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This article is about the history of passive linear analogue electronic filters. For linear filters in general see Linear filter. For electronic filters in general see Electronic filter.

Passive linear electronic analogue filters are those filters which can be described by a system of linear differential equations (linear), are composed of capacitors, inductors and, sometimes, resistors (passive) and are designed to operate on continuously varying (analogue) signals. There are many linear filters which are not analogue in implementation (digital filter), and there are many electronic filters which may not have a passive topology - both of which may have the same transfer function of the filters described in this article. 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 telecommunications, filters have been of crucial importance in a number of technological breakthroughs and have been the source of enormous profits for telecommunications companies. It should come as no surprise, therefore, that the early development of filters was intimately connected with transmission lines. Transmission line theory gave rise to filter theory, which initially took a very similar form, and the main application of filters was for use on telecommunication transmission lines. However, the arrival of network synthesis techniques greatly enhanced the degree of control of the designer.

Today, the majority of filtering is carried out in the digital domain where complex algorithms are much easier to implement, but analogue filters do still find applications, especially for low-order simple filtering tasks. Wherever possible, however, 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 filters using mechanical components which filter mechanical vibrations or acoustic waves. While there are few applications for such devices in mechanics per se, they can be used in electronics with the addition of transducers to convert to and from the electrical domain. Indeed some of the earliest ideas for filters were acoustic resonators because the electronics technology was then poorly understood. In principle, the design of such filters is completely analogous to the electronic counterpart, with kinetic energy, potential energy and heat energy corresponding to the energy in inductors, capacitors and resistors respectively.

Overview

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

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

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

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

From Submarine communications cable

See Submarine communications cable#Bandwidth problems. Key people in this were;

Early models of cable were resistance only 1823, Francis Ronalds noticed signal delay in cables Faraday noticed the capacitive effect Lord Kelvin Correct theory (but no inductance in his version - capacitance and resistance only) Oliver Heaviside Modern version of the theory

Thomson also apparently had some scheme for a generator that resonated with the cable.

From On Shannon and “Shannon’s formula”

pdf on local drive

L. Lundheim: “On Shannon and "Shannon's Formula"”, Telektronikk (special issue on "Information theory and its applications") vol. 98, no. 1-2002, pp. 20-29, ISSN 0085-7130, published by Telenor

Maxwell was first to note resonance or L/C. Used by Hertz to transmit radio

Not taken up by engineers immediately, acoustic resonance tried first as this was much better understood. Conception of bandwidth not really recognised until the telephone and Campbell's invention.

From Matthaei

Be sure to mention that Darlington was (one of?) the first to tabulate values for prototype filters.

From original article

{{mergeto|Linear filter|Talk:Linear filter#Propose merging "Analogue filter" into this article|date=May 2008}}

An analogue filter handles analogue signals or continuous-time signals, whether electric potential, sound waves, or mechanical motion directly. This is opposed to a digital filter that operates on discrete-time signals.

The design of mechanical or acoustic filters is based on similar principles to electronic linear filters.

Given a particular filter specification, Analog filters are typically designed by first selecting the overall number and arrangement of parts (the electronic filter topology) (which determines the "order" of the filter), and then calculating the specific part values (which determines the particular transfer function of that order -- pass band, transition band, stop band, cutoff frequencies, ripple, etc.).

From draft Network synthesis article plus the extra notes

Network synthesis is a method of designing electronic filters. It has produced several important classes of filter including the Butterworth filter, the Tchebyscheff filter and the Elliptic filter. It was originally intended to be applied to the design of passive linear analogue filters but its results can also be applied to implementations in active filters and digital filters. The essence of the method is to obtain the component values of the filter from a given mathematical polynomial ratio expression representing the desired transfer function.

Description of method

The method can be viewed as the inverse problem of network analysis. Network analysis starts with a network and by applying the various electric circuit theorems predicts the response of the network. Network synthesis on the other hand, starts with a desired response and its methods produce a network that outputs, or approximates to, that response.

Network synthesis was originally intended to produce filters of the kind formerly described as "wave filters" but now usually just called filters. That is, filters whose purpose is to pass waves of certain wavelengths while rejecting waves of other wavelengths. Network synthesis starts out with a specification for the transfer function of the filter, H(s), as a function of complex frequency, s. This is used to generate an expression for the input impedance of the filter (the driving point impedance) which then, by a process of continued fraction or partial fraction expansions results in the required values of the filter components. In a digital impementation of a filter, H(s) can be implemented directly.

The advantages of the method are best understood by comparing it to the filter design methodology that was used before it, the image method. The image method considers the characteristics of an individual filter section in an infinite chain (ladder topology) of identical sections. The filters produced by this method suffer from innaccuracies due to the theoretical termination impedance, the image impedance, not generally being equal to the actual termination impedance. This is not the case with network synthesis filters, the terminations are included in the design from the start. The image method also requires a certain amount of experience on the part of the designer. The designer must first decide how many sections and of what type should be used, and then after calculation, will obtain the transfer function of the filter. This may not be what is required and there can be a number of iterations. The network synthesis method, on the other hand, starts out with the required function and outputs the sections needed to build the corresponding filter.

In general, the sections of a network synthesis filter are identical topology (usually the simplest ladder type) but different component values are used in each section. By contrast, the structure of an image filter has identical values at each section - this is a consequence of the infinite chain approach - but may vary the topology from section to section to achieve various desirable characteristics. Both methods make use of low-pass prototype filters followed by frequency transformations and impedance scaling to arrive at the final desired filter.

Important filter classes

The class of a filter refers to the class of polynomials from which the filter is mathematically derived. The order of the filter is the number of filter elements present in the filters ladder implementation. Generally speaking, the higher the order of the filter, the steeper the cut-off transition between passband and stopband. Filters are often named after the mathematician or mathematics on which they are based rather than the discoverer or inventor of the filter.

Butterworth filter

Butterworth filters are described as maximally flat, meaning that the response in the frequency domain is the smoothest possible curve of any class of filter of the equivalent order.

The Butterworth class of filter was first described in a 1930 paper by the British engineer Stephen Butterworth after whom it is named. The filter response is described by Butterworth polynomials, also due to Butterworth.

Tchebyscheff filter

A Tchebyscheff filter has a faster cut-off transition than a Butterworth, but at the expense of there being ripples in the frequency response of the passband. There is a compromise to be had between the maximum allowed attenuation in the passband and the steepness of the cut-off response. This is also sometimes called a type I Tchebyscheff, the type 2 being a filter with no ripple in the passband but ripples in the stopband. The filter is named after Pafnuty Tchebyscheff whose Tchebyscheff polynomials are used in the derivation of the transfer function.

Cauer filter

Cauer filters have equal maximum ripple in the passband and the stopband. The Cauer filter has a faster transition from the passband to the stopband than any other class of network synthesis filter. The term Cauer filter can be used interchangeably with elliptical filter, but the general case of elliptical filters can have unequal ripples in the passband and stopband. An elliptical filter in the limit of zero ripple in the passband is identical to a Tchebyscheff Type 1 filter. An elliptical filter in the limit of zero ripple in the stopband is identical to a Tchebyscheff Type 2 filter. An elliptical filter in the limit of zero ripple in both passbands is identical to a Butterworth filter. The filter is named after Wilhelm Cauer and the transfer function is based on elliptic rational functions.

Bessel filter

• The Bessel filter has a maximally flat time-delay (group delay) over its passband. This gives the filter a linear phase response and results in it passing waveforms with minimal distortion. The Bessel filter has minimal distortion in the time domain due to the phase response with frequency as opposed to the Butterworth filter which has minimal distortion in the frequency domain due to the attenuation response with frequency. The Bessel filter is named after Friedrich Bessel and the transfer function is based on Bessel polynomials.

Driving point impedance

The driving point impedance is a mathematical representation of the input impedance of a filter in the frequency domain using one of a number of notations such as Laplace transform (s-domain) or Fourier transform (jω-domain). Treating it as a one-port network, the expression is expanded using continued fraction or partial fraction expansions. The resulting expansion is transformed into a network (usually a ladder network) of electrical elements. Taking an output from the end of this network, so realised, will transform it into a two-port network filter with the desired transfer function.

Not every possible mathematical function for driving point impedance can be realised using real electrical components. Wilhelm Cauer (following on from R. M. Foster) did much of the early work on what mathematical functions could be realised and in which filter topologies. The ubiquitous ladder topology of filter design is named after Cauer.

• Cauer's first form of driving point impedance consists of shunt capacitors and series inductors and leads to low-pass filters.
• Cauer's second form of driving point impedance consists of series capacitors and shunt inductors and leads to high-pass filters.
• Foster's form of driving point impedance leads to band-stop filters and band-pass filters.

Prototype filters

Main article: Prototype filter

Prototype filters are used to make the process of filter design less labour intensive. The prototype is usually designed to be a low-pass filter of unity nominal impedance and unity cut-off frequency, although other schemes are possible. The full design calculations from the relevant mathematical functions and polynomials are carried out only once. The actual filter required is obtained by a process of scaling and transforming the prototype.

Values of prototype elements are published in tables, one of the first being due to Sidney Darlington. Both modern computing power and the practice of directly implementing filter transfer functions in the digital domain have largely rendered this practice obsolete.

A different prototype is required for each order of filter in each class. For those classes in which there is attenuation ripple, a different prototype is required for each value of ripple. The same prototype may be used to produce filters which have a different bandform from the prototype. For instance low-pass, high-pass, band-pass and band-stop filters can all be produced from the same prototype.

From original article: Given a particular filter specification, Analog filters are typically designed by first selecting the overall number and arrangement of parts (the electronic filter topology) (which determines the "order" of the filter), and then calculating the specific part values (which determines the particular transfer function of that order -- pass band, transition band, stop band, cutoff frequencies, ripple, etc.).

Possible refs from Matthaei

• Van Valkenburg, M E, Introduction to Modern Network Synthesis, John Wiley & Sons, New York, 1960.
• Guillemin, E A, Synthesis of Passive Networks, John Wiley & Sons, New York, 1957.

Also (not from Matthaei), this patent from Darlington has a useful discussion on Foster and canonical forms. Also, the paper might make a useful addition to the non-linear part of Network analysis

• United States Patent US3265973

History

• Bell Labs
• Foster
• Cauer

Stuff from E Cauer paper

History

(page 1) Cauer gives the key people in the development as follows:

Basic work

• Georg Ohm (1827)
• Gustav Kirchhoff (1845-1847)

Network theory

• Hermann von Helmholtz

Don't quite get why H is included here. His article says he contributed little to elctromagentism

• James Clerk Maxwell
• Oliver Heaviside
• Charles Proteus Steinmetz
• Arthur E. Kennelly

see this article on him . He's the bad man responsible for all those nasty imaginery numbers in electrical engineering.

<snip> . . . network theory emancipated from analytical mechanics and electrophysics before 1900 as the first basic branch of electrical engineering.</snip>

• Belevitch, V, "Summary of the history of circuit theory", Proc. IRE, vol. 50, pp848-855, 1962.

Early filters

• George Ashley Campbell
• Karl Willy Wagner

wave filters with telephone applications. Ladder topology being used similar too, and inspired by, circuit models in transmission line theory.

<snip>Around 1920 electrical engineers were able to analyse the behaviour of certain filter networks and proved some theorems about the properties of attenuation curves. However, there were no results on the question of what filter characteristics are realisable and, if so, how to find a physical realisation.</snip> Is he referring to the work of Carson and Zobel here?

• Darlington, S, "A history of network synthesis and filter theory for circuits composed of resistors, inductors, and capacitors", IEEE Trans. Circuits and Systems, vol 31, pp3-13, 1984.

Network synthesis

• R. M. Foster 1924, important paper, A Reactance Theorem
• Wilhelm Cauer Immediately recognised the importance of Foster's paper
• Foster, R M, "A Reactance Theorem", Bell System Technical Journal, vol 3, pp259-267, 1924.
• Cauer, W, "Die Verwirklichung der Wechselstromwiderstände vorgeschriebener Frequenzabhängigkeit" ("The realisation of impedances of specified frequency dependence"), Archiv für Elektrotechnic, vol 17, pp355-388, 1926.

This is Cauer's doctoral thesis of 1926 in which he gives a precise mathematical analysis of the problem and proposes a program with three clear steps;

• Realisability
• Approximation
• Realisation and equivalence

of network transfer functions

• 1926 was seen (retrospectively) as the first steps towards this
• 1928 habilitation lecture in Göttingen explicitly stated
• 1930 extended version published
• Cauer W, "Die Siebschaltungen der Fernmeldetechnik", Zeitschrift f. angewandte Mathematik und Mechanik, vol 10, pp425-433, 1930.

PROGRESS TO: end section 1

(p2) 1924 after graduating, Cauer employed by Mix & Genest, Berlin, a subsidiary of the Bell Telephone Company. This made collaboration with Foster easy.

PROGRESS TO: end section 2

SECTION 3

The problem being addressed is

(p4) <snip> the inverse problem of circuit analysis: Given the external behaviour of a linear passive one-port in terms of a driving point impedance as a prescribed function of frequency, how does one find internally passive realizations for this "black-box"? </snip>

Realisability

Using modern notation rather than Cauer's symbols, starting with a driving point impedance (voltage/current transfer function) Z(s) where the complex frequency:

s=σ+iω

for a n-mesh network form the nxn matrix equation,

=s+s+ (=s)

Z,R.L and D being the matrices of impedance,resistance, inductance and elastance.

What does "quadratic forms" mean in relation to this equation?

This equation can be expressed in the form of an energy equation. Comparing this with the familiar forms of Lagrangian mechanics Cauer realised that the energies associated with L, D and R were analogous to kinetic, potential and dissipative heat energies, respectively, in a mechanical system. Analysis could now proceed by analogy with results from mechanics.

Input impedance is arrived at by the method of Lagrange multipliers;

${\displaystyle Z_{in}(s)={\frac {\det A}{sa_{pq}}}}$

where a_pq is the complement of the element ''A_pq to which the one-port is to be connected. Complement of A₁₁ here is going to be the same as the minor of A₁₁ since it is a single elemetn

By applying stability theory Cauer reached the conclusion in his 1926 thesis that the condition for realisability of Z(s) was that , and must all be positive-definite matrices. Ideal transformers may have to be admitted for this to work but any other limitation is entirely due to the choice of network topology. This conclusion is from the 1926 thesis.

.

Equivalence

In subsequent papers Cauer shows that the Zin(s) is invariant under a group of real affine transformations of , and showing the equivalence of the corresponding networks.

W. Cauer. ¨Uber eine Klasse von Funktionen, die die Stieljesschen Kettenbr¨uche als Sonderfall enth ¨alt. Jahresberichte der Dt. Mathematikervereinigung (DMV), 38:63–72, 1929.

W. Cauer. Vierpole. Elektrische Nachrichtentechnik (ENT), 6:272–282, 1929.

W. Cauer. Untersuchungen ¨uber ein Problem, das drei positiv definite quadratische Formen mit Streckenkomplexen in Beziehung setzt. Mathematische Annalen, 105:86–132, 1931.

1941, Cauer's most specific statement of this programme;

• Realisability: What classes of functions Z(s) are realisable as frequency characteristics?
• Equivalence:Which circuits are equivalent in terms of frequency characteristics?
• Approximation: How are the interpolation and approximation problems solved using functions admitted under "Realisability".?

W. Cauer. Theorie der linearen Wechselstromschaltungen, Vol.I. Akad. Verlags-Gesellschaft Becker und Erler, Leipzig, 1941. W. Cauer. Synthesis of Linear Communication Networks. McGraw-Hill, New York, 1958. <quote> . . . it is less important for the electrical engineer to solve given differential equations than to search for systems of differential equations whose solutions have a desired property.</quote> , p13 or , p49

in other words, no need to struggle with a differential equation that corresponds to a circuit which is difficult to implement. It can always be transformed into one that is more convenient.

Two-element kind networks

Foster's "Reactance Theorem" finds the conditions iff for Z(iω) (ie lossless) driving-point impedance. He showed the partial fraction expansion <quote (Cauer)> induces a canonical realization, ie a LC circuit with the minimum number of reactances.

In dissertation,

W. Cauer. Die Verwirklichung der Wechselstromwiderst ¨ande vorgeschriebener Frequenzabh¨angigkeit. Archiv f¨ur Elektrotechnik, 17:355–388, 1926.

Cauer

• extended this to Z(s)
• found canonical ladder realisations via Stieltjes' continued fraction expansion
• found an isomorphism between LC, RC and RL circuits.

Many of the topologies that naturally fall out of Cauer's theoretical work necessarily involve mutually coupled inductors and ideal transformers. However, it is his ladder network which is best known and is of most practical application.

Although it can be noted that loosely coupled tuned circuits are a common way of widening the passband of tuned RF amplifiers (no ref for this statement, but it is probably already in a wiki article somewhere).

Cauer’s program was the basis of his first monograph Siebschaltungen (filter circuits) in 1931 .

• Cauer, W, Siebschaltungen, VDI-Verlag, Berlin, 1931.

Cauer’s concept of filter synthesis was extended between 1937 and 1939 to a general systematic theory of insertion loss filter design, whereby Bader, Cauer, Cocci, Darlington, Norton and Piloty were the main contributors

The main findings on filter synthesis are included in Cauer’s secondmonograph Theorie der linearen Wechselstromschaltungen (1941)

General passive multiports

The requirement that transients remain bounded in a passive circuit imposes the necessary condition for realisability of;

${\displaystyle \Re (Z(\sigma +i\omega ))>0,\forall \sigma >0}$

(end page 5) (page 6)

Poisson integral

Lots of stuff not yet processed

The reference to Cauer's Tchebyscheff aproximation should go in the Network synthesis article. That article should be linked back here (possibly under driving-point impedance)

Further refs

http://www.quadrivium.nl/history/history.html

Stuff from Belevitch

Although Foster's proof of his reactance theorem (1924) is already a transition from the methods of analytical dynamics to those of modern network synthesis, the first paper dealing explicitly with the realization of a one-port whose impedance is a prescribed function of frequency is Cauer's 1926 contribution, based on continuous fraction expansions (also studied by Fry, 1926). With Cauer's and Foster's theorems, the synthesis problem for one ports containing two kinds of elements only was solved. The analogous problem for general one ports was solved by Brune (1931) and led to the concept of positive real function.

Another aspect of network synthesis, the approximation problem, made also its appearance during this period; the maximally flat approximation was used by Butterworth (1930) in the design of multistage amplifiers; simultaneously and independently, Cauer realized the optimal character of the Chebyshev approximation and solved the approximation problem for an important class of image-parameter filters. Finally, it should be remarked that the canonical structures obtained as solutions of the various synthesis problems made a free use of ideal transformers; the much more difficult problem of synthesis without transformers was not of paramount interest for communication applications and has only been treated recently.

The simplest network after the one port is the symmetric 2-port, which involves two frequency functions only. Geometrically symmetric 2-ports were treated by Bartlett (1927) and Brune (1932), whereas Cauer (1927) and Jaumann (1932) found a number of canonical circuits for all symmetric 2-ports. Dissymetric, and, in particular, antimetric 2-ports were studied by Cauer, who also extended Foster's theorem to LC n-ports (1931) and showed (1932-1934) that all equivalent LC networks could be derived from each other by the linear transformations considered by Howitt (1931). Certain classes of symmetric n-ports were studied by Baerwald (1931-1932).

Cauer's first book on filter design (1931) contains tables and curves for the Chebyshev approximation to a constant attenuation in the stop-band of an image parameter low-pass filter, as well as frequency transformations for other filter classes. The solution of the approximation problem involved rational functions whose extremal properties were established by Zolotareff in 1877 and which reduce to ordinary Chebyshev polynomials when elliptic functions are replaced by trigonometric functions. Cauer's presentation of his design data was based on canonic structures, practically less convenient than ladder structures, and was not accepted in engineering circles before it was realized that the statement and the solution of the approximation problem were of interest in themselves, for most of Cauer's results could easily be transferred to the ladder structure. The systematic theory of image parameter filters was further developed by Bode (1934) and Piloty (1937-1938), thus placing Zobel's earlier results in a clearer perspective.

The limitations of image-parameter theory first appeared in connection with the design of filter groups, a problem frequently encountered in carrier telephony. Zobel's procedure of x-derivation, already mentioned, was first replaced by a more systematic method of impedance correction (Bode, 1930). An image-paranmeter theory of constant impedance filter pairs was developed by Brandt (1934-1936), Cauer (1934-1937) and Piloty (1937-1939), and it was recognized that this also yielded a solution to the equivalent problem of opencircuit filter design.

See also

• Electronic filter
• Linear filter
• Composite image filter
• Network synthesis filters
• Digital filter
• Audio filter

Notes

1. ^ E. Cauer, p4
2. ^ Matthaei, pp83-84
3. ^ Matthaei et al, pp85-108
4. Butterworth, S, "On the Theory of Filter Amplifiers", Wireless Engineer, vol. 7, 1930, pp. 536-541.
5. Mathaei, p95
6. Matthaei, pp108-113
7. Foster, R M, "A Reactance Theorem", Bell System Technical Journal, vol 3, pp259-267, 1924.
8. E. Cauer, p1
9. Matthaei, p83
10. Darlington, S, "Synthesis of Reactance 4-Poles Which Produce Prescribed Insertion Loss Characteristics", Jour. Math. and Phys., Vol 18, pp257-353, September 1939.
11. See Matthaei for examples.
12. ^ Belevitch, p850
13. Belevitch, p851

References

• Matthaei, Young, Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, McGraw-Hill 1964.
• E. Cauer, W. Mathis, and R. Pauli, "Life and Work of Wilhelm Cauer (1900 – 1945)", Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June, 2000. Retrieved online 19th September 2008.
• Belevitch, V, "Summary of the history of circuit theory", Proceedings of the IRE, vol 50, Iss 5, pp848-855, May 1962.

External links

• Analog Wizard Filter - Analog Devices
• Filter Design and Analysis
• Analog Filter Design Demystified

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