Simulation Analysis of the Filter with Frequency Dependent Coupling Coefficients

Transcription

1 217 Asia-Pacific Engineering and Technology Conference (APETC 217) ISBN: Simulation Analysis of the Filter with Frequency Dependent Coupling Coefficients Gang Li ABSTRACT *This paper illustrates a general approach to the simulation of the filter with frequency dependent coupling coefficients. Based on the frequency transformation formulas from lowpass to bandpass domain, circuit parameters can be determined and schematic model can be built to analyse the effects of frequency-dependent coupling coefficients on filter. A filter example is given finally to validate the proposed approach and the simulation results are in good agreement with the theory which verifies the correctness of the theory. INTRODUCTION The microwave filter synthesis and design methods of general coupled resonator filters are generally based on narrow-band circuit models. The coupling coefficients between the resonators are assumed frequency independent under the narrowband approximation condition. The resonant frequencies cluster around the center of the pass-band. The constant frequency shifts in the diagonal elements of the coupling matrix represent deviations of the resonant frequencies from the center of the passband [1]. However, it is known that the coupling coefficients are not constant in reality and the extension of the coupling matrix concept to wide bandwidths seems to have been hampered by the fact that actual coupling coefficients in fact depend on the frequency, instead of being constant [2]. For the commensurate TEM transmission line, the frequency dependence is fully studied over a wide frequency band [3]. But, The technique is mainly limited to commensurate lines or similar structures where simple propagation models are used such as coupled uniform sections of waveguide or transmission lines [4]. Recently, the use of frequency-dependent couplings has been exploited for introducing finite transmission zeros in the frequency response[5]. Many efforts have been made on the exact synthesis of multi-coupled resonator filter with frequency-dependent couplings [6], [7]. In these works, the filters are initially synthesized while assuming ideal inverters and optimization is then used for taking Gang Li * School of Physics and Electronic Engineering, Hubei University of Arts and Science, 44153, Xiang yang, China Corresponding author: anson_ligang@163.com 157

2 into account the frequency dependent characteristic of the couplings. Coupling items in microwave structures behave as either capacitances (Electric field coupling) or inductances (Magnetic field coupling) or a combination of both. It has been shown in [8] that a multi-coupled resonator filter whose coupling coefficients are linear functions of the normalized lowpass frequency can be transformed into a filter circuit with frequency-independent coupling coefficients. However, the transformed filter networks have different topologies from the original ones. It is also difficult to make one-to-one correspondence between the equivalent circuits and physical structures. The goal of the present work is to build a circuit model simulation approach to study the effects of frequency-dependent couplings on filter response and the topology relationship between coupling matrices under frequency dependent and independent coupling conditions. CIRCUIT MODEL Fig. 1 shows the general circuit model of a cross coupled resonator filter with a wide ba-nd band-pass, whose coupling items are combination of capacitors and inductors. The coup-ling items between resonators can be either purely capacitors or inductors for electric or magnetic coupling, respectively, or both for mixed coupling. Figure 1. Circuit model of a cross coupled resonator filter with a wide passband. The filter network can be described by a standard ( N2) ( N 2) square matrix. The coupling coefficients between the ports and the different resonators are assumed frequency-independent. If the source is excited by a unit current source, the node equations take the classical form as follows: j MS1 MS 2 MSN MSL Vs 1 MS1 M11 a12 M12 a1nm1 N M L1 V 1 MS 2 a12m12 M22 a23m23 M L2 j a23m23 Vi MSN M LN MSL MLN MNL j V L This equation can be simply written in the following form: ' [ jg M B ][ V] [ A][ V] j[ e] (1) Here, M is an ( N2) ( N 2) square matrix which contains the items that do not depend on frequency. In other words, M is given by 1571

3 M M M M M M M M M M M M M M M 23 MSN M MSL MLN MNL S1 S 2 SN SL S N L1 S L2 The matrix B ' is an ( N2) ( N 2) square matrix defined by: ' B B Here, B is a square symmetric real matrix of size N N such that 1 a a a a 1 a a a13 a23 1 a34 B a34 a 1 a a1n a2n an 1, N 1 LN N N 1, N1 N1, N t The excitation vector is e [1,,...,]. Finally, the matrix G is of size ( N2) ( N 2) with all its entries null except for the first and last diagonal elements which are equal to unity, G11 GN 2, N 2 1. The scattering parameters of the circuit are given by 1 S11 12jA11 (5) 1 S ja As long as the elements of the matrix A 1 filter response remains unchanged N 2,1 (2) (3) (4) that appear in (5) are not changed, the De-normalization of the Parameters In order to establish the actual circuit model, the key is to build the frequency depen-dent J inverter and the filter parameters with frequency shifts. frequency-dependent J inverter model is: Figure 2. Circuit model of J inverter. Fig.2 is a circuit with frequency independent reactance. The ABCD matrix 1572

4 1 1 1 jj 1 ABCD jj 1 1 jj 1 j J jj It is clear to know that the circuit functions as J inverter. This J inverter is considered to have the same characteristic admittance value at any frequency, that is, it has the characteristics of non-frequency variation, so the characteristic admittance is kept constant in the process of low-pass and band-pass transformation. According to the definition of coupling coefficient, if we set the susceptance slope value b i and b is 1, characteristic admittance value formula is as follows: j i j (6) k J bb J M FBW (6) where k is de-normalized coupling coefficient, M is normalized coupling coefficient, FBW is relative bandwidth. Taking into account the frequency dependent situation and using low-pass and band-pass transformation relationship, characteristic admittance value with frequency dependent couplings between resonance is deduced as follows. J a ( f f f f) FBW M (7) Characteristic admittance value between source and resonances or load and resonances is as follows: J M FBW J M FBW (8) The filter model with lumped elements sx sx Lx Lx The filter can be modelled by lumped inductor L and capacitor C which are paralleled connected. Set susceptance slop of the filter is 1, the values of inductor L and capacitor C are all 12 f, f is resonant frequency of the filter. By using denormalized relationship, the filters resonant frequency is as follows: EXAMPLE 2 (9) fk f 1 Mii FBW 2 Mii FBW 2 A fourth-degree Chebyshev filter with 2-dB return loss is demonstrated as an example. The center frequency f is 1.5 GHz, and the BW is 3MHz. Firstly, the N+2 normalized coupling matrix M is obtained as follows M The effect caused by the different matrices B is that the strength and topology of the frequency dependent couplings is different. But, the filter response is unchanged. To explain the characteristic, we could define the matrix B by B 1. Apply the above equations reversely, We get frequency dependent coupling matrix M

5 B M Build circuit model according to above formulas. By simulation, We can get the filter responses in Fig.3 which are corresponding to the coupling matrix M and M 1. It is straightforward to know that the responses of the coupling matrices are all identical DB( S(2,1) ) corresponding to the M DB( S(1,1) ) corresponding to the M DB( S(1,1) ) corresponding to the M1 DB( S(2,1) ) corresponding to the M Frequency (GHz) Figure 3. Filter responses of the matrices. CONCLUSION In this paper, a general approach to the simulation of the filter with frequency dependent coupling coefficients was proposed. Based on the frequency transformation formulas from lowpass to bandpass domain, circuit parameters can be determined and schematic model can be built to analyse the effects of frequencydependent coupling coefficients on filter. It was shown that filters whose interresonator coupling coefficients are linear functions of the normalized lowpass frequency can be transformed into networks with frequency independent coupling coefficients. This work is supported by the Science and Technology Research Project of Department of Education of Hubei Province under project No.Q REFERENCES [1] R.J. Cameron, IEEE Trans. Microwave Theory Tech., 51, (23). [2] S. Amari, IEEE Trans. Microwave Theory Tech., 58,. ( 21). [3] R.J. Cameron, C. Kudsia, and R. Mansour,Wiley, (27). [4] W. Meng, IEEE Trans. Microwave Theory Tech., 59, (213). [5] M. Politi and A. Fossati, Proc. Eur. Microw. Conf. (EuMC), (21). [6] L. Szydlowski, A. Lamecki, and M. Mrozowski, IEEE Microw. Wireless Compon. Lett., 22, (212). [7] S. Tamiazzo and G. Macchiarella, IEEE MTT-S Int. Microw. Symp. Dig., (216). [8] S. Amari, M. Bekheit, and F. Seyfert, IEEE MTT-S Int. Microwave Symp.Dig., (28). 1574