Lattice and bridged-T equalizers are circuits which are used to correct for the amplitude and/or phase errors of a network or transmission line. Usually, the aim is to achieve an overall system performance with a flat amplitude response and constant delay over a prescribed frequency range,Stewart J.L., “Fundamentals of Signal Theory”, McGraw-Hill N.Y., 1960Tuttle D.F., “Network Synthesis, Volume1”, Wiley, N.Y., 1958 by the addition of an equalizer. In the past, designers have used a variety of techniques to realize their equalizer circuits. These include the method of complementary networks;Zobel O.J., “Distortion Correction in Electrical Circuits with Constant Resistance Recurrent Networks”, BSTJ Vol.7, No. 3, July 1928 the method of straight line asymptotes;Rounds P.W. and Lakin G.L., “Equalisation of Cables for Local Television Transmission”, BSTJ, Vol34, July 1955, (pp. 713–738) using a purpose built test-jig;Hale H.S. and MacDiarmid I.F., “Video waveform correctors for use with coaxial cable”, Proc. IEE, Vol 110, No. 8, August 1963, pp. 1319–1328 the use of standard circuit building blocks,;Bode H.W., “Network Analysis and Feedback Amplifier Design”, van Nostrand, 1945, reprinted Isha Books, India, 2013 or with the aid of computer programs.Szentirmai G., "Computer-Aided Design Methods in Filter Design: S/FILSYN and Other Packages", Chapter 3 of "CRC Handbook of Electrical Filters", ed. Taylor J.T. and Huang Q., CRC Press, N.Y., 1996, pp. 19-41. In addition, trial and error methods have been found to be surprisingly effective,Stewart J.L., "Circuit Theory and Design", Wiley, N.Y., 1956 when performed by an experienced designer.
In video or audio channels, equalization results in waveforms that are transmitted with less degradation and have sharper transient edges with reduced overshoots (ringing)Cherry C., “Pulses and transients in Communication Circuits”, Chapman & Hall, London, 1949, p.145 than before. In other applications, such as CATV distribution systemsGroffe D.E., “Bode’s Variable Equalizer”, NCTA Conference, April 1976, pp. 72–76 or frequency multiplexed telephone signalsBray J., “Innovation and the Communications Revolution”, Institution of Electrical Engineers, London, 2002 where multiple carrier signals are being passed, the aim is to equalize the transmission line so that those signals have much the same amplitude. The lattice and bridged-T circuits are favoured for passive equalizers because they can be configured as constant-resistance networks such as the Zobel network, as pointed out by Zobel and later by Bode.
The single word description “equalizer” is commonly used when the main purpose of the network is to correct the amplitude response of a system, even though some beneficial phase correction may also be achieved at same time. When phase correction is the main concern, the more explicit term "phase equalizer" or "phase corrector" is used. (In this case, the circuit is usually an all-pass network which does not alter the amplitude response at all such as the lattice phase equalizer).
When equalizing a balanced transmission line, the lattice is the best circuit configuration to use, whereas for a single-ended circuit with an earth plane, the bridged-T network is more appropriate. Although equalizer circuits, of either form, can be designed to compensate for a wide range of amplitude and phase characteristics, they can become very complicated when the compensation task is difficult, as is shown later.
A variety of methods has been used to design equalizers and some of these are described below. Several of the procedures date back to the early part of the 20th century when equalizers were needed by the rapidly expanding telephone industry. Later, with the advent of television, the equalisation of video links became very important too.
Amplitude correction The aim of an equalizer network is to correct for deficiencies in the amplitude response of a transmission line, lumped element network or amplifier chain. Equalisation is often necessary with transmission lines and lumped element delay lines which tend to have increasing loss with frequency. Without correction, waveform fidelity is lost, and rise and fall times of transients are degraded (i.e. less sharp).Terman F.E., “Voltage Amplifiers for Video Frequencies”, Electronic and Radio Engineering, Chapter 9, 4th Edition, N.Y. 1955, pp.288- 318Fink D.G., “IF Amplifiers and Detectors”, Television Engineering Handbook, Chapter 16.4, McGraw-Hill, N.Y., 1957, pp. 16.37–16.77 Sometimes amplitude correction is required for more subtle reasons, for example, in the case of analogue colour television waveforms,Fink D.G., “Television Standards”, Television Engineering Handbook, Chapter 2, McGraw-Hill, N.Y., 1957, pp. 2.1–2.54 colour errors can occur in the displayed pictures when the transmission system’s response is not flat. It is usual to choose lattice and bridged-T equalizers which are constant-resistance networks. It was pointed out by Zobel and later by Bode that such networks can be cascaded with each other and with a transmission line or with a lumped element circuit, without introducing mismatch problems. The use of constant resistance configurations has been common practice in equalizer design, for many years, and almost all of the examples presented in this article have this property.
Whatever the design method, passive equalizers always introduce additional loss into the transmission path, and this has to be made good by an amplifier or repeater.
The method of complementary networks
In some of his early work, Zobel devised a lumped element circuit to simulate the behaviour of a given long transmission line of interest. Such a device was useful in that it allowed investigatory work on a transmission system to be carried out in the convenience of the laboratory. Importantly, as was pointed out by Zobel, once such a network had been designed, it was always possible to find a realizable complementary network, which exhibited the inverse response.
An example The procedure can be illustrated by a simple example presented by Zobel, which is shown below. Here, the left hand lattice has a simple low-pass characteristic and the right hand lattice has the complementary characteristic. For this circuit R1*R2 = L1/C1 = L2/C2 = R0^2 with R1 < 2.R0 . C2 is given by
For a normalized network R0 = 1Ω. Choose R1 = 0.5Ω and L1 = 1H then R2 = 2Ω, C1 = 1F, C2 = 3F, and L2 = 3H
The responses of the individual sections and the overall response are shown in the plots for the composite network, given on the right.
This compensation process can be described mathematically, by means of the basic lattice equations given in lattice network, as follows. The transmission loss of a normalized (R0 = 1) constant resistance lattice with through arms Za and cross-diagonal arms of Zb is
Now for the low-pass circuit, given above, za is a parallel combination of a 0.5Ω resistor in parallel with 1H inductor.
and so
Next consider the high –pass section. Here za is a parallel combination of a 0.5Ω resistor in parallel with a 3F capacitor, so
and so
Finally, the overall response Ttot is
i.e. a flat response, with an overall gain of one third. This is shown in the figure above.
In this example, the overall characteristic is phaseless, i.e. full phase compensation has been achieved in addition to the amplitude correction. This is because exact correction has been achieved at all frequencies. Often correction is only successful over a limited range of frequencies, in which case the final result will have some residual phase.Rounds P.W., “Equalization of Video Cables”, Conv. Rec. IRE, Part 2, Circuit Theory, March 1954
More generally, the response of a circuit or transmission line may be difficult to reproduce precisely by means of lumped element circuits, so the usual task is to find a realizable characteristic that is an acceptable match. Sometimes trial an error methods, applied in a systematic way can prove successful
A general-purpose equalizer circuit
When the arms of a lattice network are purely reactive, constant resistance all-pass networks are possible, as shown in lattice delay network. If, however, resistors are included in the lattice arms, then different amplitude responses are possible while still retaining the constant resistance property.
One, much-used equalizer circuit is shown here, together with the equivalent bridged-T circuit. The equivalence between the lattice and bridged-T can be shown with Bartlett's bisection theorem. The circuits have constant resistance characteristic when Z1.Z2 = R02.
The transfer function of this circuit is
When normalized so that the source and terminating resistors and R0 are all unity and Z1.Z2 = 1, it becomes
and so Z1, as a function of T, is given by
The simplicity of this equation (only one impedance has to be found) makes the circuit popular in equalizer designs.
The method of straight line asymptotes
The method uses an initial trial response, made up of a sequence of straight line asymptotes, to determine the pole/zero locations of a realizable network.
The general principles of the method are as follows.
Consider the case of a simple RC low-pass circuit whose transfer function has a single pole on the real frequency axis at p = -a, as shown This response is -3 dB at ω = a and falls at 6 dB per octave at high frequencies. It is shown plotted, in decibels, versus frequency (on a log scale). Also shown, on the right, is the two-line asymptotic approximation, with break frequency also at ω = a. As can be seen, the true response and the straight line approximation are closely matched over most of the frequency range, only deviating near to ω = a.
If a zero is now added at, say, -10×a, then a new asymptote with a positive slope of 6 dB per octave is introduced, starting at ω = 10, as shown.
By extending the method, a complicated loss characteristic can be approximated by a sequence of straight line asymptotes provided the steepest slope of the loss characteristic does not exceed 6 dB per octave in the frequency range. The results are plotted on a dB scale against frequency on a log scale, as before.
In general, the transfer function is
Once an expression is deemed to meet the requirement with sufficient accuracy, the impedances Z1 and Z2 can be found from the relationships given earlier.
An example As an example, the transmission loss of a notional distribution network, with response shown by the full line in the figure on the left, is to be corrected (equalized). In this example correction is required over the frequency range from Ω1 = 0.09 r/s to Ω2 = 140 r/s. This can be approximated by a series of straight line segments as shown in the figure. Frequencies are chosen so that the various errors exhibited by the straight line approximation are all similar and as small as possible.
The straight line approximation shown in the figure has poles at ω = 0.333 r/s and 10 r/s, and zeros at ω = 1 r/s and 30 r/s.
The transfer function of a network with these poles and zeros is given by
The actual response (magnitude) of T is easily calculated and is shown, on the plot on the right, as a function of frequency, together with the target response. A closer fit would require extra poles and zeros in T(p), at a closer spacing.
The low-pass network T(p), with this response will have an L-C network for Z1 and an R-C network for Z2, when realized using the simple circuits previously described. However, of more interest here is the complementary network, since this will correct for the falling transmission loss of this response.
The transfer function of the equalizer network will be
where K = 9 so that the gain of the network never exceeds unity at any frequency.
The expression for Z1(p) can be derived from this equation using
So