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Appendix. Analysis of Resonator Properties Using General-Purpose Microwave Simulator

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Appendix. Analysis of Resonator Properties Using General-Purpose Microwave Simulator Appendix. Analysis of Resonator Properties Using General-Purpose Microwave Simulator A.I Design Parameters of Direct-Coupled Resonator BPF As previously described, bandpass filters in the RF and microwave regions are generally realized by the cascaded connection of plural resonators with uni­ form structure using appropriate coupling circuits. Starting with given filter specifications (e.g., center frequency, pass-band bandwidth, VSWR, insertion loss, attenuation at specified frequency, etc.), the first step in a filter synthe­ sis procedure is to obtain the filter design parameters, such as the required stage number n, element values 9j, and unloaded-Q, after choosing the filter response type of Chebyschev or Butterworth. In accordance with these fun­ damental parameters, the next step is to calculate the electrical parameters and/or structural parameters based on the applied resonator structure. Filter design based on lumped-element LC resonators and lumped-element coupling circuits allow direct calculation of the electrical parameters. Con­ trarily, in the high frequency regions such as RF and microwave bands, distributed-element circuits are often applied to filtering devices, and thus both electrical and structural parameters are required for filter design. Since filter design theories have been generalized and systematized based on lumped-element circuit theory, the electrical parameters are first calculated by approximating distributed circuits into lumped-element circuits, which are in turn converted into the structural parameters determining resonator design. Filter design can easily be obtained when the conversion between electrical and structural parameters is analytically derived and expressed in the form of a mathematical equation. Yet for complicated geometrical structures, gener­ alized conversion equations become difficult to derive, and thus experimental results are often combined with analytical expressions in order to obtain the structural parameters. Design methods in which filter design parameters are determined mainly by experimental results have also been established [1]. Recent CAD technology for microwave circuits has shown remarkable progress, allowing for accurate and highly reliable analysis of electromag­ netic circuits possessing three-dimensional structures. While it is possible to develop an application-specific simulator for each individual filter structure, 150 Appendix. Analysis of Resonator Properties here we discuss a generalized method of obtaining design charts [2J required for filter synthesis by calculating the electrical performances of fundamen­ tal filter structures using commercially available general-purpose microwave simulators. The intentions of this approach are to determine filter parame­ ters by replacing conventional experiment-based design processes with virtual experiments based on simulated results. Circuits parameters required for the experimental method are; resonance frequency fo, resonator unloaded-Q (Qo), external-Q (Qe) related to in­ put/output couplings, and interstage coupling coefficient k which prescribes coupling between the resonators. The theoretical background and actual pro­ cedures of this method, where design parameters are determined by simu­ lating circuit responses using a single resonator and a resonator-pair, are described as follows. A.2 Filter Design by Experimental Method Based on the initial design specifications of the BPF, center frequency fo, number of stages n, relative bandwidth w, element values 9j, and inser­ tion loss Lo(dB) are assumed to be given. Resonator structure, resonance frequency (center frequency), unloaded-Q(Qo), and external-Q(Qe) can be obtained by examining the single resonator. These parameter can also be experimentally measured. As discussed in Sect. 3.2.1, insertion loss is determined by Qo, which must be estimated under the following conditions. 4.434 n Qo> -L- L9j. (A.l) w· 0 j=1 The unloaded-Q (Qo) is determined by the geometrical structure of the distributed-element circuit, and thus the relationship between resonator structure and Qo must be thoroughly investigated because it becomes an extremely important parameter determining the total size of the BPF. The external-Q (Qe) is an important parameter which provides the rela­ tionship between the filter and the external circuit (input and output circuit), thus providing impedance matching between them. They are expressed as [1], Qel = 9091 (A.2) w Qe2 = 9n9n+l (A.3) w Qel and Qe2 indicate the external-Q at the input and output port, respec­ tively. When the input and output impedance are equal, then Qel = Qe2. The interstage coupling coefficient k has the following relationship with the basic design parameters. A.3 Determination of Q and kUsing General Purpose Microwave Simulator 151 w k· ·+1 - (A 4) J,J - "';9j9j+1 . kj,j+1indicates the coupling factor between the j-th and (j+I)-th resonator. The k value can easily be determined by obtaining the frequency response of a resonator-pair. From the above discussions, we understand that filter design can be achieved by considering experimentally measurable values of 10, Qo, Qe, and k. Consequently, it is possible to synthesize the BPF by obtaining the rela­ tionship between these parameters and the physical structure of the filter. A.3 Determination of Q and k U sing General Purpose Microwave Simulator A.3.1 Determination of Q The quality factor of a resonator is expressed as three Q values of different definition: the unloaded-Q (Qo) which indicates the figure of merit of the resonator itself; the external-Q (Qe) which expresses the coupling condition between the external circuits and the resonator; and the loaded-Q (Qd which represents the total Q including resonator and external circuits. The following relationship is recognized between these values. 1 1 1 -=-+- (A.5) QL Qe Qo This relationship implies that by obtaining two of the three Q values, the remaining can be calculated from (A.5). Experimental methods for Q measurement have been studied by Ginzton [3], and these results have laid the foundation for the reflection method or one-port method. In this section an alternative method, namely the trans­ mission method or two-port method, is employed. This method is suitable for computer simulations where two-port symmetrical circuits are utilized to simultaneously obtain Qo and Qe values, while for experimental methods obtaining these values accurate results cannot be expected due to increased error factors. Figure A.I shows a generalized two-port symmetrical circuit for the sin­ gle resonator. Measurements obtained from actual experimental circuits often lack accuracy due to processing limits which make the symmetrical circuit difficult to fabricate. Such problems can be ignored when conducting virtual experiments based on circuit simulators. In Fig. A.I, the resonator itself is in­ dicated as Y(w), while the coupling circuits are expressed by the admittance inverter Jo, which indicates a generalized coupling parameter. For practical simulations, the resonators and coupling circuits must be described as actual circuit element values or physical parameters in a descriptive language un­ derstood by the applied simulator. As in an experimental method where Qo 152 Appendix. Analysis of Resonator Properties J 0 : Inverter Y(w): Resonator Fig. A.1. Circuit configuration for determining fo, Qo and Qe and Qe are obtained by measuring frequency response, for this method the transmission response of the circuit shown in Fig. A.l must be calculated. For simplicity, Y(w) is expressed as follows. Y(w) = Go + jB(w) = Go + jbo(w/wo - wo/w) where Go : conductance of the resonator, bo : slope parameter of the resonator, Wo : angular resonance frequency. This expression illustrates a lumped-element approximation of the resonator at frequencies near resonance. Furthermore, a Go value independent of fre­ quency is assumed. Considering analysis within a sufficiently narrow fre­ quency band, the admittance inverter Jo is also assumed to be independent of frequency and is thus treated as constant, although strictly it will possess frequency-dependent characteristics. Under these conditions, we first obtain the total F-matrix [Ftl of the circuit shown in Fig. A.l, and next derive trans­ ducer loss L(w), defined as the ratio of power available from the source and power absorbed by the load. [Rl = [ 0 j/Jo] [ 1 0] [ 0 jjJo] t jJo 0 Y(w) 1 jJo 0 -1 -GojJJ - jbo(w/wo - wo/w)/JJ [ 1 1 -1 [ At Bt l. Ct Dt Using the above matrix elements, L(w) = ZI 2 IAtZo + Bt + CtZ6 + Dt Zol 2 4 0 = 4~5{(2Zo+GojJ6)2+b5(w/wO-wO/w)2/J~}. (A.6) A.3 Determination of Q and kUsing General Purpose Microwave Simulator 153 Or-----,r---------...., Fig. A.2. Frequency response of a single resonator test circuit Considering Qo = bo/Go and Qe = bo/ZoJ'5, 1 Qe 2 0 Wo L(w) = - (2 + -)2 + Qe ( --- )2 (A.7) 4 Qo Wo W Thus, as illustrated in Fig. A.2, the frequency response of L(w) will feature a single peak characteristic. L(w) attains a maximum value Lo at wo, which can be obtained from (A.7) by substituting w = Wo as, Lo = 1 + Qe + ~ (Qe)2 (A.8) Qo 4 Qo WI and W2(WI < W2) are defined as the angular frequencies showing a 3dB downfall from Lo. Since WI and W2 exist near the resonance angular frequency Wo, Wo - WI = W2 - Wo is assumed, and thus by introducing nl as, L(WI) is expressed as, L(wd = 2Lo Qe 1 (Qe) 1 2 2 (A.9) =l+ Qo +'4 00 +'4 Qenl' Considering (A.8) and (A.9), 1 2 2 Lo = '4Qe' nl · Consequently, we obtain, 2 Qe= nl~ (A. 10) Qo = VEO . (A.l1) nI(VEO -1) 154 Appendix. Analysis of Resonator Properties Structural Parameter Structural Parameter Structural Parameter (a) (b) (c) Fig. A.3. Design chart examples required for filter synthesis by experimental method. (a) Resonance frequency. (b) Unloaded-Q. (c) External-Q L:[j$B$[}Jo J 0 ' J 12 : Inverter Y(aJ): Resonator Fig. A.4.
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