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of ladder passive RF filters VAHÉ NERGUIZIAN^{1} AND CHAHÉ NERGUIZIAN^{ 2} ^{1 }Electrical Engineering École de Technologie Supérieure 1100 Notre Dame West, Montreal, Quebec, H3C1K3 CANADA ^{2 }Electrical Engineering École Polytechnique de Montréal 2500 Chemin de Polytechnique, Montreal, Quebec, H3C3A7 CANADA http://www.etsmtl.ca Abstract:  This paper presents an efficient technique to determine the redundancy of transmission zeros during the design of ladder passive filters. When the transfer function of a passive radio frequency (RF) filter is known, the twoport ladder passive filter is conceived with alternate series and parallel branches composed by dynamic elements which stop transmission at certain frequencies. In the obtained cascaded topology, several of these elements may be combined with each other modifying the expected characteristics of the filter. The identification of these element redundancies permits the determination of the exact order of the considered filter. The pedagogical goal of this paper is to show first, how to generate sections producing transmission zeros, then to identify the dynamic elements’ redundancies and finally to come up with the final cascaded filter’s order. KeyWords:  Poles and transmission zeros, Ladder passive RF filters, System stability, Thévenin theorem.
In electrical engineering field, there is a difference between analysis and synthesis of a circuit. When the components of a network and its input excitation are known, the analysis process gives usually an output solution that is unique. By contrast, when the excitation and the desired response of a circuit are given, the synthesis process consists in the determination of the network’s components and gives in general, component solutions that are not unique [1]. When the transfer function of a lumpedelement passive RF filter is given, its physical realization may be done by using reactive or dynamic elements placed in series and parallel branches blocking transmission at certain frequencies. During the design process, the two important circuit parameters are the poles and the transmission zeros. The poles dictate the behavior of the transient response and define the stability of the system. As for the transmission zeros, they correspond to the frequencies at which a zero steady state output results for a finite input at these frequencies. One of the circuit synthesis objectives is the determination of a cascadedcomponents topology which corresponds to the desired network structure. Unfortunately, several dynamic elements of the network may be combined with each other (redundancies of dynamic elements) modifying its characteristics. Consequently, a proper identification of these combinations is essential to define the exact order of the filter’s frequency response. This paper provides a technique to identify the dynamic element’s redundancy for an efficient design of ladder passive RF filters. In section 2, we give the various transmission zerosproducing sections in a ladder network. In section 3, we present our proposed methodology using the Thévenin theorem to identify the redundancy of dynamic elements in a ladder network. Finally, we close this paper with a conclusion in section 4.
2.1 Definitions The poles and the finite transmission zeros correspond to the roots of the transfer function’s denominator and numerator, respectively. For a system having n poles and m finite transmission zeros, the number of transmission zeros at infinite frequency is equal to (nm), and the order of the twoport network (n) is determined by the sum of finite and infinite transmission zeros [2], [3] and [4]. As an example, a system having a transfer function equal to will exhibit three poles at s = 1 and s = 1±1j, two finite transmission zeros at s = 0 and s = 4, and one transmission zero at infinite frequency (zero steady state response when the input frequency is very high). Transfer functions and associated poles and transmission zeros for different type of filters are given next. 2.1.1 Lowpass filter A transfer function that is equal to has one zero at s = ∞ and one pole at s =  ω_{c}. It corresponds to a first order (n = 1) lowpass filter since the steady state response for a high frequency input is equal to zero (transmission zero at s = ∞). The cutoff angular frequency of the filter is equal to ω_{c}. Lowpass filters have transmission zeros located at infinite frequency, and sometimes also at finite frequencies (due to resonances caused by component parasitic), but not at dc. 2.1.2 Highpass filter A transfer function that is equal to has one zero at s = 0 and one pole at s =  ω_{c}. It corresponds to a first order (n = 1) highpass filter since the steady state response for a low frequency input is equal to zero (transmission zero at s = 0). The cutoff angular frequency of the filter is equal to ω_{c}. Highpass filters do not have transmission zeros at infinite frequency. 2.1.3 Bandpass filter A transfer function that is equal to has two poles at (considering a damping coefficient ζless than) and two zeros at s = 0 and s = ∞. It corresponds to a bandpass filter since the steady state response for low and high frequency inputs are equal to zero (transmission zeros at s = 0 and at s = ∞). While the center angular frequency of the filter is equal to, b represents its bandwidth in radian/sec. Bandpass filters have transmission zeros both at dc and infinity. 2.1.4 Stopband filter A transfer function that is equal to has two poles at and two zeros at . It corresponds to a stopband filter since the steady state response for an input with an angular frequency of radian/sec is equal to zero (transmission zeros at ). It has to be noted that a system having as transfer function (a > 0), will exhibit a zero steady state response when the input corresponds to e^{at} (transmission zero at s = a). 2.2 General topology The structure of a passive ladder circuit is composed of serial and shunt (parallel) branch elements placed in cascade producing the desired transmission zeros (Figure 1). Fig. 1 Ladder passive circuit with elements in serial and parallel (shunt) branches On the other hand, the steady state output signal V_{2} of Figures 2a or 2b is equal to zero when Zserial = ∞ or Zshunt = 0 at an input frequency corresponding to the system’s transmission zero (series and parallel ideal reactive elements stopping transmission at certain frequencies by presenting either a series susceptance or shunt reactance of zero value). Moreover, during the design of ladder passive filters, it is essential to produce zeros at f = 0 (s = 0), at f = ∞ (s = ∞) and at a finite frequency f = f_{o} (s = ±j ω_{o}). Fig. 2a Portion of a circuit with an element in the serial branch Fig. 2b Portion of a circuit with an element in the parallel (shunt) branch The sections producing such transmission zeros are summarized in Table 1. Table 1. Summary of sections producing zeros of transmission at different frequencies 2.3 Remarks It is important to note that the addition of a resistor in series with an element Zserial in the serial branch or in parallel with an element Zshunt in the parallel branch will not affect the zeros’ locations in the complex plane. A system may have a positive transmission zero (nonminimalphase system). However, the generation of a positive transmission zero with a ladder passive network is impossible.
The sum of nonredundant transmission zeros defines the order of the network and their distribution gives an estimate of the circuit’s response through a broad range of frequencies. Moreover, when the desired filter response is not standard (i.e. not a Butterworth or Chebychev filter), the knowledge of the transmission zeros helps to have a quick estimate of the customized stopband response. Hence, a good understanding of the network order and transmission zeros is essential to an efficient synthesis of ladder passive RF filters and matching networks. The methods used for the identification of reactive elements combination (transmission zeros redundancy) consist of Thévenin impedance (Zth) application to elements producing transmission zeros at s = 0 and s = ∞ for the considered ladder passive filter. 3.1 Method applied for dynamic elements placed in serial branches For a dynamic element placed in a serial branch, one has to determine the Thévenin impedance (Zth) applied to the right side of the element and at frequency corresponding to the considered transmission zero (s = 0 or s = ∞). If the calculated Zth is different than infinity (Zth ≠ ∞), then there will be no combination of the considered dynamic element with the other ones (no redundant transmission zero). 3.2 Method applied for dynamic elements placed in parallel branches For a dynamic element placed in a parallel branch, one has to determine the Thévenin impedance (Zth) applied to the left side of the element and at a frequency corresponding to the considered zero of transmission (s = 0 or s = ∞). If the calculated Zth is different than zero (Zth ≠ 0), then there will be no combination of the considered dynamic element with the other ones (no redundant transmission zero). 3.3 Examples To show the efficiency of the proposed approach, four examples are given with the detailed computations. 3.3.1 Example 1 The RF matching network of Figure 3 is composed of five reactive elements (3 capacitors and 2 inductors). At first glance, one may think of a system with five poles. The application of the proposed methods shows that the system has only four poles. Fig. 3 Example 1 with 5 dynamic elements The demonstration is given next.
If C1 is not combined with other capacitors, it will produce a transmission zero at s = 0. To check the redundancy of the transmission zero, the computation of Zth at s = 0 is performed to the right side of C_{1}. The calculated value of the impedance is equal to infinity. Hence, C_{1} will be combined with C_{3}.
Looking at the left side of C_{2} and calculating Zth at s = ∞, the obtained value of the impedance is equal to infinity (different than zero). Hence, C_{2} will be a nonredundant transmission zero at s = ∞.
Looking at the right side of C_{3}, we compute Zth at s = 0. The calculated value of the impedance is equal to zero (different than infinity). Hence, C_{3} will produce a nonredundant transmission zero at s = 0.
Looking at the right side of L_{1}, we compute Zth at s = ∞. The calculated value of the impedance is equal to zero (different than infinity). Hence, L_{1} will produce a nonredundant transmission zero at s = ∞.
Looking at the left side of L_{2}, we compute Zth at s = 0. The calculated value of the impedance is equal to infinity (different than zero). Hence, L_{2} will produce a nonredundant transmission zero at s = 0. The transfer function of the system may be written then as follows: (K being a constant). The two zeros at s = 0 are generated by C_{3} and L_{2} providing a slope of +40dB/decade at the low frequency side, and the two zeros at s = ∞ are generated by C_{2} and L_{1} providing a gain rolloff of 40dB/decade moving toward high frequencies. The overall network corresponds to a symmetric bandpass filter. 3.3.2 Example 2 The elliptic ladder filter [5] of Figure 4 is composed of seven reactive elements (5 capacitors and 2 inductors). At first glance, one may think of a system with seven poles. The application of the proposed methods shows that the system has only five poles. Fig. 4 Example 2 with 7 dynamic elements The demonstration is given next.
Looking at the right side of C_{1} (or C_{3}) we compute Zth at s = 0. The calculated value of the impedance is equal to infinity for both cases. Hence, C_{1}, C_{3} and C_{5 }will be combined with each other to give only one transmission zero at s = 0 (produced by C_{5} because it sees at its left side a finite impedance equal to R) since the two shunt LC branches act as open circuits for s = 0. The two capacitors C_{1} and C_{3} correspond to redundant transmission zeros.
The LC combinations of the two parallel branches block the transmission at finite frequencies f_{01} and f_{02} (resonant frequencies) and cause notches in the frequency response at these frequencies. The transfer function of the system may be written then as follows: (K being a constant). The overall network corresponds to a 5th order highpass elliptic filter. 3.3.3 Example 3 Figure 5 is composed of seven dynamic elements (4 capacitors and 3 inductors). At first glance, one may think of a system with seven poles. The application of the proposed methods shows that the system has only six poles. Fig. 5 Example 3 with 7 dynamic elements The demonstration is given below.
Looking at the right side of L_{2}, we compute Zth at s = ∞. The calculated value of the impedance is equal to infinity. Hence, L_{2} will be combined with L_{3}.
Looking at the left side of L_{3}, we compute Zth at s = 0. The calculated value of the impedance is equal to infinity (different than zero). Hence, L_{3} will produce a nonredundant transmission zero at s = 0.
Looking at the right side of C_{2}, we compute Zth at s = 0. The calculated value of the impedance is equal to R_{3} (different than infinity). Hence, C_{2} will produce a nonredundant transmission zero at s = 0.
Looking at the left side of C_{4}, we compute Zth at s = ∞. The calculated value of the impedance is equal to [(R_{1}.R_{2}/R_{1}+R_{2})+R_{3}] (different than zero). Hence, C_{4} will produce a nonredundant transmission zero at s = ∞.
The transfer function of the system may be written then as follows: (K being a constant). The two transmission zeros at s = 0 are generated by C_{2} and L_{3}, and the transmission zero at s = ∞ is generated by C_{4}. 3.3.3 Example 4 The ladder RF circuit given in Figure 6a represents a capacitive coupled LC resonator filter with an equalripple in the passband region [4]. When the resonators of the circuit have unreasonable component values, network transformations (Norton, Kuroda [6], etc.) are used to obtain a final circuit (Figure 6b) with practical element values for the considered frequency range. Although the number of components of the final circuit is increased, the number of the transmission zeros remains the same. This affirmation may be shown using the proposed methodology given in this paper. Fig. 6a Example 4 with 7 dynamic elements in the original circuit Fig. 6b Example 4 with 11 dynamic elements in the final circuit
The presented paper has shown an approach to identify the combination of dynamic elements in ladder passive networks using the Thévenin theorem. This identification is important for an efficient design of ladder passive RF filters. It shows that reactive elements (capacitors or inductors) placed in serial or parallel branches of the filter’s structure may be combined with each other resulting the loss of some transmission zeros at s = 0 and at s=∞. References: [1] W.K. Chen, ‘Passive and Active Filters, Theory and Implementations’, John Wiley & Sons, 1986. [2] Ernst A. Guillemin, ‘Synthesis of passive networks, John Wiley & Sons, 1957. [3] Louis Weinberg, ‘Network Analysis and synthesis, McGrawHill, 1962. [4] L. Besser rand R. Gilmore, ‘Practical RF Circuit Design for Modern Wireless Systems: Passive Circuits and Systems, Volume 1’, Artech House, 2003. [5] http://www.ecdl.hut.fi/education/198/pdf/cc.pdf (Visited date: November 2004). [6] D. M. Pozar, ‘Microwave Engineering’, John Wiley & Sons, 3rd edition, 2005. 