19.9.2 Ladder Network Synthesis

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19.9.2 Ladder Network Synthesis

Another application of tw o-port parameters is the synthesis (or b uilding) of ladder networks, which are found frequently in practice and have

Figure 19.61 LC ladder networks for low-pass filters of: (a) odd order, (b) even order.

particular use in designing passive low-pass filters. Based on our discussion of second-order circuits in Chapter 8, the order of the filter is the order of the characteristic equation describing the filter and is determined by the number of reactive elements that cannot be combined into single elements (e.g., through series or parallel combination). Figure 19.61(a) shows an LC ladder network with an odd number of elements (to realize an odd-order filter), while Fig. 19.61(b) shows one with an even number of elements (for realizing an e ven-order filter). When either network is terminated by the load impedance ZL and the source impedance Zs, we obtain the structure in Fig. 19.62. To make the design less complicated, we will assume that Zs = 0. Our goal is to synthesize the transfer function of the LC ladder network. We begin by characterizing the ladder network by its admittance parameters, namely,

$$I_1 = y_{11}V_1 + y_{12}V_2 \tag{19.80a}$$ $$I_2 = y_{21}V_1 + y_{22}V_2 \tag{19.80b}$$

Figure 19.62 LC ladder network with terminating impedances.

(Of course, the impedance parameters could be used instead of the admittance parameters.) At the input port, V1 = Vs since Zs = 0. At the output port, V2 = Vo and I2 = −V2∕ZL = −VoYL. Thus, Eq. (19.80b) becomes

$$-\mathbf{V}_o\mathbf{Y}_L=\mathbf{y}_{21}\mathbf{V}_s+\mathbf{y}_{22}\mathbf{V}_o$$

or

$$H(s) = \frac{V_o}{V_s} = \frac{-y_{21}}{Y_L + y_{22}}$$

(19.81)

We can write this as

$$\mathbf{H}(s) = -\frac{\mathbf{y}_{21}/\mathbf{Y}_L}{1 + \mathbf{y}_{22}/\mathbf{Y}_L}$$

(19.82)

We may ignore the ne gative sign in Eq. (19.82) 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 y21 and y22 are synthesized, thereby realizing the desired transfer function. To achieve this, we tak e advantage of an important property of the LC ladder network: All z and y parameters are ratios of polynomials that contain only e ven powers of s or odd powers of s—that is, they are ratios of either Od(s)∕Ev(s) or Ev(s)∕Od(s), where Od and Ev are odd and even functions, respectively. Let

$$\mathbf{H}(s) = \frac{\mathbf{N}(s)}{\mathbf{D}(s)} = \frac{\mathbf{N}_o + \mathbf{N}_e}{\mathbf{D}_o + \mathbf{D}_e}$$

(19.83)

where N(s) and D(s) are the numerator and denominator of the transfer function H(s); No and Ne are the odd and even parts of N; Do and De are the odd and even parts of D. Given that N(s) must be either odd or even, we can write Eq. (19.83) as

$$\mathbf{H}(s) = \begin{cases} \frac{\mathbf{N}_o}{\mathbf{D}_o + \mathbf{D}_e}, & (\mathbf{N}_e = 0) \\ \frac{\mathbf{N}_e}{\mathbf{D}_o + \mathbf{D}_e}, & (\mathbf{N}_o = 0) \end{cases}$$

(19.84)

and can rewrite this as

$$\mathbf{H}(s) = \begin{cases} \frac{\mathbf{N}_o / \mathbf{D}_e}{1 + \mathbf{D}_o / \mathbf{D}_e}, & (\mathbf{N}_e = 0) \\ \frac{\mathbf{N}_e / \mathbf{D}_o}{1 + \mathbf{D}_e / \mathbf{D}_o}, & (\mathbf{N}_o = 0) \end{cases}$$

(19.85)

Comparing this with Eq. (19.82), we obtain the y parameters of the network as

$$\frac{\mathbf{y}_{21}}{\mathbf{Y}_L} = \begin{cases} \frac{\mathbf{N}_o}{\mathbf{D}_e}, & (\mathbf{N}_e = 0) \\ \frac{\mathbf{N}_e}{\mathbf{D}_o}, & (\mathbf{N}_o = 0) \end{cases}$$

(19.86)

and

$$\frac{\mathbf{y}_{22}}{\mathbf{Y}_L} = \begin{cases} \frac{\mathbf{D}_o}{\mathbf{D}_e}, & (\mathbf{N}_e = 0) \\ \frac{\mathbf{D}_e}{\mathbf{D}_o}, & (\mathbf{N}_o = 0) \end{cases}$$

(19.87)

The following example illustrates the procedure.

Design the LC ladder network terminated with a 1-Ω resistor that has the Example 19.18 normalized transfer function

$$H(s) = \frac{1}{s^3 + 2s^2 + 2s + 1}$$

(This transfer function is for a Butterworth low-pass filter.)

Solution:

The denominator sho ws that this is a third-order netw ork, so that the LC ladder netw ork is sho wn in Fig. 19.63(a), with tw o inductors and one capacitor. Our goal is to determine the v alues of the inductors and

capacitor. To achieve this, we group the terms in the denominator into odd or even parts:

$$\mathbf{D}(s) = (s^3 + 2s) + (2s^2 + 1)$$

so that

$$H(s) = (s + 2s) + (2s + 1)$$ $$H(s) = \frac{1}{(s^3 + 2s) + (2s^2 + 1)}$$

Divide the numerator and denominator by the odd part of the denominator to get

$$\mathbf{H}(s) = \frac{\frac{1}{s^3 + 2s}}{1 + \frac{2s^2 + 1}{s^3 + 2s}}$$

(19.18.1)

From Eq. (19.82), when YL = 1,

$$H(s) = \frac{-y_{21}}{1 + y_{22}} \tag{19.18.2}$$

Comparing Eqs. (19.19.1) and (19.19.2), we obtain

$$y_{21} = -\frac{1}{s^3 + 2s}$$

, $y_{22} = \frac{2s^2 + 1}{s^3 + 2s}$

Any realization of y22 will automatically realize y21, since y22 is the output driving-point admittance, that is, the output admittance of the net work with the input port short-circuited. We determine the values of L and C in Fig. 19.63(a) that will give us y22. Recall that y22 is the shortcircuit output admittance. So we short-circuit the input port as shown in Fig. 19.63(b). First we get L3 by letting

$$Z_A = \frac{1}{y_{22}} = \frac{s^3 + 2s}{2s^2 + 1} = sL_3 + Z_B$$

(19.18.3)

By long division,

$$Z_A = 0.5s + \frac{1.5s}{2s^2 + 1}$$

(19.18.4)

Comparing Eqs. (19.18.3) and (19.18.4) shows that

$$L_3 = 0.5H,$$

$Z_B = \frac{1.5s}{2s^2 + 1}$

Next, we seek to get C2 as in Fig. 19.63(c) and let

$$Y_B = \frac{1}{Z_B} = \frac{2s^2 + 1}{1.5s} = 1.333s + \frac{1}{1.5s} = sC_2 + Y_C$$

from which C2 = 1.33 F and

$$Y_C = \frac{1}{1.5s} = \frac{1}{sL_1} \qquad \Rightarrow \qquad L_1 = 1.5 \text{ H}$$

Thus, the LC ladder netw ork in Fig. 19.63(a) with L1 = 1.5 H, C2 = 1.333 F, and L3 = 0.5 H has been synthesized to provide the given transfer function H(s). This result can be confirmed by finding H(s) = V2∕V¹ in Fig. 19.63(a) or by confirming the required y21.

C2

ZB

L¹ L³

(a)

L¹ L³

C² V² 1 Ω

‒

(b)

y²² =

1 ZA

Figure 19.63 For Example 19.18.

V1 +

‒

Realize the following transfer function using an LC ladder network ter- Practice Problem 19.18 minated in a 1-Ω resistor:

r:

$$H(s) = \frac{2}{s^3 + s^2 + 4s + 2}$$

Answer: Ladder network in Fig. 19.63(a) with L1 = L3 = 1.0 H and C2 = 500 mF.