Lattice delay networks are an important subgroup of lattice networks. They are all-pass filters, so they have a flat amplitude response, but a phase response which varies linearly (or almost linearly) with frequency. All lattice circuits, regardless of their complexity, are based on the schematic shown below, which contains two series impedances, Za, and two shunt impedances, Zb. Although there is duplication of impedances in this arrangement, it offers great flexibility to the circuit designer so that, in addition to its use as delay network (as featured here) it can be configured to be a phase corrector,Stewart J.L., "Fundamentals of Signal Theory", McGraw Hill, 1960 a dispersive network,Cook C.E. and Bernfeld M., "Radar Signals", Artech House MA, 1993, , (p.413) an amplitude equalizer,Rounds P.W. and Lakin G.L., "Equalization of Cables for Local Television Transmission", BSTJ, July 1955 (pp. 713–738) or a low pass (or bandpass) filter,Guillemin E.A., Communication Networks, Vol II", Wiley N.Y., 1935 according to the choice of components for the lattice elements .
It is shown in Lattice networks that when a lattice is configured as a delay network, it has a characteristic impedance which is resistive (= Ro), its impedances Za and Zb are dual impedances, i.e. Za⋅Zb = Ro2 (or Za/Ro = Ro/Zb) and Za and Zb consist of inductors and capacitors. Such a lattice is a constant resistance network and an all-pass filter, and it has a phase response determined by the properties of Za. This makes it ideal as a delay device because it can be included in a cascade of other filter sections without affecting the overall amplitude response, nor will it create mismatch problems, but it will increase the phase slope (i.e. the delay) of the overall assembly.
In order to achieve a desired delay, it is necessary to choose specific components for Za and Zb, and the design methods to do this are given in later sections. However, regardless of the method used, networks only achieve a constant delay over a finite band of frequencies, so if an increase in bandwidth and/or delay is required, more complex solutions for Za and Zb are necessary. Normally Za and Zb are lumped element impedances, suitable for networks operating at audio or video frequencies but operation up to v.h.f. and even u.h.f. is also possible. Sometimes, the design procedures can result in Za and Zb being highly complicated networks, but it is always possible to derive a cascade of simpler lattices with identical electrical characteristics, should that be preferred.
A lattice delay section has twice the delay of a comparable ladder filter section, and this helps to mitigate concerns over component duplication. In any case, a lattice configuration can be converted to an unbalanced equivalent, which will reduce the component count and permit some relaxation of component tolerances.Bode H.W., "Network Analysis and Feedback Amplifier Design", Van Nostrand, N.Y., 1945 Consequently, lattice delay sections, or their bridged T circuit equivalents, are able to provide substantial time delays in a compact physical form and they make efficient use of their operational bandwidth. Although there are other ways of achieving signal delays, such as by a long length of coaxial cable, or by lumped element ladder networks, such solutions have either greater physical bulk, or they make inefficient use of a frequency band, or they have poor phase linearity.
Design methods for lattice delays Initially, the designs for lattice delays were based on image theoryHebb M.H., Horton C.W. and Jones F.B., "On the Design of Networks for Constant Time Delay – image theory content", Jour of Applied Physics, Vol. 20, June 1949 in which the aim was to simulate a finite length of transmission line. Later, network synthesis methods were introduced.
A commonly chosen response for the delay network is the maximally flat group delay characteristic.Thomson W.E., "Networks with Maximally-Flat Delay", Wireless Engineer, October, 1952, (pp. 256–262). This delay response is ripple free and is perfectly smooth over the passband, only deviating from the mean value as the band edge is reached. Initially, such a response might be thought to be ideal for a delay network, but it is not necessarily the most efficient and in order to achieve a wider bandwidth, for a given delay, a higher order network is required. However, some increase in bandwidth is also possible, without increasing circuit complexity, by considering alternative characteristics, where the phase and group delay responses are allowed to ripple within the passbandUlbrich E. and Piloty H., "Uber den Entwirf von Allpassen, Tiefpassen und Bandpassen mit einer im Tschebyscheffschen Sinne approximierten konstanten Gruppenlaufzeit", Arch. Eleckt. Ubertragung, Vol. 14, Oct. 1960, (pp. 457–467)’.Dewsnap G.D., The Approximation of a Time Delay", Proc. IRE (Australia), Vol.25, March 1964 (pp. 168–174)
There are several design procedures available by which a desired linear-phase approximation can be achieved, whether maximally-flat or with ripple. These methods include techniques from image theory, by the Potential Analog method, and by a Taylor expansion of a group delay, all of which are described in the following sections.
In situations where a balanced network is not appropriate, a single ended circuit operating with a ground plane is required. In such cases, the conversion of a lattice into a bridged T circuit is carried out, as described in the article Lattice network. The resulting unbalanced network has the same electrical characteristics as the balanced lattice network on which it is based. An example of this procedure is given in a later section.
Networks derived from image theory An ideal delay line characteristic has constant attenuation and linear phase variation, with frequency, i.e. it can be expressed by
where τ is the required delay.
As shown in lattice networks, the series arms of the lattice, za, are given by
More generally, for lattice circuits having a delay τ secs., with a characteristic impedance Zo, the expressions for Za and Zb are given by
As e−x and tanh(x) are not rational functions, exact solutions for za and zb are not possible, so some form of approximation has to be used.
Continued fraction approximation A continued fraction expansion of tanh(x)Abramowitz M. and Stegun I.A., "Handbook of mathematical functions", Nat. Bur. Standards 1964, reprinted by Dover Publications N.Y., 1965 (9th ed. 1972),(p.85)Weinberg L., "Network Analysis and Synthesis", McGraw-Hill, 1962 (p.193) is
So, for a network with 1sec delay, za can be written
An exact solution requires an infinite number of terms, but an nth order approximation is obtained by terminating za after n elements. (If the last component retained is a capacitor, the remainder of the network is replaced by a short circuit). So, for example, terminating this expression after six terms will give a sixth order delay, which can be synthesised directly by Cauer’s methods to give the network shown.
A circuit for zb can easily found from this solution, since it is the dual of za, and is
Although this circuit of zb was easy to derive, it is not necessarily the most ideal. If an unbalanced equivalent circuit of the lattice is ultimately required, it would be better if zb started with a series inductor (see Lattice networks). To do this it is first necessary to multiply out the continued fraction expansion for za, for this example, to give za (and zb in particular) as a ratio of polynomials in p. This is
and for the alternative Cauer I expansion proceeds as follows
and so on, until the network shown below is obtained.
Further details of lattice circuits, using these impedances, are considered in the examples section, later.
Now, as shown in Lattice networks, the transfer function of this lattice is given by
so
From this the phase plot for this sixth-order all-pass function can be calculated and is given below.
This response is the same as that of the maximally flat delay which is derived in a later section. (In fact, the derivations of za by the continued fraction method result in family of lattices all of which have a maximally flat group delay characteristic). The phase error plot (i.e. the deviation of the response from linear) of this response can be found in the section on maximally flat delay networks, where the responses of networks of several orders are given.
Networks derived using the potential analogue method
The potential analogue method was proposed by DarlingtonDarlington S., "The Potential Analogue Method of Network Synthesis", BSTJ, April 1951, (pp. 315–364) as a simple way to choose pole-zero positions for delay networks. The method allows the designer to implement a delay characteristic by locating poles and zero on the complex frequency plane intuitively, without the need for complicated mathematics or the recourse to reference tables.
Other analogue methods, which were devised to aid the designer to choose the pole zero positions for his networks, include the "rubber sheet model"Bradley W.E., "Wideband Amplification – the Elastic Sheet Model", Chapter 12 of Television Engineering Handbook, ed. D.G. Fink, McGraw-Hill, 1957.Pramanik A., "Electromagnetism Volume 1(Theory)", Chapter 5, PHI Learning Private Ltd., New Delhi, 2014. and the "electrolytic tank".Edwards R., Demetrio, T., and Johnson, D., "Resurrecting the Electrolytic Plotting Tank," Proceedings, American Society for Engineering Education Annual Conference and Exposition, AC 2011–1819, Vancouver, BC, June 2011Cherry E.C., "Application of Electrolytic Tank Techniques to Network Synthesis", from "Symposium on Modern Network Synthesis", Polytechnic Institute of Brooklyn, N.Y., 1952, (pp. 140-160) and Teledeltos paperAston University, "Field Plotting Using Teledeltos Paper", Oct. 1994. Find at: http://www-users.aston.ac.uk/~pearcecg/Teaching/PDF/TELDELT.PDF
Darlington’s procedure starts by considering the field between the two plates of a parallel plate capacitor. The field is uniform within the plates and only deviates from linear beyond the extremities of the plates. In order to increase the length over which the field is uniform, the length of the plates is increased, as required. The next step is to replace the uniform plates by uniformly spaced charged filaments, which give the same field, but may result in a ‘granularity error’ (or ripple). Finally, the equivalent electrical network is obtained by replacing the localised filament charges by poles and zeros, where the group delay characteristic corresponds to the electric field in the potential analogue.
A typical arrangement of poles and zeros to give, nominally, an electrical circuit with constant group delay follows the pattern shown in the figure below (see also Stewart). The poles and zeros lie in two lines, of finite length, parallel to the jω axis at a distance ‘a’ from it. Also, they are spaced apart at a distance ‘b’ from each other in the jω direction.
In general, Darlington showed that the group delay and the granularity effect are given by
A good approximation to a unit delay characteristic is obtained by putting a = b = 2 (a value which is easily remembered). However, the delay ripple (granularity) that results, when using these values of a and b, is rather high at ±8% and a better choice for a is 4.4 (= 1.4) which gives a ripple a lower ripple of ±2.5%. The plots shown below, are for networks with increasing numbers of poles and zeros, for a = 4.4 and b = 2. The order ‘n’ corresponds to the number of pole-zero pairs present in the network.
For frequencies beyond the end of the pole zero pattern, the group delay suffers a truncation error, but the band edge performance of a characteristic can be improved, by repositioning the outer poles and zeros slightly, to compensate for this sudden termination of the pattern. Darlington discusses this in his article.