NASNTM-2003-2 12398 

Estimation of Complex Permittivity of 

Composite Multilayer Material at Microwave 

Frequency Using Waveguide Measurements 

‘Manohar D. Deshpande and Kenneth Dudley 

Langley Research Center, Hampton, Virginia 

May 2003 

https://ntrs.nasa.gov/search.jsp?R=20030055680 2018-05-15T16:38:12+00:00Z
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NASMTM-2003-2 12398 

Estimation of Complex Permittivity of 

Composite Multilayer Material at Microwave 

Frequency Using Waveguide Measurements 

Manohar D. Deshpande and Kenneth Dudley 

Langley Research Center, Hampton, Virginia 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 2368 1-2 199 

May 2003 
The use of trademarks or names of manufacturers in the report is for accurate reporting and does not 

constitute an official endorsement, either expressed or implied, of such products or manufacturers by the 

National Aeronautics and SDace Administration. 

Available from: 

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(703) 605-6000 
Contents 

List of Figures iv 

viii 

Abstract 

I Introduction 

I1 Theory 

A. SI, and S,, Parameters 

B. S,? and SI, Parameters 

Numerical Results 

Estimation of S-Parameters( DirectFonvard Problem) 

Single Layer Composite Material 

Two Layer Composite Material 

Three Layer Composite Material 

Single Layer Composite Material 

Estimation of Dielectric Constants(Inverse Problem) 

Examples: For Known Sample Thickness 

Examples: For Unknown Sample Thickness 

Examples: For Known Sample Thickness 

Three Layer Composite Material 

Examples: For Known Sample Thickness 

Two Layer Composite Material 

IV Conclusion 

Acknowledgement 

References 

Appendix A: MatLab Code for Direct Problem 

Appendix B: MatLab Code for Inverse Problem 

1 

3 

4 

9 

10 

10 

10 

11 

15 

17 

17 

17 

25 

34 

35 

46 

46 

50 

51 

51 

52 

60 

... 

111 
List of Figures 

Figure 1 

Figure 2 

Figure 3 

Figure 4 

Figure 5 

Figure 6 

Figure 7 

Figure 8 

Figure 9 

Figure 10 

Figure 11 

Figure 12 

Figure 13 

Figure 14 

Figure 15 

Figure 16 

Geometry of rectangular waveguide loaded with multilayer composite 

material 

Geometry of rectangular waveguide loaded with acomposite material for 

estimation of S,, -- and S12 

Measured and estimated S -parameters of single Garlock slab. Thickness 

Measured and estimated S -parameters of single teflon slab. Thickness 

Measured and estimated S -parameters of two layers composite material 

(Bakelite-Teflon). Bakelite: AI = 0.33cm, zo = 0, z, = 0.33cm, 

E,. = 3.76 - j0.001,,~, = 1.0, Teflon: A, = 0.635cm, z, = 0.33cm, 

Measured and estimated values of Sz2 and S,? for composite slab 

(parameters as described in figure 5) 

Measured and estimated S -parameters of two layers composite material 

(Garlock-Nomex). Garlock: AI = 0.17cm, zo = 0, z, = 0.17cm, 

E, = 7.5 - jO.OOl,,u,. = 1.0, Nomex A? = 0.33cm, z1 = 0.17cm, 

Measured and estimated S -parameters of two layers composite material 

(Garlock-Nomex) with dimensions as shown in figure 7. 

Measured and estimated SI, and SzI parameters of composite slab 

consisting of Nano-Materialm, Garlock, and Garlock(Dimensions as 

described in text) 

Measured and estimated S,, -- and S,, parameters of composite slab 

consisting of Nano-Material#O, Garlock, and Garlock(Dimensions as 

described in text) 

Relative dielectric constant (real and imaginary) of Garlock material 

estimated using measured and ideal S-parameters. 

Differences between computed and measured values of S-parameters. 

Calculated S-parameters are determined using estimated dielectric 

constant 

Relative dielectric constant (real and imaginary) of Nan0 material #o 

estimated using measured and ideal S-parameters (thickness = 0.3099 cm) 

Relative dielectric constant (real and imaginary) of Teflon material 

estimated using measured and ideal S-parameters (thickness = 0.9398 ) 

Relative dielectric constant (real and imaginary) of Cork material 

estimated using measured and ideal S-parameters (thickness = 0.3048cm) 

Relative dielectric constant (real and imaginary) of Ceramic material 

estimated using measured and ideal S-parameters (thickness = 0.2845 cm) 

AI =0.17~m, E, =7.5-j0.001, ,u,. =l.O-jO.0 

AI = 0.635~m, E,. = 2.03 - jO.001, ,u,. = 1.0- jO.0 

z, = 0.965~m, E, = 2.03 - j0.001,,~,. = 1 

Z, = 0.50cm, E,. = 1.2 - j0.001,,~, = 1 

iv 
Figure 17 

Figure 1.8 

Figure 19 

Figure 20 

Figure 21 

Figure 22 

Figure 23 

Figure 24 

Figure 25 

Figure 26 

Figure 27 

Figure 28 

Figure 29 

Figure 30 

Figure 31 

Relative dielectric constant (real and imaginary) of Nomex Felt material 

estimated using measured and ideal S-parameters (thickness = 0.33cm) 

Relative dielectric constant (real and imaginary) of Rubylith material 

estimated using measured and ideal S-parameters (thickness = 0.3683 cm) 

Relative dielectric constant (real and imaginary) of Bakelite material 

estimated using measured and ideal S-parameters (thickness = 0.33cm 

Relative dielectric constant (real and imaginary) of Fiber Glass material 

estimated using measured and ideal S-parameters (thickness = 0.0533cm 

Relative dielectric constant (real and imaginary) of Woven Composite B 

material estimated using measured and ideal S-parameters (thickness = 

0.1625cm 

Relative dielectric constant (real and imaginary) of low density Foam 

material estimated using measured and ideal S-parameters (thickness = 

0.2921cm) 

Relative dielectric constant (real and imaginary) of Woven Composite G 

material estimated using measured and ideal S-parameters (thickness = 

0.2083 cm) 

Relative dielectric constant (real and imaginary) of Garlock material 

estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.17 cm. Thin solid line 

indicate estimated value of sample thickness., 

Differences between computed and measured values of S-parameters. 

Calculated S-parameters are determined using estimated dielectric 

constant and thickness of the slab 

Relative dielectric constant (real and imaginary) of Nan0 material #O 

estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.301 cm. Thin solid line 

indicate estimated value of sample thickness. 

Relative dielectric constant (real and imaginary) of Cork material 

estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.3048 cm. Thin solid line 

indicate estimated value of sample thickness 

Relative dielectric constant (real and imaginary) of Ceramic material 

estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.2845 cm. Thin solid line 

indicate estimated value of sample thickness. 

Relative dielectric constant (real and imaginary) of Nomex Felt material 

estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.3302 cm. Thin solid line 

indicate estimated value of sample thickness. 

Relative dielectric constant (real and imaginary) of Rubylith material 

estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.3683 cm. Thin solid line 

indicate estimated value of sample thickness. 

Relative dielectric constant (real and imaginary) of Bakelite material 

V 
Figure 32 

Figure 33 

Figure 34 

Figure 35 

Figure 36 

Figure 37 

Figure 38 

Figure 39 

Figure 40 

Figure 4 1 : 

Figure 42: 

Figure 43: 

estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.3302 cm. Thin solid line 

indicate estimated value of sample thickness. 

Relative dielectric constant (real and imaginary) of Teflon material 

estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.9398 cm. Thin solid line 

indicate estimated value of sample thickness. 

Relative dielectric constant (real and imaginary) of Fiber Glass material 

estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.0533 cm. Thin solid line 

indicate estimated value of sample thickness. 

Relative dielectric constant (real and imaginary) of Foam material 

estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.2921 cm. Thin solid line 

indicate estimated value of sample thickness. 

Relative dielectric constant (real and imaginary) of Woven Composite B 

material estimated assuming thickness unknown. Symbols indicate 

estimated values of dielectric constant using thickness = 0.1626 cm. Thin 

solid line indicate estimated value of sample thickness. 

Relative dielectric constant (real and imaginary) of Woven Composite G 

material estimated assuming thickness unknown. Symbols indicate 

estimated values of dielectric constant using thickness = 0.2083 cm. Thin 

solid line indicate estimated value of sample thickness. 

Measured and computed S-parameters for Fiber Glass Material. 

Computed S-parameters are determined using estimated values of 

dielectric constant and thickness of Fiber Glass. 

Relative dielectric constant (real and imaginary) of Bakelite-Teflon 

Composite material estimated using measured S-parameters (assuming 

thickness known 

Relative dielectric constant (real and imaginary) of Bakelite-Teflon 

Composite material estimated using ideal S-parameters (assuming 

thickness known) 

Value of error function given in (19) in estimating the dielectic constants 

of Bakelite-Teflon Composite using both measured and ideal Sparameters. 

Relative dielectric constant (real and imaginary) of Garlock-Nomex Felt 

Composite material estimated using measured S-parameters (assuming 

thickness known) 

Relative dielectric constant (real and imaginary) of Garlock - Nomex Felt 

Composite material estimated using ideal S-parameters (assuming 

thickness known) 

Value of error function given in (19) in estimating the dielectic constants 

of Garlock -Nomex Felt Composite using both measured and ideal Sparameters. 

vi 
Figure 44: 

Figure 45: 

Figure 46: 

Figure 47: 

Figure 48: 

Figure 49: 

Figure 50: 

Figure 51 

Figure 52 

Figure 53 

Figure 54 

Figure 55 

Figure 56 

Figure 57 

Figure 58 

Figure 59 

Relative dielectric constant (real and imaginary) of Garlock-Ceramic 

Composite material estimated using measured S-parameters (assuming 

thickness known) 

Relative dielectric constant (real and imaginary) of Garlock - Ceramic 

Composite material estimated using ideal S-parameters (assuming 

thickness known) 

Value of error function given in (19) in estimating the dielectric constants 

of Garlock -Ceramic Composite using both measured and ideal Sparameters. 

Relative dielectric constant (real and imaginary) of Garlock-Nan0 Material 

W Composite material estimated using measured S-parameters (assuming 

thickness known) 

Relative dielectric constant (real and imaginary) of Garlock - Nan0 

Material #o Composite material estimated using ideal S-parameters 

(assuming thickness known) 

Value of error function given in (19) in estimating the dielectic constants 

of Garlock -Nan0 Material #O Composite using both measured and ideal 

S-parameters. 

Relative dielectric constant (real and imaginary) of Garlock-Woven 

Composite G material estimated using measured S-parameters (assuming 

thickness known) 

Relative dielectric constant (real and imaginary) of Garlock Woven 

Composite G material estimated using ideal S-parameters (assuming 

thickness known) 

Value of error function given in (19) in estimating the dielectic constants 

of Garlock -Woven Composite G material using both measured and ideal 

S-parameters. 

Relative dielectric constant (real and imaginary) of Garlock-Bakelite 

Composite material estimated using measured S-parameters (assuming 

thickness known) 

Relative dielectric constant (real and imaginary) of Garlock - Bakelite 

Composite material estimated using ideal S-parameters (assuming 

thickness known) 

Value of error function given in (19) in estimating the dielectic constants 

of Garlock -Bakelite Composite material using both measured and ideal 

S-parameters. 

Estimated values of dielectric constants of individual layers from 

measured values of S-parameters of three layer composite slab (Nano 

Material W-Garlock-Garlock) 

Final values of error function in estimation of dielectric constants of 

individual layers shown in Figure 56 

Estimated values of dielectric constants of individual layers from 

measured values of S-parameters of three layer composite slab (GarlockFinal values of error function in estimation of dielectric constants of 

individual layers shown in Figure 58 

Garlock-Garlock) 

vii 
Symbols 

a 

b 

x-dimension of rectangular waveguide 

y-dimension of rectangular waveguide 

irh waveguide modal vector functions 

Transverse electric fields at zo, zI .... zN interfaces 

Transverse electric and magnetic fields in z I 0 region 

ERT ,/ERI +ER,....+ER, 

FSS Frequency Selective Surfaces 

N 

n 

Total Number Dielectric Layers 

n"' layer 

MIC Microwave Integrated Circuits 

MMIC Monolithic Microwave Integrated Circuit 

... 

Vlll 
Two port S-parameters 

Measured values of S-parameters 

Calculated values of S-parameters 

Transverse electric dominant mode 

Amplitude of j" mode at z = z, transverse plane 

irh modal admittance for the z I 0 region 

irh modal admittance for the zll-l I z I z, region 

modal admittance for for the z 2 zN region 

Location of planes for n'" layer 

Thickness of nth layer 

Relative permitivity and permeability of nrh layer 

irh mode propagation constant for the z I 0 region 

i"' mode propagation constant for the zll-I I z I zll region 

irh mode propagation constant for the z 2 zN region 

Thickness of nrh layer 

i rh 

ix 
Abstract 

A simple method is presented to estimate the complex dielectric constants of 

individual layers of a multilayer composite material. The multilayer composite material 

sample is loaded in an X-band rectangular waveguide and its two port S-parameters are 

measured as a function of frequency using the Hewlett-Packard (HP) 8510 Network 

Analyzer. Also, by applying the mode matching technique, expressions for the Sparameters of the composite material as a function of electric properties of individual 

layers are developed. Using the MatLab Optimization Tools simple MatLab scripts are 

written to search for electric properties of individual layers so as to match the measured 

and calculated S-parameters. 

A single layer composite material formed by using materials such as Bakelite, 

Nomex Felt, Fiber Glass, Woven Composite B and G, Nan0 Material #O, Cork, Garlock, 

of different thicknesses are tested using the present approach. The dielectric constants of 

these materials estimated using the present approach (assuming the thicknesses are 

known) are in good agreement with their true values. Assuming the thicknesses of 

samples unknown, the present approach is shown to work well in estimating the 

dielectric constants and the thicknesses. 

above individual materials are tested using the present approach. The values of dielectric 

constants of individual layers estimated (assuming the thickness of individual layers are 

known) using the present approach are in good agreement with their true values. 

However, the present approach could not provide estimate values close to their true 

values when the thicknesses of individual layers were assumed to be unknown. This is 

attributed to the difficulty in modelling the presence of airgaps between the layers while 

doing the measurement of S-parameters. A few example of three layer composite are 

also presented. 

A number of two layer composite materials formed by various combinations of 

I. Introduction 

Multilayer substrates are used for many practical applications such as 

Microwave Integrated Circuits (MIC), Monolithic MIC ( MMIC) [ 13 , radomes for 

protection of antennas from external environment, spatial filters for antenna beam 

shaping [2], and Frequency Selective Surfaces (FSS) [3-41. 

individual layers in a multilayer composite substrates it is possible to achieve a 

composite material with altogether new properties that were otherwise not found in the 

individual layers. Exact knowledge of the material properties such as permittivity and 

With proper choices of 
permeability of individual layers in a multilayer substrate is essential for designing spatial 

filters, radomes and composite materials for FSS applications. However, the present 

practice in the estimation of electric properties of composite materials with multi-layers 

emphasizes only the determination of overall effective properties. In this report an 

attempt is made to provide a procedure to estimate the electric properties of individual 

layers of a composite material. 

Permittivity and permeability of these composite multi-layer substrates can be 

estimated using one of the following methods: 1) free-space techniques; 2) resonant 

cavity perturbation techniques; and 3) transmission line methods. Each method has its 

own advantages and limitations. For measurement of electric properties over a wide 

frequency range, transmission line or waveguide methods [5-81 are more popular even 

though they are less accurate due to unavoidable measurement errors. In the waveguide 

measurement method, a sample of composite multilayer substrate is placed in a 

waveguide and two port S-parameters are measured using a hp-8510 Network Analyzer. 

In the earlier work [7-81, assuming that an equivalent homogeneous material occupies 

the sample space, S-parameters are estimated as a function of effective electric properties 

of the equivalent material. Using the inverse procedure, constituent parameters of the 

equivalent material are determined by matching the estimated and measured Sparameters. In these methods electric properties of individual layers are not determined. 

However, for some applications such as radomes and FSS, a knowledge of electric 

properties of individual layers is required. In this report, a waveguide measurement 

method is presented to estimate the electric properties of individual layers of a composite 

multilayer substrate. 

2 
The remainder of this report is organized as follows. The mode matching 

formulation of a waveguide loaded with multilayer composite material is developed in 

section II. Also in section II, the expression are developed to estimate 2-port Sparameters of composite material slab. Numerical and measured results on the Sparameters of numerous composite material are presented in section III for the direct 

problem where electrical properties of individual layers are assumed to be known. Also 

in section III, using the MatLab Codes given in Appendices, estimated values of 

dielectric constants of one, two, and three layer composite material are presented. The 

report concludes in section IV with remarks on the validity and usefulness of the present 

method. 

11. Theory 

In this section, the method of moments is used to determine the S-parameters of 

a rectangular waveguide loaded with a multilayer dielectric substrate as shown in figure 

1. 

Y Incident Wave pod #1 

Reflected Wave 

Figure 1 : Geometry of rectangular waveguide loaded with 

mu1 tilayer composite material. 

3 
The multilayer substrate consists of N layers, where the nrh layer has parameters 

(E,.,] ,p,.,, ) and is located between the transverse planes at z = z,-, and z = z,, . 

SI, and S,, Parameters: 

To estimate the SI, and SZ1 components of 2-port parameters, it assumed that the 

TE,, mode of unit amplitude is incident on the interface at z = 0 from the region z 5 0 . 

- 

If EI0, E,, .... EIN are the transverse electric fields on the interfaces at z = 0, z,, z1 ,...zN , 

respectively, then the transverse electric fields in the various regions of waveguide are 

obtained as 

4 
for z 2 zN. 

For uniqueness, the tangential magnetic fields at each interface must be continuous. 

Hence 

for the interface located at z = 0, 

for the interface located at z = z,, , and 

5 
for the interface located at z = zN . 

The transverse electric fields over the interfaces can be expressed in terms of vector 

modal expansion functions as: 

- J, - J,, - J iv - JO 

E,, = cTojej , E,] = CT,,Zj , E,, = zTljZj , .... E, = CTNlej ~ 

j=O j=O j=O j=O 

, 

I 

I 

where Toj ,Tj ,.... TNl are the complex unknown coefficients. Substitution of (10) into (7)- 

(9) yields 

JO 

2Y:h, = CTojij 

j=O 

I JN-I 

= CqN-lrj 

j=O 

6 
Equations (1 1) -( 13) are the required integral equations to be used to determine the 

complex amplitudes Toj ,qj ..... TN . Selecting h, as a testing function and using the 

- 

Galerkin's procedure, the equations (1 1)-( 13) are converted into a set of simultaneous 

equations: 

............... 

............... 

obtained from equation (1 1). From the continuity of magnetic field at z = z, we get 

............... 

............... 

7 
Likewise, using the continuity of magnetic fields at z = z7, z = z3 ..... z = zN-], similar sets 

of simultaneous equations are obtained. From the continuity of magnetic fields at z = zN 

we get 

......... 

......... 

Due to the orthogonal nature of vector modal functions it can be shown that the complex 

amplitudes Toj, j = 1,2,3 ... J,, Tj, j = 1,2,3 ... Jl, ...... Tw, j = 1,2,3 ... JN are all zeros. 

Hence equations (14)-( 16) can be simplified as 

................................. 

................................. 

8 
solution of above (N + 1) equations gives an estimate of complex amplitudes 

T, ,To ,TZ0 ,...... TNo from which S,, and S,, are determined as 

S,, = Too -1 and Sz1 = TNOejPOOZN 

S,, and SI, Parameters: 

The port 2 parameters, S,, and S,2 can be determined by following the procedure used 

for estimation of S,, and S,, , and reversing the locations of the layers as shown in 

figure 2. Note that the SZ2 calculated using the reference planes shown in figure 2 and 

S,, -- measured using the hp-8510 network analyzer differ by phase eipOo2" . 

Figure 2: Geometry of rectangular waveguide loaded 

with a composite material for estimation of S,, -- and SI, 

9 
111. Numerical Results 

A: Estimation of S-Parameters( Direct/Forward Problem) 

A simple MatLab code (Appendix A) is written to solve the simultaneous equations 

given in equations (1 7) and determine all four S -parameters of a composite material slab 

placed in a rectangular waveguide. In this section, assuming the properties of individual 

layers of a composite material known, the S -parameters for various composite slabs are 

computed (using the MatLab code) as a function of frequency and compared with the 

measured S -parameters. 

Single Layer Composite Material: 

Figure 3 shows S,, and S,, parameters of a composite material consisting of a 

Garlock single layer as a function of frequency. In this case a single layer of Garlock 

with thickness A, = 0.17~~~ and electric properties E, = 7.5 - jO.001 is used to form a 

composite material. Excellent agreement between measured and estimated 

0 10 11 12 -1 ' . ' ' " ' '. " '' " * ' 

FmqurneyfGHz) 

Figure 3: Measured and estimated S -parameters of single 

Garlock slab. Thickness A, = 0.17crn, E, = 7.5 - jO.001, 

p, = 1.0- jo.0 

10 
values of the S -parameters validates the MatLab code. Figure 4 shows the S,, and Szl 

parameters of a composite material formed by a single layer of Teflon material 

(E, = 2.03 - jO.001, p, = 1 .O - jO.0 ) of thickness AI = 0.635cm . A good agreement 

between the measured and estimated values of S -parameters confirms validity of the 

present method. Note that the parameters SZ2 and SI2 are expected and found to be 

identical to S,, and SZl , respectively. 

I. # 1 E 

10 11 12 Frequmcy(GHz) 

Figure 4: Measured and estimated S -parameters of single teflon slab. 

Thickness A, = 0.635cm, E, = 2.03 - jO.001, p, = 1.0 - jO.0 

Two Layer Composite Material: 

For further validation of the present method and MatLab code, the S -parameters of 

composite material formed by various combination of two layers are fabricated and 

tested. A first sample considered consists of Bakelite and Teflon layers. 

11 
The composite slab is formed by placing the bakelite layer with A, = 0.33cm, 

zo = 0, z1 = 0.33cm and the Teflon layer with A2 = 0.635crn, zI = 0.33cm, 

z2 = 0.965~~~. Measured and estimated SI, and SZ1 parameters as a function of 

frequency are shown in figure 5. For the composite material described in figure 5, the 

measured and estimated SZ2 and SI2 values are shown in figure 6. 

0.5 

&25 

9 

Eo 

I 

4. 

YP.25 

-0.5 

-0.75 

-1 

BakeWIe-Tenon lsyrn 

Tenon d = 0.635 an. er = 2.03 -pdoi BPbh d = 0.3302 cm, OT = 3.16 - 0 001 

- Rerl(S11) 

- - I - Img(S11) 

C.lcumh - Real(S21) 

9 10 11 12 

Frequency(GHz) 

Figure 5: Measured and estimated S -parameters of two layers composite 

material (Bakelite-Teflon). Bakelite: AI = 0.33cm, zo = 0, zI = 0.33cm, 

E, = 3.76 - jO.OO1,,uu, = 1.0, Teflon: AZ = 0.635cm, zI = 0.33crn, 

z7 = 0.965~m, E, = 2.03 - j0.001,,~, = 1 

12 
= - = - Imag.(S12) 

-+- Rtrl(S22) 

. lmag.(S22) 

MU.UIIIYIII -+- Rcal(S12) 

nr - 4- - Imagg.(S1 2) 

-- -- 

Frequency(GHz) 

Figure 6: Measured and estimated values of SZ2 and SI2 for composite 

slab (parameters as described in figure 5) 

A second sample of composite material considered consists of Garlock and Nomex 

materials. The composite slab is formed by placing first a Garlock slab of thickess 

A, = 0.17cm, E, = 7.5 - jO.OO1,pu, = 1.0 at zo = 0 and z1 = 0.17cm and then a Nomex 

slab of thickness A, = 0.33cm, E, = 1.2- jO.OOl,p, = 1.0 at z1 = 0.17cm and 

z2 = 0.5cm. Measured and estimated SI, , Szl , SI2 , and S2, parameters for GarlockNomex composite slab are shown in figures 7 and 8. 

From figures 7-8, estimated values of S -parameters for the Garlock-Nomex composite 

slab agrees well with measured values of the S -parameters. However, for the BakeliteTeflon combination there is small disagreement between measured and estimated S,, -- and 

S,? parameters. This disagreement may be attributed to the presence of an air gap 

between the slabs of Bakelite and Teflon. 

13 
os 

Figure 7: Measured and estimated S -parameters of two layers 

composite material (Garlock-Nomex). Galock: A, = 0.17cm, zo = 0, 

z, = 0.17cm, E, = 7.5 - jO.OOl,p, = 1.0, Nomex A, = 0.33cm, 

z, =0.17~m, z., = OSOCW~, E, = 1.2- jO.001,,~, = 1 

O=l os 

Figure 8: Measured and estimated S-parameters of two layers composite 

material (Garlock-Nomex) with dimensions as shown in Figure 7 

14 
Three Layer Composite Material: 

A third sample of composite material considered consists of Nan0 Material # 0, 

Garlock and Garlock slabs. The composite slab is formed by placing first a Nan0 

Material W slab of thickess AI = 0.31cm, E, = 2.5 - jO.OOl,,u, = 1.0 at zo = 0 and 

z1 = 0.31cm and then a Garlock slab of thickness A2 = 0.17cm, 

E, = 7.5 - jO.OOl,p, = 1 , at z1 = -.31cm, and z2 = 0.48cm. Another Garlock slab of 

thickness AI = 0.17cm, E, = 7.5 - jO.OOl,p, = 1.0 is placed at z, = 0.48cm and 

z3 = 0.65cm. Figures 9-10 show measured and estimated S -parameters of composite 

slab consisting of Nano-Mat #0-Garlock-Garlock combination. 

Cornpowre Slab. Nan, Metanl XO, Garlock. - Garlock 

Real(S11) 

= - = - Imag.(Sll) 

= 1- 11-1 Real(S21) 

1-11 lmag(S21) 

- -:,& - Real(S11) 

.. - Imag.(Sll) 

Real(S21) 

Imag.(S21) 

Frequency(GHz) 

Figure 9: Measured and estimated SI, and S, parameters of composite slab 

consisting of Nano-MaterialW, Garlock, and Garlock(Dimensions as 

described in text) 

15 
Cornpostta Slab: Nsna MaWril XO, GallodS Gs~lock 

- Real(S22) 

- - - - lmag.(S22) 

---*-B- Real(Sl2) 

1--- lmag(S12) 

- 22,- Real(S22) 

0.5 ---E- ~ lmaa.lS22) -. Real(S12) 

I lmag.(S12) 

I8 I I 1 

9 10 11 12 -1 

Frequency( GHz) 

Figure 10: Measured and estimated S,, and S1, parameters of 

composite slab consisting of Nano-MaterialW, Garlock, and 

Garlock(Dimensions as described in text) 

From figures 9-10, the trend in the values of S -parameters predicted by the computation 

is similar to the trend observed in the measured values. However, significant differences 

are observed between the measured and estimated values of S -parameters. This may be 

attributed to the sample preparation where the presence of air gaps between the 

individual slabs is unavoidable. Also even though these individual slabs may be in 

physical contact with each other, electrical contact may still not be insured. It is also 

noticeable that the disagreement between the measured and estimated S -parameters gets 

worse as more individual slabs are used to form a composite slab. 

16 
- 

B: Estimation of Dielectric Constants( Inverse Problem): 

Single Layer Composite Material: 

In this section a procedure is described for the computation of the complex 

dielectric constant of a given composite sample from the two port measured data. For a 

given composite material sample, let the two port S -parameters be ( Slln,, S21m, 

SZzm ), measured using the hp-8.510 Network Analyzer. With a prior knowledge of 

the thickness of composite slabs, the two port S -parameters can be estimated using the 

MatLab Code given in Appendix A as a function of E,. Let the estimated values of S - 

parameters be Sllc(~r),S21c(~r),S12r(~r),S22c(~r). The errors in estimated and measured 

S -parameters can then be written as ER, = real(S,,, - Sllnl) , ER, = imag.(S,,, - S,,,), 

ER, = real(S,,, - S21nl 1, ER, = img.(S2,c - S21m) , . . ... ER, = real(S,,, - SZzm) , 

ER, = imag.(S',,, - S2,,). The total mean squared error or the objective function to be 

minimized as a function of E, can be written as 

ER, = J(ER; + ER; + ER; + ER; + ER; + ER; + ER; + ER,' ) (19) 

A simple MatLab Code given in Appendix B minimizes the objective function in (19) to 

estimate the unknown value of dielectric constant E,. 

Examples (When Thickness of Sample Is Known): 

Garlock Slab: A single layer of Garlock slab of size (2.29 x 1.02 x 0.17 )cm was placed 

in an X-band rectangular waveguide. After proper calibration of the hp 851 0 Network 

Analyzer, two port S-parameters of the Garlock slab were measured as a function of 

frequency. The measured data is stored in a file R-GRLK.60. Using the MatLab Code 

17 
given in the Appendix B, the dielectric constant of the Garlock slab is estimated and 

shown in figure 1 1. 

Real and lmaginaty Parts of ReIative Dlelgtrr 

Constant of Garlock Matena) 

C 

E 

Real Part 

0 

- Real Part Estimated Uslng Ideal S-Parameters(er I 7 5 -10 001) -#-I-*- hag. Part 

hag. Part 

0 Real Part EsbmatsJ ueng hasurd s-paramas~ 

Imqnary Pan .&. 

9 10 11 12 

Frequency( GHz) 

Figure 1 1 : Relative dielectric constant (real and imaginary) of 

Garlock material estimated using measured and ideal Sparameters. 

In figure 11 solid and dash-dot lines are the estimates of dielectric constant using the 

ideal or noise free S-parameters calculated assuming the dielectric constant is known and 

equal to E, = 7.5 - jO.001. The estimates shown in figure 11 were obtained by 

considering the thickness of the slab equal to 0.170cm. To check the level of confidence 

in these estimates, from the estimated values of dielectric constant (estimated using the 

measured data), the difference between measured and computed values of S-parameters 

is plotted in Figure 12. 

18 
1. I,, I,,, 

9 10 11 12 

Frequency (GHz) 

Figure 12: Differences between computed and measured values of Sparameters. Calculated S-parameters are determined using estimated 

dielectric constant 

The differences between the computed and measured S-parameters are within the limits 

set in the optimizer. Ideally, the difference between the computed and measured Sparameters must be close to zero. However, to achieve ideal results, the optimizer would 

take longer time. 

Number of single layer composite materials of various material and thickness were 

constructed and their S-parameters were measured over the X-band frequency range. 

From these measured values of S-parameters and using the MatLab Code given in 

Appendix B, the dielectric constants of these single layer composite were estimated and 

presented in figures 13-25. 

19 
U 

-0 25 .;::I , , , I I , , , , 

1 9 10 11 12 Frequency( QHz) 

Figure 13: Relative dielectric constant (real and imaginary) of Nan0 

material #O estimated using measured and ideal S-parameters 

(thickness = 0.3099 cm) 

Real and Irn Constant 01 %Ion Material 

may Pans 01 Relative D~dedrl~ 

Frequency( GHz) 

Figure 14: Relative dielectric constant (real and imaginary) of Teflon 

material estimated using measured and ideal S-parameters (thickness = 

0.9398 ) 

20 
Real ilnd Im Constant 01 E"* Matend inary Pans of Rolati~ Did.Clr~ 

- RealParI 

I ,- ,I,_ Part 

Itnag. Part 

Eshh~ated Urng Ideal S-Parameters(n - 1 5-10 000 

2.5 Real Estimated u~ng Measured S-Parameters 

42.25 

g1.75 2k Real Part 

Imaginary Part K0.25 

-0.75 -05 L I I 1 1, 

9 10 11 12 -1 1 

Frequency( GHz) 

Figure 15: Relative dielectric constant (real and imaginary) of Cork 

material estimated using measured and ideal S-parameters (thickness = 

0.3048cm) 

Real aM Im Constant et?wamr Matvnal inary Pans of Relative Dielectric 

- RealPart 

Real part Est1mat.d Using Measuroj S-Parameters 

Itnag. Part s.25 

62 

Real Pari 

0 

0 15 

Imaginary PaIt 

Frequency( GHz) 

Figure 16: Relative dielectric constant (real and imaginary) of Ceramic 

material estimated using measured and ideal S-parameters (thickness = 

0.2845 cm) 

21 
Rsal and lmqinq PnRs 01 Rddiw Ikdeurr Constant of Nomex Felt Matend 

3tFrequency(GH2) 

Figure 17: Relative dielectric constant (real and imaginary) of 

Nomex Felt material estimated using measured and ideal Sparameters (thickness = 0.33cm) 

- RealPart I ,_, I part Esbmated Usng Ideal S-Parameten 

Real Part Estimated Using Measured S-Parameters 

Imag. Part 

I I I I 

9 10 11 12 

Frequency(GHz) 

Figure 18: Relative dielectric constant (real and imaginary) of Rubylith 

material estimated using measured and ideal S-parameters (thickness = 

0.3683 cm) 

22 
Real and Im Constant of?*llte Matend inaly Parts of Relative Dielectrr 

I I I I 

4 - 

5 

m 

u) 

- 

E3 

0 

0 

'c 

H2 

Pa,, Estimated Using Ideal S-Parameters 

Real Part Estimated Uang Measured S-Patamaterr z 

n 

- - g1 

m 

Q) 

K 

- Imaginary Pan 

0 

Figure 19: Relative dielectric constant (real and imaginary) of Bakelite 

material estimated using measured and ideal S-parameters (thickness = 

0.33cm 

Esbmated Using Ideal S-Patameerr 

Estimated Using Measured S-Parameters 

Imaginary Pan K 

-1 c 9 10 11 12 

Frequency( GHz) 

Figure 20: Relative dielectric constant (real and imaginary) of Fiber 

Glass material estimated using measured and ideal S-parameters 

(thickness = 0.0533cm 

23 
Real and Imaginary Parts of Relative Dielectrr 

Constant of JVoven Composite B Matenal 

4 'i 

1 - Rea' Part Estimated Using lded S-Parameten -l=*-*- hag. Part 

Real Part Estimated using Measured S-Parameters 

1- 

0 -' 

I I I, I 

9 10 11 12 -1 ' 

Frequency(GHz) 

Figure 21: Relative dielectric constant (real and imaginary) of Woven 

Composite B material estimated using measured and ideal S-parameters 

(thickness = 0.1625cm 

Real and Imaginary Parts oi Relabve Dietecnic Constant of Low Densty Foam Material 

- RealPart ~ ,_,_,_ Part Eshmated uvng Ideal S-Parameters 

2.5 I Real Part Estimated using Measured 5-Parameters 

hag. Part 

4.25 

tmaginay Pan 

Frequency(GHz) 

Figure 22: Relative dielectric constant (real and imaginary) of low density Foam 

material estimated using measured and ideal S-parameters (thickness = 

0.292 1 cm) 

24 
5- 

4- U 

cm 

cn 

CI 

s 3- 

- $2- 

0 

I 0- 

L 

.aI 

P) 

nImaginary Part 

Frequency(GH2) 

- RealPart I ,-, -, I Part Estimated Usng Heal S-Paramotes 

Real Part Estimated using Measured S-Parameters 

hag. Part 

mum mmDBM.Bimm 

Real Part 

Figure 23: Relative dielectric constant (real and imaginary) of Woven 

Composite G material estimated using measured and ideal S-parameters 

(thickness = 0.2083 cm) 

Examples (When Thickness of Sample Is Unknown): 

In the previous section it was assumed that the thickness of the sample is known 

apriori. These thicknesses are measured in the laboratory using an electronic 

micrometer. However, for compressible and thin samples the accuracy of these 

measurements is questionable. In this section it is assumed that the thickness of the 

sample is unknown and the optimizer is asked to estimate the thickness along with the 

dielectric constant of a single layer composite material slab. The MatLab Code used for 

estimation of sample thickness as well as dielectric constant is given in Appendix B. 

Garlock Slab: The measured data stored in a file l7-GRLK.60 is used to estimate the 

thickness of Garlock slab and dielectric constants. Using the MatLab Code given in the 

Appendix B, the dielectric constant of the Garlock slab and its thickness are estimated 

and shown in figure 24. 

25 
10 

9F 

Real and Im 

Constant of zatiock Matend (Thickness Unknown) 

lnary Patts of Relative Dleledth 

Real Part 

- Thickness in cm 

0 

-1 9 10 11 12 

Frequency( GHz) 

Figure 24: Relative dielectric constant (real and imaginary) of Garlock 

material estimated assuming thickness unknown. Symbols indicate 

estimated values of dielectric constant using thickness = 0.17 cm. Thin 

solid line indicate estimated value of sample thickness. 

In figure 24, estimated values of the dielectric constant assuming sample thickness 

unknown are in close agreement with the dielectric constant estimated assuming known 

value of sample thickness. The average value of estimated sample thickness is 0.1699 cm 

which is very close to the actual measured thickness of 0.1702cm. This validates the idea 

that apriori knowledge of sample thickness is not necessary for the inverse problem. In 

fact the thickness of the sample can be considered as one of the unknown variables along 

with the dielectric constants to optimize the error function defined in (19). Using the 

26 
4 10 11 12 4.1 f 

Fmquncy(OHz) 

Figure 25: Differences between computed and measured values of Sparameters. Calculated S-parameters are determined using estimated 

dielectric constant and thickness of the slab 

estimated values of dielectric constants and the sample thickness, the difference between 

measured and computed values of S-parameters is plotted in Figure 25. 

The differences between the computed and measured S-parameters are within the limits 

set in the optimizer. Ideally, the difference between the computed and measured Sparameters must be close to zero. However, to achieve ideal results, the optimizer would 

take a longer time. 

From the measured values of the S-parameters for a variety of single layer 

composite materials (Nano Material #O, Cork, Ceramic, Nomex Felt, Rubylith, Bakelite, 

Teflon, Fiber Glass, Woven Composite B and G, Nan0 Material ## 0) the dielectric 

constants and thicknesses of the material are estimated using the MatLab Code given in 

Appendix B. These estimated values are shown in figures 26-36. For comparison, the 

estimated values of dielectric constants using MatLab Code given in Appendix B are also 

plotted in figures 26-36. The estimated values of dielectric constants assuming the 

thickness unknown are in good agreement with the estimates obtained using known 

27 
values of dielectric constants. From the figures 26-36 it may be concluded that the 

thickness of a single layer slab can be treated as one of the unknown variables along with 

the dielectric constants. 

Real and Irna inary Parts of Relative Dmlectnc 

Constant of &no Material 1M (Thrkness Unknown) 

Thickness in cm 

9 10 11 12 

Frequency(GH2) 

Figure 26: Relative dielectric constant (real and imaginary) of Nan0 

material #O estimated assuming thickness unknown. Symbols indicate 

estimated values of dielectric constant using thickness = 0.301 cm. Thin 

solid line indicate estimated value of sample thickness. 

28 
Real and ,maginary Pans of Relarive Dielectrr Constant of Cor(c Material (Th-ss Unknown) 

-0 75 

3t 

- 

I I I 1 

Estimated Assuming Thkkners Unknown 

R Real Part Estimated Usiy Thickness - 0 3048 an 

2.25 hag. Part 

Real Part 1.5 

g.75 

o, 0.5 

9.25 

(Po 

K 

-0.25 

-0.5 

-0.75 

-1 

- 

9 10 11 12 

F mquency( G H 2) 

Figure 27: Relative dielectric constant (real and imaginary) of Cork 

material estimated assuming thickness unknown. Symbols indicate 

estimated values of dielectric constant using thickness = 0.3048 cm. 

Thin solid line indicate estimated value of sample thickness. 

29 
Real and Im 

Constant of ?omex Felt Matend (Thickness Unknown) 

inary Parts of Relative Dieledrlc 

2.75 

- Real Part 

..,I,I, I 

Part Estimated Assumirg Thekness Unknown 

~~~l part Estimated Using Thlckness I 0 3302 cm 

- 

hag. Part 

Thicknssincrn 

0 1.5 

AverageThrkness- 0 3321 cm 

Figure 29: Relative dielectric constant (real and imaginary) of Nomex Felt 

material estimated assuming thickness unknown. Symbols indicate 

estimated values of dielectric constant using thickness = 0.3302 cm. Thin 

solid line indicate estimated value of sample thickness. 

I I . .I, , I 

9 10 11 12 

Frequency(GHz) 

Figure 30: Relative dielectric constant (real and imaginary) of Rubylith 

material estimated assuming thickness unknown. Symbols indicate 

estimated values of dielectric constant using thickness = 0.3683 cm. 

Thin solid line indicate estimated value of sample thickness. 

30 
Figure 3 1 : Relative dielectric constant (real and imaginary) of Bakelite 

material estimated assuming thickness unknown. Symbols indicate 

estimated values of dielectric constant using thickness = 0.3302 cm. Thin 

solid line indicate estimated value of sample thickness. 

Red and Im !nary Pans 01 Relative Dldertnc Constant oi ?&Ion Material (llwknesa Unknown) 

3 

2 75 

25 Real Pan 2.25 

h 75 

515 

$1 3 75 

Q 05 

32 

Esbmated Usnj llnckness - 0 9388nn gi .25 

$.25 

00 

KO 25 

-0 5 

-0 75 

-1 

- 

9 10 11 12 

Frequency( GHz) 

Figure 32: Relative dielectric constant (real and imaginary) of 

Teflon material estimated assuming thickness unknown. Symbols 

indicate estimated values of dielectric constant using thickness = 

0.9398 cm. Thin solid line indicate estimated value of sample 

thickness. 

31 
i 0' 

0 

'E 

v 

.9! 

0 1- 

0 2- 

- 

I I, I4 

9 10 11 12 -J' ' ~ I 

Frequency( 0 H z) 

Figure 33: Relative dielectric constant (real and imaginary) of Fiber Glass 

material estimated assuming thickness unknown. Symbols indicate 

estimated values of dielectric constant using thickness = 0.0533 cm. Thin 

solid line indicate estimated value of sample thickness. 

4rw rn m rn rn m rn m 

Red Part 

- RealPart 

_._ ,1,- lmag. part EstimatedAssumKlg Thiclcness Unknown " Rea' Part Eshmatd Unng Thrknoss - 0 05334cm 

'-4 

hag. Part 

Thickness in an 

E 2: 

'175- 0 

1.5 

32 

" Rea' Estimated Using Thickness -0 2921 cm 08 Imag.Part 

: - Thickness in cm 

K 

-0 25 

-05 

-0.75 

-1 

lmaginiuy Part - 

- 

- 

* I I I, I I 
Real and Im inaly Pans of Relative Didodrr: Constant oi yawn Composite B Material (ThEkness Unknown) 

3 

g 

0 

- Real Pan 

3-= Real Pan - 

4 21 

51.75 - Part Esbmated AssUrmngThlckness Unknown Irnaa. Part 

RcaiPalt Estlmated Uung Thlckness - 0.1628cm 

hag. Part 

Thickness in cm 

> 

a0 

K 

-0.25 

gJ.25 

- 

lmqinary Pait 

-0.5 

-0 75 

-1 9 10 11 12 

Frequency(GHz) 

Figure 35: Relative dielectric constant (real and imaginary) of Woven Composite 

B material estimated assuming thickness unknown. Symbols indicate estimated 

values of dielectric constant using thickness = 0.1626 cm. Thin solid line 

indicate estimated value of sample thickness. 

Real and Im Constant of goven Composite G Mateiial (Thrkness Unknown) inaty Parts of Relative Dielecttr 

5r 

U e 4 F 

0 

- RealPart 

I ,I, I imago part Eshmated Assuming Thrkness Unknown 

Real Pan Eshmated Using ThEkms - 0.2083cm 

Imag. Pari 

Thickness in cm al 

m 

Q) 

U 

Averaae Thickness - 0 2186 an 

0 -* I. 

Imaginary Part 

I I.. I,,, I 

9 10 11 12 -1 

Fmquency(GHt) 

Figure 36: Relative dielectric constant (real and imaginary) of Woven 

Composite G material estimated assuming thickness unknown. Symbols 

indicate estimated values of dielectric constant using thickness = 0.2083 cm. 

Thin solid line indicate estimated value of sample thickness. 

33 
Most of estimated dielectric constants determined assuming thickness unknown agree 

well with the dielectric constants estimated using measured value of slab thickness. 

However, in figure 33, the two results disagree significantly. This is because the 

thickness of the slab is very small. In fact, the S-parameters calculated using estimated 

thickness and the dielectric constants agree very well with the measurements as shown in 

figure 37. 

= - - - Imag.(S11) 

= = - - Inrag.(SZl) 

--I- - Real(S11) 

-0.75 1 

? 12 

9 10 11 -1 

Fmquency(GHz) 

Figure 37: Measured and computed S-parameters for Fiber Glass Material. 

Computed S-parameters are determined using estimated values of dielectric 

constant and thickness of Fiber Glass. 

Two Layer Composite Material: 

In this section measured S-parameters of composite material consisting of two material 

layers are used to estimate the dielectric constants of individual layers. The error or 

objective function used for this purpose is identical to equation (19). The MatLab Code 

34 
given in Appendix B, with proper input variables is used to estimate the dielectric 

constants of two layers. 

Examples:(Thickness of Layers Known): 

A two layer composite material consisting of Bakelite and Teflon was formed by placing 

first the Bakelite layer of thickness AI = 0.33cm between the z,, = 0 and z, = 0.33cm 

planes. A Teflon layer of thickness A, = 0.635cm was then placed between z1 = 0.33cm 

and z2 = 0.965cm. The measured values of S-parameters of the Bakelite-Teflon as a 

function of frequency was stored in a file FT-BKLTEF.60. Using the MatLab Code 

given in Appendix B, the dielectric constants of the two layers were estimated and 

plotted in Figure 38. 

A4 

m 

3.5 

z 

i3 

-0.5 1- 

9 10 11 12 -1) ' I I ' ' ' ' ' I ' ' ' ' ' ' ' ' ' I ' 

Frequency (GHz) 

Figure 38: Relative dielectric constant (real and imaginary) of Bakelite-Teflon 

Composite material estimated using measured S-parameters (assuming thickness 

known) 

35 
Figure 39 shows estimated values of dielectric constants of Bakelite-Teflon Composite 

material using the ideal S-parameter values computed using the MatLab Code given in 

Appendix A. Figure 40 shows the error involved in the estimations of dielectric 

constants using measured and ideal values of S-parameters. From figure 40 it is clear that 

the error in estimation using the measured S-parameters is higher than the error involved 

in the estimation using ideal S-parameters. This may be attributed to the noise present in 

the measured data and inability of the analytical model to take into account potential air 

gaps present between the two layers. 

Two Layers Composite Maderial : BakeliiTeRon 

(Thicknesses of layers assuned known) 

Estimates are determined using ideal S-parameters 

I IlI.II I 

9 10 11 12 

Frequency (GHz) 

Figure 39: Relative dielectric constant (real and imaginary) of Bakelite-Teflon 

Composite material estimated using ideal S-parameters (assuming thickness known) 

36 
- - - I- Error Function For Using Ideal S-Parameters 

Error Function For Using Measured S-Parameters 

0.009 

0.008 

0.007 

0.006 

2 b.005 

c 

0 .- 

2 

6 0.004 - 

I I I< I, 

9 10 11 12 OtFrequency (GHr) 

Figure 40: Value of error function given in (1 9) in estimating the dielectic 

constants of Bakelite-Teflon Composite using both measured and ideal Sparameters 

For further verifications of the present approach for a two layer composite material the 

following samples were considered: 

1) Garlock-Nomex Felt: AI = 0.17cm, A2 = 0.33cm, zo = 0.0, z1 = 0.17cm, 

z2 =0.5cm 

2) Garlock-Ceramic: A, = 0.17cm, A2 = 0.2845cm, zo = 0.0, z, = 0.17cm, 

z, =0.4545cm 

3) Garlock-Nan0 Material #o : AI = 0.17cm, A, = 0.3099cm, zo = 0.0, 

z, = 0.17cm, zz = 0.4799cm 

37 
4) Garlock-Woven Composite G: A, = 0.17cm, A2 = 0.2083cm, zo = 0.0, 

z1 =0.17cm, z2 =0.3783cm 

5) Garlock-Bakelite AI = 0.17cm, A, = 0.33cm, zo = 0.0, z, = 0.17crn, 

z, = 0.50~~~. 

For these samples the relative dielectric constants of individual layer were estimated 

from the measured and ideal S-parameters. Estimated values of relative dielectric 

constants are shown in figures 41-56. 

U' 

0 

9 10 11 12 -11 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 

Frequency (GHz) 

Figure 41 : Relative dielectric constant (real and imaginary) of Garlock-Nomex 

Felt Composite material estimated using measured S-parameters (assuming 

thickness known) 

38 
Two Layers Composite Malerial : Garlock-Nomex Fell 

(Thicknesses of layers assumed known) 

Estimates are determhed using ideal S-parameters 

10 

9- 

n m 85 

m 

- E 7- 

3 6: 

K 

e 5- 

Q 

0 

0 

t 

Y U 

U 

E 4- 

- 0 3r 

ii 

$ 2: 

Q) 

- 

- RealPart 

11-1-11 Imag. Part 

--I1 Rea'Part Second$yer(NomexFetl) -11-11-11~ [mag. Part 

- Fist Layer (Garlock) 

Real Part(Garlo& Layer) 

Real Part (Nomex Fell Layer) 

IIIIIII.III)IIIIIIIIIIIIIIIII 

1' 

Frequency (GHz) 

Figure 42: Relative dielectric constant (real and imaginary) of Garlock - Nomex Felt 

Composite material estimated using ideal S-parameters (assuming thickness known) 

0-m Value of Em FunctiOn 

0'009 

0.008 0'01 E - - - - Error Function For Using Ideal S-Parameters 

Error Function For Using Measuted S-Parameters 

8 0.007 

5 0.006 - 

E 0.0°5 : 

ct! 0.004 - 

tj 0.003 - 

- 

l,,,,l< I, I, 

9 10 11 12 -0.001 1 ' ' 

Frequency (GHz) 

Figure 43: Value of error function given in (19) in estimating the dielectic constants of 

Garlock -Nomex Felt Composite using both measured and ideal S-parameters. 

39 
 est PW( (Cenmlar)rr) 

1 -- 

I, 

9 10 11 12 -lr a I ' ' ' ' I ' I I I 

Frequency (GHz) 

Figure 44: Relative dielectric constant (real and imaginary) of Garlock-Ceramic 

Composite material estimated using measured S-parameters (assuming thickness 

known) 

Figure 45: Relative dielectric constant (real and imaginary) of Garlock - Ceramic 

Composite material estimated using ideal S-parameters (assuming thickness known) 

40 
Oplmm Value of Em FuncUon 

0.009 

0.008 

c 

- 

5 - - - - I - Error Function For Using Ideal S-Parameters 

5 Error Function For Using Measured S-Parameters 

9 10 11 12 -0.001~' ' ' I ' I I ' I ' ' I' I I' ' ' I ' 

Frequency (GHz) 

B 

f 0.006 

LL E O*OQ5 

e 0.004 

Figure 46: Value of error function given in (19) in estimating the dielectic constants of 

Garlock -Ceramic Composite using both measured and ideal S-parameters. 

- 

/ 

- 

: 

7 

Img. Pall (Nan0 M.Dhlto) OW 

-1-I ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 

Figure 47: Relative dielectric constant (real and imaginary) of Garlock-Nan0 Material #o 

Composite material estimated using measured S-parameters (assuming thickness known) 

41 
Two Layen Cornposits Maifarial : Garlock-Nano Material MI 

(Thicknesses of layers assmed known) Esilmaifas are deieimlned ushg ideal S-parametem 

+ Reel Pari ( Garbck Layer) 

Figure 48: Relative dielectric constant (real and imaginary) of Garlock - Nan0 

Material #O Composite material estimated using ideal S-parameters (assuming 

thickness known) 

Oplimun Value 01 Em Function 

0.009 

0.008 - - - - - Error Function For Using Ideal S-Parameters 

Error Function For Using Measured S-Parameters 

c - 0 0.007 

0 5 0.006 

LL 

CI 

0.005 

&! 0.004 

0.003 

L 

- s 

(B g 0.002 

0.001 - 

1 I 8.1, 1 

9 10 11 12 -0.001 ' 

Frequency (GHz) 

Figure 49: Value of error function given in (19) in estimating the dielectic constants 

of Garlock -Nan0 Material #O Composite using both measured and ideal S-parameters. 

42 
- Rea'Part RntIapr(Garl0dt) 10 1.1 -1- Imag.Part 

Real Part Second lmyw(WovPnCompolilsG) 

--A--- Imeg.Part gl! 

0 11 

lmsg Pati (Woven Composite G) 

9 10 11 12 -1 

Frequency (GHz) 

Figure 50: Relative dielectric constant (real and imaginary) of Garlock-Woven 

Composite G material estimated using measured S-parameters (assuming thickness 

known) 

Figure 5 1 : Relative dielectric constant (real and imaginary) of Garlock - Woven 

Composite G material estimated using ideal S-parameters (assuming thickness known) 

43 
Opthun Valw of Em Funetbn 

- - - - - Error Function For Using Ideal S-Parameters 

Error Function For Using Measured S-Parameters 

0.07 

f 0.06 

8 0.05 

LL 

t 

0.04 

0.03 

F 

a 

- > 

9 10 11 12 -0.011' " ' I' ' ' 'I " " ' I' " 

Frequency (GHz) 

Figure 52: Value of error function given in (19) in estimating the dielectic constants of 

Garlock -Woven Composite G material using both measured and ideal S-parameters. 

Two Law Compoatio N.Prlsl: Gsllock-Balcdnp (Thlckns-otIs~urunedImorn) 

Rea'Part FbxlL.y~(GaW) -I- -1- Imag.Part 

_t_ RealPart Secondby.r(B.lcsl0) 

9 A Imag.Part 

10 

h 

m 

m8 

E7 

Q s4 

- 03 

P2 

$6 

K 

s 

r5 

0 

b 

9! 

01 

0 

-1 9 10 11 12 

Frequency (GHz) 

Figure 53: Relative dielectric constant (real and imaginary) of Garlock-Bakelite 

Composite material estimated using measured S-parameters (assuming thickness 

known) 

44 
Two Layers Compoalta Matefbl: Garlock-Baksllie 

(Thrknesses of layers assumed known) 

Estbnates 

- 

am determined usmg ideal Spramele~~ 

RealPart 

First Layer (Garlock) 911111- hag. Part 

oool Real Part Secondbyer(BakeliIa) -M-I~-~W Imag. Part 

10 

9F L 

8- Real Part (Garlock Layer) 

7- 

6- 

5- 

4- Real Part (Bakelite Layer) 

1~101010001~1m10~1m01. 

3- 

Figure 54: Relative dielectric constant (real and imaginary) of Garlock - Bakelite 

Composite material estimated using ideal S-parameters (assuming thickness 

known) 

.I - I III Error fundion For Using I&al S-Parameters 

Error Function For Using Measured S-Parameters 

t I I I 

9 10 11 12 -0.001 

Frequency (QHz) 

Figure 55: Value of error function given in (19) in estimating the dielectic 

constants of Garlock -Bakelite Composite material using both measured and 

ideal S-parameters. 

45 
From the results shown in figures 41-55, it may be concluded that the method presented 

in this report can successfully estimate dielectric constants of two layer composite 

material with apriori knowledge of the thickness of individual layers. An attempt was 

made to estimate thicknesses of individual layers along with their dielectric constants 

using the MatLab Code given in Appendix B. However, the optimizer did not estimate 

these values correctly and hence the results for such cases are not presented. 

Three Layer Composite Material: 

In this section measured S-parameters of composite material consisting of three material 

layers are used to estimate dielectric constants of individual layers. The error or 

objective function used for this purpose is identical to equation (19). The MatLab Code 

given in Appendix B, with proper input variables is used to estimate the dielectric 

constants of two layers. 

Examples:(Thickness of Layers Known): 

A three layer composite material consisting of layers of Nan0 Material #O, Garlock, and 

Garlock was formed by placing the Nan0 Material #O layer of thickness 

A, = 0.3099cm between z, = 0 and z, = 0.3099cm planes, the Garlock layer of 

thickness A, =0.17cm between z, =0.3099cm and z2 =0.4799cm planes, and 

Garlock layer of thickness A, = 0.17cm between z2 = 0.4799cm and z, = 0.6499cm 

planes. The measured values of S-parameters stored in FT-NANGRKGRK.60 were used 

to estimate dielectric constants of each individual layers using the MatLab code given in 

the Appendix B. The estimated values of dielectric constants are shown in Figure 56. 

The values of dielectric constants of each these material when estimated using single 

layer measured data were found to be E,. = 2.5 - j0.00 for the Nan0 Material #O, 

46 
E, = 7.5 - jO.00 for the Garlock material. However, when the individual dielectric 

constants are estimated using the measured data for the three layer composite material, 

these estimates come out to be little different for the earlier estimates. This may be 

attributed to the presence of airgaps between the layers. 

0 02 

0 0175 

- 3 2 0.015 

c 

0 

500125 

E 

0.01 

a 

o, E Second Layer 

- 

- 

- 

- 

- Real Part Nano Material KI Layer " 

Imag. Part 

--&- Real Part GahckLayer 

.- __ I Imag. Part 

___- Real Part Garlock Layer 

Imag. Part 

Y 

C 

m5 

tn 

0 

=3 ti 

n 

First Layer 

$2 

lmag Part for all layers 

Figure 56: Estimated values of dielectric constants of individual layers from 

measured values of S-parameters of three layer composite slab (Nano Material 

#O-Garlock-Garlock) 

Figure 57: Final values of error function in estimation of dielectric constants of 

individual layers shown in Figure 56 

47 
The final value of error function or objective function for the estimation shown in Figure 

56 is shown in Figure 57. 

A second example of three layer composite material considered was formed by three 

layers of the same Garlock slabs. The first Garlock slab of thickness A, = 0.17cm was 

placed between the z,, = 0 and z, = 0.17cm planes, second Garlock layer of thickness 

A? = 0.17cm was placed between z, = 0.17cm and z2 = 0.34~~ planes, and the third 

Garlock layer of thickness A, = 0.17cm between z1 = 0.34cm and z, = 0.5 lcm planes. 

The measured values of S-parameters stored in lT-GRKGRKGRK.60 were used to 

estimate dielectric constants of each individual layers using the MatLab code given in 

Appendix B. The estimated values of dielectric constants are shown in Figure 58. The 

values of estimated dielectric constants of each individual layers must be close to 

E,. = 7.5 - jO.00. However, the results shown in the Figure 58 are widely spread around 

the correct value of the dielectric constant. This discrepancy in the estimated values 

derived using the measured S-parameters of multilayer composite material is attributed to 

the measurement errors caused by the airgaps between the layers. The error function or 

objective function at the final values of dielectric estimates is shown in Figure 59. 

48 
13 

12 

11 

10 

9 

8 

7 

6 

.. 

10 

9 

8 

7 

6 :E 2 3

Figure 58: Estimated values of dielectric constants of individual layers from 

measured values of S-parameters of three layer composite slab (Garlock-GarlockGarlock) 

t 

Frequency (GHt) 

Figure 59: Final values of error function in estimation of dielectric constants 

of individual layers shown in Figure 58 

49 
IV Conclusion 

A simple waveguide mode matching method in conjunction with the fmiizsearch 

MatLab optimization search function has been presented to estimate the complex 

permitivity of multi-layer composite material. A multi-layer composite material is placed 

in an X-band rectangular waveguide and its two-port S-parameters are measured over the 

X-band using hp-8510 Network Analyzer. For the same composite material using the 

simple mode matching technique the two port S-parameters are calculated as a function 

of complex dielectric constants and thicknesses of each layer. The fminsearch function 

available in the MatLab Optimization Tools is then used to determine complex dielectric 

constants of each layers assuming the thicknesses of each layer are unknown. 

The present approach has been validated using number of composite material 

configurations. The composite material formed by single layers of Garlock, Cork, 

Ceramic, Nomex Felt, Woven Composite B and G, Rubylith, Bakelite, Teflon, Fiber 

Glass, Foam, Nan0 Material have been tested using the present method. The estimated 

values of dielectric constants and their thicknesses have been found to agree well with the 

true values supplied by the manufacturers. The composite materials formed by placing 

two layers of the above basic materials have been tested using the present method. The 

dielectric constants of individual layers estimated have been found to be in good 

agreement with their values specified by manufacturers. However, for the composite 

material consisting of more than one layer it was assumed that the thicknesses of 

individual layers were known. It has been observed that the present method could not 

estimate the dielectric constants of individual layers correctly when three of more layers 

were used to form a composite material. This is attributed to the fact that the airgap 

50 
present in the measurement samples could not be accurately modelled in the estimation 

model leading to an incorrect estimation of dielectric constants 

Acknowledgement 

The authors would like to acknowledge the assistance of Mr. Nima Pirzadeh, 

NASA, LARSS ,2001 summer student, in performing extensive 2-port S-parameter 

measurement on various composite materials using the hp-8510 Network Analyzer. 

References 

K. C. Gupta and P. S. Hall, “Analysis and design of integrated circuit antenna 

modules,” John Wiley & Sons, Inc., New York, 2000. 

S. Chakravarty and R. Mittra, ”Application of microgenetic algorithm to the 

design of spatial fileters with frequency selective surfaces embedded in dielectric 

media, ” IEEE Trans. On Electromagnetic Compatability, Vol. 44, No. 2 pp. 338- 

346, May 2002 

S. Chakravarty, et. al., “Application of a microgenetic algorithm (MGA) to the 

design of broad-band microwave absorbers using multiple frequency selective 

surface screens buried in dielectrics, ” IEEE Trans. On Antennas and Propagation, 

E. Michielssen, et al., “Design of light-weight broad-band microwave absorber 

using genetic algorithms, ” IEEE Trans. Microwave Theory and Techniques, Vol. 

MTT-41, pp 1024-1031, June 1993 

James Baker-Jarvis, “ Dielectric and magnetic meaasurement methods in 

transmission lines: an overview, ” Proceedings of the 1992 AMTA Workshop, 

July 25, 1992, Chicago, lllinios. 

Leo P. Ligthart, “A fast computational technique for accurate permittivity 

determination using transmission line methods, ” IEEE Trans. On Microwave 

Theory and Techniques, Vol. MTT-31, No. 3, pp. 249-254, March 1983 

R. L. Cravey et al., “Dielectric property measurements in the electromagnetic 

properties measurement laboratory, ” NASA Technical Memorandom 1 10 147, 

April 1995. 

M. D. Deshpande et al., “A new approach to estimate complex permittivity of 

dielectric materials at microwave frequencies using waveguide measurements, ” 

IEEE Trans. On Microwave Theory and Techniques, Vol. MTT-45, No. 3, pp. 

Vol. AP-50, NO.3, pp. 284-296, March 2002 

359-366, March 1997 

51 
Appendix A: MatLab code for estimation of S-parameters of multilayer substrate 

placed in a rectangular waveguide. 

% This program calculates S-parameters of 

% N-layer composite material placed in an 

% X-band rectangular waveguide as a function 

% of frequency. Waveguide dimensions (2.29 cm x 1.02 cm) 

%Select 201 frequency points over 8.2-12.4 GHz 

%Frequency band. 

delf = (12.4-8.2)/201; 

for j = 1:201 

frcj) = 8.2 +cj-l)*delf; 

end 

%Input number of layers 

nl = input('n1 = '); 

%Input dielectric constants 

for k = 1:2*nl 

%First value is real part of dielectric constant 

940 second value is imaginary part of dielectri constant 

xO(k) = input('xO(k) =I); 

end 

%Input location of interfaces 

for k = l:nl+l 

zn(k) = input('zn(k) ='); 

end 

% Set dielectric constants depending upon 

% number of layers. Only 4 layers are 

% included. If nl > 4 then add condition 

% for given nl 

if nl == 1 

er(1) = xO(1) -i*x0(2); 

end 

if nl == 2 

er(1) = xO(1) -i*x0(2); 

er(2) = xO(3) -i*x0(4); 

end 

if nl == 3 

er(1) = xO(1) -i*x0(2); 

er(2) = xO(3) -i*x0(4); 

er(3) = xO(5)-i*xO(6); 

end 

52 
if nl == 4 

er(1) = xO(1) -i*x0(2); 

er(2) = xO(3) -i*x0(4); 

er(3) = xO(5)-i*xO(6); 

er(4) = xO(7)-i*xO(8); 

end 

nlayer = nl; %number of layers redifined 

n = nlayer+l; % Matrix order 

t =zeros(n,l); % Right handside column matrix initilized 

s =zeros(n); 

ak =zeros(n,l); % propagation column matrix 

% Admittance matrix initilized 

%Set dielectric constants redifined 

%const is set to zeros (3 by 3) 

const = zeros(n); 

% Dimensions of x-band waveguide in cm 

aa = 2.29; 

bb = 1.02; %guide dimensions 

% Free space dielectric constant 

e0 = l.e-9/(36*pi); 

%f = 8.2e9 %frequency of microwave signal 

c = 3.e10; % velocity of light in meters/second 

u0 =4*pi*l.e-7; % Magnetic permeability of free space 

nO = (uO/eO)"S; % Free space impedance 

for ifr = 1:201 

freq = fr(ifr); % Frequency in GHz 

akO =2*pi*freq/30; %wave number in free space 

% This loop calculates propagation 

% constants in layers 

for kl = 1:nl 

const(k1) = akO*akO*(er(kl))-(pi/aa)A2; 

if (real(const(k1)) < 0.0) ak(k1) = -i*(-const(k1))"J; 

else ak(k1) = (const(k1))"J; 

end 

end 

% akOO and akll are propagation constants 

% for the free space regions of waveguide 

const0 = akOA2-(pi/aa)"2; 

53 
if (real(const0) e 0.0) akOO = -i*(-const0)"J; 

else akOO = (constO)A.S; 

end 

akll = akOO; 

for kl = 1:nl 

end 

addm(k1) =ak(kl)J(akO*nO); 

addm00 = akOO/(akO*nO); 

addmll = addm00; 

% Initilize s matrix 

for kl= l:nl+l 

for k2 = l:nl+l 

end 

s(kl,k2) = 0.0; 

end 

% LOOP TO CALCULATE IMPEDANCE MATRIX 

% EXCLUDING FIRST ROW AND LAST ROW 

for kl = 2:nl 

k2 = kl-1; 

argl = ak(k2)*(zn(kl)-zn(kl-l)); 

s(kl,k2) = -addm(k2)/(i*sin(argl)); 

k2 = kl; 

argl = ak(k2-1)*(zn(kl)-zn(kl-l)); 

arg2 = ak(kl)*(zn(kl+l)-zn(k1)); 

s(kl,k2) = addm(k2-l)*cos(argl)/(i*sin(argl)) + addm(kl)*cos(arg2)/(i*sin(arg2)); 

k2 = kl+l; 

arg2 = ak(kl)*(zn(kl+l)-zn(k1)); 

s( kl,k2)= -addm(kl)/( i*sin( arg2)); 

end 

% FIRST ROW OF IMPEDANCE MATRIX 

s(1,l) = addm00 + cos(ak(l)*zn(2))/(i*sin(ak( l)*zn(2)))*addm( 1); 

s( 1,2) = -addm( l)/(i*sin(ak( l)*zn(2))); 

% LAST ROW OF IMPEDANCE MATRIX 

s(nlayer+l,nlayer) = -addm(nlayer)/(i*sin(ak(nlayer) ... 

s(nlayer+l,nlayer+l) = addm(nlayer)*cos(ak(nlayer)* ... *(zn(nlayer+ 1)-zn( nlayer)))) ; 

54 
(zn( nla y er+ 1 )-zn( nla yer)))/( i *sin( ak( nlayer) * ... (zn(nlayer+l)-zn(n1ayer)))) + addmll; 

% ONLY FIRST ELEMENT OF RIGHT HAND SIDE 

% IS NON ZERO 

for j = l:nlayer+l 

tu) = 0.0 +i*O.O; 

end 

I t(1) =2*addm00; 

% INVERT S AND MULTIPLY BY t 

I T = inv(s)*t; 

% FIRST ELEMENT OF T - 1 IS REFLECTION 

% COEFFICIENT AND LAST VALUE OF T is TRANSMISSION 

% COEFFICIENT 

% SINCE MEASURED TRANSIMMSION IS REFERENCED TO 

% INPUT PLANE IT IS MULTIPLIED BY PHASE CORRECTION 

sllc(ifr) =T(l,l)-1; 

s2lc(ifr) =T(nl+l,l)*exp(i*ak00*zn(nl+l)); 

\ 

end 

% CALCULATING S22 AND S12 

%Inverse the position of interfaces 

if nl==1 

zn( 1) = zn( 1); 

zn(2) = zn(2); 

er(1) =xO(l) -i*x0(2); 

end 

if nl == 2 

zn(1) = -zn(3) + zn(3); 

zn(2) = -zn(2) + zn(3); 

55 
zn(3) = -zn(l) + zn(3); 

er(1) = xO(3) -i*x0(4); 

er(2) = xO(1) - i"xO(2); 

end 

if nl == 3 

zn(1) = -zn(4) + zn(4); 

zn(2) = -zn(3) + zn(4); 

zn(3) = -zn(2) + zn(4); 

zn(4) = -zn(l) + zn(4); 

er(1) = xO(5) -i*x0(6); 

er(2) = xO(4) -i*x0(3); 

er(3) = xO(1) -i*x0(2); 

end 

ifn==4 

zn(1) = -zn(5) + zn(5); 

zn(2) = -zn(4) + zn(5); 

zn(3) = -zn(3) + zn(5); 

zn(2) = -zn(2) + zn(5); 

zn(1) = -zn(l) + zn(5); 

er(1) = xO(7) -i*x0(8); 

er(2) = xO(5) -i*x0(6); 

er(3) = xO(3) -i*xo(4); 

er(4) = xO(1) - i"xO(2); 

end 

nlayer = nl; %number of layers redifined 

n = nlayer+l; % Matrix order 

t =zeros(n,l); % Right handside column matrix initilized 

s =zeros(n); % Admittance matrix initilized 

ak =zeros(n,l); % propagation column matrix 

%Set dielectric constants redifined 

%const is set to zeros (3 by 3) 

const = zeros(n); 

aa = 2.29; 

bb = 1.02; %guide dimensions 

% Free space dielectric constant 

e0 = l.e-9/(36*pi); 

%f = 8.2e9 %frequency of microwave signal 

c = 3.e10; % velocity of light in metershecond 

u0 4"pi"l.e-7; % Magnetic permeability of free space 

nO = (uO/eO)*S; % Free space impedance 

for ifr = 1:201 

56 
freq = fr(ifr); % Frequency in GHz 

akO =2*pi*freq/30; %wave number in free space 

% This loop calculates propagation 

% constants in layers 

for kl = 1:nl 

const(k1) = akO*akO*(er(kl))-(pia)"2; 

if (real(const(k1)) c 0.0) ak(k1) = -i*(-const(k1))"S; 

else ak(k1) = (const(k1))"J; 

end 

end 

% akOO and akll are propagation constants 

% for the free space regions of waveguide 

const0 = akOA2-(pi/aa)"2; 

if (real(const0) c 0.0) akOO = -i*(-constO)".fi; 

else akOO = (const0)"S; 

end 

akll = akOO; 

%calculate addmitince for layers: 

for kl = 1:nl 

addm( kl) =ak( kl)./(akO*nO) ; 

end 

addm00 = akOO/(akO*nO); 

addmll = addm00; 

% Initilize s matrix 

for kl= l:nl+l 

for k2 = l:nl+l 

end 

s(kl,k2) = 0.0; 

end 

% LOOP TO CALCULATE IMPEDANCE MATRIX 

% EXCLUDING FIRST ROW AND LAST ROW 

for kl = 2:nl 

k2 = kl-1; 

argl = ak(H)*(zn(kl)-zn(kl-l)); 

s( kl,k2) = -addm( k2)/( i *sin( arg 1)); 

k2 = kl; 

argl = ak(k2-l)*(zn(kl)-zn(kl-l)); 

arg2 = ak(kl)*(zn(kl+l)-zn(k1)); 

s(kl,k2) = addm(k2-l)*cos(argl)/(i*sin(argl)) + addm(kl)*cos(arg2)/(i*sin(arg2)); 

57 
k2 = kl+l; 

arg2 = ak(kl)*(zn(kl+l)-zn(k1)); 

s(kl,k2)= -addm(kl)/(i*sin(arg2)); 

end 

9% FIRST ROW OF IMPEDANCE MATRIX 

s( 1,l) = addm00 + cos(ak(l)*zn(2))/(i*sin(ak(l)*zn(2)))*addm(l); 

s(l,2) = -addm( l)/(i*sin(ak(l)*zn(2))); 

% LAST ROW OF IMPEDANCE MATRIX 

s(nlayer+l,nlayer) = -addm(nlayer)/(i*sin(ak(nlayer) ... 

s(nlayer+l,nlayer+l) = addm(nlayer)*cos(ak(nlayer)* ... *(zn(nlayer+l)-zn(n1ayer)))); 

(zn(nlayer+ 1)-zn( nlayer)))/(i*sin(ak( nlayer)" ... (zn(nlayer+l)-zn(n1ayer)))) + addmll; 

% ONLY FIRST ELEMENT OF RIGHT HAND SIDE 

9% IS NON ZERO 

for j = l:nlayer+l 

tu) = 0.0 -i*O.O; 

end 

t(1) =2*addm00; 

% INVERT S AND MULTIPLY BY t 

T = inv(s)*t; 

YO FIRST ELEMENT OF T - 1 IS REFLECTION 

% COEFFICIENT AND LAST VALUE OF T is TRANSMISSION 

96 COEFFICIENT 

% SINCE MEASURED TRANSIMMSION IS REFERENCED TO 

% INPUT PLANE IT IS MULTIPLIED BY PHASE CORRECTION 

ref =T(l,l)-1; 

s22c(ifr) = rePexp(i*ak00*2*zn(nl+l)); 

sl2c(ifr) =T(nl+l,l)*exp(i*akOO*zn(nl+l)); 

end 

ss = load('FT-GRLK.60'); 

58 
delf = (12.4-8.2)/201; 

for j = 1:201 

end 

fro') = 8.2 +cj-l)*delf; 

for j = 1:201 

sllr(j) = ss(j,l); 

slli(j) = ss(j,2); 

s2lrcj) = ss(j+201,1); 

s2li(j) = ss(j+201,2); 

sl2r(j) = sscj+402,1); 

sl2icj) = ss(j+402,2); 

s22rCj) = ss(j+603,1); 

s22iQ) = ss(j+603,2); 

end 

plot(fr,sllr,fr,slli,fr,real(sllc),fr,imag(sllc)) 

ok = input('1s plot ok ='); 

plot(fr,s21r,fr,s2li,fr,real(s2lc),fr,imag(s2lc)) 

ok e input('1s plot ok ='); 

plot(fr,sl2r,fr,s12i,fr,real(s12c),fr,imag(s12c)) 

ok = input('1s plot ok ='); 

plot(fr,s22r,fr,s22i,fr,real(s22c),fr,imag( s22c)) 

ok = input('1s plot ok =I); 

% 

59 
Appendix B: MatLab code for estimation of dielectric parameters of multilayer 

substrate placed in a rectangular waveguide. 

% This program estimates dielectric parameters of 

% multi-layer substrate. Measured data has to be loaded in 

% meas.dat file 

% X-band rectangular waveguide as a function 

% of frequency. 

%Load measured data file 

ss = load('meas.dat'); 

delf = (12.4-8.2)/201; 

for j = 1:201 

end 

frcj) = 8.2 +(j-l)*delf; 

%Input number of layers 

nl = input('n1 = '); 

%Input dielectric constants 

for k = 1:2*nl 

%First value is real part of dielectric constant 

%second value is imaginary part of dielectri constant 

xO(k) = input('xO(k) ='); 

end 

%Input location of interfaces 

for k = l:nl+l 

end 

zn(k) = input('zn(k) ='); 

for j = 1:201 

sllrCj) = sscj,l); 

sllicj) = sscj,2); 

s2lrcj) = ss(j+201,1); 

s2licj) = sscj+201,2); 

sl2rCj) = sscj+402,1); 

sl2icj) = ssCj+402,2); 

s22rCj) = sscj+603,1); 

60 
s22iCj) = ssCj+603,2); 

end 

% CHECK WHETHER SYSTEM IS LOSSLESS OR LOSSY 

for j = 1:201 

sll = sllrcj)+i*sllicj); 

s21= s2lrcj)+i*s2licj); 

s12 = sl2rCj)+i*sl2iCj); 

s22 = s22rCj)+i*s22iCj); 

freq = frCj); 

I 

xo = [ 1.0;0.0;1.0;0.0;1.0;0.0]; 

options = optimset('Display', 'iter', 'MaxFunEvals' ,lOOO, 'TolFun', 1.e-OS) 

options = optimset('TolX', le-OS) 

[x,fval,flag] = fminsearch( @obj~ref,x0,options,s1l,s21,s12,s22,freq,nl ... 

erllCj) = x(1); 

erl2cj) = x(2); 

er21Q) = x(3); 

er22Cj) = x(4); 

er31Cj) = x(5); 

er32Cj) = x(6); 

,zn); 

fglcj) = fval; 

I flag 

end 

outmat = [fr.',erll.',erl2.']; 

save(' templ','outmat','-ascii') 

outmat = [fr.',er21.',er22.']; 

save( ' temp2','outmat ' ,' -ascii ' ) 

outmat = [fr.',er31.',er32.']; 

save('temp3','outmat','-ascii') 

outmat =[fr.',fgl.']; 

save('temp4','outmat','-ascii') 

% 

61 
% This is object function 

function f = obj~ref(x0,sll,s2lgl2,s22,freq,nl,zn) 

if nl == 1 

er(1) = xO(1) - i*x0(2); 

end 

if nl == 2 

er(1) = xO(1) -i*x0(2); 

er(2) = xO(3) -i*x0(4); 

end 

if nl==3 

er(1) = xO(1) -i*x0(2); 

er(2) = xO(3) -i*x0(4); 

er(3) = xO(5) -i*x0(6); 

end 

if nl==4 

er(1) = xO(1) -i*x0(2); 

er(2) = xO(3) -i*x0(4); 

er(3) = xO(5) -i*x0(6); 

er(4) = xO(7) -i*xO(8); 

end 

nlayer = nl; %number of layers redifined 

n = nlayer+l; % Matrix order 

t =zeros(n,l); % Right handside column matrix initilized 

s =zeros(n); % Admittance matrix initilized 

ak =zeros(n,l); % propagation column matrix 

%Set dielectric constants redifined 

%const is set to zeros (3 by 3) 

const = zeros(n); 

aa = 2.29; 

bb = 1.02; %guide dimensions 

% Free space dielectric constant 

e0 = l.e-9/(36*pi); 

%f = 8.2e9 %frequency of microwave signal 

c = 3.e10; % velocity of light in meterdsecond 

u0 =4*pi*l.e-7; % Magnetic permeability of free space 

nO = (uO/eO)".5; % Free space impedance 

62 
%for ifr = 1:201 

%freq = fr(ifr); 9% Frequency in GHz 

akO =2*pi*freq/30; %wave number in free space 

% This loop calculates propagation 

% constants in three layers 

for kl = 1:nl 

const(k1) = akO*akO*(er(kl))-(pi)"2; 

if (real(const(k1)) c 0.0) ak(k1) = -i*(-const(k1))"J; 

else ak(k1) = (const(k1))"J; 

end 

end 

const0 = akOA2-(pi/aa)"2; 

if (real(const0) e 0.0) akOO = -i*(-constO)"S; 

else akOO = (const0)"S; 

end 

akll = akOO; 

% calculate addmitince for layers: 

for kl = 1:nl 

addm( kl) =ak( kl)J(akO*nO); 

end 

addm00 = akOO/(akO*nO); 

addmll = addm00; 

% Initilize s matrix 

for kl= l:nl+l 

for k2 = l:nl+l 

end 

s(kl,k2) = 0.0; 

end 

% LOOP TO CALCULATE IMPEDANCE MATRIX 

% EXCLUDING FIRST ROW AND LAST ROW 

for kl = 2:nl 

k2 = kl-1; 

argl = ak(k2)*(zn(kl)-zn(kl-l)); 

s(kl,k2) = -addm(U)/(i*sin(argl)); 

k2 = kl; 

argl = ak(k2-l)*(zn(kl)-zn(kl-l)); 

arg2 = ak(k1) *( zn( kl+l)-zn(k1)) ; 

s(kl,k2) = addm(k2-l)*cos(argl)/(i*sin(argl)) + addm(kl)*cos(arg2)/(i*sin(arg2)); 

k2 = kl+l; 

63 
arg2 = ak(kl)*(zn(kl+l)-zn(k1)); 

s( kl,k2)= -addm( kl)/(i*sin(arg2)); 

end 

% FIRST ROW OF IMPEDANCE MATRIX 

s( 1,l) = addm00 + cos(ak(l)*zn(2))/(i*sin(ak(l)*zn(2)))*addm(l); 

s(1,2) = -addm(l)/(i*sin(ak(l)*zn(2))); 

% LAST ROW OF IMPEDANCE MATRIX 

s(nlayer+l,nlayer) = -addm(nlayer)/(i*sin(ak(nlayer) ... 

s(nlayer+l,nlayer+l) = addm(nlayer)*cos(ak(nlayer)* ... *( zn(nlayer+l)-zn( nlayer)))) ; 

(zn(nlayer+l)-zn(nlayer)))/(i*sin(ak(nlayer)* ... (zn(nlayer+l)-zn(n1ayer)))) + addmll; 

% ONLY FIRST ELEMENT OF RIGHT HAND SIDE 

% IS NON ZERO 

for j = l:nlayer+l 

t(j) = O.O+i*O.O; 

end 

t( 1) =2*addm00; 

% INVERT S AND MULTIPLY BY t 

T = inv(s)*t; 

% FIRST ELEMENT OF T - 1 IS REFLECTION 

% COEFFICIENT AND LAST VALUE OF T is TRANSMISSION 

% COEFFICIENT 

% SINCE MEASURED TRANSIMMSION IS REFERENCED TO 

% INPUT PLANE IT IS MULTIPLIED BY PHASE CORRECTION 

sllc =T(l,l)-l; 

s21c =T(nl+l,l)*exp(i*akOO*zn(nl+l)); 

f = real(sl1c-sll)*real(sllc-sll) + imag(sl1c-sll)*imag(sllc-sll) ... +real(s2lc-s2l)*real(s21c-s21) + imag(s2lc-s2l)*imag(s21c-s21); 

% CALCULATING S22 AND S12 

64 
if nl == 1 

znl(1) = zn(2) - zn(2); 

znl(2) = zn(2) - zn(1); 

end 

if nl == 2 

znl(1) = zn(3) - zn(3); 

znl(2) = zn(3) - zn(2); 

znl(3) = zn(3) - zn(1); 

end 

if nl == 3 

znl(1) = zn(4) - zn(4); 

znl(2) = zn(4) - zn(3); 

znl(3) = zn(4) - zn(2); 

znl(4) = zn(4) - zn(1); 

end 

if nl == 4 

znl(1) = zn(5) - zn(5); 

znl(2) = zn(5) - zn(4); 

znl(3) = zn(5) - zn(3); 

znl(4) = zn(5) - zn(2); 

znl(5) = zn(5) - zn(1); 

end 

if nl == 1 

er(1) =xO(l) - i*x0(2); 

end 

if nl == 2 

er(2) = xO(1) - i*x0(2); 

er(1) = xO(3) - i*x0(4); 

end 

if nl==3 

er(1) = xO(5) -i*x0(6); 

er(2) = xO(3) -i*x0(4); 

er(3) = xO(1) - i*x0(2); 

end 

if nl==4 

er(1) = xO(7) -i*x0(8); 

er(2) = xO(5) -i*x0(6); 

er(3) = xO(3) -i*x0(4); 

er(4) = xO(1) - i*x0(2); 

end 

65 
nlayer = nl; %number of layers redifined 

n = nlayer+l; % Matrix order 

t =zeros(n,l); % Right handside column matrix initilized 

s =zeros(n); 

ak =zeros(n,l); % propagation column matrix 

%Set dielectric constants redifined 

% Admittance matrix initilized 

%const is set to zeros (3 by 3) 

const = zeros(n); 

aa = 2.29; 

bb = 1.02; %guide dimensions 

% Free space dielectric constant 

e0 = l.e-9/(36*pi); 

%f = 8.2e9 %frequency of microwave signal 

c = 3.e10; % velocity of light in meters/second 

u0 =4*pi*l.e-7; % Magnetic permeability of free space 

nO = (uO/eO)".5; % Free space impedance 

%for ifr = 1:201 

%freq = fr(ifr); 9% Frequency in GHz 

akO =2*pi*freq/30; %wave number in free space 

% This loop calculates propagation 

% constants in three layers 

for kl = 1:nl 

const(k1) = akO*akO*(er(kl))-(pi/aa)"2; 

if (real(const(k1)) e 0.0) ak(k1) = -i*(-const(k1))"S; 

else ak(k1) = (const(k1))"S; 

end 

end 

9% ak 

%ok = input('ok = ') 

% akOO and akll are propagation constants 

% for the free space regions of waveguide 

const0 = akOA2-(pi/aa)"2; 

if (real(const0) e 0.0) akOO = -i*(-constO)"S; 

else akOO = (const0)"S; 

end 

akll = akOO; 

% akO 

%calculate addmitince for layers: 

for kl = 1:nl 

addm(k1) =ak(kl)J(akO*nO); 

end 

66 
%addm 

%ok = input('ok =') 

addm00 = akOO/(akO*nO); 

addmll = addm00; 

% addm00 

%addm11 

%ok = input('ok =') 

% Initilize s matrix 

for kl= l:nl+l 

for k2 = l:nl+l 

end 

s(kl,k2) = 0.0; 

end 

I 

% LOOP TO CALCULATE IMPEDANCE MATRIX 

% EXCLUDING FIRST ROW AND LAST ROW 

for kl = 2:nl 

k2 = kl-1; 

argl = ak(k2)*(znl(kl)-znl(kl-l)); 

s(kl,k2) = -addm(k2)/(i*sin(argl)); 

k2 = kl; 

argl = ak( k2- 1)*( znl( k1)-znl( kl-1)) ; 

arg2 = ak( kl)*( znl (kl+l)-znl( kl)) ; 

s(kl,k2) = addm(k2-l)*cos(argl)/(i*sin(argl)) + addm(kl)*cos(arg2)/(i*sin(arg2)); 

k2 = kl+l; 

arg2 = ak(kl)*(znl(kl+l)-znl(k1)); 

s( k 1 ,k2)= -addm( k 1 )/( i*sin( arg2)) ; 

end 

I % FIRST ROW OF IMPEDANCE MATRIX 

s(1,l) = addm00 + cos(ak(l)*znl(2))/(i*sin(ak(l)*znl(2)))*addm(l); 

s( 1,2) = -addm( 1 )/( i *sin( ak( 1) *zn 1 (2))); 

% LAST ROW OF IMPEDANCE MATRIX 

s(nlayer+l,nlayer) = -addm(nlayer)/(i*sin(ak(nlayer) ... *( znl (nlayer+l)-znl( nlayer)))) ; 

s(nlayer+l,nlayer+l) = addm(nlayer)*cos(ak(nlayer)* ... (znl(nlayer+l)-znl(nlayer)))/(i*sin(ak(nlayer)* ... (znl(nlayer+l)-znl( nlayer)))) + addml 1 ; 

%S 

%ok = input('ok =') 

% form cpmplex matrix 

67 
% ONLY FIRST ELEMENT OF RIGHT HAND SIDE 

% IS NON ZERO 

for j = l:nlayer+l 

to) = 0.0 + i"O.0; 

end 

t(1) =2*addm00; 

% INVERT S AND MULTIPLY BY t 

T = inv(s)*t; 

% FIRST ELEMENT OF T - 1 IS REFLECTION 

% COEFFICIENT AND LAST VALUE OF T is TRANSMISSION 

% COEFFICIENT 

% SINCE MEASURED TRANSIMMSION IS REFERENCED TO 

% INPUT PLANE IT IS MULTIPLIED BY PHASE CORRECTION 

ref =T( 1,l)-1; 

s22c = rePexp(i*ak00*2*znl(nl+l)); 

s12c =T(nl+l,l)*exp(i*akOO*znl(nl+l)); 

f = f+ real(s22c-s22)*real(s22c-s22) +imag(s22c-s22)*imag(s22c-s22) ... + real(s 12c-s12)*real(s 12c-s12) + imag(s 12c-s12) *imag(s 12c-s 12); 

68 
I Form Approved 

OM6 No. 07040188 REPORT DOCUMENTATION PAGE 

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12a. DISTRIBUTION/AVAILABlLlN STATEMENT 

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Availability: NASA CAS1 (301) 62 1-0390 

sources gathekg and maintaining the data needed and completing and revTewing the mllechon of inlormati& Send comments &garding this obrden esiimate 07 any other aspect 01 this colleclion 01 inlormation including suggestions lor reducing this Ourden to Washington HeadqLarlers Services Directorate lor Information Operations an0 I Reports 1215 Jellerson Davis Highway Suite 1204 Arlinglon VA 22202 4302 and to the Onice 01 Management and BLdget Paperwon Reduction Project (0704 0188) 

12b. DISTRIBUTION CODE 

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED 

May 2003 Technical Memorandum 

4. TITLE AND SUBTITLE 

Estimation of Complex Permittivity of Composite Multilayer Material at 

Microwave Frequency Using Waveguide Measurements 

6. AUTHOR(S) 

Manohar D. Deshpande and Kenneth Dudley 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

NASA Langley Research Center 

Hampton. VA 2368 1-2 199 

9. SPONSORING&IONITORING AGENCY NAME@) AND ADDRESS(ES) 

National Aeronautics and Space Administration 

Washington, DC 20546-000 1 

11. SUPPLEMENTARY NOTES 

5. FUNDING NUMBERS 

706-3 1-4 1-01 -00 

8. PERFORMING ORGANIZATION 

REPORT NUMBER 

L- 18285 

10. SPONSORING/MONlTORlNG 

AGENCY REPORT NUMBER 

NASA/TM-2003-212398