Analogue filter

There are three main stages in the history of passive analogue filter development: Throughout this article the letters R, L, and C are used with their usual meanings to represent resistance, inductance, and capacitance, respectively.

Z is used for electrical impedance, any 2-terminal[note 1] combination of RLC elements and in some sections D is used for the rarely seen quantity elastance, which is the inverse of capacitance.

This was not suspected until 1826, when Felix Savary in France, and later (1842) Joseph Henry[3] in the US noted that a steel needle placed close to the discharge does not always magnetise in the same direction.

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.

[note 2] James Clerk Maxwell heard of the phenomenon from Sir William Grove in 1868 in connection with experiments on dynamos,[9] and was also aware of the earlier work of Henry Wilde in 1866.

Maxwell explained resonance[note 3] mathematically, with a set of differential equations, in much the same terms that an RLC circuit is described today.

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).

[1][17] 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.

[19] 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.

[21][note 8] 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.

[25] 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.

[26] The cut-off phenomenon is an undesirable side-effect as far as loaded lines are concerned but for telephone FDM filters it is precisely what is required.

For this application, Campbell produced band-pass filters to the same ladder topology by replacing the inductors and capacitors with resonators and anti-resonators respectively.

[note 9] 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.

[35][36] 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).

The theorems of Gustav Kirchhoff and others and the ideas of Charles Steinmetz (phasors) and Arthur Kennelly (complex impedance)[42] laid the groundwork.

Designers tend to avoid the complication of mutual inductances and transformers where possible, although transformer-coupled double-tuned amplifiers are a common way of widening bandwidth without sacrificing selectivity.

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.

[52] 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.

[55] 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.

Following an analogy with Lagrangian mechanics, 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

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.

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.

Versions of this theory are due to Sidney Darlington, Wilhelm Cauer and others all working more or less independently and is often taken as synonymous with network synthesis.

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.

[64] Darlington relates that he found in the New York City library Carl Jacobi's original paper on elliptic functions, published in Latin in 1829.

[51] 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.

An early paper on this was published during WWII by Norbert Wiener with the specific application to anti-aircraft fire control analogue computers.

Practical 1 H inductors require many turns on a high-permeability core; that material will have high losses and stability issues (e.g., a large temperature coefficient).

A 1915 example of an early type of resonant circuit known as an Oudin coil which uses Leyden jars for the capacitance.
Hutin and Leblanc's multiple telegraph filter of 1891 showing the use of resonant circuits in filtering. [ 15 ] [ note 4 ]
Ohm's model of the transmission line was simply resistance.
Lord Kelvin's model of the transmission line accounted for capacitance and the dispersion it caused. The diagram represents Kelvin's model translated into modern terms using infinitesimal elements, but this was not the actual approach used by Kelvin.
Heaviside's model of the transmission line. L, R, C and G in all three diagrams are the primary line constants. The infinitesimals δL, δR, δC and δG are to be understood as Lδ x , Rδ x , Cδ x and Gδ x respectively.
Campbell's sketch of the low-pass version of his filter from his 1915 patent [ 28 ] showing the now ubiquitous ladder topology with capacitors for the ladder rungs and inductors for the stiles. Filters of more modern design also often adopt the same ladder topology as used by Campbell. It should be understood that although superficially similar, they are really quite different. The ladder construction is essential to the Campbell filter and all the sections have identical element values. Modern designs can be realised in any number of topologies, choosing the ladder topology is merely a matter of convenience. Their response is quite different (better) than Campbell's and the element values, in general, will all be different.
Norton's mechanical filter together with its electrical equivalent circuit. Two equivalents are shown, "Fig.3" directly corresponds to the physical relationship of the mechanical components; "Fig.4" is an equivalent transformed circuit arrived at by repeated application of a well known transform , the purpose being to remove the series resonant circuit from the body of the filter leaving a simple LC ladder network. [ 67 ]