page_title: Lattice and bridged-T equalizers
section_title: An example
hierarchical_section_title: Lattice and bridged-T equalizers / Amplitude correction / The method of complementary networks / An example
caption_attribution_description: An example of a lattice network and its complement
context_page_description: 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, 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; the method of straight line asymptotes; using a purpose built test-jig; the use of standard circuit building blocks, or with the aid of computer programs. In addition, trial and error methods have been found to be surprisingly effective, 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 than before. In other applications, such as CATV distribution systems or frequency multiplexed telephone signals where multiple carrier signals are being passed, the aim is to equalize the transmission line so that those signals have much the same amplitude.
context_section_description: 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.
  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
caption: Cascade of Low-pass and High-pass Lattices
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