Modeling of Microstrip line Gap Discontinuity on Multilayer Substrate

Abstract

In this paper, the modeling of microstrip line gap discontinuity on multilayer semiconductor substrate is presented. The reflection coefficient and transmission coefficient are computed from the equivalent lumped circuit model of microstrip line gap discontinuity. The lumped circuit elements- capacitance, inductance and resistance are calculated using closed-form expressions in term of the gap discontinuity, width of microstrip line and dielectric substrate thickness. Static Spectral Domain Analysis (SDA) method is used to compute characteristic impedance of microstrip line.

Introduction

I. INTRODUCTION

Microstrip line are used in the design of microwave integrated circuits (MICs) and monolithic MICs. The microwave components such as couplers, filters, interconnecting lines and matching components are essential part of a complex microwave circuits in hybrid and monolithic MICs. Discontinuities such as open-end, step and gap discontinuity are widely used in the microwave components design. For frequency dependent analysis, Full wave methods have been used, but these methods are complex and difficult to implement. Circuit model provides fast and accurate results at low frequencies; hence circuit models are used at low frequency for circuit design. [1-9]. The circuit model for gap discontinuity has been modified by incorporating lumped circuit elements. The results are compared with full-wave spectral domain analysis [5].

For the hybrid and semiconductor based integrated circuits, it is important to analyze and characterize the microstrip line discontinuities on multilayer substrate. This paper presents analysis of microstrip line gap discontinuity on multilayer substrate. For this, the static spectral domain analysis method and single layer reduction (SLR) technique are used to obtain the effective relative permittivity and characteristic impedance of microstrip line. The equivalent circuit model has been used to compute the frequency dependent reflection coefficient and transmission coefficient of microstrip line gap discontinuity. In section-II, SDA method for multilayer microstrip line is discussed. The circuit model for gap discontinuity and calculated numerical results are discussed in section-III and Section-IV respectively.

II. ANALYSIS OF MULTILAYER MICROSTRIP LINE

In this paper, Galerkin’s technique based static SDA method have been used for computing the capacitance of multilayer microstrip line. The characteristic impedance is computed from the line capacitance. The Green’s function in Fourier domain for multilayer microstrip line structure is obtained using transverse transmission line (TTL) technique [10-12].

The cross section of a shielded multilayer microstrip line is shown in Fig.1. The width of strip is S. The thickness of lower and upper dielectric substrate is H1 and H2 respectively. The permittivity of dielectric substrate is ε1 and ε2. The capacitance is computed from following relation [10-12].

1C=L2εoQ2n=1∞ρs2βn,yGβn,y                      (1)

where Gβn,y  is Fourier transform of Green’s function, ρs is Fourier transform of charge distribution and Q is total charge present on strip. The characteristic impedance is computed from Equation (2).

Zo=1cCdCair                                                          (2)

where c is velocity of light in free space, Cd is capacitance with dielectric substrate, and Cair is capacitance when dielectric is replaced with air.

Fig.1 Multilayer Microstrip Line

III. MICROSTRIP LINE GAP DISCONTINUITY

The microstrip line gap discontinuity and its equivalent circuit model are shown in Fig.2(a) and Fig.2(b) respectively. This circuit model pi-network of admittance parameters. The closed form expression for capacitance, inductance and resistance are calculated in terms of physical parameters of microstrip line, i.e., width of strip conductor (S), gap (G) and thickness of dielectric substrate [5].

C11=1ZoA1.tanhA2 pF                                                                                                                           3(a)

A1=1.125tanh1.358SHt-0.315

A2=0.0262+0.184HtS+0.217+0.0619lnSHtGHt

C12=1ZoB1.tanhB2pF                                                                                                                            3(b)

B1=6.832tanh0.0109SHt+0.91 ,     B2=1.411+0.314HtS+GHt1.248+0.36tan-1SHt

(a)

(b)

Fig.2 Microstrip line (a) Gap Discontinuity and (b) Equivalent Circuit Model

L11=ZoD1*expD2  nH                                                                                                                           3(c)

D1=0.134+0.0436lnHtS ,   D2=-3.656+0.246HtS.GHt1.739+0.39lnSHt

L12=ZoE1+0.1827+0.00715lnSHt*exp?E2  nH                                                     3(d)

E1=0.008285tanh0.5665SHt+0.0103

E2=-5.207+1.283tanh1.656HtS.GHt0.542+0.873tan-1SHt

R1=Zo.F1*tanhF2 ohm                                                                                                                         3(e)

F1=1.024tanh2.025SHt ,     F2=0.01584+0.0187HtSGHt+0.1246+0.0394sinhSHt

C2=J1+J2.sech2.3345GHt pF                                                                                                                    3(f)

J1=0.1776+0.05104lnSHtHtG ,     J2=0.574+0.3615HtS+1.156lnSHt

L2=ZoK1.sinh2.3345GHt  nH                                                                                                                    3(g)

K1=0.00228+0.8737.52SHt+coshSHt

R2=ZoM1+M2.sinh2.3345GHt  ohm                                                                                                    3(h)

M1=-1.78+0.749SHtGHt ,       M2=1.196-0.971lnSHt

Ht=H1+H2

The admittance parameters of circuit is evaluated from the lumped parameters of equivalent circuit using Equation 3(a)-3(h). From the admittance parameters equivalent circuit, the frequency dependent reflection coefficient and transmission coefficient are computed [5]. 

IV. NUMERICAL RESULT AND DISCUSSION

Microstrip line gap discontinuity is present on two-layer dielectric substrates. First layer is silicon (Si) dielectric substrate εr1 = 11.9 and thickness H1 = 0.4mm. Above the silicon dielectric substrate, a thin silicon dioxide layer is present. The relative permittivity of silicon dioxide is εr2 = 3.9 and thickness H2 = 3μm. The width of microstrip line strip is S=0.3mm. The characteristic impedance for S=0.3mm is 52.6 ohm. For different gap (G) ranging from 10μm to 20μm, 30μm and 40μm, the lumped circuit parameters are calculated.

Fig. 3(a) and Fig. 3(b) shows the variation of reflection coefficient (mag.) and reflection coefficient (phase, deg.) with frequency for different gap (G) discontinuity.  More field is radiated from the edges of gap discontinuity with increase of frequency. Due to increase of frequency, radiation loss and surface wave loss increases. For G=10μm and G=40μm, reflection coefficients are 0.34 and 0.62 respectively at 30GHz. Furthermore, the magnitude and phase angle of transmission coefficient are also plotted in Fig. 4(a) and Fig. 4(b) respectively for four gap, G=10μm, 20μm, 30μm and 40μm. As frequency increases, the radiation loss and surface wave loss increases. The higher order modes are also generated at high frequencies. Therefore, as the operating frequency increases, the transmission coefficient also increases.

Fig. 3(a) Reflection Coefficient (Mag.)

Fig. 3(b) Reflection Coefficient (Phase, deg.)

Fig. 4(b) Transmission Coefficient (Phase, deg.)

Conclusion

In this paper, equivalent lumped circuit model of microstrip line gap discontinuity on multilayer semiconductor substrate has been analyzed. The circuit model can be used to study and compute the reflection and transmission coefficient of gap discontinuity on multilayer substrate.

References

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Copyright

Copyright © 2024 Dr. Paramjeet Singh, Dr. Y. K. Awasthi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.