EVALUATION OF COMPLEX PERMITTIVITIES OF MULTILAYER DIELECTRIC SUBSTRATES AT MICROWAVE FREQUENCIES USING WAVEGUIDE MEASUREMENTS

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1 EVALUATION OF COMPLEX PERMITTIVITIES OF MULTILAYER DIELECTRIC SUBSTRATES AT MICROWAVE FREQUENCIES USING WAVEGUIDE MEASUREMENTS R. L. Crave, M. D. Deshpande, C. J. Redd 3, and P. I. Tiemsin () NASA Langle Research Center, Hampton, Virginia 368 () ViGYAN Inc.,, Hampton, Virginia 368 (3) Hampton Universit, Hampton, Virginia 3688 ABSTRACT The techniques that are presentl available for the measurement of complex permittivities of dielectric substrates are onl applicable to single laer substrates. This paper presents a new technique which uses the Finite Element Method (FEM) to estimate the complex permittivities of individual laers from the measurement of the S-parameters of a rectangular waveguide holding a multilaer dielectric substrate sample. In this method, a network analzer is used to measure reflection and transmission coefficients of a rectangular waveguide loaded with a laered sample. Using FEM, the reflection and transmission coefficients are determined as a function of the complex permittivities of the multilaer substrate. Measured and calculated values of the reflection and transmission coefficients are then matched using the Newton-Raphson Method to estimate the complex permittivities of the laers of the sample. Ke words: Permittivit Measurements, Finite Element Method.0 INTRODUCTION For the design of multilaer microwave integrated circuits it is necessar to know the complex permittivit of each individual laer of multilaer substrate material. Simple techniques to measure the dielectric constants of single laer substrate materials exist and are described in the literature [] - [3]. These techniques involve fabrication of microstrip lines using the material under test as a substrate. In [4], the dielectric constants of substrates have been determined using a slotted waveguide into which the substrate samples are inserted. This technique, which uses measured S-parameters and the characteristic equation for the dominant mode in the guide to solve for the permittivit, requires the sample to be place along the center of the waveguide. This technique is also valid for single laer substrates onl. In the method described in this paper, a waveguide is used as a sample holder for the substrates under test. The single or multilaered samples ma be placed in an configuration in the guide. This technique is an extension of one described previousl, in which the permittivit of a single laer sample was found b placing an arbitraril shaped sample at an position in a waveguide [5]. In the present technique, the reflection and transmission coefficients of the rectangular waveguide loaded with a laered sample are measured using a network analzer. The problem geometr is then simulated with a computer-aided design program, which also produces a FEM grid with different material parameters assigned to each laer of the sample. The reflection and transmission coefficients as functions of permittivit are computed using FEM. The Newton-Raphson Method is used to determine the values of permittivit for which the measured and calculated reflection and transmission coefficients match. Section.0 of this paper contains a discussion of the theor used to compute the reflection and transmission coefficients for given material parameters. Section 3.0 describes the technique used to obtain the measured values of the S-parameters. In Section 4.0, computation of sample permittivities from S-parameter data is explained. In Section 5.0, results are presented. Section 6.0 contains conclusions..0 THEORY (DIRECT PROBLEM) In this section FEM will be used to determine the reflection and transmission coefficients of a rectangular transmission line loaded with a multilaer dielectric material. Figure shows a rectangular waveguide with a laered dielectric sample. It is assumed that the waveguide is excited b a dominant TE 0 mode from the left. The reflection coefficient is measured at the reference plane P, and the transmission coefficient is measured at the reference plane P, as shown in figure (a). For the purpose of analsis the problem is divided into three regions; Region I (z < 0), Region II (0 z L), and Region III (z > 0). Using the waveguide vector modal functions, the transverse electromagnetic fields in the regions I and III are expressed as [6] E I (x,,z) I j γ0 z e 0 e I j γ z p + a e (x, e () p p p 0

2 H I (x,,z) I j γ z I 0 h 0 Y e 0 I j γ z I p - a h (x, Y e () p p p p 0 E III (x,,z) H III (x,,z) p 0 p 0 c p e (x, p c p h (x, p III j γ z p e III jγ z III p Y p e (3) (4) T E II II k εr µ r 0 E T dv V jωµ 0 I Y T e µ r 0 0 ds jωµ 0 I Y µ r p p 0 E II e over S p T e p S In deriving equations () - (4) it is assumed that onl the dominant mode is incident on the interface P, the a p are the amplitudes of reflected modes at the z 0 plane, and the c p are the amplitudes of transmitted modes at the z L plane. Y I p and γi p appearing in equations () - (4) are the characteristic admittance and propagation constant for the p th mode and are as defined in [6]. The unknown complex modal amplitudes a p and c p ma be obtained in terms of the transverse electric field over the planes P and P as follows: + a E e 0 over S 0 ds a E e p over S p ds c p E e over S p ds S where and S are the surface areas over the planes P and P. The electromagnetic field inside Region II is obtained using the FEM formulation [7]. The vector wave equation for the E II field is given b E II II (8) µ k εr r 0 E 0 Using the weak form of the vector wave equation and some mathematical manipulation the equation (8) ma be written as (5) (6) (7) jωµ 0 III Y µ r p p 0 E II e over S p S T e p ds S (9) In order to solve equation (9), the volume enclosed b Region II is discretized b using first-order tetrahedral elements [7]. The electric field in a single tetrahedron is represented as E II 6 b W (0) m m m where b m, m,, 3,...6 are the six complex amplitudes of electric field associated with the six edges of the tetrahedron, and W m, z) is the vector basis function associated with the mth edge of the tetrahedron. A detailed derivation for the expression for W m, z) is given in reference [7]. Substituting equation (0) into equation (9), integration over the volume of one tetrahedron results in the element matrix equation S el b v where the entries of the element matrices are given b ()

3 S ( m, n) W el n V W k εr W µ m 0 m W n dv r Rectangular Waveguide Dielectric Sample jωµ 0 I Y µ r p p 0 W n e p S TE 0 Mode z W m e p S Region I Reference Plane P Region II L Region III Reference Plane P jωµ 0 III Y µ r p p 0 W n e p S W m e p () S jωµ 0 I v ( n) Y W e (3) µ r 0 n 0 s d These element matrices can be assembled over all the tetrahedral elements in the Region II to obtain a global matrix equation S b v (4) Figure (a). Longitudinal view of rectangular waveguide loaded with laered sample x a Figure (b). Cross sectional view of rectangular waveguide loaded with laered sample Figure. Geometr of rectangular waveguide excited b TE 0 mode b The solution vector b of the matrix equation (4) is then used to determine the reflection coefficient at the reference plane P as a E 0 e d over S 0 s c 0 E e over S 0 ds S (5) (6) Coax Line HP-850 Network Analzer Rectangular Waveguide With Multilaer Sample Coax to Waveguide Transition Reference Plane P Reference Plane P Figure. Rectangular waveguide measurement sstem 3

4 3.0 MEASUREMENT SETUP The rectangular waveguide measurement sstem to measure the reflection coefficient is shown in figure. Before the measurement, a full two-port calibration using a standard waveguide calibration kit is done at the reference planes P and P. The reflection and transmission coefficients a 0 and c 0 of the rectangular waveguide with a laered sample piece are then measured as functions of frequenc. If the sample piece occupied the entire cross section of the waveguide and was a single laer, an algorithm which uses the Nicholson-Ross Technique [8-9] could be used to determine the complex permittivit of the sample. However, the algorithm which is based on the Nicholson-Ross Technique cannot be used when the sample piece occupies onl part of the cross section of rectangular waveguide, or for laered samples. When the dielectric sample is of arbitrar shape or laered the procedure described in next section is used 4.0 INVERSE PROBLEM This section presents the computation of the complex dielectric constant of a given sample piece from the measurement of the reflection and transmission coefficients. From the given geometr of the sample and its position in rectangular waveguide the reflection coefficient a ε', ε'', ε', ε'' and the transmission 0 r r r r coefficient c 0 ε', ε'', ε', ε'' are calculated using r r r r the FEM for assumed values of ε', ε'' and r r ε', ε''. If and are the measured reflection r r a 0 ' c 0 ' and transmission coefficients then the error in calculated values of reflection and transmission coefficients are a ε', ε'', ε', ε'' a ' and 0 r r r r 0 c 0 ε', ε'', ε', ε''. Writing the errors in r r r r c 0 ' real and imaginar parts we get f ε', ε'', ε', ε'' r r r r real ( a a ' ) 0 0 f ε', ε'', ε', ε'' r r r r imag ( a a ' ) 0 0 f 3 ε', ε'', ε', ε'' r r r r real ( c c ' ) 0 0 f 4 ε', ε'', ε', ε'' r r r r imag ( c c ' ) 0 0 (7) (8) (9) (0) If ε', ε'', ε', ε'' are incremented b small values r r r r to ε' + dε', ε'' + dε'' r r r r, ε' + dε', ε'' + dε'' r r r r such that f, f (,, and are.) f 3 f 4 simultaneousl zero then we can write following matrix equation [0] f dε' r f dε' r dε' r dε' r f r f r r r f dε' r f dε' r dε' r dε' r f r f r r r f f f 3 f 4 dε' r dε'' r dε' r dε'' r () From the solution of equation (), new modified values of ε', ε'', ε', ε'' are obtained as r r r r () (3) (4) (5) The reflection and transmission coefficients a ( and 0.) c 0 with modified values of ε', ε'', ε', ε'' r r r r are again calculated using the FEM procedure. With the new value of a ( and computation 0.) c 0 through equations (7) - (5) are repeated. The above procedure is repeated till required convergence is obtained (i.e. dε', dε'', dε' r r r 3 ε' ε' + dε' r new r r ε'' ε'' + dε'' r new r r ε' ε' + dε' r new r r ε'' ε'' + dε'' r new r r 4

5 and dε'' where the s are preselected small r 4 quantities). The procedure described above will converge to the true value of complex permittivit if the first choice of ε', ε'', ε', ε'' is close to the true values of r r r r complex permittivities. 5.0 NUMERICAL RESULTS For numerical results, samples made from RT/ Duriod 5500 and Alumina substrate materials are considered. To validate the present method, first single laer substrate samples were constructed. An RT/Duriod sample of size (.06 x 0.45 x.06 cm) was cut and placed in a x-band rectangular waveguide as shown in figure 3. Using a Hewlett-Packard 850 network analzer the reflection coefficient a 0 ' was measured over the x-band cm.9 cm x.0cm Real Part ( ε r ) cm.9 cm x.0cm Imaginar Part (ε r ) Real Part ( ε r ) Frequenc (GHz) Imaginar Part (ε r ) Frequenc (GHz) Figure 3. Real and imaginar parts of complex permittivit of RT/Duriod 5500 substrate over x-band. Manufacturer s specification: ε r.69, ε r From the measured and estimated values of reflection coefficients the complex permittivit was obtained and is presented in figure 3 along with the manufacturer s specified value. For a sample constructed from Alumina substrate material, the sample size and its placement in the x-band rectangular waveguide was the same as given in figure 3. The estimate of its complex permittivit over the Figure 4. Real and imaginar parts of complex permittivit ofalumina substrate over x-band. Manufacturer s specification: ε r 0, ε r 0.00 x-band was determined using the procedure described here and is presented in figure 4 along with the manufacturer s specification. The estimate of complex permittivit for both samples agrees well with manufacturers specification. To determine the permittivities of multilaer substrate, a sample made from RT/Duriod and Alumina substrates is placed in a waveguide as shown in figure 5. The reflection and transmission coefficients due to presence of such a sample in a x-band rectangular waveguide are then measured. From these measurements and following the procedure described in section 4.0 the dielectric constants of two laers are estimated. The results of these calculations will be presented at the conference. 6.0 CONCLUSIONS An FEM procedure in conjunction with the Newton-Raphson Method has been presented to determine complex permittivit of a dielectric substrate material. The substrate sample is placed at the center of an X-band waveguide and reflection and transmission 5

6 coefficients are measured using a Hewlett-Packard 850 Network Analzer. For the same configuration of the x- band waveguide loaded with the sample piece, the reflection and transmission coefficients are calculated using the FEM as a function of complex dielectric constants. The Newton-Raphson Method is then used to determine the complex dielectric constant b matching the calculated values with the measured values. The computed values of complex permittivit of RT/Duriod and Alumina using the FEM method are in good agreement with the manufacturer s specifications. RT/Duriod Alumina [] Moon_Que Lee & S. Nam, An accurate broadband measurement of substrate dielectric constant, IEEE Microwave and Guided Wave Letters, Vol. 6, No. 4, pp , April 996. [] N. K. Das, et al, Two methods for the measurement of substrate dielectric constant, IEEE Trans. Microwave Theor Tech., Vol. MTT-35, No. 7, pp , Jul 987. [3] R. M. Pannell & B. W. Jervis, Two simple methods for the measurement of the dielectric permittivit of low-loss microstrip substrates, IEEE Trans. Microwave Theor Tech., Vol. MTT-9, No. 4, pp , April 98. [4] R. A. York & R. C. Compton, An automated method for dielectric constant measurements of microwave substrates, Microwave Journal, pp. 5-, March 990. [5] M. D. Deshpande, et al, Measurement of complex permittivit of dielectric material at microwave frequenc using waveguide measurements, 7th AMTA Smposium, Willx iamsburg, VA., November 3-7, 995. [6] R. F. Harrington, Time-harmonic Electromagnetic Fields, McGraw-Hill Book Compan, New York,96. [7] C. J. Redd, et al, Finite element method for eigenvalue problems in electromagnetics, NASA Technical Paper 3485, December 994. [8] Robin L. Crave, et al, Dielectric propert measurements in the electromagnetic properties measurement laborator, NASA Technical Memorandum 047, April 995. [9] Dielectric material measurement forum, Hewlett Packard, 993. [0] W. H. Press, et al, Numerical recipes, The art of scientific computing (Fortran version), Cambridge Universit Press, Cambridge, 989 (chap. 9). Rect. Waveguide Figure 5. Geometr used for measurements of reflection and transmission coefficient due to two laers substrate. REFERENCES 6