Network synthesis

The idealised prescribed function will rarely be capable of being exactly described by polynomials. It is therefore not possible to synthesise a network to exactly reproduce it.[3] A simple, and common, example is the brick-wall filter. This is the ideal response of a low-pass filter but its piecewise continuous response is impossible to represent with polynomials because of the discontinuities. To overcome this difficulty, a rational function is found that closely approximates the prescribed function using approximation theory.[4] In general, the closer the approximation is required to be, the higher the degree of the polynomial and the more elements will be required in the network.[5]

There are many polynomials and functions used in network synthesis for this purpose. The choice depends on which parameters of the prescribed function the designer wishes to optimise.[6] One of the earliest used was Butterworth polynomials which results in a maximally flat response in the passband.[7] A common choice is the Chebyshev approximation in which the designer specifies how much the passband response can deviate from the ideal in exchange for improvements in other parameters.[8] Other approximations are available for optimising time delay, impedance matching, roll-off, and many other requirements.[9]

Given a rational function, it is usually necessary to determine whether the function is realisable as a discrete passive network. All such networks are described by a rational function, but not all rational functions are realisable as a discrete passive network.[10] Historically, network synthesis was concerned exclusively with such networks. Modern active components have made this limitation less relevant in many applications,[11] but at the higher radio frequencies passive networks are still the technology of choice.[12] There is a simple property of rational functions that predicts whether the function is realisable as a passive network. Once it is determined that a function is realisable, there a number of algorithms available that will synthesise a network from it.[13]

A network realisation from a rational function is not unique. The same function may realise many equivalent networks. It is known that affine transformations of the impedance matrix formed in mesh analysis of a network are all impedance matrices of equivalent networks (further information at Analogue filter § Realisability and equivalence).[14] Other impedance transformations are known, but whether there are further equivalence classes that remain to be discovered is an open question.[15]

A major area of research in network synthesis has been to find the realisation which uses the minimum number of elements. This question has not been fully solved for the general case,[16] but solutions are available for many networks with practical applications.[17]

Foster's synthesis, in its original form, can be applied only to LC networks. A PRF represents a two-element-kind LC network if the critical frequencies of ${\displaystyle Z(s)}$ all exist on the ${\displaystyle i\omega }$ axis of the complex plane of ${\displaystyle s=\sigma +i\omega }$ (the s-plane) and will alternate between poles and zeroes. There must be a single critical frequency at the origin and at infinity, all the rest must be in conjugate pairs. ${\displaystyle Z(s)}$ must be the ratio of an even and odd polynomial and their degrees must differ by exactly one. These requirements are a consequence of Foster's reactance theorem.[55]

Example of a Foster I form realisation

Foster's first form (Foster I form) synthesises ${\displaystyle Z(s)}$ as a set of parallel LC circuits in series. For example,

${\displaystyle Z(s)={\frac {9s^{5}+30s^{3}+24s}{18s^{4}+36s^{2}+8}}}$

can be expanded into partial fractions as,

${\displaystyle Z(s)={s \over 2}+{\frac {(25+11{\sqrt {5}})s}{5(9+3{\sqrt {5}})s^{2}+20}}+{\frac {(25-11{\sqrt {5}})s}{5(9-3{\sqrt {5}})s^{2}+20}}\simeq {s \over 2}+{\frac {2.48s}{3.93s^{2}+1}}+{\frac {0.020s}{0.573s^{2}+1}}}$

The first term represents a series inductor, a consequence of ${\displaystyle Z(s)}$ having a pole at infinity. If it had had a pole at the origin, that would represent a series capacitor. The remaining two terms each represent conjugate pairs of poles on the ${\displaystyle i\omega }$ axis. Each of these terms can be synthesised as a parallel LC circuit by comparison with the impedance expession for such a circuit,[56]

${\displaystyle Z_{LC}(s)={\frac {Ls}{LCs^{2}+1}}}$

The resulting circuit is shown in the figure.

Example of a Foster II form realisation

Foster II form synthesises ${\displaystyle Z(s)}$ as a set of series LC circuits in parallel. The same method of expanding into partial fractions is used as for Foster I form, but applied to the admittance, ${\displaystyle Y(s)}$, instead of ${\displaystyle Z(s)}$. Using the same example PRF as before,

${\displaystyle Y(s)={1 \over Z(s)}={\frac {18s^{4}+36s^{2}+8}{9s^{5}+30s^{3}+24s}}}$

Expanded in partial fractions,

${\displaystyle Y(s)\simeq {1 \over 3s}+{\frac {2.498s}{0.6346s^{2}+1}}+{\frac {1.415s}{0.4719s^{2}+1}}}$

The first term represents a shunt inductor, a consequence of ${\displaystyle Y(s)}$ having a pole at the origin (or, equivalently, ${\displaystyle Z(s)}$ has a zero at the origin). If it had had a pole at infinity, that would represent a shunt capacitor. The remaining two terms each represent conjugate pairs of poles on the ${\displaystyle i\omega }$ axis. Each of these terms can be synthesised as a series LC circuit by comparison with the admittance expession for such a circuit,[57]

${\displaystyle Y_{LC}(s)={\frac {Cs}{LCs^{2}+1}}}$

The resulting circuit is shown in the figure.

Foster synthesis can be extended to any two-element-kind network. For instance, the partial fraction terms of an RC network in Foster I form will each represent an R and C element in parallel. In this case, the partial fractions will be of the form,[58]

${\displaystyle Z_{RC}(s)={\frac {R}{RCs+1}}}$

Other forms and element kinds follow by analogy. As with an LC network, The PRF can be tested to see if it is an RC or RL network by examining the critical frequencies. The critical frequencies must all be on the negative real axis and alternate between poles and zeroes, and there must be an equal number of each. If the critical frequency nearest, or at, the origin is a pole, then the PRF is an RC network if it represents a ${\displaystyle Z(s)}$, or it is an RL network if it represents a ${\displaystyle Y(s)}$. Vice versa if the critical frequency nearest, or at, the origin is a zero. These extensions of the theory also apply to the Cauer forms described below.[59]

In the Foster synthesis above, the expansion of the function is the same procedure in both the Foster I form and Foster II form. It is convenient, especially in theoretical works, to treat them together as an immittance rather than separately as either an impedance or an admittance. It is only necessary to declare whether the function represents an impedance or an admittance at the point that an actual circuit needs to be realised. Immittance can also be used in the same way with the Cauer I and Cauer II forms and other procedures.[60]

Cauer synthesis is an alternative synthesis to Foster synthesis and the conditions that a PRF must meet are exactly the same as Foster synthesis. Like Foster synthesis, there are two forms of Cauer synthesis, and both can be extended to RC and RL networks.

Example of a Cauer I form realisation

The Cauer I form expands ${\displaystyle Z(s)}$ into a continued fraction. Using the same example as used for the Foster I form,

${\displaystyle Z(s)=0.5s+{\cfrac {1}{1.5s+{\cfrac {1}{2s+{\cfrac {1}{1.5s+{\cfrac {1}{0.5s}}}}}}}}}$

or, in more compact notation,

${\displaystyle Z(s)=[0.5s;1.5s,2s,1.5s,0.5s].}$

The terms of this expansion can be directly implemented as the component values of a ladder network as shown in the figure.[61] The given PRF may have a denominator that has a greater degree than the numerator. In such cases, the multiplicative inverse of the function is expanded instead. That is, if the function represents ${\displaystyle Z(s)}$, then ${\displaystyle Y(s)}$ is expanded instead and vice versa.[62]

Example of a Cauer II form realisation

Cauer II form expands ${\displaystyle Z(s)}$ in exactly the same way as Cauer I form except that lowest degree term is extracted first in the continued fraction expansion rather than the highest degree term as is done in Cauer I form.[63] The example used for the Cauer I form and the Foster forms when expanded as a Cauer II form results in some elements having negative values.[64] This particular PRF, therefore, cannot be realised in passive components as a Cauer II form without the inclusion of transformers or mutual inductances.[65]

The essential reason that the example ${\displaystyle Z(s)}$ cannot be realised as a Cauer II form is that this form has a high-pass topology. The first element extracted in the continued fraction is a series capacitor. This makes it impossible for the zero of ${\displaystyle Z(s)}$ at the origin to be realised. The Cauer I form, on the other hand, has a low-pass topology and naturally has a zero at the origin.[66] However, the ${\displaystyle Y(s)}$ of this function can be realised as a Cauer II form since the first element extracted is a shunt inductor. This gives a pole at the origin for ${\displaystyle Y(s)}$, but that translates to the necessary zero at the origin for ${\displaystyle Z(s)}$. The continued fraction expansion is,

${\displaystyle Y(s)\simeq \left[{1 \over 3s};{1 \over 1.083s},{1 \over 0.2175s},{1 \over 1.735s}\right]}$

and the realised network is shown in the figure.

The Brune synthesis can synthesise any arbitrary PRF, so in general will result in a 3-element-kind (i.e. RLC) network. The poles and zeroes can lie anywhere in the left-hand half of the complex plane.[67] The Brune method starts with some preliminary steps to eliminate critical frequencies on the imaginary axis as in the Foster method. These preliminary steps are sometimes called the Foster preamble.[68] There is then a cycle of steps to produce a cascade of Brune sections.[69]

Poles and zeroes on the ${\displaystyle j\omega }$ axis represent L and C elements that can be extracted from the PRF. Specifically,

• a pole at the origin represents a series capacitor
• a pole at infinity represents a series inductance
• a zero at the origin represents a shunt inductor
• a zero at infinity represents a shunt capacitor
• a pair of poles at ${\displaystyle s=\pm i\omega _{c}}$ represents a parallel LC circuit of resonant frequency ${\displaystyle \omega _{c}}$ in series
• a pair of zeroes at ${\displaystyle s=\pm i\omega _{c}}$ represents a series LC circuit of resonant frequency ${\displaystyle \omega _{c}}$ in shunt

After these extractions, the remainder PRF has no critical frequencies on the imaginary axis and is known as a minimum reactance, minimum susceptance function. Brune synthesis proper begins with such a function.[70]

The essence of the Brune method is to create a conjugate pair of zeroes on the ${\displaystyle i\omega }$ axis by extracting the real and imaginary parts of the function at that frequency, and then extract the pair of zeroes as a resonant circuit. This is the first Brune section of the synthesised network. The resulting remainder is another minimum reactance function that is two degrees lower. The cycle is then repeated, each cycle producing one more Brune section of the final network until just a constant value (a resistance) remains.[71] The Brune synthesis is canonical, that is, the number of elements in the final synthesised network is equal to the number of arbitrary coefficients in the impedance function. The number of elements in the synthesised circuit cannot therefore be reduced any further.[72]

Extracting minimum resistance

A minimum reactance function will have a minimum real part, ${\displaystyle R_{\text{min}}}$, at some frequency ${\displaystyle \omega _{\text{min}}}$. This resistance can be extracted from the function leaving a remainder of another PRF called a minimum positive-real function, or just minimum function.[73] For example, the minimum reactance function

${\displaystyle Z(s)={\frac {3s^{2}+3s+6}{2s^{2}+s+2}}}$

has ${\displaystyle \omega _{\text{min}}={\sqrt {2}}}$ and ${\displaystyle R_{\text{min}}=1}$. The minimum function, ${\displaystyle Z_{1}(s)}$, is therefore,

${\displaystyle Z_{1}(s)=Z(s)-R_{\text{min}}={\frac {s^{2}+2s+4}{2s^{2}+s+2}}}$

Extraction of negative inductance

Since ${\displaystyle Z_{1}(i\omega _{0})}$ has no real part, we can write,

${\displaystyle Z_{1}(i\omega _{0})=iX\ .}$

For the example function,

${\displaystyle Z_{1}(i\omega _{0})=-i{\sqrt {2}}=iX\ .}$

In this case, ${\displaystyle X}$ is negative, and we interpret it as the reactance of a negative-valued inductor, ${\displaystyle L_{1}}$. Thus,

${\displaystyle iX=i\omega _{0}L_{1}}$ and

${\displaystyle L_{1}=-1}$

after substituting in the values of ${\displaystyle \omega _{0}}$ and ${\displaystyle iX}$. This inductance is then extracted from ${\displaystyle Z_{1}(s)}$, leaving another PRF, ${\displaystyle Z_{2}(s)}$,

${\displaystyle Z_{2}(s)=Z_{1}(s)-sL_{1}={\frac {2s^{3}+2s^{2}+4s+4}{2s^{2}+s+2}}\ .}$[74]

The reason for extracting a negative value is because ${\displaystyle -sL_{1}}$ is a PRF, which it would not be if ${\displaystyle L_{1}}$ were positive. This guarantees that ${\displaystyle Z_{2}(s)}$ will also be PRF (because the sum of two PRFs is also PRF).[75] For cases where ${\displaystyle X}$ is a positive value, the admittance function is used instead and a negative capacitance is extracted.[76] How these negative values are implemented is explained in a later section.

Extraction of a pair of zeroes

Both the real and imaginary parts of ${\displaystyle Z(i\omega _{0})}$ have been removed in previous steps. This leaves a pair of zeroes in ${\displaystyle Z_{2}(s)}$ at ${\displaystyle \pm i\omega _{0}}$ as shown by factorising the example function;[77]

${\displaystyle Z_{2}(s)={\frac {2s^{3}+2s^{2}+4s+4}{2s^{2}+s+2}}={\frac {(s^{2}+2)(2s+2)}{2s^{2}+s+2}}\ .}$

Since such a pair of zeroes represents a shunt resonant circuit, we extract it as a pair of poles from the admittance function,

${\displaystyle {\begin{aligned}Y_{2}(s)&={1 \over Z_{2}(s)}={\frac {2s^{2}+s+2}{(s^{2}+2)(2s+2)}}\\&={1 \over {2s+2}}+{\frac {s/2}{s^{2}+2}}\\&=Y_{3}(s)+{\frac {s/2}{s^{2}+2}}\\\end{aligned}}}$

The rightmost term is the extracted resonant circuit with ${\displaystyle L_{2}=2}$ and ${\displaystyle C_{2}=1/4}$.[78] The network synthesised so far is shown in the figure.

Extraction of a pole at infinity

${\displaystyle Z_{3}(s)}$ must have a pole at infinity, since one was created there by the extraction of a negative inductance. This pole can now be extracted as a positive inductance.[79]

${\displaystyle Z_{3}(s)={1 \over Y_{3}(s)}=2s+2=Z_{4}(s)+2s.}$

Thus ${\displaystyle L_{3}=2}$ as shown in the figure.

Eliminate negative inductance by using a transformer

The negative inductance cannot be implemented directly with passive components. However, the "tee" of inductors can be converted into mutually coupled inductors which absorbs the negative inductance.[80] With a coupling coefficient of unity (tightly coupled) the mutual inductance, ${\displaystyle M}$, in the example case is 2.0.

In general, ${\displaystyle Z_{4}(s)}$ will be another minimum reactance function and the Brune cycle is then repeated to extract another Brune section[81] In the example case, the original PRF was of degree 2, so after reducing it by two degrees, only a constant term is left which, trivially, synthesises as a resistance.

In step two of the cycle it was mentioned that a negative element value must be extracted in order to guarantee a PRF remainder. If ${\displaystyle X}$ is positive, the element extracted must be a shunt capacitor instead of a series inductor if the element is to be negative. It is extracted from the admittance ${\displaystyle Y_{1}(s)}$ instead of the impedance ${\displaystyle Z_{1}(s)}$. The circuit topology arrived at in step four of the cycle is a Π (pi) of capacitors plus an inductor instead of a tee of inductors plus a capacitor. It can be shown that this Π of capacitors plus inductor is an equivalent circuit of the tee of inductors plus capacitor. Thus, it is permissible to extract a positive inductance and then proceed as though ${\displaystyle Z_{2}(s)}$ were PRF, even though it is not. The correct result will still be arrived at and the remainder function will be PRF so can be fed into the next cycle.[82]

Example of Bott-Duffin synthesis steps 1 and 2

Example of Bott-Duffin synthesis steps 3 and 4

The Bott-Duffin synthesis begins as with the Brune synthesis by removing all poles and zeroes on the ${\displaystyle i\omega }$ axis. Then Richards' theorem is invoked, which states for,

${\displaystyle R(s)={\frac {kZ(s)-sZ(k)}{kZ(k)-sZ(s)}}}$

if ${\displaystyle Z(s)}$ is a PRF then ${\displaystyle R(s)}$ is a PRF for all real, positive values of ${\displaystyle k}$.[83]

Making ${\displaystyle Z(s)}$ the subject of the expression results in,[84]

${\displaystyle Z(s)=\left({\frac {R(s)}{Z(k)}}+{\frac {k}{sZ(k)}}\right)^{-1}+\left({\frac {1}{Z(k)R(s)}}+{\frac {s}{kZ(k)}}\right)^{-1}}$

An example of one cycle of Bott-Duffin synthesis is shown in the figures. The four terms in this expression are, respectively, a PRF (${\displaystyle Z_{2}(s)}$ in the diagram), an inductance, ${\displaystyle L}$, in parallel with it, another PRF (${\displaystyle Z_{1}(s)}$ in the diagram), and a capacitance, ${\displaystyle C}$, in parallel with it. A pair of critical frequencies on the ${\displaystyle i\omega }$ axis is then extracted from each of the two new PRFs (details not given here) each realised as a resonant circuit. The two residual PRFs (${\displaystyle Y_{3}(s)}$ and ${\displaystyle Z_{4}(s)}$ in the diagram) are each two degrees lower than ${\displaystyle Z(s)}$.[85] The same procedure is then repeatedly applied to the new PRFs generated until just a single element remains.[86] Since the number of PRFs generated doubles with each cycle, the number of elements synthesised will grow exponentially. Although the Bott-Duffin method avoids the use of transformers and can be applied to any expression capable of realisation as a passive network, it has limited practical use due to the high component count required.[87]

Bayard synthesis is a state-space synthesis method based on the Gauss factorisation procedure. This method returns a synthesis using the minimum number of resistors and contains no gyrators. However, the method is non-canonical and will, in general, return a non-minimal number of reactance elements.[88]

Darlington synthesis starts from a different perspective to the techniques discussed so far, which all start from a prescribed rational function and realise it as a one-port impedance. Darlington synthesis starts with a prescribed rational function that is the desired transfer function of a two-port network. Darlington showed that any PRF can be realised as a two-port network using only L and C elements with a single resistor terminating the output port.[89] The Darlington and related methods are called the insertion loss method.[90] The method can be extended to multi-port networks with each port terminated with a single resistor.[91]

The Darlington method, in general, will require transformers or coupled inductors. However, most common filter types can be constructed by the Darlington method without these undesirable features.[92]

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