Lattice network

A symmetrical lattice is a two-port electrical wave filter in which 'diagonally-crossed' shunt elements are present, a configuration which sets it apart from the familiar ladder networks. The component arrangement of the lattice is shown in the diagram below. The filter properties of this circuit were first developed using image impedance concepts but, later, the more general techniques of network analysis were applied to it.

There is duplication of components in the lattice network as the "series impedances" (the Za's) and "shunt impedances" (the Zb's) both occur twice, but the arrangement offers great flexibility to the circuit designer with a variety of responses achievable. For example, it is possible for it to have the characteristics of: a delay network,^[1] an amplitude or phase correcting network,^[2] a dispersive network ^[3] or as a linear phase filter,^[4]:412 according to the choice of components for the lattice elements.

The lattice configuration

The basic configuration of the symmetrical lattice is shown in the left-hand diagram and a commonly used short-hand version is shown on the right. In the simplified version, dotted lines are used to indicate the presence the second pair of matching impedances, to give a less cluttered schematic.

It is possible with this circuit to have the characteristic impedance specified independently of its transmission properties,^[5] a feature not available to the familiar ladder filter structures. In addition, it is possible to design the circuit to be a constant resistance network for a whole range of circuit characteristics,

The lattice structure can be converted to an unbalanced form (see below), for insertion in circuits with a ground plane. There are other benefits from doing the conversion, one of which is the reduction of the component count and another is the relaxation of component tolerances.^[6]

It is possible to redraw the lattice in the familiar Wheatstone bridge configuration,^[7] as shown in the article Zobel network. However, this is not a convenient format in which to investigate the properties of lattice filters, especially their behavior in cascade, so it is not used in articles on filter design.

Basic properties of the lattice

Results from image theory

Filter theory was developed, initially, from earlier studies of transmission lines.^[8][9] In this theory, a filter section is specified in terms of its propagation constant and image impedance (or characteristic impedance) .
Specifically for the lattice, the propagation function, γ, and characteristic impedance, Zo are defined by,^[4]:379[6] respectively

$\gamma = \ln \left[ \frac{\sqrt { \frac{Z_a}{Z_b} +1 } }{\sqrt{ \frac{Z_a}{Z_b} -1} } \right] =2\tanh^{-1} \left( \sqrt \frac{Z_a}{Z_b} \right) \qquad \text{and} \qquad Z_0 = \sqrt{Z_a Z_b} \gamma = \sqrt { \left(\frac{Z_a}{Z_b} +1 \right) }$

Once γ and Zo have been chosen, solutions can be found for Za/Zb and Za·Zb, from which the characteristics of Za and Zb can each be determined. (In practice, the choices for γ and Zo are restricted to those which result in physically realisable impedances for Za and Zb). Although a filter circuit may have one (or more) pass-bands and maybe several stop-bands, or attenuation regions, only networks with a single pass-band are considered here.

In the pass-band of the circuit, the product Za.Zb is real, i.e. Zo is resistive, and may be equated to Ro the terminating resistance of the filter. So

$Z_a Z_b = R_0^2 \qquad \text{or} \qquad \frac{Z_a}{R_0} = \frac{R_0}{Z_b} \qquad \text{(for frequencies in the passband)}$

i.e. the impedances behave as duals of each other within this frequency range.

In the attenuation range of the filter, the characteristic impedance of the filter is purely imaginary, and

$Z_a = Z_b \qquad \qquad \qquad \text{(for frequencies in the attenuation band)}$

Consequently, in order to achieve a specific characteristic, the reactances within Za and Zb are chosen so that their resonant and anti-resonant frequencies are duals of each other in the passband and match one another in the stopband. The transition region of the filter, where a change from one set of conditions to another occurs, can be made as narrow as required by increasing the complexity of Za and Zb. The phase response of the filter in the pass-band is governed by the locations (spacings) of the resonant and anti-resonant frequencies of Za and Zb.

For convenience, the normalised parameters y₀ and z₀ are defined by

$y_0 = \sqrt{\frac{Z_b}{Z_a}} = \sqrt{\frac{z_b}{z_a}} \qquad \text{and} \qquad z_0 = \sqrt{z_a z_b}$

where normalised values z_a = Z_a/Ro and z_b = Z_b/Ro have been introduced. The parameter y₀ is termed the index function and z₀ the characteristic impedance of the normalised network. The parameters y₀ and z₀ approximate unity in the attenuation and transmission regions, respectively.^[4]:383

Cascade of lattices

All high-order lattice networks can be replaced by a cascade of simpler lattices provided their characteristic impedances are all equal to that of the original and the sum of their propagation functions equals the original.^[4]:435

In the particular case of all-pass networks (i.e. networks which modify the phase characteristic only), any given network can always be replaced by a cascade of second-order lattices together with, maybe, one single first order lattice.^[6] Further details of these simple, but important, lattice sections are given later.

Whatever the filter requirements being considered, the reduction process results in simpler filter structures, with less stringent demands on component tolerances.^[6]

The shortcomings of image theory

The filter characteristics predicted by image theory require a correctly terminated network. As the necessary terminations are often impossible to achieve, resistors are commonly used as the terminations, resulting in a mismatched filter. Consequently, the predicted amplitude and phase responses of the circuit will no longer be as image theory predicts. In the case of a low pass filter, for example, where the mismatch is most severe near to the cut-off frequency, the transition from pass-band to stop-band is far less sharp than expected. The figure below illustrates the problem. A lattice filter, equivalent to two sections of constant-k low pass filter has been derived by image methods. (The network is normalized, with L = 1 and C = 1, so Ro = √(L/C) = 1 and ω_c = 2/√(L.C) = 2). The left-hand figure gives the lattice circuit and the right-hand figure gives the Insertion loss, firstly with the network terminated in its correct characteristic impedances and, secondly, when the same network is resistively terminated.

To minimise the mismatch problem, various forms of image filter end terminations were proposed by Otto Julius Zobel and others (see later), but the inevitable compromises led to the method falling out of favour. It was replaced by the more exact methods of network analysis and network synthesis.^{[10][11][12][13]}

Results derived by network analysis

Consider the general circuit for the symmetrical lattice:

By mesh analysis or nodal analysis of the circuit, its full transfer function can be found.

$\text{i.e.} \qquad \frac{v_\text{out}}{v_\text{in}} = \frac{Z_L(Zb - Za)}{(Za+Zb)(Z_S+Z_L)+2(Za Zb + Z_S Z_L)}$

The input and output impedances (Zin and Zout) of the network are given by

$Z_\text{in}= \frac{2 Za Zb + Z_L (Za+Zb)}{Za+Zb+2 Z_L} \qquad \text{and} \qquad Z_\text{out} = \frac{2 Za Zb + Z_S (Za+Zb)}{Za+Zb+2Z_S}$

These equations are exact, for all realizable impedance values, unlike image theory where the propagation function only predicts performance accurately when Z_S and Z_L are the matching characteristic impedances of the network.

The equations can be simplified by making a number of assumptions. Firstly, networks are often sourced and terminated by resistors of the same value Ro so that Z_S = Z_L = Ro and the equations become

$\frac{v_\text{out}}{v_\text{in}} = \frac{R_0(Zb - Za)}{2(Za+R_0)(Zb + R_0)} \qquad \text{and} \qquad Z_\text{in} = Z_\text{out} = \frac{2 Za\, Zb + R_0 (Za+Zb)}{Za+Zb+2R_0}$

Secondly, if the impedances Za and Zb are duals of one another, so that Za.Zb = Ro², then further simplification is possible:

$\frac{v_\text{out}}{v_\text{in}} = \frac{(R_0-Za)}{2(R_0 + Za)} = \frac{(Zb - R_0)}{2(R_0 + Zb)} \qquad \text{and} \qquad Z_\text{in} = Z_\text{out} = R_0$

so such networks are constant resistance networks.

Finally, for normalized networks, with Ro = 1,

$\frac{v_\text{out}}{v_\text{in}} = \frac{(1-z_a)}{2(1 + z_a)} = \frac{(z_b - 1)}{2(1+z_b)} \qquad \qquad \text{and} \qquad z_\text{in} = z_\text{out} = 1$

If the impedances Za and Zb (or the normalised impedances za and zb) are pure reactances, then the networks become all-pass, constant-resistance, with a flat frequency response but a variable phase response. This makes them ideal as delay networks and phase equalisers (see later).

When resistors are present within Za and Zb then, provided the duality condition still applies, a circuit will be constant resistance but have a variable amplitude response. One application for such circuits is as amplitude equalisers (see later).

Conversions and equivalences of the lattice

(See references^[4][6][14])

T to lattice

Pi to lattice

Common series element

Common parallel element

Combining two lattices into one

Lattice to T (see also the next section)

This lattice-to-T conversion only gives a realisable circuit when the evaluation of (Zb – Za/2) gives positive valued components. For other situations, the bridged-T may provide a solution, as discussed in the next section.

Unbalanced equivalents of the lattice

The lattice is a balanced configuration, which is not suitable for some applications, so it becomes necessary to convert the circuit to an electrically equivalent unbalanced form. Carrying out such a procedure gives other benefits, namely the component count is reduced and circuit tolerances are less stringent, as stated earlier. A simple conversion procedure has been given in the previous section, but this only applies in a limited set of conditions so, more generally, some form of bridged-T circuit is necessary. Many of the conversions require the inclusion of a 1:1 ideal transformer,^[14] but there are some configurations which avoid this requirement, and one example is shown below.

This conversion procedure starts by using that property of a lattice where a common series element in all arms can be taken outside the lattice as two series elements (as shown above). By repeatedly applying this property, components can be extracted from within the lattice structure. Finally, by means of Bartlett's bisection theorem,^[15][16] an unbalanced bridged-T circuit is achieved.

First consider the case where the Za arm has a shunt capacitor, Ca, and the Zb arm has a series capacitor Cb, as shown in the left-hand figure. Consequently, Za consists of Ca in parallel with Za', and Zb consists of Cb in series with Zb'. This can be developed into the unbalanced bridged-T shown, provided Ca>Cb .

(An alternative version of this circuit has the ‘T’ configuration of capacitors replaced by a ‘Pi’ (or ‘Delta’) arrangement. For this ‘T to Pi’ conversion see the equations in Attenuator (electronics)).

When Cb>Ca, an alternative procedure is necessary, where common inductors are first extracted from the lattice arms. As shown, an inductor La shunts Za’ and an inductor Lb is in series with Zb’. This leads to the alternative bridged-T circuit on the right.

If La>Lb, then the negative valued inductor can be achieved by means of mutually coupled coils. To achieve a negative mutual inductance, the two coupled inductors L1 and L2 are wound ‘series-aiding’.

So finally, the bridged-T circuit takes the form

Bridged-T circuits like these may be used in delay and phase correcting networks.

Lattice all-pass networks

(See previously quoted references to Zobel, Darlington, Bode and Guillemin. Also see Stewart.^[17] and Weinberg^[1]

All-pass networks are an important sub-class of lattice networks and they have been used as passive lumped-element delays, as phase correctors for filter networks and in dispersive networks, as mentioned earlier. They are constant resistance networks so they can be cascaded with each other and with other circuits without introducing mismatch problems.

In the case of all-pass networks, there is no attenuation region, so the impedances Za and Zb, of the lattice, are duals of each other at all frequencies and Zo is always resistive (= Ro).

i.e.

$Za\cdot Zb = R_0^2 \qquad \qquad \text{(at all frequencies)}$

For normalized networks (with Ro = 1), the transfer function T(p) can be written

$T(p) = \frac{1-z_a(p)}{1+z_a(p)} = \frac{z_b(p)-1}{z_b(p)+1}$

and so

$z_a(p) = \frac{1-T(p)}{1+T(p)} \qquad \text{and} \qquad z_b(p) = \frac{1+T(p)}{1-T(p)}$

In practice, T(p) can be expressed as a ratio of polynomials in p, and the impedances z_a and z_b are also ratios of polynomials in p. For the impedances to be realisable, they must satisfy Foster's reactance theorem .

All-pass networks of the first and second orders

The two simplest all pass networks are the first and second order lattices. These are important circuits because, as Bode pointed out,^[18] all high order all pass lattice networks can be replaced by a cascade of second order networks with, maybe, one first order network, to give the identical response.

These two simple, normalized, lattices have transfer impedances given by

$T_1(p)= \frac{-p+c}{p+c} \qquad \qquad \text{and} \qquad T_2(p) = \frac{p^2 -ap + b}{p^2 + ap +b}$

These functions have circuit schematics as shown below, in which the component values are related to the coefficients a,b and c in the equation above in the manner shown.

The locations of the pole and zero of the first order lattice on the complex frequency plane are simply located at ±c on the real axis, as shown below. For the second order network, slightly more numerical manipulation is required to related pole-zero positions to circuit values. If the poles and zeros are located at (±x ±jy) on the complex frequency plane, then the coefficients a and b can be found from

$a = 2x \qquad \qquad \text{and} \qquad b = x^2 + y^2$

The locations of the poles and zeros on the complex frequency plane are shown below.

The unbalanced versions of these networks are easily found using the procedures described earlier and circuits are shown below. There are two versions of the second order network, firstly when C1>C2 then mutually coupled coils are not required, whereas when C2>C1, mutually coupled coils become necessary.

The circuits considered above are normalized to one ohm. To convert a circuit to have terminations of Ro ohms, multiply all L values by Ro and divide all C values by Ro.

Early history of the lattice

The symmetrical lattice and the more familiar ladder networks (the constant k filter and m-derived filter), were the subject of much interest in the early part of the twentieth century.^{[4][7][19][20]} At that time, the rapidly growing telephone industry had a significant influence on the development of filter theory, as ways were sought to increase the signal carrying capacity of telephone transmission lines.^[21] George Ashley Campbell was a key contributor to this new filter theory, as was Otto Julius Zobel . They and their many colleagues worked at the laboratories of Western Electric and the American Telephone and Telegraph Co.,^[21] and their work was reported in the early editions of the Bell System Technical Journal.

Campbell discussed lattice filters in his article of 1922,^[7] while other early workers with an interest in the lattice were Johnson^[22] and Bartlett.^[23] However, Zobel's article on filter theory and design,^[19] published at about this time, mentioned lattices only briefly, with his main emphasis on ladder networks. It was only later when Zobel considered the simulation and equalization of telephone transmission lines, that he gave the lattice configuration more attention.^[24] (The telephone transmission lines of the time had a balanced-pair configuration with a nominal characteristic impedance of 600 ohms^[25]) so the lattice equaliser, with its balanced structure, was particularly appropriate for use with them). Later workers, especially Hendrik Wade Bode^[18][20] gave greater prominence to lattice networks in their filter designs.

In those early days, filter theory was based on image impedance concepts, or image filter theory, which was a design approach developed from the well-established studies of transmission lines. The filter was considered to be a lumped component version of a section of transmission line, and was one of many within a cascade of similar sections. As mentioned above, the weakness of the image filter approach was that the frequency response of a network was often not as predicted when the network was terminated resistively, instead of by the required image impedances. This was, essentially a mismatch issue and Zobel, in particular, did much work to overcome it by means of matching end sections (see: m-derived filter, mm'-type filter, General mn-type image filter, with later work by Payne^[26] and Bode.^[27]

Although lattice filters sometimes suffer from this same problem, a whole range of constant resistance networks is possible, which avoids it altogether. Some examples of the possible applications of such networks were listed in the Introduction.

During the 1930s, as techniques in network analysis and synthesis became better developed, designing ladder filters by image methods became less popular. Even so, the concepts still found relevance in some modern designs.^[28] On the other hand, lattice networks, and their circuit equivalents continued to be used in many applications.

See also

• Lattice phase equalizer
• All-pass filter
• Two-port network
• Composite image filter
• Lattice delay network

References

1. Weinberg L., "Network Analysis and Synthesis", McGraw Hill 1962, (p. 633)
2. Stewart J.L., "Fundamentals of Signal Theory", McGraw Hill, 1960, (p. 138)
3. Cook C.E. and Bernfeld M., "Radar Signals", Artech House MA, 1993, ISBN 0-89006-733-3, (p.413)
4. Guillemin E.A., Communication Networks, Vol II", Wiley N.Y., 1935
5. Zverev A.I., "Handbook of Filter Synthesis", Wiley N.Y., 1967, (p.6)
6. Bode H.W. , “Network Analysis and Feedback Amplifier Design”, Van Nostrand, N.Y., 1945
7. Campbell G.A., “Physical Theory of the Electric Wave-Filter”, BSTJ, Vol. I, No. 2, Nov. 1922, (pp. 1–32).
8. Fleming J. A.,"The Propagation of Electric Currents", 2nd edition, Constable, London, 1912.
9. Jackson W., “High Frequency Transmission Lines”, Methuen Monograph, London 1945
10. Guillemin E.A., “A Summary of Modern Methods of Network Synthesis”, Advances in Electronics and Electron Physics, Vol. 3, 1951, Ed Marton L., (pp. 261–303)
11. Darlington S., “The Potential Analogue Method of Network Synthesis”, BSTJ, April 1951 (pp. 315–364)
12. Kuo F.F.,"Network Analysis and Synthesis", Wiley, N.Y., 1962
13. Tuttle D.F., "Network Synthesis, Volume 1", Wiley N.Y., Chapman and Hall London, 1958
14. Conning S.W., "A Survey of Network Equivalences", Proc. IREE, Australia, June 1969, (pp. 166–184)
15. Bartlett A.C., “An Extension of a Property of Artificial Lines”, Phil. Mag., Vol. 4, Nov. 1927, (p. 902)
16. Bartlett A.C., "The Theory of Electrical Artificial Lines and Filters", Chapman & Hall, 1930
17. Stewart J.L., "Fundamentals of Signal Theory", McGraw-Hill, N.Y., 1960
18. Bode H.W. and Dietzold R.L., "Ideal Wave Filters", BSTJ, Vol XIV, April 1935, (pp. 215–252).
19. Zobel O.J., “Theory and Design of Uniform and Composite Electric Wave-filters”, BSTJ Vol.II, Jan 1923 (pp. 1–46)
20. Bode H.W., “A General Theory of Electric Wave Filters”, Jour. Math. & Phys. Vol. XIII, Nov. 1934, (pp. 275–362)
21. Bray J., “Innovation and the Communications Revolution”, The IEE, London, 2002.
22. Johnson K.S., "Lattice type wave filters", US Patent 1,501,667, 1924
23. Bartlett A.C., "Lattice Type Filters", British Patent 253,629
24. Zobel O.J., “Distortion Correction in Electrical Circuits with Constant Resistance Recurrent Networks”, BSTJ, Vol. 7, No. 3, July 1928, (pp. 438–534)
25. Green E.I., “The Transmission Characteristics of Open-Wire Telephone Lines”, BSTJ Vol.9, Iss. 4, Oct. 1930, (pp. 730–759)
26. Payne E.B., “Impedance Correction of Wave Filters”, BSTJ, Oct. 1930, pp.770-793.
27. Bode H.W.., “A Method of Impedance Correction”, BSTJ Vol. 9, No. 4, Oct 1930, (pp.394-835)
28. Matthaei G. L., Young L. and Jones E.M.T., “Microwave Filters, Impedance-Matching Networks, and Coupling Structures”, McGraw Hill 1964, Artech House 1980.