented by Otto Zobel,^[2] have the same bandform (ie, the same passbands) as the prototype ladder on which they are based but the transfer function is modified to improve some parameter such as impedance matching, stopband rejection or passband to stopband transition steepness. Usually, the design begins with the simple ladder topology and then some transform is applied to it. The resulting topology is ladderlike but no longer obeys the rule that shunt admittances are the dual network of series impedances. The topology invariably becomes more complex and requires higher component counts. These topologies can 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, in fact, two m-type topologies for each of the basic ladder topologies. These are 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^[3] filters which have bandforms which can be dissimilar in different parts of the frequency spectrum.

The mm'-type topology can be thought of as a double m-type design. As with the m-type, it has the same bandform but improved transfer characteristics beyond the improvements achieved by m-types. It is, however, a rarely used design due to the drawback of increased component count and complexity. It also has the disadvantage of normally requiring that basic ladder sections and m-type sections are also present in the same filter for impedance matching reasons. In other words, it would normally only ever be found in a composite filter.

Bridged-T topologies

Typical bridged-T Zobel network equaliser used to correct high end roll-off

Zobel constant resistance filters^[4] use a topology that is somewhat different from other filter types. These kinds of filters are distinguished by having a constant input resistance at all frequencies and unusually 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. The topologies that are usually associated with constant resistance filters are bridged-T and its variants, all described in the Zobel 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 to build sections intended to produce a signal delay. In the case of delay sections, there are no resistive components used in the design.

Lattice topology

Lattice topology X-section phase correction filter

Both the T-section from ladder topology and the bridge-T from Zobel topology can be transformed into a lattice topology filter section. However, in both cases this results in a filter with higher component count and complexity so it does not see much general purpose use. The most common application of lattice filters (X-sections) is in all-pass filters used for phase equalisation.^[5]

Although T and bridged-T sections can always be transformed into X-sections, the reverse is not always possible. This is 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

It has been suggested that Multiple feedback topology (electronics), State variable topology and Biquadratic topology be merged into this article or section. (Discuss)

• Sallen–Key topology
• Multiple Feedback topology
• State variable topology
• Biquadratic topology

See also

• Prototype filter
• Topology (electronics)

Notes

References