Passive analogue filter development
Encyclopedia
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, mid-range
and tweeter
loudspeaker
s; 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.
Passive linear electronic analogue filters are those filters which can be described with linear differential equation
s (linear); they are composed of capacitor
s, inductor
s and, sometimes, resistor
s (passive) and are designed to operate on continuously varying (analogue) signals. There are many linear filter
s which are not analogue in implementation (digital filter
), and there are many electronic filter
s which may not have a passive topology – both of which may have the same transfer function
of the filters described in this article. Analogue 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 analogue (continuous-time
) signals.
Analogue filters have played an important part in the development of electronics. Especially in the field of telecommunication
s, 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 line
s. 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, it is often preferred to carry out filtering in the digital domain where complex algorithms are much easier to implement, but analogue filters do still find applications, especially for low-order simple filtering tasks and are often still the norm at higher frequencies where digital technology is still impractical, or at least, less cost effective. Wherever possible, and especially at low frequencies, analogue 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 analogue mechanical filter
s 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 transducer
s 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 can be achieved entirely in terms of the electronic counterparts of mechanical quantities, with kinetic energy
, potential energy
and heat energy corresponding to the energy in inductors, capacitors and resistors respectively.
Throughout this article the letters R,L and C are used with their usual meanings to represent resistance
, inductance
and capacitance
, respectively. In particular they are used in combinations, such as LC, to mean, for instance, a network consisting only of inductors and capacitors. Z is used for electrical impedance
, any 2-terminal combination of RLC elements and in some sections D is used for the rarely seen quantity elastance, which is the inverse of capacitance.
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. 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.
Resonance was noticed early on in experiments with the Leyden jar
, invented in 1746. The Leyden jar stores electricity due to its capacitance
, and is, in fact, an early form of capacitor. When a Leyden jar is discharged by allowing a spark to jump between the electrodes, the discharge is oscillatory. This was not suspected until 1826, when Felix Savary
in France, and later (1842) Joseph Henry
in the US noted that a steel needle placed close to the discharge does not always magnetise in the same direction. They both independently drew the conclusion that there was a transient oscillation dying with time.
Hermann von Helmholtz
in 1847 published his important work on conservation of energy in part of which he used those principles to explain why the oscillation dies away, that it is the resistance of the circuit which dissipates the energy of the oscillation on each successive cycle. Helmholtz also noted that there was evidence of oscillation from the electrolysis
experiments of William Hyde Wollaston
. Wollaston was attempting to decompose water by electric shock but found that both hydrogen and oxygen were present at both electrodes. In normal electrolysis they would separate, one to each electrode.
Helmholtz explained why the oscillation decayed but he had not explained why it occurred in the first place. This was left to Sir William Thomson (Lord Kelvin) who, in 1853, postulated that there was inductance present in the circuit as well as the capacitance of the jar and the resistance of the load. This established the physical basis for the phenomenon - the energy supplied by the jar was partly dissipated in the load but also partly stored in the magnetic field of the inductor.
So far, the investigation had been on the natural frequency of transient oscillation of a resonant circuit resulting from a sudden stimulus. More important from the point of view of filter theory is the behaviour of a resonant circuit when driven by an external AC
signal: there is a sudden peak in the circuits response when the driving signal frequency is at the resonant frequency of the circuit.The resonant frequency is very close to, but usually not exactly equal to, the natural frequency of oscillation of the circuit James Clerk Maxwell
heard of the phenomenon from Sir William Grove in 1868 in connection with experiments on dynamo
s, and was also aware of the earlier work of Henry Wilde
in 1866. Maxwell explained resonanceOliver Lodge and some other English scientists tried to keep acoustic and electric terminology separate and promoted the term "syntony". However it was "resonance" that was to win the day. Blanchard, p.422 mathematically, with a set of differential equations, in much the same terms that an RLC circuit
is described today.
Heinrich Hertz (1887) experimentally demonstrated the resonance phenomena by building two resonant circuits, one of which was driven by a generator and the other was tunable
and only coupled to the first electromagnetically (i.e., no circuit connection). Hertz showed that the response of the second circuit was at a maximum when it was in tune with the first. The diagrams produced by Hertz in this paper were the first published plots of an electrical resonant response.
, Alexander Graham Bell
(1870s), Ernest Mercadier and others. Its purpose was to simultaneously transmit a number of telegraph messages over the same line and represents an early form of frequency division multiplexing (FDM). FDM requires the sending end to be transmitting at different frequencies for each individual communication channel. This demands individual tuned resonators, as well as filters to separate out the signals at the receiving end. The harmonic telegraph achieved this with electromagnetically driven tuned reeds at the transmitting end which would vibrate similar reeds at the receiving end. Only the reed with the same resonant frequency as the transmitter would vibrate to any appreciable extent at the receiving end.
Incidentally, the harmonic telegraph directly suggested to Bell the idea of the telephone. The reeds can be viewed as transducer
s converting sound to and from an electrical signal. It is no great leap from this view of the harmonic telegraph to the idea that speech can be converted to and from an electrical signal.
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
(i.e. combine) the wider bandwidth of telephone channels (as opposed to telegraph) 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 frequency response of a simple filter approaches a fall of 6 dB/octave
far from the point of resonance. This means that if telephone channels are squeezed in side-by-side into the frequency spectrum, there will be crosstalk from adjacent channels in any given channel. What is required is a much more sophisticated filter that has a flat frequency response in the required passband
like a low-Q
resonant circuit, but that rapidly falls in response (much faster than 6 dB/octave) at the transition from passband to stopband
like a high-Q resonant circuit.Q factor
is a dimensionless quantity enumerating the quality of a resonating circuit. It is roughly proportional to the number of oscillations, which a resonator would support after a single external excitation (for example, how many times a guitar string would wobble if pulled). One definition of Q factor, the most relevant one in this context, is the ratio of resonant frequency to bandwidth of a circuit. It arose as a measure of selectivity in radio receivers 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-factor, selectivity and tuning sufficed. This was all to change with the developing theory of transmission line
s 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
.Telegraph lines are typically unbalanced
with only a single conductor provided, the return path is achieved through an earth
connection which is common to all the telegraph lines on a route. Telephone lines are typically balanced
with two conductors per circuit. A telegraph signal connected common-mode
to both conductors of the telephone line will not be heard at the telephone receiver which can only detect voltage differences between the conductors. The telegraph signal is typically recovered at the far end by connection to the center tap
of a line transformer
. The return path is via an earth connection as usual. This is a form of 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 at audio frequencies. 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.
The earliest model of the transmission line
was probably described by Georg Ohm
(1827) who established that resistance in a wire is proportional to its length.At least, Ohm described the first model that was in any way correct. Earlier ideas such as Barlow's law
from Peter Barlow were either incorrect, or inadequately described. See, for example. p.603 of;
*John C. Shedd, Mayo D. Hershey, "The history of Ohm's law", The Popular Science Monthly, pp.599-614, December 1913 ISSN 0161-7370. 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.Werner von Siemens had also noted the retardation effect a few years earlier in 1849 and came to a similar conclusion as Faraday. However, there was not so much interest in Germany in underwater and underground cables as there was in Britain, the German overhead cables did not noticeably suffer from retardation and Siemen's ideas were not accepted. (Hunt, p.65.) 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 signalling 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 lines, 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 coil
s 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 led 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 filter
s to the same ladder topology by replacing the inductors and capacitors with resonator
s and anti-resonators respectively.The exact date Campbell produced each variety of filter is not clear. The work started in 1910, initially patented in 1917 (US1227113) and the full theory published in 1922, but it is known that Campbell's filters were in use by AT&T long before the 1922 date (Bray, p.62, Darlington, p.5) Both the loaded line and FDM were of great benefit economically to AT&T and this led to fast development of filtering from this point onwards.
The filters designed by CampbellCampbell has publishing priority for this invention but it is worth noting that Karl Willy Wagner
independently made a similar discovery which he was not allowed to publish immediately because World War I
was still ongoing. (Thomas H. Lee, Planar microwave engineering, p.725, Cambridge University Press 2004 ISBN 0-521-83526-7.) were named wave filters because of their property of passing some waves and strongly rejecting others. The method by which they were designed was called the image parameter methodThe term "image parameter method" was coined by Darlington (1939) in order to distinguish this earlier technique from his later "insertion-loss method" and filters designed to this method are called image filters.The terms wave filter and image filter are not synonymous, it is possible for a wave filter to not be designed by the image method, but in the 1920s the distinction was moot as the image method was the only one available The image method essentially consists of developing the transmission constants of an infinite chain of identical filter sections and then terminating the desired finite number of filter sections in the image impedance
. This exactly corresponds to the way the properties of a finite length of transmission line are derived from the theoretical properties of an infinite line, the image impedance corresponding to the characteristic impedance
of the line.
From 1920 John Carson
, also working for AT&T, began to develop a new way of looking at signals using the operational calculus
of Heaviside which in essence is working in the frequency domain
. This gave the AT&T engineers a new insight into the way their filters were working and led Otto Zobel to invent many improved forms. Carson and Zobel steadily demolished many of the old ideas. For instance the old telegraph engineers thought of the signal as being a single frequency and this idea persisted into the age of radio with some still believing that frequency modulation
(FM) transmission could be achieved with a smaller bandwidth than the baseband
signal right up until the publication of Carson's 1922 paper. Another advance concerned the nature of noise, Carson and Zobel (1923) treated noise as a random process with a continuous bandwidth, an idea that was well ahead of its time, and thus limited the amount of noise that it was possible to remove by filtering to that part of the noise spectrum which fell outside the passband. This too, was not generally accepted at first, notably being opposed by Edwin Armstrong
(who ironically, actually succeeded in reducing noise with wide-band FM) and was only finally settled with the work of Harry Nyquist
whose thermal noise power formula
is well known today.
Several improvements were made to image filters and their theory of operation by Otto Zobel. Zobel coined the term constant k filter
(or k-type filter) to distinguish Campbell's filter from later types, notably Zobel's m-derived filter
(or m-type filter). The particular problems Zobel was trying to address with these new forms were impedance matching into the end terminations and improved steepness of roll-off. These were achieved at the cost of an increase in filter circuit complexity.
A more systematic method of producing image filters was introduced by Hendrik Bode (1930), and further developed by several other investigators including Piloty (1937-1939) and Wilhelm Cauer
(1934-1937). Rather than enumerate the behaviour (transfer function, attenuation function, delay function and so on) of a specific circuit, instead a requirement for the image impedance itself was developed. The image impedance can be expressed in terms of the open-circuit and short-circuit impedancesThe open-circuit impedance of a two-port network is the impedance looking into one port when the other port is open circuit. Similarly, the short-circuit impedance is the impedance looking into one port when the other is terminated in a short circuit. The open-circuit impedance of the first port in general (except for symmetrical networks) is not equal to the open-circuit impedance of the second and likewise for short-circuit impedances of the filter as
. Since the image impedance must be real in the passbands and imaginary in the stopbands according to image theory, there is a requirement that the poles and zeroes
of Zo and Zs cancel in the passband and correspond in the stopband. The behaviour of the filter can be entirely defined in terms of the positions in the complex plane
of these pairs of poles and zeroes. Any circuit which has the requisite poles and zeroes will also have the requisite response. Cauer pursued two related questions arising from this technique: what specification of poles and zeroes are realisable as passive filters; and what realisations are equivalent to each other. The results of this work led Cauer to develop a new approach, now called network synthesis.
This "poles and zeroes" view of filter design was particularly useful where a bank of filters, each operating at different frequencies, are all connected across the same transmission line. The earlier approach was unable to deal properly with this situation, but the poles and zeroes approach could embrace it by specifying a constant impedance for the combined filter. This problem was originally related to FDM telephony but frequently now arises in loudspeaker crossover filters
.
which starts with a given network and by applying the various electric circuit theorems predicts the response of the network. The term was first used with this meaning in the doctoral thesis of Yuk-Wing Lee
(1930) and apparently arose out of a conversation with Vannevar Bush
. The advantage of network synthesis over previous methods is that it provides a solution which precisely meets the design specification. This is not the case with image filters, a degree of experience is required in their design since the image filter only meets the design specification in the unrealistic case of being terminated in its own image impedance, to produce which would require the exact circuit being sought. Network synthesis on the other hand, takes care of the termination impedances simply by incorporating them into the network being designed.
The development of network analysis needed to take place before network synthesis was possible. The theorems of Gustav Kirchhoff
and others and the ideas of Charles Steinmetz (phasors) and Arthur Kennelly (complex impedance) laid the groundwork. The concept of a port
also played a part in the development of the theory, and proved to be a more useful idea than network terminals. The first milestone on the way to network synthesis was an important paper by Ronald Foster (1924), A Reactance Theorem, in which Foster introduces the idea of a driving point impedance, that is, the impedance that is connected to the generator. The expression for this impedance determines the response of the filter and vice versa, and a realisation of the filter can be obtained by expansion of this expression. It is not possible to realise any arbitrary impedance expression as a network. Foster's reactance theorem stipulates necessary and sufficient conditions for realisability: that the reactance must be algebraically increasing with frequency and the poles and zeroes must alternate.
Wilhelm Cauer
expanded on the work of Foster (1926) and was the first to talk of realisation of a one-port impedance with a prescribed frequency function. Foster's work considered only reactances (i.e., only LC-kind circuits). Cauer generalised this to any 2-element kind one-port network, finding there was an isomorphism between them. He also found ladder realisations which is the best known of the filter topologies. It is for this reason that ladder topology is often referred to as Cauer topology (the forms used earlier by Foster are quite different) even though ladder topology had long since been in use in image filter design of the network using Thomas Stieltjes' continued fraction expansion. This work was the basis on which network synthesis was built, although Cauer's work was not at first used much by engineers, partly because of the intervention of World War II, partly for reasons explained in the next section and partly because Cauer presented his results using topologies that required mutually coupled inductors and ideal transformers. Although on this last point, it has to be said that transformer coupled double tuned amplifiers are a common enough way of widening bandwidth without sacrificing selectivity.
The computational difficulty of the network synthesis method was addressed by tabulating the component values of a prototype filter
and then scaling the frequency and impedance and transforming the bandform to those actually required. This kind of approach, or similar, was already in use with image filters, for instance by Zobel, but the concept of a "reference filter" is due to Sidney Darlington
. Darlington (1939), was also the first to tabulate values for network synthesis prototype filters, nevertheless it had to wait until the 1950s before the Cauer-Darlington elliptic filter first came into use.
Once computational power was readily available, it became possible to easily design filters to minimise any arbitrary parameter, for example time delay or tolerance to component variation. The difficulties of the image method were firmly put in the past, and even the need for prototypes became largely superfluous. Furthermore, the advent of active filter
s eased the computation difficulty because sections could be isolated and iterative processes were not then generally necessary.
, Cauer formed the matrix equation,
where [Z],[R],[L] and [D] are the nxn matrices of, respectively, impedance
, resistance
, inductance
and elastance of an n-mesh
network and s is the complex frequency operator
. Here [R],[L] and [D] have associated energies corresponding to the kinetic, potential and dissipative heat energies, respectively, in a mechanical system and the already known results from mechanics could be applied here. Cauer determined the driving point impedance by the method of Lagrange multipliers
;
where a11 is the complement of the element A11 to which the one-port is to be connected. From stability theory
Cauer found that [R], [L] and [D] must all be positive-definite matrices
for Zp(s) to be realisable if ideal transformers are not excluded. Realisability is only otherwise restricted by practical limitations on topology. This work is also partly due to Otto Brune (1931), who worked with Cauer in the US prior to Cauer returning to Germany. A well known condition for realisability of a one-port rationalA rational impedance is one expressed as a ratio of two finite polynomials in s, that is, a rational function
in s. The implication of finite polynomials is that the impedance, when realised, will consist of a finite number of meshes with a finite number of elements impedance due to Cauer (1929) is that it must be a function of s that is analytic in the right halfplane (σ>0), have a positive real part in the right halfplane and take on real values on the real axis. This follows from the Poisson integral representation of these functions. Brune coined the term positive-real for this class of function and proved that it was a necessary and sufficient condition (Cauer had only proved it to be necessary) and they extended the work to LC multiports. A theorem due to Sidney Darlington
states that any positive-real function Z(s) can be realised as a lossless two-port terminated in a positive resistor R. No resistors within the network are necessary to realise the specified response.
As for equivalence, Cauer found that the group of real affine transformation
s,
is invariant in Zp(s), that is, all the transformed networks are equivalents of the original.
, becoming ever closer to the ideal the higher the order of the polynomial. The first to address this problem was Stephen Butterworth
(1930) using his Butterworth polynomials. Independently, Cauer (1931) used Chebyshev polynomials
, initially applied to image filters, and not to the now well-known ladder realisation of this filter.
must be a ratio of polynomial functions. The order of a filter is the order of the highest order polynomial of the two and will equal the number of elements (or resonators) required to build it. Usually, the higher the order of a filter, the steeper the roll-off of the filter will be. In general, the values of the elements in each section of the filter will not be the same if the order is increased and will need to be recalculated. This is in contrast to the image method of design which simply adds on more identical sections of filters due to Stephen Butterworth
(1930) which are now recognised as being a special case of Cauer's elliptic filter
s. Butterworth discovered this filter independently of Cauer's work and implemented it in his version with each section isolated from the next with a valve amplifier
which made calculation of component values easy since the filter sections could not interact with each other and each section represented one term in the Butterworth polynomials. This gives Butterworth the credit for being both the first to deviate from image parameter theory and the first to design active filters. It was later shown that Butterworth filters could be implemented in ladder topology without the need for amplifiers, possibly the first to do so was William Bennett (1932) in a patent which presents formulae for component values identical to the modern ones. Bennett, at this stage though, is still discussing the design as an artificial transmission line and so is adopting an image parameter approach despite having produced what would now be considered a network synthesis design. He also does not appear to be aware of the work of Butterworth or the connection between them.
, Wilhelm Cauer and others all working more or less independently and is often taken as synonymous with network synthesis. Butterworth's filter implementation is, in those terms, an insertion-loss filter, but it is a relatively trivial one mathematically since the active amplifiers used by Butterworth ensured that each stage individually worked into a resistive load. Butterworth's filter becomes a non-trivial example when it is implemented entirely with passive components. An even earlier filter which influenced the insertion-loss method was Norton's dual-band filter where the input of two filters are connected in parallel and designed so that the combined input presents a constant resistance. Norton's design method, together with Cauer's canonical LC networks and Darlington's theorem that only LC components were required in the body of the filter resulted in the insertion-loss method. However, ladder topology proved to be more practical than Cauer's canonical forms.
Darlington's insertion-loss method is a generalisation of the procedure used by Norton. In Norton's filter it can be shown that each filter is equivalent to a separate filter unterminated at the common end. Darlington's method applies to the more straightforward and general case of a 2-port LC network terminated at both ends. The procedure consists of the following steps:
Darlington additionally used a transformation found by Hendrik Bode that predicted the response of a filter using non-ideal components but all with the same Q. Darlington used this transformation in reverse to produce filters with a prescribed insertion-loss with non-ideal components. Such filters have the ideal insertion-loss response plus a flat attenuation across all frequencies.
in their transfer function as an approximation to the ideal filter response and the result is called a Chebyshev approximation. This is the same Chebyshev approximation technique used by Cauer on image filters but follows the Darlington insertion-loss design method and uses slightly different elliptic functions. Cauer had some contact with Darlington and Bell Labs before WWII (for a time he worked in the US) but during the war they worked independently, in some cases making the same discoveries. Cauer had disclosed the Chebyshev approximation to Bell Labs but had not left them with the proof. Sergei Schelkunoff
provided this and a generalisation to all equal ripple problems. Elliptic filters are a general class of filter which incorporate several other important classes as special cases: Cauer filter (equal ripple in passband and stopband
), Chebyshev filter (ripple only in passband), reverse Chebyshev filter (ripple only in stopband) and Butterworth filter (no ripple in either band).
Generally, for insertion-loss filters where the transmission zeroes and infinite losses are all on the real axis of the complex frequency plane (which they usually are for minimum component count), the insertion-loss function can be written as;
where F is either an even (resulting in an antimetric
filter) or an odd (resulting in an symmetric filter) function of frequency. Zeroes of F correspond to zero loss and the poles of F correspond to transmission zeroes. J sets the passband ripple height and the stopband loss and these two design requirements can be interchanged. The zeroes and poles of F and J can be set arbitrarily. The nature of F determines the class of the filter;
A Chebyshev response simultaneously in the passband and stopband is possible, such as Cauer's equal ripple elliptic filter.
Darlington relates that he found in the New York City library Carl Jacobi's original paper on elliptic functions, published in Latin in 1829. In this paper Darlington was surprised to find foldout tables of the exact elliptic function transformations needed for Chebyshev approximations of both Cauer's image parameter, and Darlington's insertion-loss filters.
Edward Norton
, around 1930, designed a mechanical filter for use on phonograph
recorders and players. Norton designed the filter in the electrical domain and then used the correspondence of mechanical quantities to electrical quantities to realise the filter using mechanical components. Mass
corresponds to inductance
, stiffness
to elastance and damping
to resistance
. The filter was designed to have a maximally flat frequency response.
In modern designs it is common to use quartz crystal filter
s, especially for narrowband filtering applications. The signal exists as a mechanical acoustic wave while it is in the crystal and is converted by transducer
s between the electrical and mechanical domains at the terminals of the crystal.
filters are not usually associated with passive implementations but the concept can be found in a Wiener and Lee patent from 1935 which describes a filter consisting of a cascade of all-pass section
s. The outputs of the various sections are summed in the proportions needed to result in the required frequency function. This works by the principle that certain frequencies will be in, or close to antiphase, at different sections and will tend to cancel when added. These are the frequencies rejected by the filter and can produce filters with very sharp cut-offs. This approach did not find any immediate applications, and is not common in passive filters. However, the principle finds many applications as an active delay line implementation for wide band discrete-time filter applications such as television, radar and high-speed data transmission.
(S/N) at the expense of pulse shape. Pulse shape, unlike many other applications, is unimportant in radar while S/N is the primary limitation on performance. The filters were introduced during WWII (described 1943) by Dwight North and are often eponymously referred to as "North filters
".
with the specific application to anti-aircraft fire control analogue computers. Rudy Kalman (Kalman filter
) later reformulated this in terms of state-space
smoothing and prediction where it is known as the linear-quadratic-Gaussian control
problem. Kalman started an interest in state-space solutions, but according to Darlington this approach can also be found in the work of Heaviside and earlier.
s, became cheaply available. The reason for the change is that wound components (the usual method of manufacture for inductors) are far from ideal, the wire adding resistance as well as inductance to the component. Inductors are also relatively expensive and are not "off-the-shelf" components. On the other hand, the function of LC ladder sections, LC resonators and RL sections can be replaced by RC components in an amplifier feedback loop (active filters). These components will usually be much more cost effective, and smaller as well. Cheap digital technology, in its turn, has largely supplanted analogue implementations of filters. However, there is still an occasional place for them in the simpler applications such as coupling where sophisticated functions of frequency are not needed.
Filter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
are a basic building block of signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
much used in electronics
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
. Amongst their many applications are the separation of an audio signal before application to bass, mid-range
Mid-range speaker
A loudspeaker driver that produces the frequency range from approximately 300–5000 hertz is known as a mid-range.Midrange drivers are usually cone types or, less commonly, dome types, or compression horn drivers...
and tweeter
Tweeter
A tweeter is a loudspeaker designed to produce high audio frequencies, typically from around 2,000 Hz to 20,000 Hz . Some tweeters can manage response up to 65 kHz...
loudspeaker
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...
s; the combining and later separation of multiple telephone conversations onto a single channel; the selection of a chosen radio station
Radio station
Radio broadcasting is a one-way wireless transmission over radio waves intended to reach a wide audience. Stations can be linked in radio networks to broadcast a common radio format, either in broadcast syndication or simulcast or both...
in a radio receiver and rejection of others.
Passive linear electronic analogue filters are those filters which can be described with linear differential equation
Linear differential equation
Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
s (linear); they are composed of capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s, inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s and, sometimes, resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s (passive) and are designed to operate on continuously varying (analogue) signals. There are many linear filter
Linear filter
Linear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity.This results from systems composed solely of components classified as having a linear response....
s which are not analogue in implementation (digital filter
Digital filter
In electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is...
), and there are many electronic filter
Electronic filter
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
s which may not have a passive topology – both of which may have the same transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
of the filters described in this article. Analogue 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 analogue (continuous-time
Continuous signal
A continuous signal or a continuous-time signal is a varying quantity whose domain, which is often time, is a continuum . That is, the function's domain is an uncountable set. The function itself need not be continuous...
) signals.
Analogue filters have played an important part in the development of electronics. Especially in the field of telecommunication
Telecommunication
Telecommunication is the transmission of information over significant distances to communicate. In earlier times, telecommunications involved the use of visual signals, such as beacons, smoke signals, semaphore telegraphs, signal flags, and optical heliographs, or audio messages via coded...
s, 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 line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
s. 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
Network synthesis filters
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...
techniques greatly enhanced the degree of control of the designer.
Today, it is often preferred to carry out filtering in the digital domain where complex algorithms are much easier to implement, but analogue filters do still find applications, especially for low-order simple filtering tasks and are often still the norm at higher frequencies where digital technology is still impractical, or at least, less cost effective. Wherever possible, and especially at low frequencies, analogue filters are now implemented in a filter topology
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....
which is active in order to avoid the wound components required by passive topology.
It is possible to design linear analogue mechanical filter
Mechanical filter
A mechanical filter is a signal processing filter usually used in place of an electronic filter at radio frequencies. Its purpose is the same as that of a normal electronic filter: to pass a range of signal frequencies, but to block others. The filter acts on mechanical vibrations which are the...
s using mechanical components which filter mechanical vibrations or acoustic
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
waves. While there are few applications for such devices in mechanics per se, they can be used in electronics with the addition of transducer
Transducer
A transducer is a device that converts one type of energy to another. Energy types include electrical, mechanical, electromagnetic , chemical, acoustic or thermal energy. While the term transducer commonly implies the use of a sensor/detector, any device which converts energy can be considered a...
s 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 can be achieved entirely in terms of the electronic counterparts of mechanical quantities, with kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
, potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
and heat energy corresponding to the energy in inductors, capacitors and resistors respectively.
Historical overview
There are three main stages in the history of passive analogue filter development:- Simple filters. The frequency dependence of electrical response was known for capacitors and inductors 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 telegraphyTelegraphyTelegraphy is the long-distance transmission of messages via some form of signalling technology. Telegraphy requires messages to be converted to a code which is known to both sender and receiver...
, 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 domainFrequency domainIn electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
nature of signals. - Image filtersComposite image filterA composite image filter is an electronic filter consisting of multiple image filter sections of two or more different types.The image method of filter design determines the properties of filter sections by calculating the properties they have in an infinite chain of such sections. In this, the...
. Image filter theory grew out of transmission line theory and the design proceeded in a similar manner to transmission line analysis. For the first time filters could be produced that had precisely controllable passbandPassbandA passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated.A bandpass filtered signal , is known as a bandpass signal, as opposed to a baseband signal....
s 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 analogue telecommunications has declined. Their immediate application was the economically important development of frequency division multiplexing for use on intercity and international telephonyTelephonyIn telecommunications, telephony encompasses the general use of equipment to provide communication over distances, specifically by connecting telephones to each other....
lines. - Network synthesis filtersNetwork synthesis filtersNetwork 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...
. The mathematical bases of network synthesis were laid in the 1930s and 1940s. After the end of World War IIWorld War IIWorld War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
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 boxBlack boxA black box is a device, object, or system whose inner workings are unknown; only the input, transfer, and output are known characteristics.The term black box can also refer to:-In science and technology:*Black box theory, a philosophical theory...
terms.
Throughout this article the letters R,L and C are used with their usual meanings to represent resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
, inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
and capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
, respectively. In particular they are used in combinations, such as LC, to mean, for instance, a network consisting only of inductors and capacitors. Z is used for electrical impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
, any 2-terminal combination of RLC elements and in some sections D is used for the rarely seen quantity elastance, which is the inverse of capacitance.
Resonance
Early filters utilised the phenomenon of resonanceResonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
to filter signals. Although electrical resonance
Electrical resonance
Electrical resonance occurs in an electric circuit at a particular resonance frequency where the imaginary parts of circuit element impedances or admittances cancel each other...
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
Acoustic resonance
Acoustic resonance is the tendency of an acoustic system to absorb more energy when it is forced or driven at a frequency that matches one of its own natural frequencies of vibration than it does at other frequencies....
(which in turn, can be explained in terms of the even more familiar mechanical resonance
Mechanical resonance
Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies...
) found its way into filter design ahead of electrical resonance. 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.
Electrical resonance
Leyden jar
A Leyden jar, or Leiden jar, is a device that "stores" static electricity between two electrodes on the inside and outside of a jar. It was invented independently by German cleric Ewald Georg von Kleist on 11 October 1745 and by Dutch scientist Pieter van Musschenbroek of Leiden in 1745–1746. The...
, invented in 1746. The Leyden jar stores electricity due to its capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
, and is, in fact, an early form of capacitor. When a Leyden jar is discharged by allowing a spark to jump between the electrodes, the discharge is oscillatory. This was not suspected until 1826, when Felix Savary
Félix Savary
Félix Savary, who was born on October 4, 1797 in Paris and died on July 15, 1841 in Estagel, was a French astronomer.He studied at the École Polytechnique, where he was later a professor of astronomy...
in France, and later (1842) Joseph Henry
Joseph Henry
Joseph Henry was an American scientist who served as the first Secretary of the Smithsonian Institution, as well as a founding member of the National Institute for the Promotion of Science, a precursor of the Smithsonian Institution. During his lifetime, he was highly regarded...
in the US noted that a steel needle placed close to the discharge does not always magnetise in the same direction. They both independently drew the conclusion that there was a transient oscillation dying with time.
Hermann von Helmholtz
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...
in 1847 published his important work on conservation of energy in part of which he used those principles to explain why the oscillation dies away, that it is the resistance of the circuit which dissipates the energy of the oscillation on each successive cycle. Helmholtz also noted that there was evidence of oscillation from the electrolysis
Electrolysis
In chemistry and manufacturing, electrolysis is a method of using a direct electric current to drive an otherwise non-spontaneous chemical reaction...
experiments of William Hyde Wollaston
William Hyde Wollaston
William Hyde Wollaston FRS was an English chemist and physicist who is famous for discovering two chemical elements and for developing a way to process platinum ore.-Biography:...
. Wollaston was attempting to decompose water by electric shock but found that both hydrogen and oxygen were present at both electrodes. In normal electrolysis they would separate, one to each electrode.
Helmholtz explained why the oscillation decayed but he had not explained why it occurred in the first place. This was left to Sir William Thomson (Lord Kelvin) who, in 1853, postulated that there was inductance present in the circuit as well as the capacitance of the jar and the resistance of the load. This established the physical basis for the phenomenon - the energy supplied by the jar was partly dissipated in the load but also partly stored in the magnetic field of the inductor.
So far, the investigation had been on the natural frequency of transient oscillation of a resonant circuit resulting from a sudden stimulus. More important from the point of view of filter theory is the behaviour of a resonant circuit when driven by an external AC
Alternating current
In alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....
signal: there is a sudden peak in the circuits response when the driving signal frequency is at the resonant frequency of the circuit.The resonant frequency is very close to, but usually not exactly equal to, the natural frequency of oscillation of the circuit James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
heard of the phenomenon from Sir William Grove in 1868 in connection with experiments on dynamo
Dynamo
- Engineering :* Dynamo, a magnetic device originally used as an electric generator* Dynamo theory, a theory relating to magnetic fields of celestial bodies* Solar dynamo, the physical process that generates the Sun's magnetic field- Software :...
s, and was also aware of the earlier work of Henry Wilde
Henry Wilde (engineer)
Henry Wilde was a wealthy individual from Manchester, England who used his self-made fortune to indulge his interest in electrical engineering. He invented the dynamo-electric machine, or self-energising dynamo, an invention for which Werner von Siemens is more usually credited and, in fact,...
in 1866. Maxwell explained resonanceOliver Lodge and some other English scientists tried to keep acoustic and electric terminology separate and promoted the term "syntony". However it was "resonance" that was to win the day. Blanchard, p.422 mathematically, with a set of differential equations, in much the same terms that an RLC circuit
RLC circuit
An RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively...
is described today.
Heinrich Hertz (1887) experimentally demonstrated the resonance phenomena by building two resonant circuits, one of which was driven by a generator and the other was tunable
Tuner (radio)
A radio tuner is a subsystem that receives radio broadcasts and converts them into audio-frequency signals which can be fed into an amplifier driving a loudspeaker. FM tuner, AM tuner, Digital Audio Broadcasting DAB tuner, etc. are types of radio tuner dealing with transmissions using different...
and only coupled to the first electromagnetically (i.e., no circuit connection). Hertz showed that the response of the second circuit was at a maximum when it was in tune with the first. The diagrams produced by Hertz in this paper were the first published plots of an electrical resonant response.
Acoustic resonance
As mentioned earlier, it was acoustic resonance that inspired filtering applications, the first of these being a telegraph system known as the "harmonic telegraph". Versions are due to Elisha GrayElisha Gray
Elisha Gray was an American electrical engineer who co-founded the Western Electric Manufacturing Company...
, Alexander Graham Bell
Alexander Graham Bell
Alexander Graham Bell was an eminent scientist, inventor, engineer and innovator who is credited with inventing the first practical telephone....
(1870s), Ernest Mercadier and others. Its purpose was to simultaneously transmit a number of telegraph messages over the same line and represents an early form of frequency division multiplexing (FDM). FDM requires the sending end to be transmitting at different frequencies for each individual communication channel. This demands individual tuned resonators, as well as filters to separate out the signals at the receiving end. The harmonic telegraph achieved this with electromagnetically driven tuned reeds at the transmitting end which would vibrate similar reeds at the receiving end. Only the reed with the same resonant frequency as the transmitter would vibrate to any appreciable extent at the receiving end.
Incidentally, the harmonic telegraph directly suggested to Bell the idea of the telephone. The reeds can be viewed as transducer
Transducer
A transducer is a device that converts one type of energy to another. Energy types include electrical, mechanical, electromagnetic , chemical, acoustic or thermal energy. While the term transducer commonly implies the use of a sensor/detector, any device which converts energy can be considered a...
s converting sound to and from an electrical signal. It is no great leap from this view of the harmonic telegraph to the idea that speech can be converted to and from an electrical signal.
Early multiplexing
John Stone Stone
John Stone Stone was an American mathematician, physicist and inventor. He labored as an early telephone engineer, was influential in developing wireless communication technology, and holds dozens of key patents in the field of "space telegraphy".-Early years:Stone was born in Dover, now Manakin...
with similar ideas, priority eventually being awarded to Pupin. However, no scheme using just simple resonant circuit filters can successfully multiplex
Multiplexing
The multiplexed signal is transmitted over a communication channel, which may be a physical transmission medium. The multiplexing divides the capacity of the low-level communication channel into several higher-level logical channels, one for each message signal or data stream to be transferred...
(i.e. combine) the wider bandwidth of telephone channels (as opposed to telegraph) 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 frequency response of a simple filter approaches a fall of 6 dB/octave
Octave (electronics)
In electronics, an octave is a doubling or halving of a frequency. The term is derived from the musical octave which similarly describes such frequency ratios, but the prefix octa-, denoting eight, has no significance in physics...
far from the point of resonance. This means that if telephone channels are squeezed in side-by-side into the frequency spectrum, there will be crosstalk from adjacent channels in any given channel. What is required is a much more sophisticated filter that has a flat frequency response in the required passband
Passband
A passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated.A bandpass filtered signal , is known as a bandpass signal, as opposed to a baseband signal....
like a low-Q
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....
resonant circuit, but that rapidly falls in response (much faster than 6 dB/octave) at the transition from passband to stopband
Stopband
A stopband is a band of frequencies, between specified limits, through which a circuit, such as a filter or telephone circuit, does not allow signals to pass, or the attenuation is above the required stopband attenuation level...
like a high-Q resonant circuit.Q factor
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....
is a dimensionless quantity enumerating the quality of a resonating circuit. It is roughly proportional to the number of oscillations, which a resonator would support after a single external excitation (for example, how many times a guitar string would wobble if pulled). One definition of Q factor, the most relevant one in this context, is the ratio of resonant frequency to bandwidth of a circuit. It arose as a measure of selectivity in radio receivers 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-factor, selectivity and tuning sufficed. This was all to change with the developing theory of transmission line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
s 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
Phantom circuit
In telecommunication and electrical engineering, a phantom circuit is an electrical circuit derived from suitably arranged wires with one or more conductive paths being a circuit in itself and at the same time acting as one conductor of another circuit....
.Telegraph lines are typically unbalanced
Unbalanced line
In Electrical engineering, an unbalanced line is a transmission line, usually coaxial cable, whose conductors have unequal impedances with respect to ground; as opposed to a balanced line.Microstrip and single-wire lines are also unbalanced lines....
with only a single conductor provided, the return path is achieved through an earth
Ground (electricity)
In electrical engineering, ground or earth may be the reference point in an electrical circuit from which other voltages are measured, or a common return path for electric current, or a direct physical connection to the Earth....
connection which is common to all the telegraph lines on a route. Telephone lines are typically balanced
Balanced line
In telecommunications and professional audio, a balanced line or balanced signal pair is a transmission line consisting of two conductors of the same type, each of which have equal impedances along their lengths and equal impedances to ground and to other circuits. The chief advantage of the...
with two conductors per circuit. A telegraph signal connected common-mode
Common-mode signal
Common-mode signal is the component of an analog signal which is present with one sign on all considered conclusions. In electronics where the signal is transferred with differential voltage use, the common-mode signal is called a half-sum of voltages:...
to both conductors of the telephone line will not be heard at the telephone receiver which can only detect voltage differences between the conductors. The telegraph signal is typically recovered at the far end by connection to the center tap
Center tap
In electronics, a center tap is a connection made to a point half way along a winding of a transformer or inductor, or along the element of a resistor or a potentiometer....
of a line transformer
Repeating coil
In telecommunications, a repeating coil is a voice-frequency transformer characterized by a closed magnetic core, a pair of identical balanced primary windings, a pair of identical but not necessarily balanced secondary windings, and low transmission loss at voice frequencies...
. The return path is via an earth connection as usual. This is a form of phantom circuit
Phantom circuit
In telecommunication and electrical engineering, a phantom circuit is an electrical circuit derived from suitably arranged wires with one or more conductive paths being a circuit in itself and at the same time acting as one conductor of another circuit....
An LC filter
LC circuit
An LC circuit, also called a resonant circuit or tuned circuit, consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C...
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 at audio frequencies. The first of these systems in the UK was a Siemens and Halske
Siemens
Siemens may refer toSiemens, a German family name carried by generations of telecommunications industrialists, including:* Werner von Siemens , inventor, founder of Siemens AG...
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
Carrier wave
In telecommunications, a carrier wave or carrier is a waveform that is modulated with an input signal for the purpose of conveying information. This carrier wave is usually a much higher frequency than the input signal...
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
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
was probably described by Georg Ohm
Georg Ohm
Georg Simon Ohm was a German physicist. As a high school teacher, Ohm began his research with the recently-invented electrochemical cell, invented by Italian Count Alessandro Volta. Using equipment of his own creation, Ohm determined that there is a direct proportionality between the potential...
(1827) who established that resistance in a wire is proportional to its length.At least, Ohm described the first model that was in any way correct. Earlier ideas such as Barlow's law
Barlow's law
Barlow's law was an incorrect physical law proposed by Peter Barlow in 1824 to describe the ability of wires to conduct electricity. It said that the conductance of a wire varies inversely with the square root of its length and directly with the square root of its cross-sectional area, or:where G...
from Peter Barlow were either incorrect, or inadequately described. See, for example. p.603 of;
*John C. Shedd, Mayo D. Hershey, "The history of Ohm's law", The Popular Science Monthly, pp.599-614, December 1913 ISSN 0161-7370. 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
Dispersion relation
In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....
but then called retardation, and Michael Faraday
Michael Faraday
Michael Faraday, FRS was an English chemist and physicist who contributed to the fields of electromagnetism and electrochemistry....
(1853) established that this was due to the capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
present in the transmission line.Werner von Siemens had also noted the retardation effect a few years earlier in 1849 and came to a similar conclusion as Faraday. However, there was not so much interest in Germany in underwater and underground cables as there was in Britain, the German overhead cables did not noticeably suffer from retardation and Siemen's ideas were not accepted. (Hunt, p.65.) 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
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
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 signalling 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
Symbol rate
In digital communications, symbol rate is the number of symbol changes made to the transmission medium per second using a digitally modulated signal or a line code. The Symbol rate is measured in baud or symbols/second. In the case of a line code, the symbol rate is the pulse rate in pulses/second...
. The mathematical model of the transmission line reached its fullest development with Oliver Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...
. Heaviside (1881) introduced series inductance and shunt conductance into the model making four distributed elements
Distributed element model
In electrical engineering, the distributed element model or transmission line model of electrical circuits assumes that the attributes of the circuit are distributed continuously throughout the material of the circuit...
in all. This model is now known as the telegrapher's equation and the distributed elements are called the primary line constants
Primary line constants
The primary line constants are parameters that describe the characteristics of copper transmission lines in terms of the physical electrical properties of the line. The primary line constants are only relevant to copper lines and are to be contrasted with the secondary line constants, which can...
.
From the work of Heaviside (1887) it had become clear that the performance of telegraph lines, and most especially telephone lines, could be improved by the addition of inductance to the line. George Campbell
George Ashley Campbell
George Ashley Campbell was a pioneer in developing and applying quantitative mathematical methods to the problems of long-distance 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...
at AT&T
American Telephone & Telegraph
AT&T Corp., originally American Telephone and Telegraph Company, is an American telecommunications company that provides voice, video, data, and Internet telecommunications and professional services to businesses, consumers, and government agencies. AT&T is the oldest telecommunications company...
implemented this idea (1899) by inserting loading coil
Loading coil
In electronics, a loading coil or load coil is a coil that does not provide coupling to any other circuit, but is inserted in a circuit to increase its inductance. The need was discovered by Oliver Heaviside in studying the disappointing slow speed of the Transatlantic telegraph cable...
s 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
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...
. This was a result of the loading coils and the line capacitance forming a low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
, an effect that is only apparent on lines incorporating lumped components
Lumped element model
The lumped element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions...
such as the loading coils. This naturally led 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 filter
Band-pass filter
A band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range.Optical band-pass filters are of common usage....
s to the same ladder topology by replacing the inductors and capacitors with resonator
Resonator
A resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others. The oscillations in a resonator can be either electromagnetic or mechanical...
s and anti-resonators respectively.The exact date Campbell produced each variety of filter is not clear. The work started in 1910, initially patented in 1917 (US1227113) and the full theory published in 1922, but it is known that Campbell's filters were in use by AT&T long before the 1922 date (Bray, p.62, Darlington, p.5) Both the loaded line and FDM were of great benefit economically to AT&T and this led to fast development of filtering from this point onwards.
Image filters
Karl Willy Wagner
Karl Willy Wagner was a German pioneer in the theory of electronic filters. He is noted by Hendrik Bode as being one of two Germans whose;The other German being referred to is Wilhelm Cauer...
independently made a similar discovery which he was not allowed to publish immediately because World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
was still ongoing. (Thomas H. Lee, Planar microwave engineering, p.725, Cambridge University Press 2004 ISBN 0-521-83526-7.) were named wave filters because of their property of passing some waves and strongly rejecting others. The method by which they were designed was called the image parameter methodThe term "image parameter method" was coined by Darlington (1939) in order to distinguish this earlier technique from his later "insertion-loss method" and filters designed to this method are called image filters.The terms wave filter and image filter are not synonymous, it is possible for a wave filter to not be designed by the image method, but in the 1920s the distinction was moot as the image method was the only one available The image method essentially consists of developing the transmission constants of an infinite chain of identical filter sections and then terminating the desired finite number of filter sections in the image impedance
Image impedance
Image impedance is a concept used in electronic network design and analysis and most especially in filter design. The term image impedance applies to the impedance seen looking in to the ports of a network. Usually a two-port network is implied but the concept is capable of being extended to...
. This exactly corresponds to the way the properties of a finite length of transmission line are derived from the theoretical properties of an infinite line, the image impedance corresponding to the characteristic impedance
Characteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...
of the line.
From 1920 John Carson
John Renshaw Carson
John Renshaw Carson , who published as J. R. Carson, was a noted transmission theorist for early communications systems...
, also working for AT&T, began to develop a new way of looking at signals using the operational calculus
Operational calculus
Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation.-History:...
of Heaviside which in essence is working in the frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
. This gave the AT&T engineers a new insight into the way their filters were working and led Otto Zobel to invent many improved forms. Carson and Zobel steadily demolished many of the old ideas. For instance the old telegraph engineers thought of the signal as being a single frequency and this idea persisted into the age of radio with some still believing that frequency modulation
Frequency modulation
In telecommunications and signal processing, frequency modulation conveys information over a carrier wave by varying its instantaneous frequency. This contrasts with amplitude modulation, in which the amplitude of the carrier is varied while its frequency remains constant...
(FM) transmission could be achieved with a smaller bandwidth than the baseband
Baseband
In telecommunications and signal processing, baseband is an adjective that describes signals and systems whose range of frequencies is measured from close to 0 hertz to a cut-off frequency, a maximum bandwidth or highest signal frequency; it is sometimes used as a noun for a band of frequencies...
signal right up until the publication of Carson's 1922 paper. Another advance concerned the nature of noise, Carson and Zobel (1923) treated noise as a random process with a continuous bandwidth, an idea that was well ahead of its time, and thus limited the amount of noise that it was possible to remove by filtering to that part of the noise spectrum which fell outside the passband. This too, was not generally accepted at first, notably being opposed by Edwin Armstrong
Edwin Armstrong
Edwin Howard Armstrong was an American electrical engineer and inventor. Armstrong was the inventor of modern frequency modulation radio....
(who ironically, actually succeeded in reducing noise with wide-band FM) and was only finally settled with the work of Harry Nyquist
Harry Nyquist
Harry Nyquist was an important contributor to information theory.-Personal life:...
whose thermal noise power formula
Johnson–Nyquist noise
Johnson–Nyquist noise is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage...
is well known today.
Several improvements were made to image filters and their theory of operation by Otto Zobel. Zobel coined the term constant k filter
Constant k filter
Constant k filters, also k-type filters, are a type of electronic filter designed using the image method. They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive components...
(or k-type filter) to distinguish Campbell's filter from later types, notably Zobel's m-derived filter
M-derived filter
m-derived filters or m-type filters are a type of electronic filter designed using the image method. They were invented by Otto Zobel in the early 1920s. This filter type was originally intended for use with telephone multiplexing and was an improvement on the existing constant k type filter...
(or m-type filter). The particular problems Zobel was trying to address with these new forms were impedance matching into the end terminations and improved steepness of roll-off. These were achieved at the cost of an increase in filter circuit complexity.
A more systematic method of producing image filters was introduced by Hendrik Bode (1930), and further developed by several other investigators including Piloty (1937-1939) and Wilhelm Cauer
Wilhelm Cauer
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...
(1934-1937). Rather than enumerate the behaviour (transfer function, attenuation function, delay function and so on) of a specific circuit, instead a requirement for the image impedance itself was developed. The image impedance can be expressed in terms of the open-circuit and short-circuit impedancesThe open-circuit impedance of a two-port network is the impedance looking into one port when the other port is open circuit. Similarly, the short-circuit impedance is the impedance looking into one port when the other is terminated in a short circuit. The open-circuit impedance of the first port in general (except for symmetrical networks) is not equal to the open-circuit impedance of the second and likewise for short-circuit impedances of the filter as
. Since the image impedance must be real in the passbands and imaginary in the stopbands according to image theory, there is a requirement that the poles and zeroesZero (complex analysis)
In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...
of Zo and Zs cancel in the passband and correspond in the stopband. The behaviour of the filter can be entirely defined in terms of the positions in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
of these pairs of poles and zeroes. Any circuit which has the requisite poles and zeroes will also have the requisite response. Cauer pursued two related questions arising from this technique: what specification of poles and zeroes are realisable as passive filters; and what realisations are equivalent to each other. The results of this work led Cauer to develop a new approach, now called network synthesis.
This "poles and zeroes" view of filter design was particularly useful where a bank of filters, each operating at different frequencies, are all connected across the same transmission line. The earlier approach was unable to deal properly with this situation, but the poles and zeroes approach could embrace it by specifying a constant impedance for the combined filter. This problem was originally related to FDM telephony but frequently now arises in loudspeaker crossover filters
Audio crossover
Audio crossovers are a class of electronic filter used in audio applications. Most individual loudspeaker drivers are incapable of covering the entire audio spectrum from low frequencies to high frequencies with acceptable relative volume and lack of distortion so most hi-fi speaker systems use a...
.
Network synthesis filters
The essence of network synthesis is to start with a required filter response and produce a network that delivers that response, or approximates to it within a specified boundary. This is the inverse of network analysisNetwork analysis (electrical circuits)
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, every component in the network. There are a number of different techniques for achieving this...
which starts with a given network and by applying the various electric circuit theorems predicts the response of the network. The term was first used with this meaning in the doctoral thesis of Yuk-Wing Lee
Yuk-Wing Lee
Yuk-Wing Lee was a Professor of Electrical Engineering at the Massachusetts Institute of Technology. He is best known for adapting and popularizing the pioneering work of Norbert Wiener and for his own research on statistical communication theory. John Costas and Amar Bose were his students at MIT....
(1930) and apparently arose out of a conversation with Vannevar Bush
Vannevar Bush
Vannevar Bush was an American engineer and science administrator known for his work on analog computing, his political role in the development of the atomic bomb as a primary organizer of the Manhattan Project, the founding of Raytheon, and the idea of the memex, an adjustable microfilm viewer...
. The advantage of network synthesis over previous methods is that it provides a solution which precisely meets the design specification. This is not the case with image filters, a degree of experience is required in their design since the image filter only meets the design specification in the unrealistic case of being terminated in its own image impedance, to produce which would require the exact circuit being sought. Network synthesis on the other hand, takes care of the termination impedances simply by incorporating them into the network being designed.
The development of network analysis needed to take place before network synthesis was possible. The theorems of Gustav Kirchhoff
Gustav Kirchhoff
Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...
and others and the ideas of Charles Steinmetz (phasors) and Arthur Kennelly (complex impedance) laid the groundwork. The concept of a port
Two-port network
A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...
also played a part in the development of the theory, and proved to be a more useful idea than network terminals. The first milestone on the way to network synthesis was an important paper by Ronald Foster (1924), A Reactance Theorem, in which Foster introduces the idea of a driving point impedance, that is, the impedance that is connected to the generator. The expression for this impedance determines the response of the filter and vice versa, and a realisation of the filter can be obtained by expansion of this expression. It is not possible to realise any arbitrary impedance expression as a network. Foster's reactance theorem stipulates necessary and sufficient conditions for realisability: that the reactance must be algebraically increasing with frequency and the poles and zeroes must alternate.
Wilhelm Cauer
Wilhelm Cauer
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...
expanded on the work of Foster (1926) and was the first to talk of realisation of a one-port impedance with a prescribed frequency function. Foster's work considered only reactances (i.e., only LC-kind circuits). Cauer generalised this to any 2-element kind one-port network, finding there was an isomorphism between them. He also found ladder realisations which is the best known of the filter topologies. It is for this reason that ladder topology is often referred to as Cauer topology (the forms used earlier by Foster are quite different) even though ladder topology had long since been in use in image filter design of the network using Thomas Stieltjes' continued fraction expansion. This work was the basis on which network synthesis was built, although Cauer's work was not at first used much by engineers, partly because of the intervention of World War II, partly for reasons explained in the next section and partly because Cauer presented his results using topologies that required mutually coupled inductors and ideal transformers. Although on this last point, it has to be said that transformer coupled double tuned amplifiers are a common enough way of widening bandwidth without sacrificing selectivity.
Image method versus synthesis
Image filters continued to be used by designers long after the superior network synthesis techniques were available. Part of the reason for this may have been simply inertia, but it was largely due to the greater computation required for network synthesis filters, often needing a mathematical iterative process. Image filters, in their simplest form, consist of a chain of repeated, identical sections. The design can be improved simply by adding more sections and the computation required to produce the initial section is on the level of "back of an envelope" designing. In the case of network synthesis filters, on the other hand, the filter is designed as a whole, single entity and to add more sections (i.e., increase the order) the designer would have no option but to go back to the beginning and start over. The advantages of synthesised designs are real, but they are not overwhelming compared to what a skilled image designer could achieve, and in many cases it was more cost effective to dispense with time-consuming calculations. This is simply not an issue with the modern availability of computing power, but in the 1950s it was non-existent, in the 1960s and 1970s available only at cost, and not finally becoming widely available to all designers until the 1980s with the advent of the desktop personal computer. Image filters continued to be designed up to that point and many remained in service into the 21st century.The computational difficulty of the network synthesis method was addressed by tabulating the component values of a prototype filter
Prototype filter
Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or transformed. They are most often seen in regards to...
and then scaling the frequency and impedance and transforming the bandform to those actually required. This kind of approach, or similar, was already in use with image filters, for instance by Zobel, but the concept of a "reference filter" is due to Sidney Darlington
Sidney Darlington
Sidney Darlington was an electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair...
. Darlington (1939), was also the first to tabulate values for network synthesis prototype filters, nevertheless it had to wait until the 1950s before the Cauer-Darlington elliptic filter first came into use.
Once computational power was readily available, it became possible to easily design filters to minimise any arbitrary parameter, for example time delay or tolerance to component variation. The difficulties of the image method were firmly put in the past, and even the need for prototypes became largely superfluous. Furthermore, the advent of active filter
Active filter
An active filter is a type of analog electronic filter that uses an amplifier stage. Amplifiers included in a filter design can be used to improve the performance, stability and predictability of a filter. An amplifier prevents the impedance of source or load stages from affecting the...
s eased the computation difficulty because sections could be isolated and iterative processes were not then generally necessary.
Realisability and equivalence
Realisability (that is, which functions are realisable as real impedance networks) and equivalence (which networks equivalently have the same function) are two important questions in network synthesis. Following an analogy with Lagrangian mechanicsLagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
, Cauer formed the matrix equation,
where [Z],[R],[L] and [D] are the nxn matrices of, respectively, impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
, resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
, inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
and elastance of an n-mesh
Mesh analysis
Mesh analysis is a method that is used to solve planar circuits for the currents at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis can be applied to any circuit, planar...
network and s is the complex frequency operator
. Here [R],[L] and [D] have associated energies corresponding to the kinetic, potential and dissipative heat energies, respectively, in a mechanical system and the already known results from mechanics could be applied here. Cauer determined the driving point impedance by the method of Lagrange multipliersLagrange multipliers
In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
;
where a11 is the complement of the element A11 to which the one-port is to be connected. From stability theory
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...
Cauer found that [R], [L] and [D] must all be positive-definite matrices
Positive-definite matrix
In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....
for Zp(s) to be realisable if ideal transformers are not excluded. Realisability is only otherwise restricted by practical limitations on topology. This work is also partly due to Otto Brune (1931), who worked with Cauer in the US prior to Cauer returning to Germany. A well known condition for realisability of a one-port rationalA rational impedance is one expressed as a ratio of two finite polynomials in s, that is, a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
in s. The implication of finite polynomials is that the impedance, when realised, will consist of a finite number of meshes with a finite number of elements impedance due to Cauer (1929) is that it must be a function of s that is analytic in the right halfplane (σ>0), have a positive real part in the right halfplane and take on real values on the real axis. This follows from the Poisson integral representation of these functions. Brune coined the term positive-real for this class of function and proved that it was a necessary and sufficient condition (Cauer had only proved it to be necessary) and they extended the work to LC multiports. A theorem due to Sidney Darlington
Sidney Darlington
Sidney Darlington was an electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair...
states that any positive-real function Z(s) can be realised as a lossless two-port terminated in a positive resistor R. No resistors within the network are necessary to realise the specified response.
As for equivalence, Cauer found that the group of real affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s,
- where,
is invariant in Zp(s), that is, all the transformed networks are equivalents of the original.
Approximation
The approximation problem in network synthesis is to find functions which will produce realisable networks approximating to a prescribed function of frequency within limits arbitrarily set. The approximation problem is an important issue since the ideal function of frequency required will commonly be unachievable with rational networks. For instance, the ideal prescribed function is often taken to be the unachievable lossless transmission in the passband, infinite attenuation in the stopband and a vertical transition between the two. However, the ideal function can be approximated with a rational functionRational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
, becoming ever closer to the ideal the higher the order of the polynomial. The first to address this problem was Stephen Butterworth
Stephen Butterworth
Stephen Butterworth was a British physicist who invented the Butterworth filter, a class of electrical circuits that are used to separate different frequencies of electrical signals....
(1930) using his Butterworth polynomials. Independently, Cauer (1931) used Chebyshev polynomials
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
, initially applied to image filters, and not to the now well-known ladder realisation of this filter.
Butterworth filter
Butterworth filters are an important classA class of filters is a collection of filters which are all described by the same class of mathematical function, for instance, the class of Chebyshev filters are all described by the class of Chebyshev polynomials. For realisable linear passive networks, the transfer functionTransfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
must be a ratio of polynomial functions. The order of a filter is the order of the highest order polynomial of the two and will equal the number of elements (or resonators) required to build it. Usually, the higher the order of a filter, the steeper the roll-off of the filter will be. In general, the values of the elements in each section of the filter will not be the same if the order is increased and will need to be recalculated. This is in contrast to the image method of design which simply adds on more identical sections of filters due to Stephen Butterworth
Stephen Butterworth
Stephen Butterworth was a British physicist who invented the Butterworth filter, a class of electrical circuits that are used to separate different frequencies of electrical signals....
(1930) which are now recognised as being a special case of Cauer's elliptic filter
Elliptic filter
An elliptic filter is a signal processing filter with equalized ripple behavior in both the passband and the stopband...
s. Butterworth discovered this filter independently of Cauer's work and implemented it in his version with each section isolated from the next with a valve amplifier
Valve amplifier
A valve amplifier or tube amplifier is a type of electronic amplifier that makes use of vacuum tubes to increase the power and/or amplitude of a signal. Low to medium power valve amplifiers for frequencies below the microwaves were largely replaced by solid state amplifiers during the 1960s and...
which made calculation of component values easy since the filter sections could not interact with each other and each section represented one term in the Butterworth polynomials. This gives Butterworth the credit for being both the first to deviate from image parameter theory and the first to design active filters. It was later shown that Butterworth filters could be implemented in ladder topology without the need for amplifiers, possibly the first to do so was William Bennett (1932) in a patent which presents formulae for component values identical to the modern ones. Bennett, at this stage though, is still discussing the design as an artificial transmission line and so is adopting an image parameter approach despite having produced what would now be considered a network synthesis design. He also does not appear to be aware of the work of Butterworth or the connection between them.
Insertion-loss method
The insertion-loss method of designing filters is, in essence, to prescribe a desired function of frequency for the filter as an attenuation of the signal when the filter is inserted between the terminations relative to the level that would have been received were the terminations connected to each other via an ideal transformer perfectly matching them. Versions of this theory are due to Sidney DarlingtonSidney Darlington
Sidney Darlington was an electrical engineer and inventor of a transistor configuration in 1953, the Darlington pair...
, Wilhelm Cauer and others all working more or less independently and is often taken as synonymous with network synthesis. Butterworth's filter implementation is, in those terms, an insertion-loss filter, but it is a relatively trivial one mathematically since the active amplifiers used by Butterworth ensured that each stage individually worked into a resistive load. Butterworth's filter becomes a non-trivial example when it is implemented entirely with passive components. An even earlier filter which influenced the insertion-loss method was Norton's dual-band filter where the input of two filters are connected in parallel and designed so that the combined input presents a constant resistance. Norton's design method, together with Cauer's canonical LC networks and Darlington's theorem that only LC components were required in the body of the filter resulted in the insertion-loss method. However, ladder topology proved to be more practical than Cauer's canonical forms.
Darlington's insertion-loss method is a generalisation of the procedure used by Norton. In Norton's filter it can be shown that each filter is equivalent to a separate filter unterminated at the common end. Darlington's method applies to the more straightforward and general case of a 2-port LC network terminated at both ends. The procedure consists of the following steps:
- determine the poles of the prescribed insertion-loss function,
- from that find the complex transmission function,
- from that find the complex reflection coefficients at the terminating resistors,
- find the driving point impedance from the short-circuit and open-circuit impedances,
- expand the driving point impedance into an LC (usually ladder) network.
Darlington additionally used a transformation found by Hendrik Bode that predicted the response of a filter using non-ideal components but all with the same Q. Darlington used this transformation in reverse to produce filters with a prescribed insertion-loss with non-ideal components. Such filters have the ideal insertion-loss response plus a flat attenuation across all frequencies.
Elliptic filters
Elliptic filters are filters produced by the insertion-loss method which use elliptic rational functionsElliptic rational functions
In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic electronic filters...
in their transfer function as an approximation to the ideal filter response and the result is called a Chebyshev approximation. This is the same Chebyshev approximation technique used by Cauer on image filters but follows the Darlington insertion-loss design method and uses slightly different elliptic functions. Cauer had some contact with Darlington and Bell Labs before WWII (for a time he worked in the US) but during the war they worked independently, in some cases making the same discoveries. Cauer had disclosed the Chebyshev approximation to Bell Labs but had not left them with the proof. Sergei Schelkunoff
Sergei Alexander Schelkunoff
Dr. Sergei Alexander Schelkunoff , who published as S. A. Schelkunoff, was a distinguished mathematician and electromagnetism theorist who made noted contributions to antenna theory.-Biography:...
provided this and a generalisation to all equal ripple problems. Elliptic filters are a general class of filter which incorporate several other important classes as special cases: Cauer filter (equal ripple in passband and stopband
Stopband
A stopband is a band of frequencies, between specified limits, through which a circuit, such as a filter or telephone circuit, does not allow signals to pass, or the attenuation is above the required stopband attenuation level...
), Chebyshev filter (ripple only in passband), reverse Chebyshev filter (ripple only in stopband) and Butterworth filter (no ripple in either band).
Generally, for insertion-loss filters where the transmission zeroes and infinite losses are all on the real axis of the complex frequency plane (which they usually are for minimum component count), the insertion-loss function can be written as;
where F is either an even (resulting in an antimetric
Antimetric (electrical networks)
An antimetric electrical network is one that exhibits anti-symmetrical electrical properties. The term is often encountered in filter theory, but it applies to general electrical network analysis...
filter) or an odd (resulting in an symmetric filter) function of frequency. Zeroes of F correspond to zero loss and the poles of F correspond to transmission zeroes. J sets the passband ripple height and the stopband loss and these two design requirements can be interchanged. The zeroes and poles of F and J can be set arbitrarily. The nature of F determines the class of the filter;
- if F is a Chebyshev approximation the result is a Chebyshev filter,
- if F is a maximally flat approximation the result is a passband maximally flat filter,
- if 1/F is a Chebyshev approximation the result is a reverse Chebyshev filter,
- if 1/F is a maximally flat approximation the result is a stopband maximally flat filter,
A Chebyshev response simultaneously in the passband and stopband is possible, such as Cauer's equal ripple elliptic filter.
Darlington relates that he found in the New York City library Carl Jacobi's original paper on elliptic functions, published in Latin in 1829. In this paper Darlington was surprised to find foldout tables of the exact elliptic function transformations needed for Chebyshev approximations of both Cauer's image parameter, and Darlington's insertion-loss filters.
Other methods
Darlington considers the topology of coupled tuned circuits to involve a separate approximation technique to the insertion-loss method, but also producing nominally flat passbands and high attenuation stopbands. The most common topology for these is shunt anti-resonators coupled by series capacitors, less commonly, by inductors, or in the case of a two-section filter, by mutual inductance. These are most useful where the design requirement is not too stringent, that is, moderate bandwidth, roll-off and passband ripple.Mechanical filters
Edward Lawry Norton
Edward Lawry Norton was an accomplished Bell Labs engineer and scientist famous for developing the concept of the Norton equivalent circuit. He attended the University of Maine for two years before transferring to M.I.T. and received a S.B. degree in 1922. He received an M.A...
, around 1930, designed a mechanical filter for use on phonograph
Phonograph
The phonograph record player, or gramophone is a device introduced in 1877 that has had continued common use for reproducing sound recordings, although when first developed, the phonograph was used to both record and reproduce sounds...
recorders and players. Norton designed the filter in the electrical domain and then used the correspondence of mechanical quantities to electrical quantities to realise the filter using mechanical components. Mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
corresponds to inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
, stiffness
Stiffness
Stiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.-Calculations:...
to elastance and damping
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
to resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
. The filter was designed to have a maximally flat frequency response.
In modern designs it is common to use quartz crystal filter
Crystal filter
A crystal filter is a special form of quartz crystal used in electronics systems, in particular communications devices. It provides a very precisely defined centre frequency and very steep bandpass characteristics, that is a very high Q factor—far higher than can be obtained with conventional...
s, especially for narrowband filtering applications. The signal exists as a mechanical acoustic wave while it is in the crystal and is converted by transducer
Transducer
A transducer is a device that converts one type of energy to another. Energy types include electrical, mechanical, electromagnetic , chemical, acoustic or thermal energy. While the term transducer commonly implies the use of a sensor/detector, any device which converts energy can be considered a...
s between the electrical and mechanical domains at the terminals of the crystal.
Transversal filters
TransversalTransversal
In geometry , when two coplanar lines exist such that a third coplanar line passes thru both lines. This third line is named the Transversal....
filters are not usually associated with passive implementations but the concept can be found in a Wiener and Lee patent from 1935 which describes a filter consisting of a cascade of all-pass section
All-pass filter
An all-pass filter is a signal processing filter that passes all frequencies equally, but changes the phase relationship between various frequencies. It does this by varying its propagation delay with frequency...
s. The outputs of the various sections are summed in the proportions needed to result in the required frequency function. This works by the principle that certain frequencies will be in, or close to antiphase, at different sections and will tend to cancel when added. These are the frequencies rejected by the filter and can produce filters with very sharp cut-offs. This approach did not find any immediate applications, and is not common in passive filters. However, the principle finds many applications as an active delay line implementation for wide band discrete-time filter applications such as television, radar and high-speed data transmission.
Matched filter
The purpose of matched filters is to maximise the signal-to-noise ratioSignal-to-noise ratio
Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. It is defined as the ratio of signal power to the noise power. A ratio higher than 1:1 indicates more signal than noise...
(S/N) at the expense of pulse shape. Pulse shape, unlike many other applications, is unimportant in radar while S/N is the primary limitation on performance. The filters were introduced during WWII (described 1943) by Dwight North and are often eponymously referred to as "North filters
Matched filter
In telecommunications, a matched filter is obtained by correlating a known signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template...
".
Filters for control systems
Control systems have a need for smoothing filters in their feedback loops with criteria to maximise the speed of movement of a mechanical system to the prescribed mark and at the same time minimise overshoot and noise induced motions. A key problem here is the extraction of Gaussian signals from a noisy background. An early paper on this was published during WWII by Norbert WienerNorbert Wiener
Norbert Wiener was an American mathematician.A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.Wiener is regarded as the originator of cybernetics, a...
with the specific application to anti-aircraft fire control analogue computers. Rudy Kalman (Kalman filter
Kalman filter
In statistics, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated...
) later reformulated this in terms of state-space
State space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...
smoothing and prediction where it is known as the linear-quadratic-Gaussian control
Linear-quadratic-Gaussian control
In control theory, the linear-quadratic-Gaussian control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, having incomplete state information and undergoing control subject to quadratic costs...
problem. Kalman started an interest in state-space solutions, but according to Darlington this approach can also be found in the work of Heaviside and earlier.
Modern practice
LC passive filters gradually became less popular as active amplifying elements, particularly operational amplifierOperational amplifier
An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...
s, became cheaply available. The reason for the change is that wound components (the usual method of manufacture for inductors) are far from ideal, the wire adding resistance as well as inductance to the component. Inductors are also relatively expensive and are not "off-the-shelf" components. On the other hand, the function of LC ladder sections, LC resonators and RL sections can be replaced by RC components in an amplifier feedback loop (active filters). These components will usually be much more cost effective, and smaller as well. Cheap digital technology, in its turn, has largely supplanted analogue implementations of filters. However, there is still an occasional place for them in the simpler applications such as coupling where sophisticated functions of frequency are not needed.
See also
- Electronic filterElectronic filterElectronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
- Linear filterLinear filterLinear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity.This results from systems composed solely of components classified as having a linear response....
- Composite image filterComposite image filterA composite image filter is an electronic filter consisting of multiple image filter sections of two or more different types.The image method of filter design determines the properties of filter sections by calculating the properties they have in an infinite chain of such sections. In this, the...
- Network synthesis filtersNetwork synthesis filtersNetwork 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...
- Digital filterDigital filterIn electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is...
- Audio filterAudio filterAn audio filter is a frequency dependent amplifier circuit, working in the audio frequency range, 0 Hz to beyond 20 kHz. Many types of filters exist for applications including graphic equalizers, synthesizers, sound effects, CD players and virtual reality systems.Being a frequency dependent...