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 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 a duplication of components in the lattice network as the "series impedances" and "shunt impedances" both occur twice, an arrangement that offers increased flexibility to the circuit designer with a variety of responses achievable. It is possible for the lattice network to have the characteristics of: a delay network, an amplitude or phase correcting network, a dispersive network or as a linear phase filter, according to the choice of components for the lattice elements.

Configuration

The basic configuration of the symmetrical lattice is shown in the left-hand diagram. A commonly used short-hand version is shown on the right, with dotted lines indicating the presence the second pair of matching impedances.
It is possible with this circuit to have the characteristic impedance specified independently of its transmission properties, a feature not available to ladder filter structures. In addition, it is possible to design the circuit to be a constant-resistance network for a range of circuit characteristics.
The lattice structure can be converted to an unbalanced form, for insertion in circuits with a ground plane. Such conversions also reduce the component count and relax component tolerances.
It is possible to redraw the lattice in the Wheatstone bridge configuration. However, this is not a convenient format in which to investigate the properties of lattice filters, especially their behavior in cascade.

Basic properties

Results from image theory

Filter theory was initially developed from earlier studies of transmission lines. In this theory, a filter section is specified in terms of its propagation constant and image impedance.
Specifically for the lattice, the propagation function,, and characteristic impedance, _o, are defined by,
Once and _o have been chosen, solutions can be found for and from which the characteristics of _a and _b
can each be determined.
Although a filter circuit may have one or more pass-bands and possibly several stop-bands, only networks with a single pass-band are considered here.
In the pass-band of the circuit, the product is real and may be equated to _o, the terminating resistance of the filter. So
That is, 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
Consequently, in order to achieve a specific characteristic, the reactances within _a and _b 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 _a and _b . The phase response of the filter in the pass-band is governed by the locations of the resonant and anti-resonant frequencies of _a and _b .
For convenience, the normalised parameters _o and _o are defined by
where normalised values and have been introduced. The parameter _o is termed the index function and _o is the normalised characteristic impedance of the network. The parameter _o is approximately in the attenuation region; _o is approximately in the transmission region.

Cascaded 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. In the particular case of all-pass networks, any given network can always be replaced by a cascade of second-order lattices together with, possibly, one single first order lattice.
Whatever the filter requirements being considered, the reduction process results in simpler filter structures, with less stringent demands on component tolerances.

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 the cut-off frequency, the transition from pass-band to stop-band is far less sharp than expected.
The figure below illustrates the issue: A lattice filter, equivalent to two sections of constant low-pass filter, has been derived by image methods. resistively, and in its correct characteristic impedances.
To minimise the mismatch problem, various forms of image filter end terminations were proposed by Zobel and others, 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.

Results derived by network analysis

This diagram shows the general circuit for the symmetrical lattice:
Through mesh analysis or nodal analysis of the circuit, its full transfer function can be found. That is,
The input and output impedances of the network are given by
These equations are exact, for all realisable impedance values, unlike image theory where the propagation function only predicts performance accurately when and 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 so that and the equations become
Secondly, if the impedances and are duals of one another, so that, then further simplification is possible:
so such networks are constant-resistance networks.
Finally, for normalised networks, with,
If the impedances and 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.
When resistors are present within and 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.

Conversions and equivalences

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 gives positive valued components. For other situations, the bridged-T may provide a solution, as discussed in the next section.

Unbalanced equivalents

The lattice is a balanced configuration which is not suitable for some applications. In such cases it is necessary to convert the circuit to an electrically equivalent unbalanced form. This provides benefits, including reduced component count and relaxed circuit tolerances. The simple conversion procedure shown in the previous section can only be applied in a limited set of conditions - generally, some form of bridged-T circuit is necessary. Many of the conversions require the inclusion of a 1:1 ideal transformer, but there are some configurations which avoid this requirement, and one example is shown below.
This conversion procedure starts by using the property of a lattice where a common series element in all arms can be taken outside the lattice as two series elements. By repeatedly applying this property, components can be extracted from within the lattice structure. Finally, by means of Bartlett's bisection theorem, an unbalanced bridged-T circuit is achieved.
In the left-hand figure, the Z_a arm has a shunt capacitor, C_a, and the Z_b arm has a series capacitor, C_b. Consequently, Z_a consists of C_a in parallel with Z_a′, and Z_b consists of C_b in series with Z_b′. This can be developed into the unbalanced bridged-T shown, provided.
arrangement. For this T to Pi conversion, see the equations in Attenuator.
When, an alternative procedure is necessary, where common inductors are first extracted from the lattice arms. As shown, an inductor L_a shunts Z_a′ and an inductor L_b is in series with Z_b′. This leads to the alternative bridged-T circuit on the right.
If, 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.
Another lattice configuration, containing resistors, is shown below. It has shunt resistors Ro across the Z_a’s and series resistors Ro as part of the Z_b's, as shown in the left hand figure. It is easily converted to an unbalanced bridged-T circuit, as shown on the right.
When Z₁Z₂ = R₀² it becomes a constant resistance network, which has an insertion loss given by
When normalized to 1ohm, the source, load and R₀ are all unity, so Z₁.Z₂ = 1, and the insertion loss becomes
In the past, circuits configured in this way were very popular as amplitude equalisers. For example, they were used to correct for the high frequency losses in telephone cables and in long runs of coaxial cable for television installations.
An example, showing the design procedure for a simple equaliser, is given in the section on synthesis, later.