Optimized Design Method of Microstrip Parallel-Coupled Bandpass Filters with Compensation for Center Frequency Deviation

Transcription

1 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August Optimized Design Method of Microstrip Parallel-Coupled Bandpass Filters with Compensation for Center Frequency Deviation H. L. Gan, D. X. Yang, D. W. Lou Zhejiang University, China Abstract An optimized design method is presented to compensate for the open-end effect in parallel-coupled microstrip bandpass filters. The analysis of the relationship between center frequency deviation and microstrip open-end effect is given. Based on the theoretical analysis, a design example of 10GHz bandpass filter is presented, which has proved the validity of this method. Introduction With the rapid development of modern wireless communication technology, microstrip filters with high quality and low cost have been widely used in microwave circuits. Consisting of a cascade of coupled stage, the configuration of parallel-coupled microstrip bandpass filters is an attractive form. The traditional design method [1] begins with lowpass prototype, includes frequency transformation and calculation for electrical parameters of each coupled stage, and ends with the solution of physical microstrip dimensions. There are, however, two open ends in each coupled stage. The edge field effects of these open ends are modeled as parasitic capacitances, which result in resonant frequency deviation. In practice, it is common to introduce an equivalent end extra line with length l, which is cut off at each open end so as to bring the center frequency back to the desired value [1-3]. The value of l is often estimated by experience or approximately calculated from empirical formulations [1, 2], which could result in complex calculation and unavoidable error. In addition, the values of l under even and odd modes are different, and the formulations to solve them respectively are quite lengthy [4]. Consequently, it is common practice to depend on EM simulators to determine the exact dimensions of the microstrip structures. In this paper, it is shown in section II that the real center frequency (f i ) of the filter can be significantly lower than its expected center frequency (f i ). Multiple-port network analysis is carried out in section III to obtain the relationship between equivalent capacitances at open ends and the resonant frequency of coupled stages. The characteristics of f i f i curve are also discussed in this section. In section IV, a new multi-step design method that no longer requires the value of l is described, and a design example of a 10GHz bandpass filter is given to show its validity. Resonant Frequency of Coupled Stages The basic structure of coupled-line stages in parallel-coupled microstrip bandpass filters is shown in Fig. 1. It has an electrical length of θ, and even and odd mode characteristic impedances Z 0e and Z 0o, respectively. When several stages are cascaded to form a multi-stage bandpass filter, the two ports of each stage should be matched to adjacent ports of its previous and next stages. Figure 1: Basic structure of a coupled-line stage. There are, however, two open ends in the coupled-line stage shown in Fig. 1. The edge field effects of openend microstrip lines are commonly modeled as equivalent capacitances C f, which can be further substituted by extra microstrip lines with length of l, as shown in Fig. 2. Empirical formulations [2] point out that the value

2 2 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August Figure 2: Equivalent circuit for an open-end microstrip line, (a) open-end microstrip line, (b) equivalent capacitance, and (c) equivalent extra length of microstrip line. of C f depends only on the width of microstrip line and the height of substrate, and does not vary as frequency or electrical length changes. Assume that ideal coupled-line stage (no open ends) with length of l 0 resonates at f 0, while after open-end equivalent capacitances are taken into consideration, the length of the stage increases to l 0 + l at f 0, and it would resonate at f 0 rather than f 0. When resonating, θ = π/2, and f 0 can be solved as f 0 = [1 2 arctan(c f Z 0 f 0) ]f 0. (1) π It is shown in (1) that, f 0 < f 0, and if f 0 is plotted with f 0 as the variable, the slope of the curve would decrease as f 0 increases. This fact indicates that the f 0 f 0 curve performs as a convex function. The analysis above is based on the assumption that open-ended coupled-line stages with a length of l 0 are equivalent to ideal (without open ends) coupled-line stages with a length of l 0 + l. However, the two extra lines with length of l are only equivalent lines, and they are positioned at opposite ports, which means that they are not really coupled to each other, and should not be included in the total length of coupled-line stage. In that case, multiple-port network analysis is required to accurately examine the deviation between f 0 and f 0 caused by open-end equivalent capacitances. Equivalent Network Parameters Analysis In order to derive its network parameters, the coupled-line stage is considered as a four-port network, as shown in Fig. 3, where Z c is the impedance of C f, and it is connected at both ports 2 and 4. According to [1], the impedance matrix of the coupled-line stage can be written as [ Z1 Z 2 Z 3 Z 4 ] = j 2 [ (Z0e + Z 0o ) cot θ (Z 0e Z 0o ) cot θ (Z 0e Z 0o ) csc θ (Z 0e + Z 0o ) csc θ ]. (2) In addition, at ports 2 and 4, V 0 = Z c I 0, V 4 = Z c I 4. The current and voltage equations at ports 1 and 3 can be derived as [ ] [ ] [ ] V1 Z = 11 Z 13 I1 V 3 Z 31 Z 33 (3) I 3 where, Z 11 = Z 33 = Z 1 + (Z2 2 + Z 2 4)(Z 1 + Z C ) 2Z 2 Z 3 Z 4 Z 2 3 (Z C Z 1 ) 2 (4) Z 13 = Z 31 = Z 3 + 2Z 2Z 4 (Z 1 Z C ) Z 3 (Z Z2 4 ) Z 2 3 (Z C Z 1 ) 2 (5) These are the 2-port impedance matrix elements of the coupled-line stage with open-end effect. At several GHz, Z C Z i (i = 1, 2, 3, 4), the input impedance of the coupled-line stages with open-end effect can be again derived as Z in = [Z 1 + Z 3 + Y C (Z 2 + Z 4 ) 2 ] [Z 1 Z 3 + Y C (Z 2 Z 4 ) 2 ] (6) where Y C = 1/Z C = jωc f. When θ = π/2, (2) and (6) give 1 Z in (π/2) = 4 (Z 0 e Z 0o ) ω2 Cf 2(Z 0 e + Z 0o ) 4 (7)

3 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August It is shown in (7) that if C f does not exist, the input impedance is Z in = (Z 0e Z 0o )/2, which is exactly the port transmission lines characteristic impedance Z 0, and the coupled-line stage resonates exactly at θ = π/2. However, the second term in the square root sign exhibits the effect of C f. The electrical length at the real resonant frequency f 0 could be assumed to be θ, and Z in (θ ) = Z 0. Since Z in (θ ) Z in (π/2) increases because Z in (π/2) decreases as frequency goes up, it can be easily shown that θ π/2 also increases as frequency increases. On the other hand, f 0 = θ ν p 2πl 0 = 2θ π f 0, (8) where ν 0 is the phase velocity. Equation (8) shows that the slope of f 0 f 0 curve decreases when frequency increases, which proves the convex function property of this relationship. And this conclusion is in well agreement with that drawn in section II. Figure 3: Network analysis model of a coupled-line stage. Figure 4: The simulated f 0 f 0 curve, compared with a line of unity slope. New Design Method and Example Microstrip parallel-coupled bandpass filters are designed with traditional procedures without correction of physical dimensions, to obtain relative bandwidth of 0.03, and center frequencies at every 1GHz between 5GHz and 10GHz. The simulated results f 0 are plotted in Fig. 4, along with the expected center frequency f 0. The simulator is EMSight [6]. The f 0 f 0 curve shown in Fig. 4 proves that f 0 is always lower than f 0, and numerical details also show its convex function property, which agrees with the theoretical analysis. If f 0 is set to equal the desired center frequency, and f 0 is solved from (1), an optimized design frequency f 0 = f 1 can be derived. Directly designing the filter with center frequency at f 1 can ensure the real center frequency locates exactly at that expected, and no effort is needed for corrections of physical dimensions. However, this approach still requires the value of C f in the solution for f 1. With the help of modern EM simulators, this procedure can be further simplified. A multi-step design procedure is introduced next. Fig. 5 shows the method. The essence of this method is to raise the target center frequency in traditional design procedures, so that the real center frequency will be a bit lower than that, and locate exactly at where desired. In (5), the vertical axis denotes real center frequency f 0, the horizontal axis denotes designed center frequency f 0, and bold line represents the f 0 f 0 curve. Take the design procedures of a bandpass filter with center frequency at 10GHz for example. The goal is to obtain the design value f i (C), which will be used as the design goal in traditional procedures and the consequent filter has its center frequency at f i (C)=10GHz. In Fig. 5, the intersection point of f 0 f 0 curve and f =10GHz horizontal line indicates the point of C. The detailed steps are as follows: 1) Design the filter at 10GHz with traditional procedures. 2) Use EM simulator to determine its real (simulated) center frequency f 0(A), which is shown as point A on the bold line in Fig. 5. 3) Connect the origin point and point A, and extrapolate this line to intersect with horizontal line f = 10GHz. The abscissa value of this intersection point is f 1 (B), and this is the second design frequency. Design the filter at this frequency in traditional procedures again, and EM simulator is used again to obtain the

4 4 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August second real (simulated) center frequency f 1(B). f 1 (B) and f 1(B) determine point B. As f 0 f 0 curve shows convex function property, point B is bound to be between point A and C. 4) If f 1(B) is still away from 10GHz, then points A and B can be connected and extrapolated to intersect with f = 10GHz horizontal line, with the abscissa value of intersection point as a new frequency for designing. Then EM simulator decides real (simulated) center frequency again. Repeating this procedure, new points D, E, and so on can be determined, and they are bound to locate between their previous one and C, and getting closer and closer to the goal point C, due to the convex function property. The design equations are summarized in (9) and (10), where f denotes center frequencies to design at with traditional procedures, while the apostrophe indicates its corresponding simulated value. The subscript 0 is for point A, 1 for B, and i for any subsequent point. f 1 = f 2 0 /f 0 (9) f i+1 = f i + (f 0 f i )(f i f i 1 ) (f i f i 1 ) (10) This method was used to design a parallel-coupled bandpass filter with relative bandpass of 0.03, center frequency at 10GHz, and ripple of 0.01dB. The lowpass prototype has N=5, and substrate parameters are ε r =9.8, H=1.2mm. Traditional design procedures are taken without any correction to microstrip physical dimensions, and EMSight [6] gives the simulated result of center frequency at 8.85GHz, as shown in Fig. 6. According to (9), the next frequency to design is 11.30GHz. Traditional procedures are taken again, resulting in a simulated center frequency at 10.0GHz, which is fairly satisfying, as shown in Fig. 6. Figure 5: Multi-step design method based on f 0 f 0 curve. Figure 6: Simulated results using multi-step design method. Conclusions This paper begins with the resonant condition of microstrip coupled-line stages. The relationship between f 0 and f 0, equation (1), is then established. Network analysis is then carried out to prove the convex function property of f 0 f 0 curve. Based on these theoretical discussions, a multi-step design method is presented, which could circumvent the difficulties encountered in the design of microstrip parallel-coupled bandpass filters when determining open-end parameters C f and l, and compensate for the center frequency deviation with the help of EM simulators. Its validity is proved by a design example of a filter at 10GHz with relative bandwidth of This new method presents a new practical way to design microstrip parallel-coupled bandpass filters with higher efficiency and accuracy. Acknowledgements This work was supported by NSFC under Grant No D. X. Yang s address is yangdx@zju.edu.cn.

5 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August REFERENCES 1. Tsinghua University, Microstrip Circuits, Beijing: Post & Telecom Press, Edwards, T. C., Foundations for Microstrip Circuit Design, Chichester: J. Wiley & Sons, 2nd Ed., Wang, A. and Q. Lin, A Design Procedure for Band-pass Filter Using Parallel Coupled Microstrip Lines with Equal Width, Journal of Circuits and Systems, Vol. 5, 43-47, March Gupta, K. C., R. Garg, I. Bahl and P. Bhartia, Microstrip Lines and Slotlines, Boston: Artech House, 2nd Ed., Ludwig, R. and P. Bretchko, RF Circuit Design: Theory and Applications, Beijing: Pub. House of Electronics Industry, Applied Wave Research, Inc., EMSight, 2001.