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100Analog Filter

Analog Filter

About[]

Passive linear electronic analog filters are those filters which can be described by a system of linear differential equations (linear), are composed of capacitors, inductors and, sometimes, resistors (passive) and are designed to operate on continuously varying (analog) signals. There are many linear filters which are not analog in implementation (digital filter), and there are many electronic filters which may not have a passive topology - both of which may have the same transfer function of the filters described in this article. Analog filters are most often used in wave filtering applications, that is, where it is required to pass particular frequency components and to reject others from Analog (continuous-time) signals.

Analog filters have played an important part in the development of electronics. Especially in the field of telecommunications, filters have been of crucial importance in a number of technological breakthroughs and have been the source of enormous profits for telecommunications companies. It should come as no surprise, therefore, that the early development of filters was intimately connected with transmission lines. Transmission line theory gave rise to filter theory, which initially took a very similar form, and the main application of filters was for use on telecommunication transmission lines. However, the arrival of network synthesis techniques greatly enhanced the degree of control of the designer.

Today, the majority of filtering is carried out in the digital domain where complex algorithms are much easier to implement, but Analog filters do still find applications, especially for low-order simple filtering tasks. Wherever possible, however, Analog filters are now implemented in a filter topology which is active in order to avoid the wound components required by passive topology.

It is possible to design linear Analog mechanical filters using mechanical components which filter mechanical vibrations or acoustic waves. While there are few applications for such devices in mechanics per se, they can be used in electronics with the addition of transducers to convert to and from the electrical domain. Indeed some of the earliest ideas for filters were acoustic resonators because the electronics technology was poorly understood at the time. In principle, the design of such filters is completely analogous to the electronic counterpart, with kinetic energy, potential energy and heat energy corresponding to the energy in inductors, capacitors and resistors respectively.

Overview[]

There are three main stages in the history of the development of the Analog filter;

1. Simple filters. The frequency dependence of capacitors and inductors was known about from very early on. The resonance phenomenon was also familiar from an early date and it was possible to produce simple, single-branch filters with these components. Although attempts were made in the 1880s to apply them to telegraphy, these designs proved inadequate for successful frequency division multiplexing. Network analysis was not yet powerful enough to provide the theory for more complex filters and progress was further hampered by a general failure to understand the frequency domain nature of signals.

2. Image filters. Image filter theory grew out of transmission line theory and the design proceeds in a similar manner to transmission line analysis. For the first time filters could be produced that had precisely controllable passbands and other parameters. These developments took place in the 1920s and filters produced to these designs were still in widespread use in the 1980s, only declining as the use of Analog telecommunications has declined. Their immediate application was the economically important development of frequency division multiplexing for use on city-to-city and international telephony lines.

3. Network synthesis filters. The mathematical bases of network synthesis were laid in the 1930s and 1940s. After the end of World War Two network synthesis became the primary tool of filter design. Network synthesis put filter design on a firm mathematical foundation, freeing it from the mathematically sloppy techniques of image design and severing the connection with physical lines. The essence of network synthesis is that it produces a design that will (at least if implemented with ideal components) accurately reproduce the response originally specified in black box terms.

Throughout this article the letters R,L and C are used with their usual meanings to represent resistance, inductance and capacitance. In particular they are used in combination's, such as LC, to mean, for instance, a network consisting only of inductors and capacitors. Z is used for electrical impedance, any 2-pole combination of RLC elements and in some sections D is used for elasticity, the inverse of capacitance.

Resonance[]

Early filters utilized the phenomenon of resonance to filter signals. Although electrical resonance had been investigated by researchers from a very early stage, it was at first not widely understood by electrical engineers. Consequently, the much more familiar concept of acoustic resonance (which in turn, can be explained in terms of the even more familiar mechanical resonance) found its way into filter design ahead of electrical resonance.[1] Resonance can be used to achieve a filtering effect because the resonant device will respond to frequencies at, or near, to the resonant frequency but will not respond to frequencies far from resonance. Hence frequencies far from resonance are filtered out from the output of the device.

Early multiplexing[]

Hutin and Leblanc's multiple telegraph filter of 1891 showing the use of resonant cicuits in filtering.

By the 1890s electrical resonance was much more widely understood and had become a normal part of the engineer's toolkit. In 1891 Hutin and Leblanc patented an FDM scheme for telephone circuits using resonant circuit filters. Rival patents were filed in 1892 by Michael Pupin and John Stone Stone with similar ideas, priority eventually being awarded to Pupin. However, no scheme using just simple resonant circuit filters can successfully multiplex the wider bandwidth of telephone (as opposed to telegraph) channels without either an unacceptable restriction of speech bandwidth or a channel spacing so wide as to make the benefits of multiplexing uneconomic.

The basic technical reason for this difficulty is that the response of a simple filter approaches a fall of 6dB/octave far from the point of resonance. This means that if telephone channels are squeezed in side-by-side there will be crosstalk from adjacent channels in any given channel. What is required is a much more sophisticated filter that has a flat response in the required passband like a low-Q resonant circuit, but that rapidly falls in response (much faster than 6dB/octave) at the transition from passband to stopband like a high-Q resonant circuit. Obviously, these are contradictory requirements to be met with a single resonant circuit. The solution to these needs was founded in the theory of transmission lines and consequently the necessary filters did not become available until this theory was fully developed. At this early stage the idea of signal bandwidth, and hence the need for filters to match to it, was not fully understood, indeed, it was as late as 1920 before the concept of bandwidth was fully established. For early radio, the concepts of Q, selectivity and tuning sufficed. This was all to change with the developing theory of transmission lines on which image filters are based, as explained in the next section.

At the turn of the century as telephone lines became available, it became popular to add telegraph on to telephone lines with an earth return phantom circuit. An LC filter was required to prevent telegraph clicks being heard on the telephone line. From the 1920s onwards, telephone lines, or balanced lines dedicated to the purpose, were used for FDM telegraph in the audio band. The first of these systems in the UK was a Siemens and Halske installation between London and Manchester. GEC and AT&T also had FDM systems. Separate pairs were used for the send and receive signals. The Siemens and GEC systems had six channels of telegraph in each direction, the AT&T system had twelve. All of these systems used electronic oscillators to generate a different carrier for each telegraph signal and required a bank of band-pass filters to separate out the multiplexed signal at the receiving end.

Transmission line theory[]

Ohm's model of the transmission line was simply resistance. Lord Kelvin's model of the transmission line accounted for capacitance and the dispersion it caused Heaviside's model of the transmission line.

The earliest model of the transmission line could be said to be due to Georg Ohm (1827) who established that resistance in a wire is proportional to its length. The Ohm model thus included only resistance. Latimer Clark noted that signals were delayed and elongated along a cable, an undesirable form of distortion now called dispersion but then called retardation, and Michael Faraday (1853) established that this was due to the capacitance present in the transmission line. Lord Kelvin (1854) found the correct mathematical description needed in his work on early transatlantic cables; he arrived at an equation identical to the conduction of a heat pulse along a metal bar. This model incorporates only resistance and capacitance, but that is all that was needed in undersea cables dominated by capacitance effects. Kelvin's model predicts a limit on the telegraph signaling speed of a cable but Kelvin still did not use the concept of bandwidth, the limit was entirely explained in terms of the dispersion of the telegraph symbols. The mathematical model of the transmission line reached its fullest development with Oliver Heaviside. Heaviside (1881) introduced series inductance and shunt conductance into the model making four distributed elements in all. This model is now known as the telegrapher's equation and the distributed elements are called the primary line constants.

From the work of Heaviside (1887) it had become clear that the performance of telegraph, and most especially telephone, lines could be improved by the addition of inductance to the line. George Campbell at AT&T implemented this idea (1899) by inserting loading coils at intervals along the line. Campbell found that as well as the desired improvements to the line's characteristics in the passband there was also a definite frequency beyond which signals could not be passed without great attenuation. This was a result of the loading coils and the line capacitance forming a low-pass filter, an effect that is only apparent on lines incorporating lumped components such as the loading coils. This naturally lead Campbell (1910) to produce a filter with ladder topology, a glance at the circuit diagram of this filter is enough to see its relationship to a loaded transmission line. The cut-off phenomenon is an undesirable side-effect as far as loaded lines are concerned but for telephone FDM filters it is precisely what is required. For this application, Campbell produced band-pass filters to the same ladder topology by replacing the inductors and capacitors with resonators and anti-resonators respectively. Both the loaded line and FDM were of great benefit economically to AT&T and this lead to fast development of filtering from this point onwards.

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