Broadband and compact multi-pole microstrip bandpass filters using ground plane aperture technique

L. Zhu, H. Bu and K. Wu

Abstract: A gound plane aperture technique is developed for effective enhancement of the capacitive coupling factor in the parallel-coupled microstrip line (PCML). By applying a so-called ‘short-open calibration’ (SOC) scheme in the fullwave method of moments (MOM) algorithm, ths PCML with two external lines is characterised by an equivalent J-inverter network with its susceptance and two electrical line lengths. Extracted parameters indicate that the coupling factor appears to be frequency-dependent and its maximum value rises rapidly as the aperture is widened. With the introduction of a single microstrip line section between two identical PCMLs, a broadband and compact multi-pole microstrip bandpass filter is proposed for the first time, and its electrical behaviour is studied and optimised on the basis of its equivalent circuit network. The network-based optimised results are confirmed by an EM simulation of the entire filter layout, featuring ultra-broadband and four-pole bandpass behaviour. Further, a single capacitively loaded line section is utilised to formulate a multi-pole bandpass filter, and its electrical effects are also discussed for filter design. The predicted and measured results confirm attractive properties of the proposed multi-pole filter with B W= 60%. /SI I < - 16dB and 220% wide upper stop-band.

1 Introduction

The microstrip bandpass filter has been studied and developed as an important building block in the design of microwave circuits and systems [l]. With its easily achievable design-specified coupling factor, the parallel- coupled microstrip line (PCML) has widely been used in multi-stage bandpass filters as a capacitive coupling element between two adjacent line resonators [2, 31. To realise a multi-pole and broad bandpass filter with a deep out-of- band rejection, the usual procedure is to reduce both its strip and slot widths in order to acheve a tight coupling, and a large number of line resonators are required in this case. This may lead to a degradation of its filtering behaviour, namely, low Q-factor and high insertion loss. Also, it may introduce some difficulties into the design procedure and fabrication process due to its sensitivity to the strip/slot widths and conductor thickness/configuration. The filter size and fabrication cost are usually not desirable as the required number of line resonators increases.

With the rapid development of three-dimensional (3D) microwave- and millimetre-wave integrated circuit proces- sing techniques, much attention has been directed to the use of a high-quality multilayer planar circuit that allows for an additional degree of design freedom along the vertical

0 IEE, 2002 IEE Proceedings online no. 20010145 DOE IO. 104Y/ip-map:20010145 Paper first received 3rd July 2001 and in revised form 10th December 2001 L. Zhu is with the School of Electrical & Electronic Engineering, Nanyang Technological University, 639798, Singapore H. Bu is with the Amplifier Design Group, Mitec Telecom Inc., Pointe Claire, Montreal, QC, H9R 528, Canada K. Wu is with the Department of Electrical Engineering, Ecole Polytechnique, CP. 6079. Sum. Centre-Ville, Montreal, QC, H3C 3A7, Canada

orientation [4]. To meet the requirement of capacitive tight coupling, an overlap-gap coupling structure has been developed in a two-layered structure for designing a broadband microstrip bandpass filter [5 ] . In [6], a high-Q and broadband inductor was proposed by removing a partial ground plane of the spiral circuit based on a 3D Si- MMIC technology. By forming a backside aperture in the ground plane, a novel parallel-coupled microstrip line (PCML) has also been developed and characterised in [7], to show its potential in effective enhancement of capacitive coupling required in the design of a broadband microstrip bandpass filter.

In recent years, ultra-wideband technologies have stimulated interest in communication and radar applica- tions [8]. Nevertheless, it is difficult to design broadband active and passive circuits with a bandwidth >20%0. In fact, the filter design procedure available to date was essentially established with (quasi-)lumped elements as described in [ 11, and its design formulas were developed over a narrow frequency range around the centre frequency. It therefore seems difficult to apply this procedure in the design of bandpass filters with BW>20%. This is because all the basic elements such as the line resonator and coupling section are strongly frequency dependent in this case. In other words, their electrical characteristics are too frequency dependent over a wide frequency range.

In ths work, a ground plane aperture technique is proposed and developed for effective enhancement of a tight coupling over the frequency range of interest, and realisation of periodic frequency-dependent coupling char- acteristics over a wide frequency range. This is achieved by forming a wide aperture on the ground plane of the PCML. With the use of a so-called ‘short-open calibration’ (SOC) scheme [9], that is self-contained in our fullwave method of moments (MOM) algorithm [lo], the two-port PCML with external lines is generally characterised as an equivalent J-

IEE Proc.-Microw. Antennu Propug., Vol. l49> No. I , February 2002 71

inverter network. Consequently, a novel multi-pole and broadband microstrip bandpass filter with a single line resonator is originated for the first time by attaching a uniform line section between the two PCML sections with a backside aperture. A closed-form equation is established to demonstrate the operating mechanism of the proposed filter. It is shown that the multi-pole bandpass behaviour is generated by the first-/second-order resonant modes of the line resonator and the J-inverter susceptance with the same value as the characteristic admittance of the microstrip line in the PCML. To further realise design specifications such as low return loss, adjustable broad bandwidth and wide out-of-band rejection, a pair of capacitive open-ended stubs is introduced into the central location of the line resonator that is used to shift downward its second-order resonant frequency [I I]. Two filter layouts are optimally designed and the optimised electrical performances are verified by our predicted results through field simulations of the entire layout and our experiments.

2 aperture

Fig. la shows a 3D geometry of the PCML, in which a wide aperture is formed over its ground plane. A tight coupling of this PCML can be readily acheved by reducing its ground-to-strip coupling with such a backside aperture. To investigate its coupling behaviour, this PCML with two external lines is initially modelled by using our 3D admittance-type MOM algorithm, as detailed in [lo]. Fig. lb shows a scheme arranged for its MOM characterisation, in whch a pair of impressed electrical fields (E , and E2) is introduced to formulate a deterministic MOM at two ports (PI and P2) far away from the reference planes (Rl and R2). To accurately de-embed circuit parameters of this PCML from the MOM calculation, the SOC procedure [9] is deployed to remove error terms involved in the algorithm that allows extracting an equivalent circuit model at the reference planes ( R 1 and R2). Fig. I C gives a circuit description of Fig. 16, where the entire layout is partitioned into two identical error terms [X, ] and an equivalent J- inverter network.

As detailed in [9], these error terms stand for the approximation of source excitation and inconsistency of 2D and 3D MOM-based impedance definitions. They can effectively be evaluated and removed with the help of two numerical calibration standards, namely, short and open elements. As such, the circuit network of the PCML can be explicitly extracted as a general-purpose two-port admit- tance matrix that accounts for all of its discontinuity effects. The equivalence of two networks allows the transforming of ths admittance matrix into a J-inverter network that consists of a susceptance (4 and two equivalent electrical line lengths (8/2), as shown in Fig. IC. For a symmetric two- port network, the relationship can be simplified from (2) in [12] as follows:

- tan(e/2) + Bll J = - ( l a )

Unifield circuit model of PCML with backside

B12 tan(e/2)

in which J = J/Yo, B11 = Bl,/Yo = B22/Yo, = B I ~ / Y o = Bzl/Yo, and Yo is the characteristic admittance of the microstrip line.

Fig. 2 shows the SOC-calibrated normalised J-inverter susceptance (J) and equivalent electrical length (e/2) of a

aperture Y.

h - 7 ground a

c

Fig. 1 microstrip line (PCML) with u ground plune uperture u Geometrical layout b Full-wave MOM modelling c Equivalent circuit network

Topological view and characterisation of parallel-coupled

PCML with different aperture widths W over a wide frequency range. It indicates that the parameter J varies in a periodical manner with frequency for all three cases, thus exhibiting a frequency dispersion behaviour. It can be interpreted that the coupling between the two strip conductors depends strongly on the electrical length of this PCML section, e.g. Lly0. The peak coupling essentially appears around the frequency of 4L/,,-, = (2n- 1) and the null J is close to that of 4L/@ = 2n, where n = 1 or 2 in t h s case. The peak value of J increases significantly from 0.6 to 1.2 as W is widened from 1.4 to 3.0" , showing an effective enhancement of its coupling factor by a backside aperture. Fig. 2b illustrates that the electrical length (8/2) increases from 15" to 260" in an approximately linear manner as the frequency increases from 1.0 to 16.0GHz. From Figs. 2a and b, we can further elicit that the peak J occurs in the proximity of the frequency at which 0/2 = 90" or 270", while the null J is around a frequency correspond- ing to Q/2= 180".

On the other hand, this frequency dependent periodicity of 1, as shown in Fig. 2a, provides us with a hint for constructing an alternative bandpass filter, whose bandpass behaviour can be formulated by a tight coupling factor, e.g. J M .l. Interestingly, the frequency of I= 1 strictly corresponds to the pole location over the bandpass range. Looking into the J-inverter network as in Fig. IC, the return loss I SI I can analytically be reduced to (2) in a closed form of J . Fig. 3 gives predicted return/insertion losses of the PCML with W= 2.2" and 3.0" based on the obtained J-inverter parameters in Fig. 2a. It shows the bandpass behaviour of one pole and two poles, respectively. This initial result demonstrates that the frequency-dependent J- inverter susceptance itself can generate its own pass-band around its maximum value. In the following, our main interests are focused on the proposal and investigation of a new class of miniaturised, multi-pole and broadband

IEE Proc -Microw Antenrius Propay, Vol 149, No 1, February 2002 12

1.2

1 .o t" ,; 0.8 U ul ._ 3 0.6

0.4

5 ,2

0.2

0 2 4 6 8 10 12 14 16

frequency t GHz a

300

J

2 4 6 8 10 12 14 16

frequency f, GHz b

Fig. 2 with dijfevent uperture widths W a Normalised J-inverter susceptance (J/ Yo) b Equivalent electrical line length (0/2)

SOC-calibrated J-inverter network parameters of PCML

microstrip bandpass filter. This is acheved by utilising the PCML's J-inverter susceptance as well as the first- and second-order resonant modes in a microstrip line resonator.

3 verification

Prototype bandpass filter: concept and

Fig. 4a shows the schematic layout of a prototype micro- strip bandpass filter proposed for achieving ultra-broad- band and multi-pole bandpass behaviour. A microstrip line is used to link two identical PCML sections with a backside aperture. Fig. 46 presents its complete equivalent circuit topology, arranged for gaining insight into its operating mechanism that also allows an efficient optimisa- tion. The PCML section is characterised as a J-inverter susceptance (J) and two electrical lengths (Oj2) that represent its series capacitive coupling and equivalent phase shifts, respectively. On the other hand, the central uniform

0

-5

-1 0

-15

-20 - - z -25

4 - - -30

-35

-40

-45

-50 2 4 6 8 10 12 14 16

frequency f, GHz

Predictedfrequency response of P C M L with W=2.2 and Fig. 3 3.0mm us shown in Fig. 2

Fig. 4 pole bandpass filter with single unfijrm line resonator a Schematic layout b Equivalent circuit network model

Top view und equivalent circuit topology of proposed multi-

line can be perceived as an additional phase factor 4, so that the total electrical length di should be made up of three separate parts, i.e. @ = 6/2 + + Qj2. This equivalent line resonator is formulated to generate two additional bandpass poles from its first- and second-order resonant modes.

To investigate its electrical behaviour, let us start with the characterisation of its equivalent cascaded circuit topology as shown in Fig. 4b. On the basis of a transmission line theorem, the normalised input admittance (T2,, = Kn/ y0) at one termination (# l), looking into its opposite termination (# l'), can be easily deduced and expressed as a closed-form function of 7 and di such that

Accordingly, its reflection coefficient (SI1) at # 1 can be further simplified as follows to provide a better under-

1 3 IEE Proc-Microw. Antennas Propug., Vol. 149, No. I , February 2002

standing of its fundamental filtering performance:

Considering the fact that the pole is usually defined as a frequency point where IS11 I = 0 or ISl] I achieves a minimum value, it is understood from (4) that there exist multiple poles of frequency where @ = 180", @ = 360" and J = 1. The former two poles correspond to the first- and second-order resonant frequencies, and we can find in the following that the frequency spacing between them basically forms a desired ultra-broad bandwidth of the filter. The latter one or two poles are contributed by a highly enhanced J-inverter susceptance ( J = l ) , and they are arranged around the central operating frequency through a suitable choice of the line length L in Fig. 4a.

Figs. 5a and b show our predicted frequency responses of the filter with L = 4.9" over the frequency range (1 .0 to 10.0GHz) obtained from (4) via three groups of J-inverter network parameters in Fig. 2. In the case of a narrow aperture width (W=1.4mm), only two poles can be observed, and they appear at the half- and full-wavelength

0 - :; -.--------- ; I : I

-10

a, W = 2.2 mm ...........

-35

-40 I 1 2 3 4 5 6 7

frequency f, GHz a

-60 I 2 3 4 5 6 7 8 9 1 0

frequency f, GHz b

Fig. 5 Fig. 4 with di$erent aperture widths W u Insertion loss I S2, I b Return loss 1 SI 1 I

14

Predictedfrequency response of$lter luyout us described in

1 8 9 10

resonant frequencies, marked by Fol and Fo2, which correspond to those of @ = 180" and @ = 360", respectively. The maximum J in this case is about 0.8, from Fig. 2a, indicating a relatively weak coupling. Ths leads to a worse bandpass behaviour with a return loss of I SI I 1 = -6dB between two resonant frequencies (FOI and F02) in the absence of any additional pole from the condition J z 1.

As W is enlarged to 2.2mm, 1 SI I rapidly decreases in such a way that its entire pattern is separated into two parts by an additional pole (minimum value) around the central location, while IS2, 1 gradually increases to close to its 0dB level between FOl and Fo2. With reference to Fig. 2 4 th s additional pole is physically generated by the maximum value of its normalised J-inverter susceptance where J = 1. As W increases further to 0.3mm, it can be observed from Fig. 5b that one pole is split into two poles at its two sides withn the pass band. Due to the maximum J > 1 in this case, the two frequency points where J = 1 can be found from Fig. 2a, and they lead to the emergence of these two additional poles via (4). Unfortunately, the return loss I SI I shifts up slightly in the central location, which degrades its whole bandpass behaviour due to its extremely enhanced coupling degree, e.g. J >> 1. Nevertheless, thls problem can be solved by suitably adjusting the backside aperture width of PCML to meet the requirement of its J-inverter susceptance in the optimisation procedure.

So far, the operating mechanism of the proposed multi- pole bandpass filter with a single line resonator has been explained on the basis of equivalent J-inverter circuit theory, and its filtering behaviour has been characterised by a closed-form equation. Next, our interest is to optimally design its frequency response in order to obtain a low return loss over the passband. Fig. 6a gives the optimised results of the filter with the same dimension as in Fig. 4a, except for a different aperture width (W= 2.5"). In this case, we find that J = 1.05 (maximum) effectively reduces the return loss IS11 I to -17dB from about -9.0dB as illustrated in Fig. 5b for the case of W=3.0". Similarly, the four poles also appear withn the pass band, allowing the design of ultra- broad and flat bandpass with an insertion loss of

Over a wide frequency range (1 .0 to 16.0GHz), a parasitic harmonic pass-band begins to emerge aroundf= 11.8GHz, close to the pass-band of interest. This leads to an unwanted degradation of its pass-band performance and a very narrow upper stop-band. To verify our predicted results from its equivalent circuit model, the commercial software HP Momentum is used to carry out an independent simulation of the entire filter layout. Fig. 6b illustrates the simulated results showing a very similar filtering behaviour to Fig. 6a, including the parasitic effects. In the following, we shall look into the possibility of suppressing the spurious harmonic band by attaching a capacitive stub into the line resonator.

IS21 I > -0.2dB.

4 experiment

Harmonic resonant frequencies of a uniform line resonator are approximately two or three times its fundamental resonant frequency [l]. To enable the change of resonant frequency spectrum, the non-uniform line resonator was found to be a powerful candidate decades ago [ 131. In [ 1 11, a specific non-uniform line resonator, called a 'stepped impedance resonator' (SIR), was presented for the design of a parallel-coupled stripline bandpass filter with good harmonic suppression. Ths line resonator is made up of

Improved bandpass filter: design and

IEE Proc-Microw Antennus Propay., Vol. 149, No. I . February 2002

0

-10

%!

rz" 7 - 6

-20 - .-

- -30

-40

-50 2 4 6 8 10 12 14

frequency f, GHz a

0

-1 0

m - " -20

rz" F

5

U -

7 -30

-40

-50 I 2 4 6 8 10 12 14

frequency f, GHz

b

Fig. 6 Optimisedand predicted jiequency responses for $Iter layout in Fig. 4 u Equivalent circuit model b Fullwave EM simulation

cascaded line sections with different characteristic impe- dance to shift up its first-order harmonic resonant frequency in terms of its impedance ratio. As pointed out in [l 13, this SIR allows the realisation of a broad rejection-band between its fundamental and spurious harmonic pass-bands in the filter.

Following the above description, the first harmonic passband in our filter is attributed to the third-order resonant frequency. Therefore, it is critical to build up a non-uniform line resonator with a large difference between its second- and third-order resonant frequencies. Fig. 7a shows the schematic layout of an improved bandpass filter with a non-uniform line resonator. This resonator is constructed by attaching a pair of open-ended stubs at its central location. Fig. 76 is its equivalent circuit topology, in which such a pair of stubs can be perceived as an equivalent frequency-dependent capacitance C, (a). In this case, the central location represents perfect short- and open-circuit terminals for the first/thrd-order resonant modes and for the second/fourth-order resonant modes, respectively. The attachment of such stubs is expected to lower the second/ fourth-order resonant frequencies, whde the first-/third- order resonant frequencies remain almost unchanged.

,

HI2 912 dl2 012

~~~ C*(4

#1 #2 #2 #I b

Fig. 7 posed bundpass j l t er with single cupacitive-loaded line resonator a Geometrical descnption with added stub tuner b Equivalent circuit model

Schematiclayout and equivalent circuit topology of pro-

Figs. 8a and b show the predicted frequency response of the filter layout with a fixed aperture, as shown in Fig. 7 4 for three different stub lengths. They are obtained on the basis of an equivalent circuit topology as in Fig. 7b for a relatively weak coupling, arranged to demonstrate the complete frequency spectrum of the first four resonant modes for the non-uniform line resonator. We observe that the first and third (odd) resonant frequencies remain stationary, even though the stub length is extended to a great extent. It is valid in theory because the central location corresponds to a perfect short circuit with an extremely large admittance. The second and fourth (even) resonant frequencies are found to simultaneously decrease towards the first- and third-order counterparts as the stub length increases. As a result, the rejection band of interest can be successfully tuned from 4.2GHz to 7.0GHz while the fundamental pass bandwidth drops down significantly from 3.2GHz to 0.8GHz. Thus, the attachment of such stubs raises the possibility of realising a broad bandpass filter with a good harmonic suppression, as well as an easy-to-tune broad frequency bandwidth ranging from 15% to 70% estimated from Fig. 8. Further, we can predict from its low return loss characteristic I SI 1 I < -20dB in a weakly coupling case of S= 4.8" that the capacitive shunt stubs may allow one to greatly alleviate the strict requirement of an extremely tight coupling. It provides us with a possibility for the development of a broad bandpass filter in the complete absence of the backside aperture with relatively weak coupling.

Now let us turn to the design of a four-pole bandpass filter with a single stub-loaded resonator through the equivalent circuit, as in Fig. 7b. Fig. 9a presents its optimised frequency response over a wide frequency range (1 .O to 1 S.OGHz), showing an excellent bandpass behaviour with bandwidth about 60% and return loss 1 SI I 1 < -30dB. Also, the four-poles with a minimum value of IS, I I can be observed from Fig. 90. A spurious harmonic band appears at f= 14.0GHz, exhibiting a much wider upper stop-band BW= 220% with reference to its central operating frequency fo = 6.OGHz. To verify such attractive features, the HP Momentum software is further used to simulate the entire filter layout through slight adjustment of line/aperture dimensions (L, Wand S), and a bandpass filter sample is

1 5 IEE Proc.-Microw. Antennas Propug., Vol. 149. No. I , February 2002

0

-1 0

8 - x

9

- - -20 - U m = -30

-40

-50

O r

' ..-,. ' . i

-I0 t g -20 1

I W = 1.4 mm I

/ ~ = 4 . 9 m m 1

-60 -50 tl 6 8 10 12 14 2 4

frequency f , GHz

a

2 4 6 8 10 12 14 16 frequency f , GHz

a

0

-10

-20

m * -30

-40

- - N

u) - c -50 ._ 4-

$ -60 ._

-70

-80

-90 2 4 6 8 10 12 14 16

frequency f , GHz b

Fig. 8 dijj7erent stub length S a Retum loss I SI I b Insertion loss IS,, 1

Optimisedfrequency response ofj l ter layout in Fig. 7 with

0

-Y -1 0

8 - - -20 - x - U

-30 5 g -40

-50 2 4 6 8 10 12 14

frequency f, GHz

b

Fig. 9 Fig. 7 a Equivalent circuit model b Simulations and experiments

Predictedand measured frequency responses of filter in

behaviour with two poles in the case of J L 1. Subse- quently, a uniform line section is brought between the two PCML sections to construct a prototype layout of the proposed bandpass filter. The operating mechanism of this filter has been explained and its frequency response optimally designed to exhibit its multi-pole and broad filtering property. The multiple poles are conceptually contributed by the first- and second-order resonant modes of the line resonator as well as a tight coupling of the two PCMLs, e.g. J = 1. Our results show a ultra-broadband and 4-pole bandpass behaviour with BW> 70% and I SI I < - 15dB. Furthermore, a pair of open-stubs is attached at the central location of this line resonator to effectively tune resonant frequencies of the second- and fourth-order harmonics. An improved version of the bandpass filter with a single capacitive-loaded line resonator has been designed to show its attractive features such as low return loss (< -30dB), wide upper stop-band (> 220%) and broad bandwidth (60%). Our measured results are in an excellent agreement with our predicted results over a wide frequency range (1.0 to 15.OGHz), and the fabricated filter shows I S1 1 < - 16dB and IS,, I > -0.6dB within the pass-band BW= 60%.

,

IEE Proc.-Microw. Antennas Propug.. Vol. 149, No. I , February 2002

then fabricated and measured. Fig. 9b describes simulated and measured frequency responses that are almost identical over the whole frequency range. Fig. 9b shows that both insertion losses IS,, I can be made with -0.6dB withn the pass band, wlule the simulated return loss I SI I is lower than -20dB against the measured ISll I lower than -16dB. On the other hand, the total length of this stub-loaded bandpass filter (D+L+D= 13.36") is about 0.7, at fo = 6. lGHz against 1.25,,,, for a conventional 4-pole parallel-coupled microstrip hne bandpass filter as in [ 1-31, demonstrating its attractive miniaturisation and low cost of fabrication.

5 Conclusions

In t h s work, a ground plane aperture technique has been originated in the PCML for effective enhancement of a tight frequency-dependent coupling, and further applied for innovative design of a new class of miniaturised, multi-pole and broadband microstrip bandpass filter. With our SOC de-embedding (calibration) procedure, the PCML can generally be characterised as an equivalent J-inverter network, allowing one to gain a better understanding of its coupling behaviour. The PCML itself exhibits its filtering

16

6 References

1 MATTHAEI, G.L., YOUNG, L., and JONES, E.M.T.: ‘Microwave filters, impedance-matchng networks, and coupling structures’, (Artech House, Nonvood, 1980)

2 COHN, S.B.: ‘Parallel-coupled transmission-line-resonator filter’, IRE Trans. Microw. Theory Tech., 1958, 6, (4), pp. 223-231

3 SCHWAB, W.: ‘Full-wave design of parallel-coupled microstrip band- pass filters with aligned input and output lines’. Proceedings of 26th European Microwave Conference, 1996, pp. 418421 HIRANO, M., NISHIKAWA, K., TOYODA, I., AOYAMA, S., SUGITANI, S., and YAMASAKI, K.: ‘Three-dimensional passive circuit technology for ultra-compact MMIC‘s’, IEEE Trans. Microw. Theory Tech., 1995, 45, (12), pp. 2845-2850

5 FORDHAM, Q., TSAI, M.J., and ALEXOPOULOS, N.G.: ‘Electromagnetic systhesis of overlap-gap-coupled microstrip filters’. IEEE MlT-S International Microwave Symposium Design, 1995, pp. 1199-1202

SU, T., and TANAKA, M.: ‘A novel high-Q and wide-frequency- range inductor using SI 3D MMIC technology’, IEEE Microw. Guid. Wave Lett., 1999, 9, (I), pp. 16-18

7 ZHU, L., and WU, K.: ‘Multilayered coupled-microstrip lines technique with aperture compensation for innovative planar filter

4

6 KAMOGAWA, K., NISHIKAWA, K., TOYODA, I., TOKUMIT-

design’. Proceedings of Asia-Pacific Microwave Conference, 1999, pp,

8 WORKSHOP, WMK: ‘Ultrawideband systems and application’. Proceedings of IEEE MIT-S Intemational Microwave Symposium, 2000

9 ZHU, L., and WU, K.: ‘Unified equivalent-circuit model of planar discontinuities suitable for field theory-based CAD and optimization of M(H)MIC‘, IEEE Trans. Microw. Theory Tech., 1999,47, (9), pp. 158S1602

10 ZHU, L., and WU, K.: ‘Characterization of unbounded multiport microstrip passive circuits using an explicit network-based method of moments’, IEEE Trans. Microw. Theory Tech., 1997, 45, (12), pp. 2 1 14-2 124 MAKIMOTO, M., and YAMASHITA, S.: ‘Bandpass filters using parallel coupled stripline stepped impedance resonator’, IEEE Trans. Microw. Theory Tech., 1980, 28, (12), pp. 141>1417

12 ZHU, L., and WU, K.: ‘Accurate circuit model of interdigital capacitor and its application to design of new quasi-lumped miniaturized filters with suppression of harmonic resonance’, IEEE Trans. Microw. Theory Tech., 2000, 48, (3), pp. 347-356

13 WOMACK. C.P.: ‘The use of exponential transmission lines in microwave components’, IRE Trans. Microw. Theory Tech., 1962,10, (3), pp. 124-132

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