Approximate Design Equation for Iris WidthCalculation of Iris Substrate Integrated Waveguide

(SIW) Bandpass FiltersRamanand Sagar Sangam

Department of Electronics and Electrical EngineeringIndian Institute of Technology Guwahati

Guwahati 781039, IndiaEmail: ramanand@iitg.ernet.in

Rakhesh Singh Kshetrimayum, Senior Memeber, IEEEDepartment of Electronics and Electrical Engineering

Indian Institute of Technology GuwahatiGuwahati 781039, IndiaEmail: krs@iitg.ernet.in

Abstract—This paper presents the approximate design equa-tion for iris width calculation of Iris Substrate IntegratedWaveguide (SIW) bandpass filter. The design equation hasbeen derived by using curve fitting techniques followed by de-embedding method and it is valid for narrowband band passfilter design; the fractional bandwidth should not be more than10%. A generalized procedure has been presented to calculatethe inverter parameters of the iris window. At 60 GHz frequency,the design equation has been developed for its validation. Thetheoretical and calculated normalized reactance values are ingood agreement. An iris SIW bandpass filter at 60 GHz has alsobeen designed using the developed procedure.

Index Terms—Substrate Integrated Waveguide, Iris Waveg-uide, Bandpass Filter, De-embedding.

I. INTRODUCTION

Substrate Integrated Waveguide (SIW) is a relatively newtechnology, which can be used to design microwave and mil-limeter wave components like filters, couplers and antennas[1]. The technology provides bridge between bulky metallicwaveguide and lossy planar technology.

The SIW filters design based on iris cavity are equivalentto shunt-inductance-coupled waveguide filters design [2]. Inrecent years, iris SIW filters have been reported for nar-rowband applications where the width of iris is calculatedfrom equivalent K-inverter. But it has not been reported inthe literatures how one can find the width of the iris fromthe K-inverter parameters for a given order. In this paper,a generalized procedure has been presented which can beutilized to design any narrowband iris SIW filters with FBWnot more than 10%. Finally, for 60 GHz band, the designequation is derived using curve fitting techniques. Also, aniris SIW filter at 60 GHz has been designed following thesame procedure.

II. DESIGN TECHNIQUES

Substrate integrated waveguide is made of two periodic ar-rays of metal cavity which connects the top and ground planeand works as side walls of conventional metallic waveguide.The longitudinal spacing between them is kept below a certainlimit so that radiation leakages are minimized [3].

The characteristics of electromagnetic fields in SIW struc-ture and its equivalent rectangular waveguide are similar.

Hence, a relation has been developed to calculate the effectivewidth aeff of the equivalent rectangular waveguide [4]

aeff = a− d2

0.95p(1)

where, d is the diameter of the metal cavity, a is the width ofSIW, p is the spacing between adjacent via holes and aeff isthe width of the equivalent rectangular waveguide, see Fig. 1.

After defining the dimensions of SIW structure, now itsequivalent to a conventional waveguide, so we can use theiris structure of waveguide to design iris SIW band pass filter.The resonators of band pass filter are cascaded via K-inverterswhere each iris is equivalent to a K-inverter; their normalizedimpedances of the inverters can be calculated from [5] as

K0,1

Z0=

√π

2

ωλ

g0g1(2)

Kj,j+1

Z0=

π

2

ωλ√gjgj+1

(3)

Kn,n+1

Z0=

√π

2

ωλ

gngn+1(4)

where Z0 is the waveguide impedance and ωλ is the guided-wavelength fractional bandwidth at the edges λg1, λg2 andmid-band λg0 as defined below

ωλ =λg1 − λg2

λg0(5)

The low pass prototype element values g for a given orderis found from [6]. Hence, one can find the K-inverterstheoretically.

Using the theoretically calculated values of K-inverters,we can find the shunt reactance Xj,j+1 for purely lumpedinductance discontinuities as follows [5]

Xj,j+1

Z0=

Kj,j+1

Z0

1−(

Kj,j+1

Z0

)2 (6)

Fig. 1. SIW structure.

Fig. 2. Model of inductive window and its equivalent circuit.

III. APPROXIMATE DESIGN EQUATION

Once the normalized impedances of the inverters are cal-culated, the physical dimensions of corresponding irises needto be found.

Since, each iris is equivalent to K-inverter with correspond-ing reactance X as shown in Fig. 2. The K-inverters canbe found from S parameters (from the simulation results)approximately as [2]

jX

Z0=

2S21

(1− S11)2 − S2

21

(7)

ϕ = − tan−1

(2X

Z0

)(8)

K

Z0=

∣∣∣∣tan ϕ

2

∣∣∣∣ (9)

Once the inverter parameters are known from equations(7)-(9), the distance between each iris windows (li) for filterdesign can be found as

li =θiλg0

2π(10)

θi = π +1

2(ϕi + ϕi+1) (11)

Steps followed for iris width calculation:Step 1: De-embedding: Using full wave simulator like

HFSS, we can chose de-embedding technique to find out Sparameters for different values of iris width instead of momentmethod [7]. The length of the SIW section is chosen about 0.5-1 wavelength or twice the width of the equivalent waveguidefrom each side of the iris. Then, the reference plane is movedto the centre of the iris for each port in order to de-embed

TABLE ICALCULATED AND THEORETICAL NORMALIZED INVERTER

PARAMETERS FOR DIFFERENT IRIS WIDTH

CalculatedXZ0

CalculatedKZ0

Calculatedϕ (radians)

Iris widthw (mm)

TheoreticalXZ0

0.1125 0.1111 0.2213 1.1 0.11250.12 0.11723 0.2356 1.12 0.11890.1324 0.13 0.259 1.15 0.13220.1541 0.15 0.3 1.2 0.15350.1774 0.172 0.341 1.25 0.17720.204 0.196 0.3874 1.3 0.20380.23128 0.22 0.43325 1.35 0.23120.261 0.2453 0.4811 1.4 0.26100.2923 0.271 0.529 1.45 0.29250.3043 0.2804 0.5467 1.47 0.30430.325 0.2965 0.574 1.5 0.32510.34625 0.3124 0.6057 1.53 0.34260.39868 0.35 0.673 1.6 0.39890.4777 0.4 0.7626 1.7 0.47620.5042 0.41667 0.78958 1.73 0.50420.521736 0.42673 0.80667 1.75 0.52170.548 0.4413 0.8312 1.78 0.54800.5658 0.451 0.847 1.8 0.56620.5858 0.4612 0.86425 1.82 0.58580.5944 0.4656 0.87144 1.83 0.59450.6131 0.4749 0.8867 1.85 0.61320.644 0.4896 0.9106 1.88 0.64400.6723 0.5 0.931 1.91 0.66670.765 0.541 0.992 2 0.76490.8712 0.579 1.05 2.1 0.87101.0986 0.6436 1.1437 2.3 1.0987

the phase changes occurred in the waveguide. Finally, extractthe S parameters at the mid-frequency.

Step 2: Substitute the S parameters into above equationsand get the inverter parameters.

Step 3: Repeat the procedure for different values of the iriswidth.

Step 4: Applying curve-fitting technique, plot the graphbetween inverter parameters (K,ϕ) and iris width (w).

Step 5: Obtain the approximate iris width corresponding tonormalized inverter parameters.

The specifications for the design are:• Solution Frequency : 60 GHz• Substrate: Taconic TacLamPLUS (ϵr = 2.1, h = 0.2 mm)The diameter of each metal cavity is set to 0.25 mm and

spacing between adjacent via holes is 0.4 mm. The width ofSIW is 2.8 mm.

By using curve fitting technique (Fig. 3), an approximate3rd order polynomial equation has been derived for iris widthscalculation corresponding to each normalized K-inverter

w=0.7734+3.4266K

Z0−4.6881

(K

Z0

)2

+4.7287

(K

Z0

)3

(12)

Table I lists the calculated and theoretical normalizedinverter parameters and also the approximate iris width cor-responding to them.

The theoretical normalized reactance values obtained fromequation (6) are very close to calculated normalized reactancefrom simulation, equation (7), validating the above procedureand approximate design equation. However, the design pro-cedure is valid only for narrowband filter but can be appliedfor any desired frequency band.

Fig. 3. Plot of iris width variation corresponding to normalized inverterparameter K.

IV. IRIS SIW BANDPASS FILTER DESIGN

Following the above procedure, a 3rd order Chebyshevfilter for V-band applications (9 GHz unlicensed bandwidthbetween 57 to 66 GHz) [2] has been designed. The insertionloss is 0.5 dB in the middle of the bandwidth and return loss(RL) is better than 15 dB in whole 57-66 GHz passband. Thenormalized inverter values are calculated first from equations(2)-(4) theoretically as K01 = K34 = 0.48 and K12 = K23

=0.28. Then dependencies of normalized inverter parameters(K,ϕ) with iris width (w) from equation (12) are used. Theiris width values corresponding to the calculated K-values areobtained as w1 =1.87 mm and w2 = 1.47 mm. Using the phasechange occurred (ϕi) from equation (8) into equation (11), thedistance between each iris windows from equation (10) canbe calculated as l1 = l3 = 1.68 mm and l2 = 1.81 mm. Thesevalues are considered as a first approximation for filter designand then tuned to achieve better passband return loss.

The schematic design of the filter has been shown in Fig.4. Their final dimensions are l1 = l3 = 1.65 mm, l2 = 1.78mm, w1 =1.83 mm and w2 = 1.51 mm.

The simulation has been performed using FEM based HFSSsoftware. The frequency response of the designed iris SIWBPF is shown in Fig. 5. The insertion loss is less than 0.5 dBand RL is better than 15 dB in whole designated passband.The performance of the filter is in good agreement with [2].However, there is not good rejection roll-off properties atthe higher frequencies which can be improved by using themethod as proposed in [2].

V. CONCLUSIONS

A design procedure has been presented for calculatingiris width of an inductive window of narrowband iris SIWfilter followed by de-embedding technique; the fractionalbandwidth is about 10%. At 60 GHz, a design equationhas been developed for calculating different width of iriswindows by using curve fitting technique and an iris SIW BPFhas also been designed following the proposed method. Thetheoretical and calculated normalized reactance parameters arewell matched which validate the same.

Fig. 4. 3rd order iris SIW bandpass filter design.

Fig. 5. Simulated frequency response of the designed iris SIW BPF.

REFERENCES

[1] M. Bozzi, A. Georgiadis, and K. Wu, “Review of substrate-integratedwaveguide circuits and antennas,” IET Microwaves Antennas Prop., Vol.5, No. 8, pp. 909-920, 2011.

[2] D. Zelenchuk and V. Fusco, “Low insertion loss substrate integratedwaveguide quasi-elliptic filters for V-band wireless personal area networkapplications,” IET Microwaves Antennas Prop., Vol. 5, No. 8, pp. 921-927, 2011.

[3] R. Garg, I. Bahl, and M. Bozzi, Microstrip Lines and Slotlines, 3rd

edition, Artech House, 2013.[4] Y. Cassivi, L. Perregrini, P. Arcioni, M. Bressan, K. Wu, and G.

Conciauro, “Dispersion characteristics of substrate integrated rectangularwaveguide,” IEEE Microwave and Wireless Components Letters, Vol. 12,No. 9, pp 333-335, 2002.

[5] D. Deslandes and K. Wu, “Millimeter-wave substrate integrated waveg-uide filters,” IEEE CCECE Canadian Conference on Electrical andComputer Engineering, Vol. 3, pp. 1917-1920, 2003.

[6] J.-S Hong and M. J. Lancaster, Microstrip Filters for RF/MicrowaveApplications, New York: Wiley, 2001.

[7] Y. Leviatan, P. G. Li, A. T. Adams, and J. Perini, “Single-post inductiveobstacle in rectangular waveguide,” IEEE Trans. Microwave TheoryTech., Vol. MTT-31, pp. 806-812, Oct. 1983.