Lattice Network | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

Introduction

The lattice network is a four-arm, symmetrical two-port network commonly used in network analysis and filter design. Each of the two opposite series arms contains an impedance denoted by Z_A, and each of the two diagonal (or shunt) arms contains an impedance denoted by Z_B. The idealised schematic of a lattice network is shown in the figure below. Because of its symmetry the lattice is often transformed into an equivalent bridge form for analysis; the bridge representation simplifies the use of basic circuit laws to obtain characteristic parameters such as characteristic impedance and propagation constant.

The following sections derive expressions for the characteristic impedance Z₀ and the propagation constant γ of a symmetrical lattice network. The derivations are carried out conveniently using the bridge form of the lattice.

Characteristic Impedance (Z₀)

The bridge form of the lattice network is convenient for applying Kirchhoff's laws. The labelled bridge diagram used in the derivation is shown below.

Consider the bridge diagram and the closed path 1-2-2′-1′-1. Applying Kirchhoff's voltage law (KVL) around this loop yields the first equation of the analysis:

Consider the other closed path 1-2′-2-1′-1. Applying KVL around this loop gives the second equation:

From equation (1) we obtain the following relation:

From equation (2) we obtain the following relation:

Equating the two expressions obtained from the two KVL equations gives the next relation used to eliminate intermediate variables:

Using the property of a symmetrical network, when the network is terminated in its characteristic impedance, the input impedance seen at the port equals Z₀. Let the port voltage be E and the port current be I_s so that the input impedance is E / I_s. Thus we write:

Substituting the previously obtained expression for E / I_s into the above equation leads to:

In terms of open- and short-circuit impedances

It is often useful to express characteristic impedance in terms of open-circuit and short-circuit port impedances. For this purpose the lattice (bridge) may be re-arranged and the open- and short-circuit impedances calculated as shown below.

Considering the arrangement in Fig. (a):

Considering the arrangement in Fig. (b):

Multiplying the expressions obtained for the open-circuit and short-circuit impedances gives the following compact relation:

From these relations one may obtain the characteristic impedance Z₀ in terms of measurable open- and short-circuit impedances. These relations are particularly useful in practical measurements and in synthesis problems.

Propagation Constant (γ)

For any symmetrical two-port network, the propagation constant γ of an equivalent uniform transmission-line section can be related to network element values. The propagation constant for the lattice can be written in the general form:

Using the previously derived current relations (equations (3) and (4) from the KVL analysis), the currents in the bridge form are related by:

Recall that for a network terminated in its characteristic impedance we have E = I_s · Z₀. Using this condition and the algebraic relations obtained earlier we obtain the following expressions:

These expressions permit elimination of the intermediate variables and yield an explicit expression for γ in terms of the arm impedances Z_A and Z_B together with the characteristic impedance Z₀.

Impedances Z_A and Z_B in terms of Z₀ and γ

It is frequently required to express the lattice-arm impedances in terms of the equivalent section's characteristic impedance and propagation constant. Starting from the relation between network voltages and currents, consider the following fundamental equation:

The algebraic manipulations produce the following forms for the arm impedances:

Similarly, the complementary expression is

Hence, a lattice network whose arms are chosen according to these expressions is equivalent to a transmission-line section characterised by Z₀ and γ. The ladder showing the lattice with arm impedances expressed in terms of Z₀ and γ is shown below.

Remarks and applications

The lattice network is important in several contexts:

• The lattice provides a convenient way to realise all-pass and phase-shift networks used in analogue signal processing and in some filter structures.
• Its symmetry makes it suitable for deriving characteristic parameters (Z₀, γ) and for practical measurement using open- and short-circuit tests.
• By choosing Z_A and Z_B as frequency-dependent impedances (for example inductances and capacitances), lattice sections can implement desired amplitude and phase responses over frequency.

After a detailed analysis of important symmetrical lattice sections, one next considers asymmetrical networks where the two series arms or the two diagonal arms are not identical. Asymmetry complicates analysis and generally requires solution of more general two-port parameters (ABCD-parameters or chain parameters) or direct circuit-equation methods.

Optional brief summary: The lattice is a symmetric four-arm two-port whose bridge representation facilitates derivation of characteristic impedance and propagation constant. These characteristic parameters may be expressed directly from lattice arm impedances or obtained from open- and short-circuit port measurements. Selecting frequency-dependent arm impedances allows realisation of useful filter and phase networks.