Chapter 2 Theory of Microwave Resonator Filters and Distributed Circuit
2.2 Distributed Circuits with Transmission Line Elements
2.2.3 Transmission Line Approximating Functions and Synthesis
balun is inherently a band-pass network. Thus, filter theorem can be adopted to synthesize the Marchand balun. The key point is to determine the polynomial of reflection coefficient for a cascade of unit elements and prototype LC distributed elements. Table 2.2 provides each of the distributed L, C and U.E. and its corresponding ABCD matrices. For showing how to obtain the reflection coefficient, a prototype circuit shown in Fig. 2.26 is taken as an example. The first is to utilize the ABCD parameter in Table 2.2 and obtain the overall cascaded ABCD matrices
Then, the reflection coefficient is given by the well-known formula
( ) ( ) ( ) ( ) ( )
Actually, in [132]-[136] distributed transmission line elements have been studied and some approximating functions are obtained to be suitable to the corresponding circuit networks. For the concerned prototype circuits shown in Fig. 2.26, the generalized magnitude squared high- and low-pass transfer functions with Chebyshev responses are described and summarized by Horton and Wenzel [135]. The characteristic functions associated with the high- and low-pass prototype circuits are given by
C
C
1 1
0 1
SC
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
1 0 1
⎡ SL ⎤
⎢ ⎥
⎣ ⎦
1 0
1 1
SL
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
1 0 1 SC
⎡ ⎤
⎢ ⎥
⎣ ⎦
0
2 0
1 1 1 1
Z S S Y S
⎡ ⎤
⎢ ⎥
− ⎣ ⎦
Table 2.2. ABCD matrices for distributed LC ladder and a unit element.
(a)
(b)
Fig. 2.26. (a) A possible high-pass prototype circuit. (b) A possible low-pass prototype circuit
and
respectively, where T xn
( )
and U xn( )
are the unnormalized Chebyshev polynomials of the first and second kind of order n:( )
cos(
cos 1)
T xn = n − x
U xn
( )
=sin(
ncos−1x)
(2.78) , SC = Ω , j C Ω is the cutoff frequency which occurs at half power (-3dB), and m is C the number of a mixed cascade ladder elements and n is the number of unit elements.Furthermore, it is recalled that the scattering parameters are related to the characteristic function K S( )2 and the ripple level ε of the circuit by The next step is to derive the input impedance function Zin
( )
S shown in Fig.2.26. The source resistance is assumed to be unity. The relationship between S11
( )
S and Zin( )
S is expressed asFinally, the circuit networks shown in Fig. 2.26 are synthesized using the method of standard element extraction which can be found in [135]-[137].
Chapter 3 Quarter-Wave Stepped-Impedance Resonator Filters with Quadruplet and Canonical Form Responses
In this chapter, compact microstrip quarter-wave stepped-impedance resonator (SIR) bandpass filters with quadruplet and canonical form responses are proposed.
The proposed quadruplet filter can be designed to have a pair of transmission zeros to achieve sharp selectivity. In addition, by applying an extra source-load coupling, two additional transmission zeros on both side of passband are created to further enhance the selectivity. Because the quarter-wave SIRs are adopted, the circuit size of the filters can be largely reduced and the upper stopband can be extended. Two generalized Chebyshev filters corresponding to quadruplet and canonical form coupling schemes are fabricated. Simulated and measured results are matched very well.
3.1 Introduction
Compact and high-performance microstrip bandpass filters are important building blocks in wireless and mobile communications due to their small size, ease of fabrication and light weight. In many literatures, half-wave or quarter-wave resonators are utilized to design microstrip bandpass filters. The conventional parallel-coupled filters using half-wave resonators have a large circuit size, and they don’t exhibit generalized Chebyshev responses for high selectivity [69], [97]. The hairpin and quarter-wave resonators with cross couplings are proposed to solve the problems [72], [75]-[77]. However, these filters still suffer from spurious responses due to the distributed nature. The filters with half-wave resonators depict spurious responses at twice of the center frequency, and the filters with quarter-wavelength resonators have spurious responses at three times of the center frequency.
Recently, many efforts focus on the half-wave and the quarter-wave SIRs to design cross-coupled filters [78]-[80]. In [78], the fourth order filter utilizing both λ/2 and λ/4 resonators shows a good rejection bandwidth and high selectivity. However, control of the physical parameters of the open stubs of the λ/2 SIR-like resonators (resonator 1 and 4) are not easy because they not only suppress the high-order resonances but also implement the external coupling to maintain the appropriate passband performance. In [79], although a good selectivity is obtained, both the coupling routes and the physical layout are too complicated so that the synthesis of the coupling matrix and final fine tuning of the physical layout are too time consuming. In [80], the λ/4 SIR filter shows good selectivity. However, to design the locations of the transmission zeros is very complicated and too much depends on experience.
In this chapter, we propose two microstrip λ/4 SIR filters both with generalized Chebyshev responses. The proposed filters can overcome the above problems. Fig.
3.1(a) and Fig. 3.1(b) show two fourth-order filters with cross coupling that one is without source-load coupling and the other is with source-load coupling. The filter in Fig. 3.1(a) has a pair of transmission zeros due to quadruplet coupling scheme whereas the filter in Fig. 3.1(b) has two pairs of transmission zeros due to canonical form coupling scheme. A coupling enhancement line between source and load in Fig.
3.1(b) is used to control the source-load coupling strength. Thus, with the coupling enhancement line, two extra transmission zeros can be controlled easily with a little influence on passband performance. In addition, the λ/4 stepped impedance resonators are used to obtain compact size and to push the high order spurious frequencies to as high as possible. Therefore, the proposed filters are suitable for bandpass filters with small size, wide stopband and sharp selectivity in a modern communication system.
(a)
(b)
Fig. 3.1. The circuit layouts of the proposed microstrip quarter-wave SIR filters. (a) The fourth-order quadruplet filter. (b) The fourth-order quadruplet filter with source-load coupling.