Question 19.18: Design the LC ladder network terminated with a 1-Ω resistor ...
Design the LC ladder network terminated with a 1-Ω resistor that has the normalized transfer function
\pmb{H} (s)=\frac{1}{s^3+2 s^2+2 s+1}
(This transfer function is for a Butterworth low-pass filter.)
The denominator shows that this is a third-order network, so that the LC ladder network is shown in Fig. 19.63(a), with two inductors and one capacitor. Our goal is to determine the values of the inductors and capacitor. To achieve this, we group the terms in the denominator into odd or even parts:
D(s) = (s³ + 2s) + (2s² + 1)
so that
\pmb{H} (s)=\frac{1}{\left(s^3+2 s\right)+\left(2 s^2+1\right)}
Divide the numerator and denominator by the odd part of the denominator to get
\pmb{H} (s)=\frac{\frac{1}{s^3+2 s}}{1+\frac{2 s^2+1}{s^3+2 s}} (19.18.1)
From Eq. (19.82), when \pmb{Y_L} = 1,
(19.82): \pmb{H} (s)=-\frac{\pmb{ y _{21} / Y _L}}{1+ \pmb{y _{22} / Y _L}}
\pmb{H} (s)=\frac{-y_{21}}{1+y_{22}} (19.18.2)
Comparing Eqs. (19.19.1) and (19.19.2), we obtain
y_{21}=-\frac{1}{s^3+2 s}, \quad y_{22}=\frac{2 s^2+1}{s^3+2 s}
Any realization of y_{22} will automatically realize y_{21}, since y_{22} is the output driving-point admittance, that is, the output admittance of the network with the input port short-circuited. We determine the values of L and C in Fig. 19.63(a) that will give us y_{22}. Recall that y_{22} is the short-circuit output admittance. So we short-circuit the input port as shown in Fig. 19.63(b). First we get L_{3} by letting
Z_A=\frac{1}{y_{22}}=\frac{s^3+2 s}{2 s^2+1}=s L_3+Z_B (19.18.3)
By long division,
Z_A=0.5 s+\frac{1.5 s}{2 s^2+1} (19.18.4)
Comparing Eqs. (19.18.3) and (19.18.4) shows that
L_3=0.5 H , \quad Z _B=\frac{1.5 s}{2 s^2+1}
Next, we seek to get C_2 as in Fig. 19.63(c) and let
Y_B=\frac{1}{Z_B}=\frac{2 s^2+1}{1.5 s}=1.333 s+\frac{1}{1.5 s}=s C_2+Y_C
from which C_2 = 1.33 F and
Y_C=\frac{1}{1.5 s}=\frac{1}{s L_1} \quad \Rightarrow \quad L_1=1.5 H
Thus, the LC ladder network in Fig. 19.63(a) with L_1 = 1.5 H, C_2 = 1.333 F, \text{ and } L_3 = 0.5 H has been synthesized to provide the given transfer function H(s). This result can be confirmed by finding \pmb{H}(s) = \pmb{V_2/V_1} in Fig. 19.63(a) or by confirming the required y_{21}.
