Application of the Two Port Networks to the Ladder Network Synthesis

Another application of two-port parameters is the synthesis (or building) of ladder networks which are found frequently in practice and have particular use in designing passive lowpass filters. Figure 1(a) shows an $ L C $ ladder network with an odd number of elements (to realize an odd-order filter), while Fig. 1(b) shows one with an even number of elements (for realizing an even-order filter). When either network is terminated by the load impedance $ Z_{L} $ and the source impedance $ Z_{s} $, we obtain the structure in Fig. 2. To make the design less complicated, we will assume that $ Z_{s}=0 $. Our goal is to synthesize the transfer function of the $ L C $ ladder network. We begin by characterizing the ladder network by its admittance parameters, namely, (Of course, the impedance parameters could be used instead of the admittance parameters.) At the input port, $ \mathbf{V}_{1}=\mathbf{V}_{s} $ since $ \mathbf{Z}_{s}=0 $. At the output port, $ \mathbf{V}_{2}=\mathbf{V}_{o} $ and $ \mathbf{I}_{2}=-\mathbf{V}_{2} / \mathbf{Z}_{L}=-\mathbf{V}_{o} \mathbf{Y}_{L} $. Thus Eq. (1.b) becomes or We can write this as We may ignore the negative sign in Eq. (2) because filter requirements are often stated in terms of the magnitude of the transfer function. The main objective in filter design is to select capacitors and inductors so that the parameters $ \mathbf{y}_{21} $ and $ \mathbf{y}_{22} $ are synthesized, thereby realizing the desired transfer function. To achieve this, we take advantage of an important property of the $ L C $ ladder network: all $ z $ and $ y $ parameters are ratios of polynomials that contain only even powers of $ s $ or odd powers of $ s $-that is, they are ratios of either $ \mathrm{Od}(s) / \operatorname{Ev}(s) $ or $ \operatorname{Ev}(s) / \operatorname{Od}(s) $, where $ \mathrm{Od} $ and $ \mathrm{Ev} $ are odd and even functions, respectively. Let where $ \mathbf{N}(s) $ and $ \mathbf{D}(s) $ are the numerator and denominator of the transfer function $ \mathbf{H}(s) ; \mathbf{N}_{o} $ and $ \mathbf{N}_{e} $ are the odd and even parts of $ \mathbf{N} ; \mathbf{D}_{o} $ and $ \mathbf{D}_{e} $ are the odd and even parts of $ \mathbf{D} $. Since $ \mathbf{N}(s) $ must be either odd or even, we can write Eq. (3) as and can rewrite this as Comparing this with Eq. (2), we obtain the $ y $ parameters of the network as and The following example illustrates the procedure.