Microstrip bộ lọc cho các ứng dụng lò vi sóng RF (P5)

Microstrip Filters for RF/Microwave Applications Jia-Sheng Hong, M J Lancaster Copyright © 2001 John Wiley & Sons, Inc ISBNs: 0-471-38877-7 (Hardback); 0-471-22161-9 (Electronic) CHAPTER Lowpass and Bandpass Filters Conventional microstrip lowpass and bandpass filters such as stepped-impedance filters, open-stub filters, semilumped element filters, end- and parallel-coupled half-wavelength resonator filters, hairpin-line filters, interdigital and combline filters, pseudocombline filters, and stub-line filters are widely used in many RF/microwave applications It is the purpose of this chapter to present the designs of these filters with instructive design examples 5.1 LOWPASS FILTERS In general, the design of microstrip lowpass filters involves two main steps The first one is to select an appropriate lowpass prototype, such as one as described in Chapter The choice of the type of response, including passband ripple and the number of reactive elements, will depend on the required specifications The element values of the lowpass prototype filter, which are usually normalized to make a source impedance g0 = and a cutoff frequency
c = 1.0, are then transformed to the L-C elements for the desired cutoff frequency and the desired source impedance, which is normally 50 ohms for microstrip filters Having obtained a suitable lumped-element filter design, the next main step in the design of microstrip lowpass filters is to find an appropriate microstrip realization that approximates the lumpedelement filter In this section, we concentrate on the second step Several microstrip realizations will be described 5.1.1 Stepped-Impedance, L-C Ladder Type Lowpass Filters Figure 5.1(a) shows a general structure of the stepped-impedance lowpass microstrip filters, which use a cascaded structure of alternating high- and lowimpedance transmission lines These are much shorter than the associated guided109 110 LOWPASS AND BANDPASS FILTERS (a) (b) FIGURE 5.1 (a) General structure of the stepped-impedance lowpass microstrip filters (b) L-C ladder type of lowpass filters to be approximated wavelength, so as to act as semilumped elements The high-impedance lines act as series inductors and the low-impedance lines act as shunt capacitors Therefore, this filter structure is directly realizing the L-C ladder type of lowpass filters of Figure 5.1(b) Some a priori design information must be provided about the microstrip lines, because expressions for inductance and capacitance depend upon both characteristic impedance and length It would be practical to initially fix the characteristic impedances of high- and low-impedance lines by consideration of 앫 Z0C < Z0 < Z0L, where Z0C and Z0L denote the characteristic impedances of the low and high impedance lines, respectively, and Z0 is the source impedance, which is usually 50 ohms for microstrip filters 앫 A lowerZ0C results in a better approximation of a lumped-element capacitor, but the resulting line width WC must not allow any transverse resonance to occur at operation frequencies 앫 A higher Z0L leads to a better approximation of a lumped-element inductor, but Z0L must not be so high that its fabrication becomes inordinately difficult as a narrow line, or its current-carrying capability becomes a limitation In order to illustrate the design procedure for this type of filter, the design of a three-pole lowpass filter is described in follows The specifications for the filter under consideration are Cutoff frequency fc = GHz Passband ripple 0.1 dB (or return loss  –16.42 dB) Source/load impedance Z0 = 50 ohms 5.1 LOWPASS FILTERS 111 A lowpass prototype with Chebyshev response is chosen, whose element values are g0 = g4 = g1 = g3 = 1.0316 g2 = 1.1474 for the normalized cutoff
c = 1.0 Using the element transformations described in Chapter 3, we have
c g = 8.209 × 10 冢 冣冢  2f 冣 Z0 L1 = L3 =  g0 冢 冣冢 g0 C2 =  Z0 –9 H c
c  g2 = 3.652 × 10–12 F 2fc 冣 (5.1) The filter is to be fabricated on a substrate with a relative dielectric constant of 10.8 and a thickness of 1.27 mm Following the above-mentioned considerations, the characteristic impedances of the high- and low-impedance lines are chosen as Z0L = 93 ohms and Z0C = 24 ohms The relevant design parameters of microstrip lines, which are determined using the formulas given in Chapter 4, are listed in Table 5.1, where the guided wavelengths are calculated at the cutoff frequency fc = 1.0 GHz Initially, the physical lengths of the high- and low-impedance lines may be found by gL cL lL =  sin–1  Z0L 2 冢 冣 (5.2) gC lC =  sin–1(cCZ0C) 2 which give lL = 11.04 mm and lC = 9.75 mm for this example The results of (5.2) not take into account series reactance of the low-impedance line and shunt susceptance of the high-impedance lines To include these effects, the lengths of the highand low-impedance lines should be adjusted to satisfy lC 2lL cL = Z0L sin  + Z0C tan  gC gL 冢 冣 冢 冣 1 2lC lL cC =  sin  + ×  tan  Z0C gC Z0L gL 冢 冣 冢 冣 (5.3) TABLE 5.1 Design parameters of microstrip lines for a stepped-impedance lowpass filter Characteristic impedance (ohms) Guided wavelengths (mm) Microstrip line width (mm) Z0C = 24 gC = 105 WC = 4.0 Z0 = 50 g0 = 112 W0 = 1.1 Z0L = 93 gL = 118 WL = 0.2 112 LOWPASS AND BANDPASS FILTERS where L and C are the required element values of lumped inductors and capacitor given above This set of equations is solved for lL and lC, resulting in lL = 9.81 mm and lC = 7.11 mm A layout of this designed microstrip filter is illustrated in Figure 5.2(a), and its performance obtained by full-wave EM simulation is plotted in Figure 5.2(b) 5.1.2 L-C Ladder Type of Lowpass Filters Using Open-Circuited Stubs The previous stepped-impedance lowpass filter realizes the shunt capacitors of the lowpass prototype as low impedance lines in the transmission path An alternative realization of a shunt capacitor is to use an open-circuited stub subject to (a) (b) FIGURE 5.2 (a) Layout of a three-pole, stepped-impedance microstrip lowpass filter on a substrate with a relative dielectric constant of 10.8 and a thickness of 1.27 mm (b) Full-wave EM simulated performance of the filter 5.1 LOWPASS FILTERS 2 C =  tan  l Z0 g 冢 冣 for l < g/4 113 (5.4) where the term on the left-hand side is the susceptance of shunt capacitor, whereas the term on the right-hand side represents the input susceptance of open-circuited stub, which has characteristic impedance Z0 and a physical length l that is smaller than a quarter of guided wavelength g The following example will demonstrate how to realize this type of microstrip lowpass filter For comparison, the same prototype filter and the substrate for the previous design example of stepped-impedance microstrip lowpass filter is employed Also, the same high-impedance (Z0L = 93 ohms) lines are used for the series inductors, while the open-circuited stub will have the same low characteristic impedance as Z0C = 24 ohms Thus, the design parameters of the microstrip lines listed in Table 5.1 are valid for this design example To realize the lumped L-C elements, the physical lengths of the high-impedance lines and the open-circuited stub are initially determined by cL gL lL =  sin–1  = 11.04 mm Z0L 2 冢 冣 gC lC =  tan–1(cCZ0C) = 8.41 mm 2 To compensate for the unwanted susceptance resulting from the two adjacent highimpedance lines, the initial lC should be changed to satisfy 2lC lL cC =  tan  + ×  tan  Z0C gC Z0L gL 冢 冣 冢 冣 (5.5) which is solved for lC and results in lC = 6.28 mm for this example Furthermore, the open-end effect of the open-circuited stub must be taken into account as well According to the discussions in Chapter 4, a length of l = 0.5 mm should be compensated for in this case Therefore, the final dimension of the open-circuited stub is lC = 6.28 – 0.5 = 5.78 mm The layout and EM-simulated performance of the designed filter are given in Figure 5.3 Comparing to the filter response to that in Figure 5.2, both filters show a very similar filtering characteristic in the given frequency range, which is expected, as they are designed based on the same prototype filter However, one should bear in mind that the two filters have different realizations that only approximate the lumped elements of the prototype in the vicinity of the cutoff frequency, and hence, their wide-band frequency responses can be different, as shown in Figure 5.4 The filter using an open-circuited stub exhibits a better stopband characteristic with an attenuation peak at about 5.6 GHz This is because at this frequency, the open-cir- 114 LOWPASS AND BANDPASS FILTERS (a) (b) FIGURE 5.3 (a) Layout of a 3-pole microstrip lowpass filter using open-circuited stubs on a substrate with a relative dielectric constant of 10.8 and a thickness of 1.27 mm (b) Full-wave EM simulated performance of the filter cuited stub is about a quarter guided wavelength so as to almost short out a transmission, and cause the attenuation peak To obtain a sharper rate of cutoff, a higher degree of filter can be designed in the same way Figure 5.5(a) is a seven-pole, lumped-element lowpass filter with its microstrip realization illustrated in Figure 5.5(b) The four open-circuited stubs, which have the same line width WC, are used to approximate the shunt capacitors; and the three narrow microstrip lines of width WL are for approximation of the series inductors The lowpass filter is designed to have a Chebyshev response, with a passband ripple of 0.1 dB and a cutoff frequency at 1.0 GHz The lumped element values in Figure 5.5(a) are then given by 5.1 LOWPASS FILTERS 115 FIGURE 5.4 Comparison of wide-band frequency responses of the filters in Figure 5.2(a) and Figure 5.3(a) Z0 = 50 ohm C1 = C7 = 3.7596 pF L2 = L6 = 11.322 nH C3 = C5 = 6.6737 pF L4 = 12.52 nH The microstrip filter design uses a substrate having a relative dielectric constant r = 10.8 and a thickness h = 1.27 mm To emphasize and demonstrate that the microstrip realization in Figure 5.5(b) can only approximate the ideal lumped-element filter in Figure 5.5(a), two microstrip filter designs that use different characteristic impedances for the high-impedance lines are presented in Table 5.2 The first design (Design 1) uses the high-impedance lines that have a characteristic impedance Z0L = 110 ohms and a line width WL = 0.1 mm on the substrate used The second design (Design 2) uses a characteristic impedance Z0L = 93 ohms and a line width WL = 0.2 mm The performance of these two microstrip filters is shown in Figure 5.5(c), as compared to that of the lumped-element filter As can be seen, the two microstrip filters behave not only differently from the lumped-element one, but also differently from each other The main difference lies in the stopband behaviors The microstrip filter (Design 1) that uses the narrower inductive lines (WL = 0.1 mm) has a better matched stopband performance This is because that the use of the inductive lines with the higher characteristic impedance and the shorter lengths (referring to Table 5.2) achieves a better approximation of the lumped inductors The other microstrip filter (Design 2) with the wider inductive lines (WL = 0.2 mm) exhibits an unwanted transmission peak at 2.86 GHz, which is due to its longer inductive lines being about half-wavelength and resonating at about this frequency 116 LOWPASS AND BANDPASS FILTERS L2 Z0 L4 C1 L6 C3 C5 C7 Z0 (a) WC WL l1 l2 l3 l4 l5 l6 l7 (b) (c) FIGURE 5.5 (a) A seven-pole, lumped-element lowpass filter (b) Microstrip realization (c) Comparison of filter performance for the lumped-element design and the two microstrip designs given in Table 5.2 5.1.3 Semilumped Lowpass Filters Having Finite-Frequency Attenuation Poles The previous two types of microstrip lowpass filter realize the lowpass prototype filters having their frequencies of infinite attenuation at f = In order to obtain an even sharper rate of cutoff for a given number of reactive elements, it is desirable to 5.1 LOWPASS FILTERS 117 TABLE 5.2 Two microstrip lowpass filter designs with open-circuited stubs Substrate (r = 10.8, h = 1.27 mm) WC = mm l1 = l7 (mm) l2 = l6 (mm) l3 = l5 (mm) l4 (mm) Design (WL = 0.1 mm) Design (WL = 0.2 mm) 5.86 5.39 13.32 16.36 9.54 8.67 15.09 18.93 use filter structures giving infinite attenuation at finite frequencies A prototype of this type may have an elliptic function response, as discussed in Chapter Figure 5.6(a) shows an elliptic function lowpass filter that has two series-resonant branches connected in shunt that short out transmission at their resonant frequencies, and thus give two finite-frequency attenuation poles Note that at f = these two branches have no effect, and the inductances L1, L3, and L5 block transmission by having infinite series reactance, whereas the capacitance C6 shorts out transmission by having infinite shunt susceptance A microstrip filter structure that can realize, approximately, such a filtering characteristic is illustrated in Figure 5.6(b), which is much the same as that for the stripline realization in [1] Similar to the stepped-impedance microstrip filters described in Section 5.1, the lumped L-C elements in Figure 5.6(a) are to be approximated by use of short lengths of high- and low-impedance lines, and the actual dimensions of the lines are determined in a similar way to that discussed previously For demonstration, a design example is described below L1 Z0 L3 L5 L2 L4 C2 C4 C6 Z0 (a) (b) FIGURE 5.6 (a) An elliptic-function, lumped-element lowpass filter (b) Microstrip realization of the elliptic function lowpass filter 118 LOWPASS AND BANDPASS FILTERS The element values for elliptic function lowpass prototype filters may be obtained from Table 3.3 or from [2] and [3] For this example, we use the lowpass prototype element values g0 = g7 = 1.000 gL4 = g = 0.7413 gL1 = g1 = 0.8214 gC4 = g4 = 0.9077 gL2 = g = 0.3892 gL5 = g5 = 1.1170 gC2 = g2 = 1.0840 gC6 = g6 = 1.1360 gL3 = g3 = 1.1880 where we use gLi and gCi to denote the inductive and capacitive elements, respectively This prototype filter has a passband ripple LAr = 0.18 dB and a minimum stopband attenuation LAs = 38.1 dB at
s = 1.194 for the cutoff
c = 1.0 [2] The microstrip filter is designed to have a cutoff frequency fc = 1.0 GHz and input/output terminal impedance Z0 = 50 ohms Therefore, the L-C element values, which are scaled to Z0 and fc, can be determined by Li = Z0gLi 2fc 1 Ci =  gCi 2fc Z0 (5.6) This yields L1 = 6.53649 nH L2 = 3.09716 nH L3 = 9.45380 nH C2 = 3.45048 pF L5 = 8.88880 nH L4 = 5.89908 nH C6 = 3.61600 pF C4 = 2.88930 pF (5.7) The two finite-frequency attenuation poles occur at fp1 =  = 1.219 GHz 2兹L 苶苶 苶 4C4 (5.8) fp2 =  = 1.540 GHz 2 兹L 苶苶 苶 2C2 For microstrip realization, a substrate with a relative dielectric constant of 10.8 and a thickness of 1.27 mm is assumed All inductors will be realized using high-impedance lines with characteristic impedance Z0L = 93 ohms, whereas the all capacitors ... filter (b) Microstrip realization (c) Comparison of filter performance for the lumped-element design and the two microstrip designs given in Table 5.2 5.1.3 Semilumped Lowpass Filters Having Finite-Frequency... L4 = 12.52 nH The microstrip filter design uses a substrate having a relative dielectric constant r = 10.8 and a thickness h = 1.27 mm To emphasize and demonstrate that the microstrip realization... width WL = 0.2 mm The performance of these two microstrip filters is shown in Figure 5.5(c), as compared to that of the lumped-element filter As can be seen, the two microstrip filters behave