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Solution
Now f 1 = 950 MHz, f 2 = 1,050 MHz, and f 0 = 998.7 MHz. From Table 7.1 we can read the element values for the lowpass filter prototype: g 0 =
R G = 1, g 1 = C1 = 1, g 2 = L 2 = 2, g 3 = C3 = 1, g 4 = R L = 1. Let us transform the lowpass prototype into a bandpass filter using the equations
presented in Figure 7.13: C1′ = C1 /Dv 0 = C1 /(v2 − v1 ) = 1.592 × 10−9 = C3′, L 2′ = L 2 /(v2 − v 1 ) = 3.183 × 10−9. L 1′, C2′, and L 3′ are obtained
from condition L k′Ck′ = 1/v02. L 2′ and C2′ form a series resonant circuit in series. Let us transform this into a shunt element using a J -inverter. We can
choose R aG = 1 and J = 1. From Table 7.4 we obtain Ca1 = C1′, Ca2 = L 2′, Ca3 = C3′, and R aL = 1. The parallel resonant circuits can be realized with
short-circuited stubs with |
a length of lg /4. From equation |
z k = (p/4) √L ak /Cak = p /(4v 0 Cak ) we obtain the normalized characteristic |
|
impedances of the lines: z 1 = z 3 |
= 0.079 and z 2 = 0.039. The characteristic |
impedances of the short-circuited parallel stubs are Z 1 = Z 3 = 4.0V and |
|
Z 2 = 2.0V. Between these stubs there are inverters with a 50-V characteristic |
|
impedance and a lg /4 length. The short-circuited stubs may also be 3l g /4 long and then the characteristic impedances are three times these values. However, they are still extremely hard to realize and the response differs even more from the response obtained with a filter formed with lumped elements.
7.2.3 Practical Microwave Filters
A large number of different possible structures can be used to realize a synthesized filter. In the following we briefly describe some commonly used filter structures. Other filters and their design rules are presented in the literature [6].
A simple lowpass filter can be realized by coupling short transmission line sections in series, with alternating low and high characteristic impedances. A short (electrical length b l < p/4) line with a high characteristic impedance Z 0 corresponds approximately to a series reactance of
X ≈ Z 0 b l |
(7.47) |
Accordingly, a short line with a low characteristic impedance corresponds to a parallel susceptance of
B ≈ bl /Z 0 |
(7.48) |
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167 |
From the element values of the prototype filter we can calculate the required electrical lengths of the lines. The ratio of the high and low impedance should be as high as possible. The longer the line sections, the further from the ideal the response will be. Usually there are also passbands at higher frequencies. Figure 7.17 shows a microstrip layout of such a steppedimpedance filter with six line sections. One must also remember that an abrupt change in the microstrip line width causes a fringing capacitance, which must be taken into account in determining the lengths of the sections.
The bandpass and bandstop filters consist of series and parallel resonant circuits. In the microwave region the resonators can be realized in various ways, and then filters can be constructed from them in a number of ways. Figure 7.18(a) shows a bandpass filter consisting of open-circuited l g /2 long microstrip lines that are side-coupled over a length of lg /4 to each other [12]. This structure is suitable for filters with a passband width less than 20%. Figure 7.18(b) shows a microstrip bandpass filter consisting of endcoupled lg /2 resonators. The small gap between the line ends corresponds to a series capacitance. This capacitance is necessarily rather small due to
Figure 7.17 Microstrip layout of a stepped-impedance lowpass filter.
Figure 7.18 Microstrip layouts of bandpass filters: (a) side-coupled line resonators; and
(b) capacitive-gap coupled (end-coupled) line resonators.
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limitations in fabricating small gaps using microstrip techniques, and therefore the resulting bandwidth is rather small.
A comb-line filter consists of resonators with a length less than l g /4, which are grounded in one end and capacitively loaded in the other end. Figure 7.19(a) shows a comb-line filter consisting of rectangular or circular metal posts in a rectangular metal box. By adding a metal plate to the end of each capacitively loaded resonator, the end capacitance can be increased. A fine-tuning of the resonance frequency may be done with tuning screws.
An interdigital filter also consists of lg /4-long resonators, which are grounded in one end and open in the other end, as shown in Figure 7.19(b). This structure is suitable for a broad bandpass filter. The comb-line and interdigital filters can also be manufactured using microstrip medium.
Figure 7.20 shows a bandpass filter made in a metal waveguide from iris-coupled resonators. The resonator cavities are lg /2 long and separated by thin metal walls, each containing a rectangular or circular inductive iris. The resonance frequency of each individual resonator can be tuned with a tuning screw through the broad waveguide wall in the middle of the resonator.
A waveguide-cavity filter, a comb-line filter, or an interdigital filter can be made much smaller if the metal housing is filled with a ceramic of
Figure 7.19 (a) Comb-line filter; and (b) interdigital filter.
Figure 7.20 Bandpass filter made from waveguide cavities coupled through inductive irises.
Resonators and Filters |
169 |
high permittivity (er = 10 to 100). Such a ceramic |
interdigital filter at |
5 GHz with a 100 MHz passband may have dimensions of 10 × 4 × 3 mm3, and its insertion loss is typically 1 dB. Ceramic filters are used also in mobile phones as duplex filters, with dimensions like 10 × 5 × 2 mm3, but in this application such dimensions are considered large, and the tendency is to move to using surface acoustic wave (SAW) or bulk acoustic wave (BAW) filters [13, 14].
A SAW filter is based on an acoustic wave propagating on the surface of a piezoelectric medium. The SAW filters are bandpass filters of a small size. The electric signal is transformed into an acoustic wave with an interdigital transducer and vice versa, as shown in Figure 7.21. The SAW filters are usable to frequencies over 2 GHz. A 2-GHz SAW filter chip has an area of 1 × 1 mm2, but with the package it requires a somewhat larger area. Insertion loss is typically 2 dB to 3 dB. A BAW filter is even smaller and provides somewhat higher frequency selectivity because of higher Q compared to SAW.
Although the basic filter theory has been around for a long time, there are continuing advances in the synthesis methods. In particular, there are continuous technological advances in SAW and BAW filters, in MEMS tuning of micromechanical microwave filters, and in the more conventional filter structures where high-temperature superconductors are now successfully utilized.
Figure 7.21 SAW filter.
References
[1]Nguyen, C. T.-C., ‘‘Frequency-Selective MEMS for Miniaturized Low-Power Communication Devices,’’ IEEE Trans. Microwave Theory and Techniques, Vol. 47, No. 8, 1999, pp. 1486–1503.
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[2]Ramo, S., J. Whinnery, and T. van Duzer, Fields and Waves in Communication Electronics, New York: John Wiley & Sons, 1965.
[3]Collin, R. E., Foundations for Microwave Engineering, 2nd ed., New York: IEEE Press, 2001.
[4]Kajfez, D., and P. Guillon, (eds.), Dielectric Resonators, Oxford, MS: Vector Fields, 1990.
[5]Pozar, D. M., Microwave Engineering, 2nd ed., New York: John Wiley & Sons, 1998.
[6]Matthaei, G. L., L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Dedham, MA: Artech House, 1980.
[7]Fano, R. M., and A. W. Lawson, ‘‘The Theory of Microwave Filters,’’ G. L. Ragan, (ed.), Microwave Transmission Circuits, Radiation Laboratory Series, New York: McGraw-Hill, 1948.
[8]Zverev, A. I., Handbook of Filter Synthesis, New York: John Wiley & Sons, 1967.
[9]Richards, P. I., ‘‘Resistor-Transmission-Line Networks,’’ Proc. IRE, Vol. 36, February 1948, pp. 217–220.
[10]Ozaki, H., and J. Ishii, ‘‘Synthesis of a Class of Stripline Filters,’’ IRE Trans. on Circuit Theory, Vol. CT-5, June 1958, pp. 104–109.
[11]Matsumoto, A., Microwave Filters and Circuits, New York: Academic Press, 1970.
[12]Cohn, S. B., ‘‘Parallel-Coupled Transmission-Line-Resonator Filter,’’ IRE Trans. on Microwave Theory and Techniques, Vol. 6, April 1958, pp. 223–231.
[13]Campbell, C. K., ‘‘Application of Surface Acoustic and Shallow Bulk Acoustic Wave Devices,’’ Proc. IEEE, Vol. 77, No. 10, 1989, pp. 1453–1484.
[14]Special Issue on Modeling, Optimization, and Design of Surface and Bulk Acoustic Wave Devices, IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control,
Vol. 48, No. 5, 2001, pp. 1161–1479.