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8 FILTER DESIGN A circuit designer frequently requires ®lters to extract the desired frequency spectrum from a wide variety of electrical signals. If a circuit passes all signals from dc through a frequency o c but stops the rest of the spectrum, then it is known as a low-pass ®lter. The frequency o c is called its cutoff frequency. Conversely, a high-pass ®lter stops all signals up to o c and passes those at higher frequencies. If a circuit passes only a ®nite frequency band that does not include zero (dc) and in®nite frequency, then it is called a band-pass ®lter. Similarly, a band-stop ®lter passes all signals except a ®nite band. Thus, band-pass and band-stop ®lters are speci®ed by two cutoff frequencies to set the frequency band. If a ®lter is designed to block a single frequency, then it is called a notch-®lter. The ratio of the power delivered by a source to a load with and without a two-port network inserted in between is known as the insertion loss of that two-port. It is generally expressed in dB. The fraction of the input power that is lost due to re¯ection at its input port is called the return loss. The ratio of the power delivered to a matched load to that supplied to it by a matched source is called the attenuation of that two-port network. Filters have been designed using active devices such as transistors and operational ampli®ers, as well as with only passive devices (inductors and capacitors only). Therefore, these circuits may be classi®ed as active or passive ®lters. Unlike passive ®lters, active ®lters can amplify the signal besides blocking the undesired frequencies. However, passive ®lters are economical and easy to design. Further, passive ®lters perform fairly well at higher frequencies. This chapter presents the design procedure of these passive circuits. There are two methods available to synthesize passive ®lters. One of them is known as the image parameter method and the other as the insertion-loss method. The former provides a design that can pass or stop a certain frequency band but its 295 Radio-Frequency and Microwave Communication Circuits: Analysis and Design Devendra K. Misra Copyright # 2001 John Wiley & Sons, Inc. ISBNs: 0-471-41253-8 (Hardback); 0-471-22435-9 (Electronic) frequency response cannot be shaped. The insertion-loss method is more powerful in the sense that it provides a speci®ed response of the ®lter. Both of these techniques are included in this chapter. It concludes with a design overview of microwave ®lters. 8.1 IMAGE PARAMETER METHOD Consider the two-port network shown in Figure 8.1. V 1 and V 2 represent voltages at its ports. Currents I 1 and I 2 are assumed as indicated in the ®gure. It may be noted that I 1 is entering port-1 while I 2 is leaving port-2. Further, Z in is input impedance at port-1 when Z i2 terminates port-2. Similarly, Z o is output impedance with Z i1 connected at port-1. Z i1 and Z i2 are known as the image impedance of the network. Following the transmission parameter description of the two-port, we can write V 1 AV 2 BI 2 8:1:1 and, I 1 CV 2 DI 2 8:1:2 Therefore, impedance Z in at its input can be found as Z in V 1 I 1 AV 2 BI 2 CV 2 DI 2 AZ i2 B CZ i2 D 8:1:3 Alternatively, (8.1.1) and (8.1.2) can be rearranged as follows after noting that AD À BC must be unity for a reciprocal network. Hence, V 2 DV 1 À BI 1 8:1:4 and, I 2 ÀCV 1 AI 1 8:1:5 Therefore, output impedance Z o is found to be Z o À V 2 I 2 À DV 1 À BI 1 ÀCV 1 AI 1 DZ i1 B CZ i1 A 8:1:6 Figure 8.1 A two-port network with terminations. 296 FILTER DESIGN Note that Z i1 À V 1 I 1 For Z in Z i1 and Z o Z i2 , equations (8.1.3) and (8.1.6) give Z i1 CZ i2 DAZ i2 B 8:1:7 and, Z i2 CZ i1 ADZ i1 B 8:1:8 Subtracting equation (8.1.8) from (8.1.7), we ®nd that Z i2 D A Z i1 8:1:9 Now, substituting (8.1.9) into (8.1.7), we have Z i1 AB CD r 8:1:10 Similarly, substituting (8.1.10) into (8.1.9) Z i2 BD AC r 8:1:11 Transfer characteristics of the network can be formulated as follows. From (8.1.1), V 1 V 2 A B I 2 V 2 A B Z i2 or, V 1 V 2 A D r AD p BC p For a reciprocal two-port network, AD À BC is unity. Therefore, V 2 V 1 D A r AD À BC AD p BC p D A r AD p À BC p 8:1:12 IMAGE PARAMETER METHOD 297 Similarly, from (8.1.2), I 2 I 1 A D r AD p À BC p 8:1:13 Note that equation (8.1.12) is similar to (8.1.13) except that the multiplying coef®cient in one is the reciprocal of the other. This coef®cient may be interpreted as the transformer turn ratio. It is unity for symmetrical T and Pi networks. Propagation factor g (equal to a jb, as usual) of the network can be de®ned as e Àg AD p À BC p Since AD À BC1, we ®nd that coshg AD p 8:1:14 Characteristic parameters of p- and T-networks are summarized in Table 8.1; corresponding low-pass and high-pass circuits are illustrated in Table 8.2. TABLE 8.1 Parameters of T and Pi Networks p-Network T-Network ABCD parameters A 1 Z 1 2Z 2 A 1 Z 1 2Z 2 B Z 1 B Z 1 Z 2 1 4Z 2 C 1 Z 2 Z 1 4Z 2 2 C 1 Z 2 D 1 Z 1 2Z 2 D 1 Z 1 2Z 2 Image impedance Z ip Z 1 Z 2 1 Z 1 4Z 2 v u u u t Z 1 Z 2 Z iT Z iT Z 1 Z 2 1 Z 1 4Z 2 s Propagation constant, g coshg1 Z 1 2Z 2 coshg1 Z 1 2Z 2 298 FILTER DESIGN For a low-pass T-section as illustrated in Table 8.2, Z 1 joL 8:1:15 and, Z 2 1 joC 8:1:16 Therefore, its image impedance can be found from Table 8.1 as follows: Z iT L C r 1 À o 2 LC 4 0:5 8:1:17 In the case of a dc signal, the second term inside the parentheses will be zero and the resulting image impedance is generally known as the nominal impedance, Z o . Hence, Z o L C r Note that the image impedance goes to zero if o 2 LC=4 1. The frequency that satis®es this condition is known as the cutoff frequency, o c . Hence, o c 2 LC p 8:1:18 TABLE 8.2 Constant-k Filter Sections Filter Type T-Section p-Section Low-pass High-pass IMAGE PARAMETER METHOD 299 Similarly, for a high-pass T-section, Z 1 1 joC 8:1:19 and, Z 2 joL 8:1:20 Therefore, its image impedance is found to be Z iT L C r 1 À 1 4o 2 LC 0:5 8:1:21 The cutoff frequency of this circuit will be given as follows. o c 1 2 LC p 8:1:22 Example 8.1: Design a low-pass constant-k T-section that has a nominal impedance of 75 O and a cut-off frequency of 2 MHz. Plot its frequency response in the frequency band of 100 kHzto 10 MHz. Since the nominal impedance Z o must be 75 O, Z o 75 L C r The cutoff frequency o c of a low-pass T-section is given by (8.1.18). Therefore, o c 2p  2  10 6 2 LC p These two equations can be solved for the inductance L and the capacitance C,as follows. L 11:9366 mH and, C 2:122 nF This circuit is illustrated in Figure 8.2. Note that inductance L calculated here is twice the value needed for a T-section. Propagation constant g of this circuit is 300 FILTER DESIGN determined from the formula listed in Table 8.1. Transfer characteristics are then found as e Àg . The frequency response of the designed circuit is shown in Figure 8.3. The magnitude of the transfer function (ratio of the output to input voltages) remains constant at 0 dB for frequencies lower than 2 MHz. Therefore, the output magnitude will be equal to the input in this range. It falls by 3 dB if the signal frequency approaches 2 MHz. It falls continuously as the frequency increases further. Note that the phase angle of the transfer function remains constant at À180 beyond 2 MHz. However, it shows a nonlinear characteristic in the pass-band. Figure 8.2 A low-pass constant-k T-section. Figure 8.3 Frequency response of the T-section shown in Figure 8.2. IMAGE PARAMETER METHOD 301 The image impedance Z iT of this T-section can be found from (8.1.17). Its characteristics (magnitude and phase angle) with frequency are displayed in Figure 8.4. The magnitude of Z iT continuously reduces as frequency is increased and becomes zero at the cutoff. The phase angle of Z iT is zero in pass-band and it changes to 90 in the stop-band. Thus, image impedance is a variable resistance in the pass-band, whereas it switches to an inductive reactance in the stop-band. Frequency characteristics illustrated in Figures 8.3 and 8.4 are representative of any constant-k ®lter. There are two major drawbacks to this kind of ®lter: 1. Signal attenuation rate after the cutoff point is not very sharp. 2. Image impedance is not constant with frequency. From a design point of view, it is important that it stays constant at least in its pass-band. Figure 8.4 Image impedance of the constant-k ®lter of Figure 8.2 as a function of frequency. 302 FILTER DESIGN These problems can be remedied using the techniques described in the following section. m-Derived Filter Sections Consider two T-sections shown in Figure 8.5. The ®rst network represents the constant-k ®lter that is considered in the preceding section, and the second is a new m-derived section. It is assumed that the two networks have the same image impedance. From Table 8.1, we can write Z H iT Z iT Z H 1 Z H 2 1 Z H 1 4Z H 2 s Z 1 Z 2 1 Z 1 4Z 2 s It can be solved for Z H 2 as follows. Z H 2 1 Z H 1 Z 1 Z 2 Z 2 1 À Z H2 1 4 8:1:23 Now, assume that Z H 1 mZ 1 8:1:24 Substituting (8.1.24) in to (8.1.23) we get Z H 2 Z 2 m 1 À m 2 4m Z 1 8:1:25 Thus, an m-derived section is designed from values of components determined for the corresponding constant-k ®lter. The value of m is selected to sharpen the attenuation at cutoff or to control image impedance characteristics in the pass-band. Figure 8.5 A constant-k T-section (a) and an m-derived section (b). IMAGE PARAMETER METHOD 303 For a low-pass ®lter, the m-derived section can be designed from the correspond- ing constant-k ®lter using (8.1.24) and (8.1.25), as follows: Z H 1 jomL 8:1:26 and, Z H 2 1 À m 2 4m  joL 1 jomC 8:1:27 Now we need to ®nd its propagation constant g and devise some way to control its attenuation around the cutoff. Expression for the propagation constant of a T-section is listed in Table 8.1. In order to ®nd g of this T-section, we ®rst divide (8.1.26) by (8.1.27): Z H 1 Z H 2 À o 2 m 2 LC 1 À 1 À m 2 4 o 2 LC À 4o 2 m 2 o 2 c 1 À1À m 2 o 2 o 2 c 8:1:28 where, o c 2 LC p 8:1:29 Using the formula listed in Table 8.1 and (8.1.28), propagation constant g is found as follows: coshg1 Z H 1 2Z H 2 1À 2 mo o c 2 1 À1 À m 2 o o c 2 or, coshg o 2 c À o 2 Àmo 2 o 2 c À1 À m 2 o 2 8:1:30 Hence, the right-hand side of (8.1.30) is dependent on frequency o.Itwillgoto in®nity (and therefore, g) if the following condition is satis®ed. o o c 1 À m 2 p o I 8:1:31 304 FILTER DESIGN [...]... not exist and a compromise is needed to design the ®lters The image parameter method described in the preceding section provides a simple design procedure However, transfer characteristics of this circuit cannot be shaped as desired On the other hand, the insertion loss method provides ways to shape pass- and stop-bands of the ®lter although its design theory is much more complex INSERTION LOSS METHOD... For scaling the frequency from 1 to oc , divide all normalized g values that represent capacitors or inductors by the desired cutoff frequency expressed in radians per second Resistors are excluded from this operation Impedance scaling: To scale g0 and gn1 to X O from unity, multiply all g values that represent resistors or inductors by X On other hand, divide those g values representing capacitors... p-section with m 0:6 remains almost constant over 90 per cent of the pass-band If this network is bisected to connect on either side of the cascaded constant-k and m-derived sections, then it should provide the desired impedance characteristics In order to verify these characteristics, let us consider a bisected p-section as shown in Figure 8.10 Its transmission parameters can be easily found following... illustrated in Figure 8.9 indicate that the m-derived ®lter has a sharp change at cutoff frequency of 2 MHz However, the output signal rises to À4dB in its stop-band On the other hand, the constant-k ®lter provides higher attenuation in its stop-band For example, the m-derived ®lter characteristic in Figure 8.8 An m-derived T-section for Example 8.2 308 FILTER DESIGN Figure 8.9 Frequency response of the m-derived... designed Several approximations to these characteristics are available that can be physically synthesized Two of these are presented below Maximally Flat Filter As its name suggests, this kind of ®lter provides the ¯attest possible pass-band response However, its transition from pass-band to stop-band is Figure 8.17 Characteristics of an ideal low-pass ®lter INSERTION LOSS METHOD 317 gradual This is also... 1010 À 1 o log10 log10 B oc 8:2:9 Chebyshev Filter A ®lter with sharper cutoff can be realized at the cost of ¯atness in its pass-band Chebyshev ®lters possess ripples in the pass-band but provide a sharp transition into the stop-band In this case, Chebyshev polynomials are used to represent the insertion loss Mathematically, 1 jG oj p ; 2 1 BTm o m 1; 2; 3;... corresponding constant-k T-section has an attenuation of more than 30 dB at this frequency, as depicted in Figure 8.3 Composite Filters As demonstrated through the preceding examples, the m-derived ®lter provides an in®nitely sharp attenuation right at its cutoff However, the attenuation in its stopband is unacceptably low Contrary to this, the constant-k ®lter shows higher attenuation in its stop-band although... log10 1010 À 1 1 log10 103 À 1 3 2  log 4 log 1:9953 À 1 40 10 10 log10 log10 1010 À 1 10 0:5  3 % 2:5 0:6 Since the number of elements must be an integer, selecting n 3 will provide more than the speci®ed attenuation at 40 MHz From (8.2.15) and (8.2.16), g0 g 4 1 p g1 2 sin 1 6 p g2 2 sin 2 2 INSERTION LOSS METHOD Figure 8.22 325 A maximally ¯at low-pass ®lter obtained... considered a part of this circuit In other words, this represents voltage across RL with respect to source voltage, not to input of the circuit Since source resistance is equal to the load, there is equal division of source voltage that results in À6 dB Another 3-dB loss at the band edge shows a total of about 9 dB at 10 MHz This characteristic shows that there is an insertion loss of over 40 dB at 40 MHz... "s# 100:5 À 1 cosh 100:001 À 1 À1 m coshÀ1 4 2 327 328 FILTER DESIGN Since we want a symmetrical ®lter with 75 O on each side, we select m 3 (an odd number) It will provide more than 5 dB of insertion loss at 400 MHz The g values can now be determined from (8.2.17) to (8.2.24), as follows From (8.2.23), a1 sin p 0:5 6 p a2 sin 1 2 and, a3 sin 5p 0:5 . have been designed using active devices such as transistors and operational ampli®ers, as well as with only passive devices (inductors and capacitors only) response cannot be shaped. The insertion-loss method is more powerful in the sense that it provides a speci®ed response of the ®lter. Both of these techniques