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2.6: Element Extraction

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    46101
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    In the previous two sections the mathematical responses of Butterworth and Chebyshev filters were derived for various orders. In this section it will be shown how these filters can be implemented with inductors and capacitors using what is called ladder synthesis [4].

    2.6.1 Ladder Synthesis

    To obtain the element values yielding the desired transfer function, an impedance or admittance function must first be obtained. The impedance or admittance function can be readily obtained from the input reflection coefficient of a network, but for now the focus is on synthesizing a given impedance function. A general impedance function can be expressed as

    \[\label{eq:1}Z(s)=\frac{a_{n}(s^{2} +\omega_{1}^{2})(s^{2} + \omega_{3}^{2})(s^{2} + \omega_{5}^{2})\cdots}{b_{m}s (s^{2} +\omega_{2}^{2}) (s^{2} + \omega_{4}^{2}) (s^{2} +\omega_{6}^{2})\cdots} \]

    where \(a_{n}\) and \(b_{m}\) are constants. This can be realized using \(L\) and \(C\) elements in a network terminated by a resistor, provided that the degree of the numerator and denominator differ by no more than unity (i.e., \(|m−n|\leq 1\)). In

    clipboard_e76f462f4832a79099a4df43c6403871a.png

    Figure \(\PageIndex{1}\): Extraction of a network \(X\) to reduce an impedance \(Z_{\text{in, }i}\) to a lower-order impedance \(Z_{\text{in, }i+1}\).

    clipboard_e03bf0794a9bc8b9a7ab8177b50e3cd46.png

    Figure \(\PageIndex{2}\): Synthesis of impedance and admittance functions. Starting with an impedance function \(Z(s)\): (a) extraction of a series capacitor; (c) extraction of a series inductor; and (e) extraction of a series parallel \(LC\) block \(i\). Starting with an admittance function \(Y(s)\): (b) extraction of a shunt inductor; (d) extraction of a shunt capacitor; and (f) extraction of a shunt series \(LC\) block.

    the case of a doubly terminated network, this resistor is the load. The element extraction procedure, shown in Figure \(\PageIndex{1}\), involves extracting a network \(X\) from \(Z_{\text{in, }i}\), leaving a reduced-order impedance \(Z_{\text{in, }i+1}\).

    The extraction of inductors and capacitors is illustrated in Figure \(\PageIndex{2}\). Thus, following the extraction of an element or a pair of elements an impedance, \(Z_{\text{rem}}\), or admittance, \(Y_{\text{rem}}\), remains that can be similarly simplified. For example, and referring to Figure \(\PageIndex{2}\)(a), \(Z(s)=1/(sC)+Z_{\text{rem}}\). So a pole of \(Z(s)\) at DC requires the extraction of a series capacitor of value (see Figure \(\PageIndex{2}\)(a))

    \[\label{eq:2}C_{0}=\left.\frac{1}{sZ(s)}\right|_{s=0} \]

    while a pole at infinity requires the extraction of a series inductor of value (see Figure \(\PageIndex{2}\)(c))

    \[\label{eq:3}L_{\infty}=\left.\frac{Z(s)}{s}\right|_{s=\infty} \]

    Another possibility is a pole at a finite frequency (call this \(\omega_{0}\)), which requires the extraction of a series parallel \(LC\) block, as shown in Figure \(\PageIndex{2}\)(e), with elements of value

    \[\label{eq:4}C_{i}=\left.\frac{s}{(s^{2}+\omega_{2}^{2})Z(s)}\right|_{s=\jmath\omega_{0}}\quad\text{and}\quad L_{i}=\frac{1}{\omega_{0}^{2}C_{i}} \]

    The extraction process can also be carried out on an admittance basis. First

    \[\label{eq:5}Y(s)=\frac{b_{m}s (s^{2} +\omega_{2}^{2})(s^{2} +\omega_{4}^{2})(s^{2} + \omega_{s}^{2})\cdots}{a_{n} (s^{2} +\omega_{1}^{2}) (s^{2} +\omega_{3}^{2}) (s^{2} +\omega_{5}^{2})\cdots} \]

    Now a pole at zero requires the extraction of a shunt inductor of value (see Figure \(\PageIndex{2}\)(b))

    \[\label{eq:6}L_{0}=\left.\frac{1}{sY(s)}\right|_{s=0} \]

    and a pole at infinity requires the extraction of a shunt capacitor of value (see Figure \(\PageIndex{2}\)(d))

    \[\label{eq:7}C_{\infty}=\left.\frac{Y(s)}{s}\right|_{s=\infty} \]

    A pole at a finite frequency requires the extraction of a shunt-series \(LC\) block (as shown in Figure \(\PageIndex{2}\)(f)) with values

    \[\label{eq:8}L_{i}=\left.\frac{s}{(s^{2}+\omega_{0}^{2})Y(s)}\right|_{s=\jmath\omega_{0}}\quad\text{and}\quad C_{i}=\frac{1}{\omega_{0}^{2}L_{i}} \]

    Many aspects of filter synthesis can seem abstract when presented in full generality. Consequently it is common to illustrate filter synthesis concepts using examples. Following this time-honored tradition, an example is now presented.

    Example \(\PageIndex{1}\): Element Extraction for a Third-Order Lowpass Filter

    A third-order maximally flat filter has the reflection coefficient

    \[\label{eq:9}\Gamma_{1}(s)=\frac{s^{3}}{(s+1)(s^{2}+s+1)} \]

    Synthesize this filter as a doubly terminated network.

    Solution

    The reflection coefficient function (Equation \(\eqref{eq:9}\)) has all its poles located at infinity, so the corresponding network realization must be made of simple \(L\) or \(C\) elements and terminated in a resistor. Hence, referring to Figure 2.2.1 and considering a \(1\:\Omega\) system,

    \[\label{eq:10}Z_{\text{in, }1}(s)=\frac{1+\Gamma_{1}(s)}{1-\Gamma_{1}(s)}=\frac{2s^{3}+2s^{2}+2s+1}{2s^{2}+2s+1} \]

    Note that the input impedance approaches infinity as the frequency goes to infinity, hence a series inductor must be extracted. The value of this inductor is

    \[\label{eq:11}L_{\infty 1}=\left.\frac{Z_{\text{in, }1}(s)}{s}\right|_{s=\infty}=1\text{ H} \]

    The filter is developed by extracting one element at a time. Following the extraction of the first element, the second-stage impedance is left. Now the impedance function is

    \[\begin{aligned} Z_{\text{in, }2}(s)&=Z_{\text{in, }1}(s)-sL_{\infty 1}=\frac{2s^{3}+2s^{2}+2s+1}{2s^{2}+2s+1}-sL_{\infty 1} \\ &=\frac{2s^{3}+2s^{2}+2s+1-s(2s^{2}+2s+1)}{2s^{2}+2s+1}=\frac{s+1}{2s^{2}+2s+1}\end{aligned}\nonumber \]

    Note that the stage impedance above, \(Z_{\text{in, }2}\), approaches zero as the frequency goes to infinity. There is not a single series element that would cause this. However, the stage admittance function,

    \[\label{eq:12}Y_{\text{in, }2}(s)=\frac{1}{Z_{\text{in, }2}(s)}=\frac{2s^{2}+2s+1}{s+1} \]

    goes to infinity as the frequency approaches infinity and so a shunt capacitor is extracted:

    \[\label{eq:13}Y_{\text{in, }3}(s)=Y_{\text{in, }2}(s)-sC_{\infty 2}=\frac{1}{s+1} \]

    where

    \[\label{eq:14}C_{\infty 2}=2F \]

    So sometimes it is more convenient to consider extraction of an admittance and sometimes it is better to consider extraction of an impedance.

    By examining the remaining stage impedance, it is seen that a pole exists at infinity, and so a series inductor, \(L_{\infty 3}\), is extracted. The value of this inductor comes from

    \[\label{eq:15}Z_{\text{in, }3}=\frac{1}{Y_{\text{in, }3}}=s+1\:\Omega \]

    and so the inductor value is

    \[\label{eq:16}L_{\infty 3}=\left.\frac{s+1}{s}\right|_{S=\infty}=1\text{ H} \]

    The final step is to extract a load of value 1 as follows:

    \[\label{eq:17}Z_{\text{in, }4}=Z_{\text{in, }s}-sL_{\infty 3}=1\:\Omega \]

    This example synthesized a doubly terminated network. The resulting network, called a ladder circuit, is shown in Figure \(\PageIndex{3}\). The left-most \(1\:\Omega\) resistor is part of the source.

    This circuit has a dual form consisting of two shunt capacitors separated by a series inductor. The dual circuit derives from realizing the admittance function obtained from the reflection coefficient. Other network extraction techniques are presented in Scanlan and Levy [4, 5] and Matthaei et al. [1].

    clipboard_ed33b46e02cdf52ad1d0a0701856c8e66.png

    Figure \(\PageIndex{3}\): Synthesized maximally flat network with a thirdorder lowpass reflection response.

    2.6.2 Summary

    The input impedance function of a lumped-element circuit can always be expressed as the ratio of two polynomials in \(s\) and the order of the numerator and the denominator polynomials can differ by at most one [5]. If the orders differ by one, then a single inductor or capacitor can always be extracted, however, the remaining impedance function may not be realizable. This indicates that a more complex \(LC\) (and possibly \(R\)) combination is required. To be able to systematically extract arbitrarily complex circuits, a long list of possible functions, such as those shown in Figure \(\PageIndex{2}\), is required. For most of the circuits of interest the \(LC\) combinations shown in Figure \(\PageIndex{2}\) are sufficient. The next example describes impedance function extraction that requires an \(LC\) combination.

    Example \(\PageIndex{2}\): Element Extraction of an Impedance Function

    Realize the impedance function \(Z_{w}=\frac{4s^{3}+4s^{2}+2s+2}{4s^{2}+2s+1}\).

    Solution

    The order of the numerator is \(1\) greater than the order of the denominator and this indicates that a series inductor is perhaps present. The series inductance is

    \[\label{eq:18} L_{1}=\left.\frac{Z_{w}(s)}{(s)}\right|_{z=\infty}=1\text{ H} \]

    The remaining impedance is

    \[\label{eq:19}Z_{\text{in, }2}=Z_{w}-sL_{1}=\frac{4s^{3}+4s^{2}+2s+2}{4s^{2}+2s+1}-s=\frac{2s^{2}+s+2}{4s^{2}+2s+1} \]

    The numerator and denominator of \(Z_{\text{in, }2}\) have the same order. Therefore a simple \(L\) or \(C\) element cannot be used to reduce the complexity of the impedance function. Thus an initial series inductor was not the right choice and the extraction must backtrack.

    Figure \(\PageIndex{2}\) shows several element combinations that can be used to reduce the complexity of an impedance function. Insight into which alternative to choose comes from factoring \(z_{w}\), and note that real roots are required, thus

    \[\label{eq:20}Z_{w}=\frac{4s^{3}+4s^{2}+2s+2}{4s^{2}+2s+1}=\frac{(2s^{2}+1)(2s+2)}{4s^{2}+2s+1} \]

    Examination of Figure \(\PageIndex{2}\) reveals that a ready fit to \(Z_{w}\) is not found. Instead consider the admittance function

    \[\label{eq:21}Y_{w}=\frac{1}{Z_{w}}=\frac{4s^{2}+2s+1}{(2s^{2}+1)(2s+2)} \]

    So the reduction shown in Figure \(\PageIndex{2}\)(f) looks like the right candidate. The general choice for the element is

    \[\label{eq:22}y_{x}=\frac{as}{bs^{2}+1} \]

    Choosing \(b = 2\) now reduces complexity (since part of the factored denominator of \(Y_{w}\) now occurs), so

    \[\begin{align}\label{eq:23}Y_{w}&=\frac{as}{2s^{2}+1}+\left(\frac{4s^{2}+2s+1}{(2s^{2}+1)(2s+2)}-\frac{as}{2s^{2}+1}\right) \\ \label{eq:24}&=\frac{as}{2s^{2}+1}+\left(\frac{(4-2a)s^{2}+(2-2a)s+1}{(2s^{2}+1)(2s+2)}\right)\end{align} \]

    Choose \(a = 1\),

    \[\begin{align}\label{eq:25}Y_{w}&=\frac{s}{(2s^{2}+1)}+\frac{2s^{2}+1}{(2s^{2}+1)(2s+2)}=\frac{s}{2s^{2}+1}+\frac{1}{2s+2} \\ \label{eq:26}&=\frac{s}{2s^{2}+1}+Y_{\text{in, }2}\end{align} \]

    So \(C_{1}L_{1} = b = 1,\: C_{1} = a = 1\text{ F},\: L_{1} = 2\text{ H}\), and

    \[\label{eq:27}Y_{\text{in, }2}=\frac{1}{(2s+2)}\quad\text{or}\quad Z_{\text{in, }2}=\frac{1}{Y_{\text{in, }2}}=2s+2 \]

    The final network is

    clipboard_e73bc7ea3fba03fa2093df846313f8308.png

    Figure \(\PageIndex{4}\)


    2.6: Element Extraction is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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