Continued fraction approximation

A continued fraction expansion of tanh(x)^{[1][4][10][11]} is

${\displaystyle \tanh(x)={\cfrac {1}{{\cfrac {1}{x}}+{\cfrac {1}{{\cfrac {3}{x}}+{\cfrac {1}{{\cfrac {5}{x}}+{\cfrac {1}{{\cfrac {7}{x}}+\cdots }}}}}}}}}$

So, for a network with 1sec delay, z_a can be written

${\displaystyle z_{a}(p)=\tanh \left({\frac {p}{2}}\right)={\cfrac {1}{1\cdot {\cfrac {2}{p}}+{\cfrac {1}{3\cdot {\cfrac {2}{p}}+{\cfrac {1}{5\cdot {\cfrac {2}{p}}+{\cfrac {1}{\cdots }}}}}}}}}$

An exact solution requires an infinite number of terms, but an nth order approximation is obtained by terminating z_a 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^[4][11] to give the network shown.

A circuit for z_b can easily found from this solution, since it is the dual of z_a, and is

Although this circuit of z_b 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 z_b started with a series inductor (see Lattice networks). To do this it is first necessary to multiply out the continued fraction expansion for z_a, for this example, to give z_a (and z_b in particular) as a ratio of polynomials in p. This is

${\displaystyle z_{a}(p)={\frac {24p^{5}+10080p^{3}+332640p}{p^{6}+840p^{4}+75600p^{2}+665280}}={\frac {1}{z_{b}(p)}}}$

and for the alternative Cauer I expansion proceeds as follows

${\displaystyle z_{b}={\frac {1}{42}}p+Z_{1}(p)\quad {\text{where}}\quad Y_{1}(p)={\frac {1}{Z_{1}(p)}}={\frac {42p^{5}+10080p^{3}+332640p}{600p^{4}+67680p^{2}+665280}}}$

${\displaystyle Y_{1}(p)={\frac {42}{600}}p+Y_{2}(p)\quad {\text{where}}\quad Z_{2}(p)={\frac {1}{Y_{2}(p)}}={\frac {600p^{4}+67680p^{2}+665280}{5342.4p^{3}+286070.4p}}}$

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

${\displaystyle T(p)={\frac {1-z_{a}}{1+z_{a}}}}$

so

${\displaystyle T(p)={\frac {p^{6}-42p^{5}+840p^{4}-10080p^{3}+75600p^{2}-332640p+665280}{p^{6}+42p^{5}+840p^{4}+10080p^{3}+75600p^{2}+332640p+665280}}}$

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 z_a 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 Darlington^[12] 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"^[13][14] and the "electrolytic tank".^[15][16] and Teledeltos paper^[17]

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^[1]). 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

${\displaystyle {\text{GD}}=-{\frac {d\phi }{d\omega }}={\frac {2\pi }{b}}\pm {\frac {4\pi }{b}}\exp \left(-{\frac {2\pi a}{b}}\right)\cos \left({\frac {2\pi \omega }{b}}\right)}$

A good approximation to a unit delay characteristic is obtained by putting a = b = 2π (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 (= 1.4π) 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 = 2π. 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.^[12]

The networks can be realised as a cascade of second order lattices (or their bridged-T equivalents) by allocating a complex conjugate quad of poles and zeros to each section of the cascade (as outlined in Lattice networks). The current example does not have a pole-zero pair located on the real axis, so a first order network is not required.

Delay networks derived with a Chebyshev ripple

Details of pole positions for low pass networks that have group delay with a "Chebyshev ripple" characteristic across the passband, for various orders of filter and various levels of ripple have been calculated and published by Ulbrich et al.^[8] and by MacNee.^[27] The tables below, based on this data, are for all-pass networks. A filter of given order can achieve more delay and/or bandwidth if more passband phase ripple is permitted.

Pole-zero position for all-pass networks with unit mean delay and 1% group delay ripple:

Pole-zero position for all-pass networks with unit mean delay and 2% group delay ripple:

Pole-zero position for all-pass networks with unit mean delay and 5% group delay ripple:

Pole-zero position for all-pass networks with unit mean delay and 10% group delay ripple:

A delay network can be conveniently made up of a cascade of second order lattice networks, allocating a quad of poles and zeros, from the tables above, to each section. An example of a fourth order network, with 10% group delay ripple is considered later.

Delay ripple by using infinite product approximations

An alternative form of group delay ripple, preferable to the equal amplitude Chebyshev ripple, has low amplitude ripples at low frequencies but ripples of increasing amplitude as frequency increases. This characteristic is more desirable than the Chebyshev one because the phase errors are small at low frequencies, (where the spectrum of a typical waveforms has high energy content) but can be high at higher frequencies (where the energy content of the spectrum is lower).

A suitable ripple characteristic is obtained by taking power series approximations of sinh(x) and cosh(x),^[1][10] rather than deriving the continued fraction expansion of tanh(x), as was done earlier. Typically, with this procedure, the ripple on the phase characteristic deviates by ±5% from the mean (linear) value.

These results are similar to those obtained by the ‘Forced Ripple Method’,^[9][28] where a technique of curve fitting, at a finite number of frequencies of the phase response, is employed.

For normalized networks (Zo = 1) with unit time delay, the equations for za and zb can be written

${\displaystyle z_{a}=\tanh \left({\frac {p}{2}}\right)={\frac {\sinh(p/2)}{cosh(p/2)}}\qquad {\text{and}}\qquad z_{b}=\coth \left({\frac {p}{2}}\right)={\frac {\cosh(p/2)}{sinh(p/2)}}}$

sinh(x) and cosh(x) can be represented by infinite products,^[1][10] and these are

${\displaystyle \sinh(x)=x\prod _{k=1}^{\infty }\left(1+{\frac {x^{2}}{k^{2}\pi ^{2}}}\right)\qquad {\text{and}}\qquad \cosh(x)=\prod _{k=1}^{\infty }\left[1+{\frac {4.x^{2}}{(2k-1)^{2}\pi ^{2}}}\right]}$

So, for a unit delay network

${\displaystyle z_{a}={\frac {p}{2}}{\frac {\left(1+{\frac {p^{2}}{4\pi ^{2}}}\right)\left(1+{\frac {p^{2}}{16.\pi ^{2}}}\right)\left(1+{\frac {p^{2}}{36\pi ^{2}}}\right)\cdots }{\left(1+{\frac {p^{2}}{\pi ^{2}}}\right)\left(1+{\frac {p^{2}}{9\pi ^{2}}}\right)\left(1+{\frac {p^{2}}{25\pi ^{2}}}\right)\cdots }}}$

Terminating the series after a finite number of terms gives a limited bandwidth approximation for 1sec delay. So, for example, an expression to including terms up to p⁴ will give a fourth order delay network. In this case, z_a is

${\displaystyle z_{a}=K_{4}{\frac {4\pi ^{2}p+p^{3}}{9\pi ^{4}+10.\pi ^{2}p^{2}+p^{4}}}\qquad {\text{where}}\qquad K_{4}={\frac {1\times 9\times \pi ^{2}}{2\times 4}}={\frac {9.\pi ^{2}}{8}}}$

which can be realized as a ladder network using Cauer’s procedure,^[4] to give the circuit below for z_a. As before, the dual network, z_b, is easily obtained by inspection.

As already stated, the transfer function of a normalized lattice all-pass network is given by

${\displaystyle T(p)={\frac {1-z_{a}}{1+z_{a}}}}$

so for the fourth order network containing the impedance za, derived by the power series expansions, is

${\displaystyle T(p)={\frac {p^{4}-K_{4}p^{3}+80\pi ^{2}p^{2}-K_{4}.4\pi ^{2}p+9\pi ^{4}}{p^{4}+K_{4}p^{3}+80.\pi ^{2}p^{2}+K_{4}\cdot 4\pi ^{2}p+9\pi ^{4}}}\qquad {\text{where}}\qquad K_{4}={\frac {9\pi ^{2}}{8}}}$

This has an all-pass magnitude characteristic, with the phase response shown in the figure below.

A collection of results, for even order networks with n = 2 to 10 are given below. (As with results given earlier, the polynomials are not presented in full, only the coefficients are listed).

In these results, ${\displaystyle T(p)={\frac {N(p)}{D(p)}}}$ where the coefficients for the numerator and denominator polynomials are listed. For the denominator, D(p), all coefficients are positive, whereas for the numerator, N(p), the negative values are taken where indicated.

The pole and zero locations in the complex frequency plane, for these responses, are as follows.

The phase error responses for the even order networks from n = 2 to n = 10 have been plotted in the accompanying figure.

Comparing the bandwidths of networks with passband ripple to those with a maximally flat response, an increase of approximately 50% is achieved.

Circuits having a single lattice

The first example gives the circuit for a 6th order maximally flat delay. Circuit values for z_a and z_b for a normalized lattice (with z_b the dual of z_a) were given earlier. However, in this example the alternative version of z_b is used, so that an unbalanced alternative can be easily produced. The circuit is

where component values for a normalised 1 ohm network, with 1 second delay at low frequencies, are:

Using the procedures of Lattice networks, this can be converted to an unbalanced form, to give

Circuits with a cascade of low-order lattices

It is often desirable to decompose a lattice into a cascade of lower order networks, because component tolerances can be relaxed.

To carry out the procedure take the three sets of pole-zero data from the table for maximally flat functions, for n = 6, and use the methods in Lattice networks

These component values are used in the circuit shown below.

The phase characteristic of this three section cascade is, of course, identical to that of the single complex lattice, given earlier.

This cascade of second order lattices can be converted to an unbalanced configuration by the methods of Lattice networks, and the resulting circuit is shown.

Chebyshev, 4th order with 10% GD ripple

From the tables of Chebyshev data, given above, find the pole-zero positions:

So use these values in the circuit below.

Circuit for the 4th-order forced ripple approximation

From the tables for power product approximation, given above, find the pole-zero positions:

Use these values in the circuit shown above.

Both 4th order networks can be converted to unbalanced form using the procedures of Lattice networks