Direct Synthesis Technique (DST) for Complex General Chebyshev Filters

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS Int. J. Circ. Theor. Appl. (2017) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cta.2368

Direct synthesis technique (DST) for complex general Chebyshev filters

Fei Xiao*,†

School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

SUMMARY Recently, we discussed the concept of direct synthesis technique (DST), in which real‐coefficient filtering polynomials containing all information of the filters to be synthesized are derived directly for realization, and they could find applications in the design of lumped‐element LC filters, active RC filters, and infinite impulse response digital filters. In this paper, another DST for complex general Chebyshev bandpass filters is discussed, which is based on a complex mapping relation and featured by complex‐coefficient filtering polynomials. It is called as complex DST in this paper. Compared with real‐coefficient filtering polynomials whose polarities are determined by the number of their zeros at zero frequency, the polarities of complex‐ coefficient filtering polynomials can be easily changed by multiplying imaginary unit j. Such advantage might make their realization more flexible. The analysis shows that conventional coupling matrix could be considered as narrow‐band approximation of network matrix derived by complex DST in the normalized frequency domain. In order to demonstrate the validity of complex DST in this paper, it is applied in the design of classic parallel‐coupled microstrip bandpass filters. Compared with conventional synthesis techniques, complex DST could find out better dimensions and provide more choices for realization and synthesize both even‐order and odd‐order parallel‐coupled microstrip bandpass filters. Copyright © 2017 John Wiley & Sons, Ltd.

Received 7 December 2016; Revised 25 April 2017; Accepted 27 April 2017

KEY WORDS: complex mapping relation; complex direct synthesis technique; direct synthesis technique; complex general Chebyshev ﬁlter; network matrix

1. INTRODUCTION

Filter synthesis has gone through at least 100‐year development since Karl Willy Wagner and Campbell independently proposed their filters in 1915 [1, 2]. Up to now, there arise various filter types such as Butterworth filters, Chebyshev filters, and Bessel filters [3–5]. Conventional filter synthesis techniques usually start from the derivation of filtering polynomials of lowpass prototypes according to the specifications of the filters to be synthesized. Lowpass prototypes are defined as the ones whose element values are scaled to make source/load resistance/conductance and the cutoff frequency equal to be unity. They are usually realized in ladder network forms. After applying lowpass to bandpass or other frequency transformations on ladder networks of lowpass prototypes, realization networks can be presented for the filters to be synthesized. Although conventional techniques are applied successfully in practice, there are still some problems with them. For example, they are only effective for the cases that frequency response of lowpass prototypes is symmetric about zero frequency. Otherwise, it is unlikely for them to be realized in

*Correspondence to: Fei Xiao, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, China. †E‐mail: [email protected]; [email protected]

Copyright © 2017 John Wiley & Sons, Ltd. F. XIAO lumped LC networks [4]. Recently, we discussed the concept of direct synthesis technique (DST); there, we derive filtering polynomials containing all information of a filter in its own frequency domain for realization [6–11]. We therefore will call conventional synthesis techniques as indirect synthesis technique (IST) for comparison. However, they can also be extended to DST, as performed in [7]. DST for general Chebyhsev filters is characterized by its capability to place transmission zeros anywhere in order to control the performance of filters. Ordinary Chebyshev filters are actually a special case with all transmissions at zero frequency or infinite frequency. Meanwhile, Elliptic filters are also the special cases of general Chebyhsev filters with transmission zero at specific frequencies. At high frequency such as RF/microwave/optical frequency ranges, filters are realized through some transmission‐line structures with distributed‐element effect, such as waveguide, microstrip, stripline, suspended stripline, slotline, coplanar waveguide, and finline. Undoubtedly, electromagnetic theory can be used to describe them accurately but only certain simple solutions exist. In practice, lumped‐ element filter synthesis techniques are usually used to calculate initial dimensions. For example, the concept of coupling matrix is discussed in [12–15], which is used to describe filters in the same coupling topologies and provides the information of Q factor and coupling coefficients for the determination of their initial dimensions. In essence, these parameters are derived in the normalized frequency domain and should be transformed into the bandpass domain for use. Unfortunately, such approach is only accurate for narrow‐band filters [16]. In this paper, another DST for complex general Chebyshev bandpass filters is proposed. It is different from the one in [7] in that it is based on a real mapping relation, derives real‐coefficient filtering polynomials, and can be realized through lumped‐element LC networks. The DST in this paper is based on a complex mapping relation and is called as complex DST for distinction. Thus, its filtering polynomials have complex coefficients, and they can be represented by lumped‐elements CB or LX models where C is capacitor, B is frequency‐invariant susceptance, L is inductor, and X is frequency‐invariant reactance. The polarities of complex‐coefficient filtering polynomials can be easily changed by multiplying imaginary unit j, which makes their realization more flexible. A network matrix could be constructed to represent coupling schemes of a filter. Conventional coupling matrix is considered as narrow‐band approximation of network matrix obtained by complex DST. To demonstrate the validity of the proposed complex DST, it is applied in the design of classic parallel‐coupled microstrip bandpass filters. Several examples with different order and specifications are presented.

2. FILTERING FUNCTIONS

Note that only a lossless network is considered in this paper. The following scaled s‐plane to z‐plane complex mapping relation is used [17].

s−jω z2 ¼ u ; (1) s−jωd where z is a temporal complex variable. ωu and ωd are the upper and lower passband edge frequencies of the filter to be synthesized. For simplicity, a characteristic frequency ωc is used for scaling. Thus, s ¼ s=ωc ¼ jω=ωc, ωu ¼ ωu=ωc, and ωd ¼ ωd=ωc. Here, only transmission zeros at positive frequencies are taken into consideration. If there are Np transmission zeros at zero frequency, Nm at finite frequencies and Nl at infinity, the total number of transmission zeros considered should be

N ¼ N p þ N m þ N l: (2)

This is also deﬁned as the order of the ﬁlter to be synthesized. These transmission zeros can be mapped in the z‐plane by using (1). For example, the kth transmission zero sk ¼ jωk is mapped to the point zk in the z‐plane. If the procedure in [7] is followed, a prototype function exhibiting the

Copyright © 2017 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2017 DOI: 10.1002/cta DST FOR COMPLEX GENERAL CHEBYSHEV FILTERS required features is constructed in the z‐plane and mapped into the s‐plane through (1). Finally, ﬁltering polynomials such as reﬂection polynomial FðÞs , transfer polynomialPðÞs , and common polynomial EðÞs are derived through the following equations.

N N 2 2k Ev ∏ ðÞzk −z ¼ ∑ d2k z ; (3) k¼1 k¼0

N k N−k β ⋅ FðÞ¼s ∑ d2k ðÞs − jωu ðÞs − jωd ; (4) k¼0

N m N p ε⋅PðÞ¼s P0ðÞ¼s s ⋅ ∏ ðÞs − sk ; (5) k¼1

where Ev means the operation of taking even part of a function. zk is the corresponding point of the kth ‐ fi ∏N ðÞ− 2 β transmission zero in the z plane. d2k is the coef cient obtained by expanding Ev k¼1 zk z . is used to make the coefficient of the highest‐order term in FðÞs to be unity. ε can be determined by the specified ripple or return loss at the edge frequency of the passband. The common polynomial EðÞs can be obtained through the following equality owing to the fact that only a lossless network is considered in the paper.

P⋅P þ F⋅F ¼ E⋅E: (6)

3. TRANSVERSAL NETWORK

We discussed how to derive complex‐coefficient filtering polynomials of a complex general Chebyshev bandpass filter in the previous section. In this section, how to realize them through lumped‐element networks will be addressed. Mathematically speaking, it is an inverse problem. Up to now, there are three approaches available, that is, the element‐based synthesis method, the block‐ based synthesis method, and the whole synthesis method [10]. Here, the whole synthesis method in [15] is applied. In Figure 1, a canonical N‐resonator transversal network consisting of lumped‐ element CB resonators and admittance inverters J is presented, which is called the CB model for simplicity. Its dual network is depicted in Figure 2, which consists of lumped‐element LX resonators and impedance inverters K. The network in Figure 2 can be called the LX model for simplicity. In the following, only the CB model in Figure 1 is discussed and its dual network in Figure 2 can be derived in a similar way. The CB model consists of N + 1 paths between the source and the load. One is the direct source/load coupling realized through the inverter JSL, and each of the N paths left is composed of a resonator consisting of a capacitor Ck and a frequency‐invariant susceptance jBk, an admittance inverter JSk between the source and the resonator, and an admittance inverter JLk between the resonator and the load. The susceptances BS and BL allow for some special cases such as antimetric cases. Different from conventional synthesis techniques in [13–15] that are actually the representation in the normalized frequency domain, the networks in Figures 1 and 2 are the representations in the bandpass domain. In addition, they are also different from that in [7], in which the LC model is used. If the circuit in Figure 1 is fed by a current source iS with internal conductance GS and the load conductance is GL, the relation of the voltage across the resonators vk, the voltage at the source vS, and the voltage at the load vL can be described by the nodal equations.

Figure 1. N‐resonator transversal network consisting of lumped‐element CB resonators and admittance inverters J.

Figure 2. N‐resonator transversal network consisting of lumped‐element LX resonators and impedance inverters K.

N ðÞþ þ ∑ þ ¼ ; GS BS vS jJ Sk vk jJ SLvL iS k¼1 ðÞþ þ þ ¼ ; ¼ ; … ; sCk jBk vk jJ Sk vS jJ LkvL 0 k 1 N (7) N þ ∑ þ ðÞþ ¼ ; jJ SLvS jJ Lk vk GL BL vL 0 k¼1 where s is complex frequency variable. These equations can be rewritten in matrix form

½A ⋅½¼v −j ⋅ ½i : (8)

Here, [v] is the voltage vector, [i] the current vector, and [A] the network matrix.

On the other hand, it is easy to obtain the admittance matrix of the transversal network.

2 3 N 2 N þ ∑ J Sk þ ∑ J Sk J Lk 6 BS jJ SL 7 6 ¼ jðÞω−ωk ¼ jðÞω−ωk 7 ½y ¼ 6 k 1 k 1 7: (9) C 4 N N 2 5 þ ∑ J Sk J Lk þ ∑ J Lk jJ SL ðÞω−ω BL ðÞω−ω k¼1 j k k¼1 j k

Here, the capacitance value of the kth resonator is set to be unit for simplicity and thus the susceptance value Bk is determined by the resonant frequency ωk of the kth resonator.

4. SYNTHESIS PROCEDURE

According to [7], the polarities of both P(s) and F(s) should be opposite when a filter is to be realized in symmetric form. For DST in [7], F(s) is always even. The polarity of P(s) is determined by the number of its zeros at zero frequency if only imaginary transmission zeros are considered. When there is even number of transmission zeros at zero frequency, both P(s) and F(s) are even. Thus, it is unlikely for them to be realized through symmetric networks. Fortunately, filtering polynomials derived by complex DST in this paper are complex‐coefficient, which means that their polarities can be changed through the multiplication of the imaginary unit j. Then, we have the following notes to support these comments.

Note 1: If a function Q(s) is expressed as

n QsðÞ¼a ∏ ðÞs−jωk ; (10) k¼1 where s is complex frequency variable and both a and ωk are real, the polarity of Q(s) is determined by n. In other words, Q(s) is even if n is even or odd if n is odd.

Note 2: The polarity of Q(s) can be changed inversely by multiplying imaginary unit j.

Complex DST can be applied in the design of transmission‐line filters, and the synthesis flowchart is shown in Figure 3. According to filter specifications, the location of transmission zeros and return loss in passband are determined first. Then, filtering polynomials are derived by using (3)–(5). If symmetric networks are used to realize the desired frequency response, the polarities of both reflection and transfer polynomials should be opposite; otherwise, they can be changed by multiplying imaginary unit j in practice. Further, the corresponding admittance parameters can be derived, written in the form of partial fraction expansion and compared with those in (9) derived from the CB or LX models. Accordingly, we can determine those unknown parameters such as the resonant frequencies, and the values of the inverters. Finally, the following scaled transversal network matrix for the CB model is written as

2 3 −jGðÞS þ BS J S1 ⋯ J SN J SL 6 7 6 ω−ω ⋯ 7 6 J S1 1 0 J L1 7 6 7 A ¼ 6 ⋮ 0 ⋱ 0 ⋮ 7: (11) 6 7 4 5 J SN ⋮ 0 ω−ωN J LN

J SL J L1 ⋯ J LN −jGðÞL þ BL

Figure 3. Synthesis ﬂowchart of complex direct synthesis technique.

Similarly, the following transversal network matrix for the LX model in Figure 2 can be obtained. 2 3 −jRðÞS þ X S KS1 ⋯ KSN KSL 6 7 6 ω−ω ⋯ 7 6 KS1 1 0 KL1 7 6 7 A ¼ 6 ⋮ 0 ⋱ 0 ⋮ 7: (12) 6 7 4 5 KSN ⋮ 0 ω−ωN KLN

KSL KL1 ⋯ KLN −jRðÞL þ X L

The prototype network matrix is defined as the one with source/load resistance/conductance equal to be unity and scaled by a characteristic frequency ωc. This matrix is represented by global eigenresonances. By applying matrix operation such as similarity transformation, it can be transformed into the desired topology corresponding to transmission‐line filter to be deigned. Therefore, the equivalence between transmission‐line filter to be designed and its lumped‐element network is set up, and then we can calculate initial dimensions of the transmission‐line filter according to its specifications. Finally, its performance can be optimized to meet specification requirement by adjusting its dimensions in vicinity of their initial values.

5. SYNTHESIS EXAMPLES

In this section, we will present an example to demonstrate the synthesis procedure of complex DST discussed in the previous sections. It is a fourth‐order general Chebyshev bandpass filter whose passband covers the range from 4 to 6 GHz (40% fractional bandwidth) and return loss in the passband of −20 dB. Two transmission zeros are at zero frequency, the third one at 6.5 GHz, and the fourth one at infinity. The synthesis process starts with the derivation of filtering polynomials. Generally speaking, the characteristic frequency ωc could be chosen arbitrarily if only it can help simplification. 9 Here, ωc =2π ×10 rad/s. Then, the four transmission zeros ωzk (k = 1, 2, 3, and 4) are mapped into the z‐plane by (1), and then we can obtain

4 2 8 6 4 2 Ev ∏ ðÞz−zzk ¼ z þ 26:1686z þ 56:9607z þ 18:0370z þ 0:45: (13) k¼1

By using (4) and (5), we can obtain PðÞs , FðÞs , and EðÞs as follows.

FðÞ¼s s 4 −20:1799 j⋅s3 −151:6734⋅s2 þ 503:1035 j⋅s þ 621:3091; (14a)

PðÞ¼s −0:0388⋅s3 þ 0:2521 j⋅s2; (14b)

EðÞ¼s s 4 þ ðÞ2:0991− j20:1799 ⋅s3 þ ðÞ−149:4712− j31:9230 ⋅s2 : (14c) þ ðÞ−159:0200− j480:5466 ⋅s þ ðÞ564:8060 þ j258:8808

Because PðÞs is odd and FðÞs is even, this fourth‐order bandpass ﬁlter can be realized through a symmetrical network. The derivation of transversal network matrix is similar to that in [15], and then we can determine the prototype network matrix of this example. 2 3 −jGðÞþ B J J J J J 6 S S S1 S2 S3 S4 SL 7 6 ω−ω 7 6 J S1 1 000 J L1 7 6 7 6 J 0 ω−ω 00 J 7 A ¼ 6 S2 2 L2 7: (15) 6 ω−ω 7 6 J S3 00 3 0 J L3 7 6 7 4 J S4 000ω−ω4 J L4 5

J SL J L1 J L2 J L3 J L4 −jGðÞL þ BL where GS = GL =1,BS = BL =0,J S1 ¼ 0:4153,J S2 ¼ 0:6413, J S3 ¼ 0:6017, J S4 ¼ 0:3224, ω1 ¼ 3:6709, ω2 ¼ 4:4465, ω3 ¼ 5:8243, ω4 ¼ 6:2381, J L1 ¼ 0:4153, J L2 ¼ −0:6413, J L3 ¼ 0:6017, J L4 ¼ −0:3224, and J SL ¼ 0. The frequency response of the network matrix is shown in Figure 4, which is indistinguishable from that obtained from the filtering polynomials. Interestingly, the bandpass frequency response is only formed in the positive frequency range, which is different from those in [7]. Thus, the filters synthesized by complex DST in this paper could be called complex filters while those by the DST in [7] could be called real filters, according to [17–19].

6. RELATION BETWEEN THE COMPLEX DST AND THE IST

In this section, we will compare complex DST with IST in [13–15] to reveal some interesting features underlying them. According to [13–15], conventional coupling matrix can be written as the following form.

Figure 4. Frequency response of the fourth‐order complex general Chebyshev bandpass ﬁlter.

Copyright © 2017 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2017 DOI: 10.1002/cta F. XIAO 2 3 − ⋯ jGS M S1 M SN M SL 6 7 6 M Ω þ M M ⋯ M 7 6 S1 11 12 L1 7 6 7 ½¼M 6 ⋮ M 12 ⋱ M 1N ⋮ 7; (16) 6 7 4 M SN ⋮ M 1N Ω þ M NN M LN 5 ⋯ − M SL M L1 M LN jGL

where Ω is the angular frequency variable in the normalized frequency domain and Mij the normalized coupling coefﬁcient. The entries in this matrix are derived from lowpass prototypes and effective in the normalized frequency domain. In practice, they should be transformed into the bandpass domain for practical use by using the conventional frequency transformation.

1 ω ω jΩ ¼ j − 0 ; (17) FBW ω0 ω

pffiffiffiffiffiffiffiffiffiffi where ω0 ¼ ωuωd and FBW = (ωu − ωd)/ω0. We can follow the procedure in [10] to reveal the relation between network matrix obtained by complex DST in this paper and coupling matrix obtained by IST. At first, ω in (11) can be expressed in terms of Ω by using (17). Then, we substitute it into the diagonal entries in (11), expand them into Taylor series around Ω = 0 and only keep the first two terms, while higher order terms are neglected. Finally, these diagonal entries in (11) containing frequency variable are expressed as

Γk ðÞ¼Ω ω−ωk ≈bk ⋅Ω þ ak; (18) where

1 b ¼ FBW⋅ω ; (19a) k 2 0

ak ¼ ω0 −ωk : (19b)

The analysis shows that truncation error is mainly related to fractional bandwidth. The larger is fractional bandwidth, the larger is truncation error. After these approximation expressions are substituted into network matrix (11), the lowpass representation of the network matrix can be obtained. The lowpass matrix derived from network matrix (11) is very similar to coupling matrix (16) derived by IST. Thus, we draw a conclusion that network matrix obtained by complex DST represents bandpass response, while coupling matrix obtained by IST can be regarded as the lowpass approximation of the former. In order to demonstrate the relation between IST and complex DST proposed in this paper, they applied in the synthesis of a fully canonical fourth‐order bandpass filter with −22 dB return loss and four transmission zeros at finite frequencies, that is, f1 = 4.15 GHz, f2 = 4.57 GHz, f3 = 5.41 GHz, and f4 = 6.78 GHz. For simplicity, GS and GL are set to be unity. If we use IST to synthesize this filter, the coupling matrix of the corresponding lowpass prototype should be synthesized first, which is exactly the example in [13]. Here, its transversal representation is given by (20). If complex DST in this paper is used to synthesize this filter, the network matrix is given by (21). The frequency response of the filter obtained by both techniques is presented in Figure 5. As we can see, the results of both techniques coincide very well.

Figure 5. Frequency response of a fully canonical fourth‐order complex general Chebyshev bandpass ﬁlter synthesized by complex direct synthesis technique (DST) and indirect synthesis technique (IST) respectively.

2 3 −j 0:3641 −0:6537 0:6677 −0:3433 0:0151 6 7 6 : Ω þ : : 7 6 0 3641 1 3142 0 0 0 0 3641 7 6 7 6 −0:6537 0 Ω þ 0:7831 0 0 0:6537 7 ½¼6 7; M 6 7 (20) 6 0:6677 0 0 Ω−0:8041 0 0:6677 7 6 7 4 −0:3433 0 0 0 Ω−1:2968 0:3433 5 0:0151 0:3641 0:6537 0:6677 0:3433 −j 2 3 −j 0:1739 0:3387 0:3222 0:1791 0:0145 6 7 6 : ω− : − : 7 6 0 1739 5 3360 0 0 0 0 1739 7 6 7 6 0:3387 0 ω−5:2044 0 0 0:3387 7 ¼ 6 7: A 6 7 (21) 6 0:3222 0 0 ω−4:8074 0 −0:3222 7 6 7 4 0:1791 0 0 0 ω−4:6836 0:1791 5 0:0145 −0:1739 0:3387 −0:3222 0:1791 −j

Then, we can obtain the lowpass representation of (21), that is, (22). Interestingly, (22) and (20) are very similar, which veriﬁes the conclusion that coupling matrix obtained by IST could be considered as the lowpass approximation of the network matrix synthesized by complex DST. 2 3 −j 0:3582 −0:6444 0:6774 −0:3478 0:0145 6 7 6 : Ω þ : : 7 6 0 3582 1 2656 0 0 0 0 3582 7 6 7 6 −0:6444 0 Ω þ 0:7704 0 0 0:6444 7 A ¼ 6 7: (22) L 6 7 6 0:6774 0 0 Ω−0:8176 0 0:6774 7 6 7 4 −0:3478 0 0 0 Ω−1:3440 0:3478 5 0:0145 0:3582 0:6444 0:6774 0:3478 −j

7. APPLICATION OF COMPLEX DST IN DESIGN OF PARALLEL‐COUPLED MICROSTRIP BANDPASS FILTERS

In the RF/microwave frequency range, ﬁlters are usually realized through transmission‐line structures with distributed‐element effect, such as waveguide, microstrip, stripline, suspended stripline, slotline,

Copyright © 2017 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2017 DOI: 10.1002/cta F. XIAO coplanar waveguide, and finline. Up to now, people have found out a great number of filter variations with different coupling topologies, which exhibit different frequency response such as narrow‐band, wide‐band, or multi‐band response. Electromagnetic theory can be used to analyze them accurately, in which distributed‐element effect is usually characterized by trigonometric functions or other special functions. One important feature might be periodicity, which exhibits usually in form of harmonic for distributed‐element resonators. In practice, it might be easier to use lumped‐element network synthesis techniques to determine dimensions of transmission‐line filters according to the specifications such as passband, ripple in passband, return loss in passband, out‐of‐band performance, and so on. It is because that network parameters in lumped‐element network synthesis are expressed in form of polynomial or rational polynomials, which are simpler than trigonometric functions or other special functions for operation. Undoubtedly, the key point of applying lumped‐ element network synthesis techniques in the design of distributed‐element filters is to find out accurate equivalence between them. In practice, the fabrication of a filter must meet the requirement of actual fabrication precision. For example, microstrip line width and gap should be no less than 0.12 mm for our current fabrication. When we design microstrip filters, such restriction should be obeyed, which also set limitation to the capacity of a filter such as what is widest or most narrow bandwidth it might achieve. Among various RF/microwave filters, parallel‐coupled microstrip bandpass filter is classic and widely‐used [3,20]. In this section, complex DST will be applied to the design of some parallel‐ coupled microstrip bandpass filters with different order.

7.1. Lumped‐element equivalent network of coupled‐line section To start, the lumped‐element equivalent network of open‐circuited coupled‐line section should be examined. In Figure 6(a), the schematic of coupled‐line section is shown. Its distributed‐element equivalent network is shown in Figure 6(b), which is represented by one transmission‐line section with two open‐circuited stubs at each end. As for transmission‐line section, it can be represented by its mixed‐element equivalent network in Figure 7(b). If the transmission‐line section in Figure 6(b) is replaced by its mixed‐element equivalent network and adjacent open‐circuited stubs are merged, the ﬁnal mixed‐element equivalent network of coupled‐line section can be obtained. Finally, we can obtain the lumped‐element LX equivalent network for coupled‐line section shown in Figure 8.

π Z þ Z L ¼ 0e 0o ; (23a) ω0 4

Z þ Z X ¼ −π 0e 0o ; (23b) 4 where ω0 is the resonant angular frequency. In addition, the impedance inverter is approximated as

Figure 6. Open‐circuited coupled‐line section and its distributed‐element equivalent network. (a) Open‐ circuited coupled‐line section. (b) Distributed‐element equivalent network.

Figure 7. Transmission‐line section and its mixed‐element equivalent network. (a) Transmission‐line section. (b) Mixed‐element equivalent network.

Figure 8. Lumped‐element equivalent network of open‐circuited coupled‐line section.

Z −Z Z −Z K ¼ 0e 0o ≈ 0e 0o : (24) 2 sinθ 2 In Figure 9(a), general structure of coupled‐line microstrip bandpass ﬁlters is shown, which consists of n coupled‐line sections where n is an integer and n ≥ 3. Conventional synthesis techniques only deal with odd‐order cases, that is, n being an even integer. In this paper, it will be demonstrated that complex DST is suitable for not only the cases of n being an even integer but also the cases of n being an odd integer. If each coupled‐line section in Figure 9(a) is replaced with its lumped‐element LX equivalent network, we can obtain the fully lumped‐element LX equivalent network of coupled‐line microstrip bandpass ﬁlter as shown in Figure 9(b).

Figure 9. General structure of parallel‐coupled microstrip bandpass ﬁlter and its lumped‐element LX equivalent network. (a) General structure of parallel‐coupled microstrip bandpass ﬁlter. (b) Its lumped‐ element LX equivalent network.

7.2. Odd‐order parallel‐coupled microstrip bandpass filter In this subsection, an example of third‐order parallel‐coupled microstrip bandpass filter is discussed. Its passband covers 2.88 to 3.12 GHz, namely, 8% fractional bandwidth at center frequency 3.0 GHz. The return loss in the passband is to be lower than −20 dB. In order to demonstrate the advantage of the DST, conventional synthesis technique is firstly applied to synthesize this filter [3]. According to this, the lowpass prototype parameters are calculated as

¼ : ; ¼ : ; ¼ : ; ¼ : : g1 0 8535 g2 1 1039 g3 0 8535 and g4 1 0000

These design parameters derived by conventional synthesis techniques are listed in Table I. If Rogers RT/duroid 4350 substrate with a relative permittivity of 3.66 and a dielectric height of 0.508 mm is used in this paper, this result in the microstrip design listed in Table II. However, the minimum gap is as narrow as 0.08 mm, which is not suitable for actual fabrication. This is because that the minimum gap and line width is no less than 0.12 mm in our actual fabrication. Furthermore, conventional synthesis techniques only provide single set of microstrip design parameters, which is inflexible in practice. If complex DST is used here, a third‐order complex general Chebyshev bandpass filter can be synthesized, realized in forms of lumped‐element LX network and finally matched with that of the third‐order coupled‐line microstrip bandpass filter. The impedance inverters in the lumped‐element LX equivalent network of coupled‐line microstrip bandpass filter is considered frequency‐invariant in this paper, which corresponds to a third‐order complex general Chebyshev bandpass filter with all its transmission zeros at infinity. Then, the filtering polynomials in the bandpass domain are derived by complex DST according to (3)–(5).

PðÞ¼s 0:004298; (25a)

FðÞ¼s s3 − j9:0000⋅s2 − 26:9892⋅s þ j26:9767; (25b)

EðÞ¼s s3 þ ðÞ0:2812−j9:0000 ⋅s2 þ ðÞ−26:9497−j1:6873 ⋅s þ ðÞ−2:5266 þ j26:8490 : (25c)

Table I. Design parameters of the third‐order parallel‐coupled microstrip bandpass ﬁlter derived by conventional synthesis technique. iJi, i +1/Y0 (Z0e)i, i +1 (Z0o)i, i +1 1 0.3837 76.5463 38.1763 2 0.1295 57.3135 44.3635

Table II. Microstrip design parameters of the third‐order parallel‐coupled microstrip bandpass ﬁlter derived by conventional synthesis technique. ili (mm) wi (mm) si (mm) 1 15.02 0.79 0.08 2 14.64 1.06 0.49

Then, the following prototype network matrix can be obtained. 2 3 −j 0:1875 0:2652 0:1875 0 6 7 6 0:1875 ω−3:1748 0 0 0:1875 7 6 7 6 7 A ¼ 6 0:2652 0 ω−3:00−0:2652 7: (26) 6 7 4 0:1875 0 0 ω−2:8252 0:1875 5 00:1875 −0:2652 0:1875 −j

After applying similarity transformation on it, various network matrices can be obtained. We could choose those suitable for fabrication. For example, 2 3 − jRS KS1 000 6 7 6 K ωL þ X K 007 6 S1 1 1 12 7 6 7 ½¼A 6 0 K12 ωL2 þ X 2 K23 0 7; (27) 6 7 4 00 K23 ωL3 þ X 3 KL3 5 − 00 0 KL3 jRL

−8 where RS = RL =50Ω, KS1 = KL3 = 21.3257Ω, K12 = K23 = 7.5000Ω, L1 = L3 = 1.0295 ⋅ 10 H, −9 L2 = 9.0561 ⋅ 10 H, X1 = X3 = − 194.0618Ω, and X2 = − 170.7043Ω. This network matrix corresponds to the following lumped‐element network in Figure 10. The network in Figure 10 looks similar to the equivalent network of the third‐order coupled‐line microstrip bandpass filter with n = 4 except that the some elements L1, jX1, L4, and jX4 are neglected. Therefore, the equivalence between the network matrix derived by complex DST and the structural dimensions of the third‐order coupled‐line microstrip bandpass filter is established. Note that two adjacent coupled‐line sections will contribute to a same LX resonator in Figure 10. In practice, we could allocate their contribution in a reasonable way to avoid some extreme cases for fabrication, which demonstrates the flexibility of complex DST. The dimensions of the third‐order coupled‐line microstrip bandpass filter can be determined through the comparison between its lumped‐element equivalent network in Figure 9 with n =4 and that of the third‐order complex general Chebyshev bandpass filter synthesized by complex DST in Figure 10. For example, the second LX resonator in Figure 10 or the network matrix (27) is written as

−9 ω⋅L2 þ X 2 ¼ ω⋅0:90561⋅10 −170:7043; (28) which might be realized through the contribution of both the second and third coupled‐line sections. Here, their contribution is equal if the ﬁlter structure is considered symmetric. We choose the ‐ ω⋅ 1 þ 1 second coupled line section to contribute a half of the second LX resonator, that is, 2 L2 2 X 2 . By using (23a), the following equation can be obtained for the second coupled‐line section

4ω 1 Z þ Z ¼ 0 0:9056⋅10−9 ¼ 108:6732; (29) 2e 2o π 2

Figure 10. Lumped‐element LX network of a third‐order complex general Chebyshev bandpass ﬁlter synthesized by complex direct synthesis technique.

9 where ω0 =2π ×3×10 rad/s. In addition,

Z −Z K ¼ 7:5 ¼ 2e 2o : (30) 12 2

By using these two equations, Z2e = 61.8366 Ω and Z2o = 46.8366 Ω. If Rogers RT/duroid 4350 substrate is used here, we can determine dimensions for the second coupled‐line section, that is, l2 = 14.62 mm, w2 = 0.95 mm, and s2 = 0.46 mm, respectively. For the ﬁrst LX resonator in Figure 10 or the network matrix (27), that is,

−8 ω ⋅ L1 þ X 1 ¼ ω ⋅ 1:0295 ⋅ 10 −194:0618; (31) which might be realized through the contribution of both the ﬁrst and second coupled‐line sections. As ‐ ω ⋅ 1 þ 1 ﬁ discussed earlier, the second coupled line section willÀÁ contribute ÀÁ2 L2 2 X 2 and then the rst ‐ ω⋅ c − 1 c þ c − 1 c coupled line section has to contribute the left, that is, L1 2 L2 X 1 2 X 2 . According to (23a), 4ω 1 Z þ Z ¼ 0 Lc − Lc ¼ 138:4068 Ω: (32) 1e 1o π 1 2 2

In addition,

Z −Z K ¼ 1e 1o ¼ 21:3257: (33) S1 2

By using these two equations, we find Z1e = 90.5294 Ω and Z1o = 47.8774 Ω. If Rogers RT/duroid 4350 substrate is used here, we can determine initial dimensions for the first coupled‐line section, that is, l1 = 14.89 mm, w1 = 0.58 mm, and s1 = 0.15 mm, respectively. Electromagnetic simulation is applied on the third‐order parallel‐coupled microstrip bandpass filter with these calculated dimensions, and the simulated frequency response is presented in Figure 11. The simulated |S21| is very close to the synthesized one except some small differences. In practice, some dimensions of the third‐order parallel‐coupled microstrip bandpass filter are adjusted to optimize its performance. For example, a set of dimensions very close to the calculated dimensions are listed in Table III. In Figure 11, the comparison among the frequency response synthesized by complex DST, the simulated ones with the calculated dimensions and the simulated ones with the adjusted dimensions is presented. This set of the adjusted dimensions meets the requirement of our fabrication limitation and then is chosen for final fabrication. The photo of the fabricated third‐order coupled‐line microstrip bandpass filter is shown in Figure 12. Both the simulated and measured

Figure 11. Frequency response comparison between the synthesized ones, the simulated ones with the calculated dimensions and the simulated ones with adjusted dimensions. (a) |S21|. (b) |S11|.

Table III. Microstrip design parameters of the third‐order parallel‐coupled microstrip bandpass ﬁlter obtained by complex DST.

l1 w1 s1 l2 w2 s2 Calculated (mm) 14.89 0.58 0.15 14.62 0.95 0.46 Adjusted (mm) 14.89 0.58 0.18 14.65 0.95 0.46 DST, direct synthesis technique.

Figure 12. Photo of the fabricated third‐order parallel‐coupled microstrip bandpass filter. frequency response is presented in Figure 13, which coincides well in wide frequency range. The measurement is performed without any tuning. In other words, it demonstrates that the calculated dimensions obtained by complex DST are good. In complex DST, it is easy to apply matrix transformation on prototype network matrix so that various network matrices are available for use. Those suitable for fabrication can be chosen, which will undoubtedly provide great flexibility for practical use. For example, we can derive another network matrix to realize the same third‐order general Chebyshev bandpass frequency response earlier. The entries of the network matrix are RS = RL =50 Ω, KS1 = KL3 = 23.9974Ω, −8 −9 K12 = K23 = 7.5000Ω, L1 = L3 = 1.3037 ⋅ 10 H, L2 = 7.1519 ⋅ 10 H, X1 = X3 = − 245.7323 Ω, and X2 = − 134.8101 Ω, respectively. We can determine the initial dimensions as performed previously. Finally, the calculated dimensions and the adjusted dimensions are listed in Table IV, which are very close to each other. The photo of the third‐order parallel‐coupled microstrip bandpass filter with the adjusted dimensions is shown in Figure 14. The simulated and measured frequency response is presented in Figure 15.

Figure 13. The simulated and measured results of the fabricated third‐order parallel‐coupled microstrip bandpass ﬁlter.

Table IV. Another set of dimensions for the third‐order parallel‐coupled microstrip bandpass ﬁlter.

l1 w1 s1 l2 w2 s2 Calculated (mm) 15.42 0.18 0.39 14.35 1.35 0.26 Adjusted (mm) 15.27 0.18 0.41 14.42 1.35 0.26

Figure 14. Photo of the fabricated third‐order parallel‐coupled microstrip bandpass ﬁlter.

7.3. Even‐order parallel‐coupled microstrip bandpass filter Conventional synthesis techniques are only suitable for odd‐order parallel‐coupled microstrip bandpass filters, that is, the case of n being even integer. Complex DST proposed in this paper is also applicable to even‐order parallel‐coupled microstrip bandpass filters, that is, the case of n being odd integer. In this subsection, a second‐order parallel‐coupled microstrip bandpass filter is considered, that is, n = 3. Without losing generality, its passband covers 1.96 to 2.04 GHz, namely, about 4% fractional bandwidth at center frequency 2.0 GHz. The return loss in passband is set lower than −20 dB. According to these specifications, the following network matrix is

2 3 −j50 15:0420 0 0 6 − 7 6 15:0420 ω⋅1:2003⋅10 8−150:8278 5 0 7 ½¼A 6 7: (34) 4 05ω⋅1:2003⋅10−8−150:8278 15:0420 5 00 15:0420 −j50

This network matrix corresponds to the lumped‐element LX network in Figure 16. Following the procedure discussed earlier, we can obtain a set of calculated dimensions and a set of adjusted

Figure 15. The simulated and measured results of the fabricated third‐order parallel‐coupled microstrip bandpass ﬁlter.

Figure 16. Lumped‐element network representation of the second‐order general Chebyshev bandpass filter. dimensions for second‐order parallel‐coupled microstrip bandpass filter, which are listed in Table V. It has been fabricated, and the photo of the fabricated filter is presented in Figure 17. Both the simulated and measured frequency response is presented in Figure 18, which coincides well in wide frequency range. Further, an example of fourth‐order parallel‐coupled microstrip bandpass filter is presented. Its passband covers 3.76 to 4.24 GHz, namely, 12% fractional bandwidth at center frequency 4.0 GHz. The return loss in passband is set lower than −20 dB. According to these specifications, we obtain one network matrix as follows.

Table V. Microstrip design parameters of the second‐order parallel‐coupled microstrip bandpass ﬁlter obtained by complex DST.

l1 w1 s1 l2 w2 s2 Calculated (mm) 22.29 0.72 0.22 21.30 1.93 0.25 Adjusted (mm) 22.26 0.72 0.22 21.30 1.93 0.25 DST, direct synthesis technique.

Figure 17. Photo of the fabricated second‐order parallel‐coupled microstrip bandpass ﬁlter.

Figure 18. Simulated and measured frequency response of the fabricated second‐order parallel‐coupled microstrip bandpass ﬁlter.

Copyright © 2017 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2017 DOI: 10.1002/cta F. XIAO 2 3 −jR KS1 00 00 6 S 7 6 ω⋅ þ 7 6 KS1 L1 X 1 K12 0007 6 7 6 0 K12 ω⋅L2 þ X 2 K23 007 ½¼A 6 7; (35) 6 ω⋅ þ 7 6 00 K23 L3 X 3 K34 0 7 6 ω⋅ þ 7 4 00 0 K34 L4 jX 4 KL4 5 − 00 0 0 KL4 jRL

where RS = RL =50 Ω, KS1 = KL4 = 23.0000 Ω, K12 = K34 = 7.0000 Ω, K23 = 4.1845 Ω, −9 −9 L1 = L4 = 6.5475 ⋅ 10 H, L2 = L3 = 3.9693 ⋅ 10 H, X1 = X4 = − 164.5500 Ω, X2 = X3 = − 99.7630 Ω. Following the procedure discussed earlier, we can obtain a set of calculated dimensions and a set of adjusted dimensions for the fourth‐order parallel‐coupled microstrip bandpass ﬁlter, which are listed in Table VI. As can be seen, they are very close, which also demonstrates the accuracy of complex DST.

Table VI. Microstrip design parameters of the 4th‐order parallel‐coupled microstrip bandpass ﬁlter obtained by complex DST.

l1 w1 s1 l2 w2 s2 l3 w3 s3 Calculated (mm) 11.11 0.56 0.13 10.51 1.78 0.14 10.34 2.40 0.20 Adjusted (mm) 11.05 0.54 0.17 10.54 1.79 0.15 10.63 2.35 0.21 DST, direct synthesis technique.

Figure 19. Photo of the fabricated fourth‐order parallel‐coupled microstrip bandpass ﬁlter.

Figure 20. Simulated and measured frequency response of the fabricated fourth‐order parallel‐coupled microstrip bandpass ﬁlter.

A photo of the fabricated ﬁlter is presented in Figure 19. Both the simulated and measured frequency response is presented in Figure 20, which coincides well in wide frequency range.

8. CONCLUSION

In this paper, a complex DST based on a complex mapping relation is proposed for complex general Chebyshev bandpass filter. Compared with the technique based on a real mapping relation, filtering polynomials of complex DST in this paper are complex coefficients and are represented by lumped‐element CB or LX networks. Owing to such feature, their polarities could be easily changed by multiplying the imaginary unit j, which makes their realization more flexible. The analysis shows that conventional coupling matrix could be considered as narrow‐ band approximation of network matrix derived by complex DST in this paper. Further, complex DST is applied in the design of classic parallel‐coupled microstrip bandpass filters. The lumped‐ element LX equivalent network of coupled‐line section is discussed. The relation between lumped‐element values and structural dimensions is presented. By comparing lumped‐element LX equivalent network of the filter to be synthesized with network matrix derived by complex DST, good initial dimensions of transmission‐line filters can be determined for further optimization. Through matrix operation, complex DST could provide more than one set of dimensions, from which designers could choose those suitable for fabrication. Apart from odd‐order parallel‐ coupled microstrip bandpass filters, complex DST can also synthesize even‐order ones that conventional synthesis techniques are unable to synthesize.

ACKNOWLEDGEMENTS The authors would like thank Miss Rudan Xiao for her whole‐hearted support! This work was supported by National Natural Science Foundation of China (Project No. 61671111).

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