Material Characterization Improvement in High Temperature Rectangular Waveguide Measurements

The scattering parameters of a single-top gap partially filled rectangular waveguide (PFW) are calculated using mode-matching of the transverse fields. This is accomplished by including in the calculations the complex power contained in higher-order TM modes scattered by a material sample in the waveguide. Furthermore, use of the forward and reverse scattering parameters eliminates the dependence of the placement of the sample with respect to a reference or calibration plane. A combination of the higher-order mode-matching and the reference plane independent (RPI) technique provides the best method for overcoming errors associated with high temperature waveguide measurements. Experimental results are presented to validate the analysis.


Acknowledgements
Tremendous thanks go out to several people: My wonderful wife who always loved me and tried her best to keep me on track through these 18 months -you'll have many more chances in the future. To my son -you're so wonderful! How hard it was for me to leave the house for work when I wanted to spend time with you! Thank you for being so well behaved.
To Dr. Havrilla, my advisor and best professor, who put up with a lot of my procrastination, but was (and is) always patient with me, and made me realize the importance of thinking physically and caring about details. Thanks for the neverending encouragement. The support of Dr. Crittenden was invaluable, and I owe a large part of this work to his guidance. To Dr. Baldwin and Chaplain Srode, who arranged for Mass at AFIT every week -Thanks be to God! What a blessing that has been. RJ Barton, who always challenged me to never give less than my very best. I've learned a lot from your dedication to excellence -thanks for keeping me honest.
Most of all, thanks be to God for all the blessings he has bestowed upon his

I. Introduction
Electromagnetic characterization of materials quantifies the response to applied electric and magnetic fields through the use of the complex quantities permittivity (ǫ) and permeability (µ), respectively. The real part of each complex number is related to energy storage in the material, while the imaginary part accounts for conduction losses, manifested as thermal energy.
In applied electromagnetics, the parameters ǫ and µ are routinely sought by microwave engineers, since these parameters determine the behavior of fields. Materials can be either non-magnetic dielectrics or magnetic materials with both electric and magnetic losses. It is common for certain types of materials to be used to absorb and dissipate electromagnetic field energy, such as that transmitted by a radar.
These shielding materials are effective at reducing the echo area or radar cross section (RCS) [20]. The materials used to achieve this reduction typically vary in weight, composition, effective frequency band, and durability, as well as method of application.
A highly relevant topic in RCS engineering is the reduction of scattering from the exhaust cavity of aircraft engines. Shaping, the primary tool for RCS reduction, cannot be used on existing legacy aircraft. Therefore, radar absorbing shielding material must be applied to needed areas. It is necessary that the absorber be able to withstand the extremely high temperatures of the exhaust environment, typically 2500 • F, without performance degradation. Ceramic shielding material is commonly used, because of its durability under heat.
1 As mentioned before, knowledge of ǫ and µ of the shielding material is necessary to determine the response of the electromagnetic fields and the effective RCS, that is, whether the shielding material will meet the requirement for RCS reduction. Microwave measurements are used to calculate the electromagnetic parameters of heatresistant ceramic shielding material, but must be performed at elevated temperatures to properly simulate the performance environment. This requirement introduces several complications into the measurement process, which would otherwise be trivial.
Thermal expansion of the metal waveguide, which is normally greater than that of the ceramic shielding material, introduces air gaps between the sample and the waveguide walls, leading to the excitation of higher order modes; this situation will be referred to as a partially filled waveguide (PFW). If a gap is nonexistent, then the situation is a fully filled waveguide (FFW).
The use of closed form algorithms, such as Nicolson-Ross-Weir (NRW), does not account for the complex power loss to higher-order and evanescent modes [23,27].
These must be considered if an accurate extraction is desired. Occasionally, due to the extreme heat or imprecise placement, the sample will shift longitudinally away from (or towards) the calibration plane, introducing a phase shift. If the phase shift is not compensated, an extraction of the electromagnetic parameters will be flawed.
This thesis uses modal analysis techniques to overcome the problem of an air gap and is combined with a new method for reference plane independent measurements in a rectangular waveguide containing magnetic material.

Problem Statement
A need exists to accurately characterize magnetic shielding materials at ex-  [17,21,29]. In addition, the reflection measurements are extremely sensitive to the axial placement of the sample in the waveguide. Any displacement from the calibration plane along the waveguide axis incurs a two-way phase delay or advance which, without compensation, yields inaccurate results.
This thesis presents a mode-matching analysis using TM y modes to model a PFW system containing a single air gap between the top of the sample and the waveguide. Theoretical S-parameters, namely S thy 11 and S thy 21 , are calculated and compared to the experimental S-parameters obtained from the network analyzer. A complex 2-D Newton-Raphson root search algorithm solves for the values of permittivity and permeability that minimize the difference, which are assumed to be the actual values of the material parameters. It is shown numerically to be sufficient to enforce continuity of two of the three transverse field components at the boundary between the empty and PFW regions.
To counteract the error introduced by shifting the sample in the waveguide, the respective forward and reverse theoretical S-parameters are multiplied together and compared to the experimental forward and reverse products. A complex 2-D Newton-Raphson root search algorithm iteratively solves for permittivity and permeability based on these new functions. This reference plane independent (RPI) formulation is based on the mode-matching technique described above.
The combination of these components (mode matching, RPI formulation, and use for magnetic materials) has not, to the best knowledge of the author, been presented before. 3

Scope
The standard industry method for materials characterization uses the microwave stripline. The availability of appropriately sized samples, as well as ease of use and performing calculations, contribute greatly to this fact. Other popular test mechanisms include the focused beam, cavity resonator, coaxial waveguide, and single-probe waveguide [21]. All of these methods can be applied in high temperature situations, and perhaps the focused beam system gives the best performance (no higher order modes are excited). Since this thesis presents a solution specifically for the rectangular waveguide, other test setups are not discussed. This exclusion is appropriate, since many facilities, including the sponsor of this research, do not possess every apparatus on this list.
In general, materials characterization will need to be performed across a wide range of frequencies. However, only S-band (2.6 -3.95 GHz) and X-band (8.2 -12.4 GHz) measurements are taken throughout this research. The mode-matching technique can easily be applied to rectangular waveguides at other frequency bands.
Also, only air gaps in the y direction (short dimension) of the waveguide are addressed; this is the more dramatic problem, since the transverse T E 10 E-field is non-zero at these gaps. For gaps in the x direction (long dimension), the tangential (and transverse) E-field is identically zero along the boundary, so small gaps have less effect on the measurement, at least for dielectrics [21].

Thesis Organization
Chapter 2 gives an overview of previous research done in the area of partially filled waveguides, and highlights the contributions of this work. Chapter 3 presents an review of electromagnetic field theory necessary for the research, including the use of vector potentials. The derivation of the mode-matching solution to the problem, as well as the inclusion of the reference plane independent analysis, is discussed in

II. Previous Efforts
Microwave measurement problems considering partially filled waveguides have been thoroughly treated over the years. Several texts, such as those by Collin [10], Harrington [16] and Marcuvitz [22], address the issue to varying degrees of complexity. This chapter will review some of these methods.

Modal Methods
Wexler [28] has presented a general analysis of scattered modes in discontinuous waveguides, without regard to the type of fields present. Boundary conditions and continuity of transverse fields are satisfied by an infinite series of modes on each side of the obstacle or junction. As in this thesis, the objective is to determine the distribution of complex power among the the scattered modes. Wexler uses the orthogonality relation CS e n × h m ·û z dCS = 0 on the non-degenerate modes. Although [28] mentions only PEC obstacles, the modal formulation may be used on dielectric and/or magnetic material obstacles as well.
The problem of finding cutoff frequencies in a waveguide partially filled with an exponentially varying dielectric has been addressed by Gonzalez [15]; his solution does not, however, explicitly solve for the propagation constants of higher order modes.
The problem discussed by Jarem et al. [19] is similar to that treated in this research, although with notable differences. Using TM y modes, Jarem uses a method of moments analysis to calculate the theoretical reflection and transmission coefficients, and uses least-squares curve fitting to match them to the measured S-parameter data.
The axial propagation wave number γ bn is the solution of the eigenvalue equation where d is the sample thickness, b is the height of the waveguide, and γ zin is the y-directed wave number in the partially filled region. The index i = 1, 2 refers the subregions of material (ǫ 1 , µ 1 ) and (ǫ 2 , µ 2 ) respectively. Similarly, Catala-Civera et al. [8] present a method for extracting complex permittivity of a dielectric using PFW theory. An uncertainty study of the procedure is also given. The material sample is discontinuous inx (i.e. long transverse dimension).
An iterative material perturbation technique is used to search for the correct axial wave number γ of the PFW, using where ∆ǫ and ∆µ are the material perturbations and the subscript "0" refers to the unperturbed values [6,16]. Depending on the height of the material sample, the initial unperturbed material parameters those of either the empty or full waveguide. The uncertainty analysis revealed greater error for low-loss samples, as well as samples of short axial length. Additionally, the importance of precise sample alignment with the calibration plane is stressed.
Bogle [7] has also developed a similar solution to the partially filled waveguide characterization problem, although he uses a slightly different formulation. The PFW used in his analysis has left-right gaps, with the sample in the center. Havrilla has used a perturbational method to compensate for small gaps in a PFW, but has included only the dominant TE 10 mode in the solution, excluding higher-order modes [17].

Variational Methods
While modal methods can provide an exact solution for wave propagation in an partially filled waveguide, variational calculus may also be used to obtain an approximate solution. Berk [6] outlined a general variational procedure for obtaining the complex propagation coefficients in a waveguide partially filled with a dielectric slab. Collin and Vaillancourt [11] have successfully used the Rayleigh-Ritz method to obtain approximate eigenfunctions and eigenvalues in a waveguide partially filled with dielectric in the y-axis. The piecewise function κ (y) corresponds to the electric permittivity ǫ as a function of position. The magnetic vector potential is equal to where ψ En (y) is one of infinitely many solutions of the Sturm-Liouville equation having corresponding eigenvalues γ 2 n . It can be shown that the solutions ψ En form an orthogonal set with respect to the weighting function κ −1 . Additionally, the equation is a variational expression for the true propagation constant γ 2 [2]. Collin proposes the set of eigenfunctions in the empty waveguide, where ǫ is Neumann factor, to use in the extremisation of (2.2). This method is used to match the tangential components of the fields at the junction between the empty and partially filled waveguide [11]. While this technique is by its nature approximate, it has the advantage of avoiding the solution of a transcendental equation.
The use of the Rayleigh-Ritz method was improved by Vander Vorst and Govaerts, who computerized the algorithm for use in a variation-iteration method [26].
The exact solution for wave propagation in a waveguide containing E -plane slabs of dielectric has been given by Gardiol [14].

Other Methods
Fehlen [13], also using a modal field expansion, developed a rigorous PFW analysis for samples in a coaxial test fixture, with future application to high-temperature measurements in mind.
Seeking to improve on the transmission/reflection method of materials characterization, Baker-Jarvis et al. developed an method to correct errors of the NRW algorithm (found in Appendix A) in low-loss samples with thicknesses approximately integer multiples of half-wavelengths [3]. Relevant to this thesis is the presentation of a family of equations that are independent with respect to both the reference plane and the sample thickness itself. These equations isolate the s-parameters in terms of other known quantities, to be used in a minimization equation as part of a root search for the correct permittivity. The paper does not test magnetic materials, and is specific to measurements using fully-filled waveguides.
Wilson [29] and Champlin [9] have analyzed the effect of an air gap on calculation of complex permittivity from transmission and reflection measurements. Wilson, using Wexler's formulation with a material slab discontinuity, successfully derived field expansions in the empty and PFW regions. It is suggested that a conducting paste be applied in the gap as a correction, and this is done with great success. However, such a solution is not applicable in a high-temperature PFW situation, since 9 the paste cannot withstand the high temperature [21]. Champlin, recognizing the difficulty of obtaining uniformity in the sample, gives an analytic correction for small gaps to be used in characterization of non-magnetic materials. He does not use modal analysis.

Summary
Notable contributions concerning microwave measurements of a PFW system were reviewed in this chapter. Several works have been published investigating different PFW scenarios, including those containing variable dielectric material. Solution techniques are commonly modal or variational, with computational results often given as verification.

III. Electromagnetics Fundamentals
An understanding of basic electromagnetics and guided wave theory is necessary to develop the solution for the problem at hand. Presented in this chapter is an explanation of vector potentials, the proper use as they pertain to field construction, and a review of guided wave theory as it pertains to this thesis. The analysis of this chapter borrows extensively from [4], [16] and [21]. A e jωt time dependence is assumed and suppressed throughout. Readers familiar with these texts may feel comfortable moving to the next chapter.

Maxwell's Equations
Electric and magnetic field behavior in simple media, defined as linear, isotropic, homogeneous, and dispersive, can be described by the coupled vector form of Maxwell's equations: The following auxiliary relations, also for propagation in simple media, apply,: where ω is the radian frequency, E and H are the electric and magnetic fields respectively, B is the magnetic flux density, D is the electric flux density, J is the electric current density, M is the magnetic current density, ρ e is the electric charge density, ρ m is the magnetic charge density, ǫ is the complex electric permittivity, µ is the complex magnetic permeability, and σ is the conductivity of the material. The real and imaginary components of permittivity and permeability are denoted with single (′) and double (′′) prime notation, respectively. Relative electric permittivity ǫ r (or magnetic permeability µ r ) is defined as the ratio of the material permittivity (permeability) to the permittivity (permeability) of free space, Maxwell's equations describe the coupling of electric and magnetic fields as energy propagates in space or through a material. The constitutive parameters ǫ, µ, and σ determine the field response in the material to the application of an electromagnetic field [21]. The real and imaginary parts of ǫ and µ represent energy stored in the material and loss mechanisms of the material, respectively.

Vector Potentials
It is useful to define auxiliary functions to aid in the solutions of problems involving Maxwell's equations, such as the partially filled waveguide problem of this thesis.
3.2.1 Magnetic Vector Potential. The fields generated by an electric current in a region free of magnetic sources (i.e. M = 0) must satisfy Gauss's Law, ∇ · B = 0.
Since in a source free region B is always solenoidal, this implies it has a magnetic vector potential A, such that or, by (3.2b), which can also be written as By identity, this implies that E A + jωA has a scalar potential, that is or, where the electric scalar potential φ e is a function of position.
By taking the curl of both sides of (3.4) and using the vector identity ∇ × ∇ × Using (3.1b) leads to Substituting (3.8) into (3.10) obtains where k 2 = ω 2 ǫµ. Having previously defined the curl of A in (3.3), the divergence of A, which is independent of the curl, may be defined as which is known as the Lorentz gauge condition. Substituting into (3.11) results in This is known as the wave equation for A. The electric vector field due to the magnetic vector potential A, using (3.13) in (3.8), can be written as Thus knowledge of A enables the finding of E A and H A from (3.15) and (3.4), respectively. The components of the fields due to an electric current density J having been found, it remains necessary to calculate the fields due to a magnetic current M.

Electric Vector Potential.
The fields generated by an equivalent magnetic current in a region free of electric sources (i.e. J = 0) must satisfy Gauss's Law, Observing that D is also solenoidal, this implies it has an electric vector potential F such that 16) or, by (3.2a), which can also be written By identity, this implies that H F + jωF has a scalar potential, that is where the magnetic scalar potential φ m is a function of position.
By taking the curl of both sides of (3.17) and using the vector identity ∇ × ∇ × Using (3.1a) leads to Substituting (3.21) into (3.23) obtains Having previously defined the curl of F in (3.16), the divergence of F, which is independent of the curl, may be defined as which is known as the Lorentz gauge condition. Substituting this into (3.24) results in This is the wave equation for F. The magnetic vector field due to the electric vector potential F, using (3.25) in (3.21) can be written as Thus knowledge of F enables the determination of E F and H F from (3.17) and (3.27), respectively.

Summary.
It has been shown that the electromagnetic field components sustained by an electric current density J or magnetic current density M can be calculated through use of the magnetic vector potential A or electric vector potential F. When both sources are present, the principle of superposition may be applied to determine the total fields, namely and Using vector potentials to construct solutions to Maxwell's equations is the subject of the next section.

Field Construction
The simplest solutions to Maxwell's equations are Transverse ElectroMagnetic In this chapter, only solutions in a rectangular coordinate system are considered.

TEM Modes.
In a source free region, both J and M do not exist.
Equations (3.14) and (3.26) become the homogeneous differential equations and Since (3.30) and (3.31) are of the same form, the solutions will also be of the same form. Therefore, the following development is only be presented for A. Letting The total electric field is given by the sum of (3.15) and (3.17), Applying (3.32) to (3.34), the total E field can be written in component form By the Duality Principle the total magnetic field likewise is given by which which can also be expanded in components as To generate a TEM z mode, both A and F must be used as generating functions.
Requiring A =ẑA z and F =ẑF z to be non-zero while A x = A y = F x = F y = 0 will, by (3.34) and (3.36), produce the required TEM z field components.

TM,TE modes.
While the use of TEM modes is sufficient for many applications (such as wave propagation in free space), other boundary conditions require the use of TM and TE modes. This nomenclature indicates that either the magnetic or electric field components lie in a plane transverse to a given direction.
For example, a TE y field configuration implies that E y = 0; the remaining electric field components, and all the magnetic field components, may or may not exist.
To generate modes TM (or TE) to a given direction, it is sufficient to allow the magnetic vector potential A (or electric vector potential F) to have a single non-zero component in the direction in which the fields are desired to be transverse [4]. For example, the generating potential for TM z field components is where the vector potential A must satisfy the wave equation (3.14) with J = 0. Using the separation of variables technique, solution is assumed to be in the form where the functions f, g and h must be chosen to satisfy the wave equation (3.33c) and boundary conditions of the problem. For a rectangular waveguide oriented on the z-axis it is easiest to apply the boundary conditions if the solution is written in the form where the complex exponentials represent traveling waves and the sine and cosine functions represent standing waves. The separation of variables also obtains the constraint equation In (3.40), the coefficients C, D, and B are amplitude constants determined upon satisfaction of the boundary conditions. The wave numbers k x and k y are spatial constants that describe field variation in x and y respectively, while the wavenumber k is dependent on both frequency and the fundamental properties of the medium itself. The wavenumber γ z describes field behavior in the propagation axis, and consists of a real and imaginary part, such that γ = α + jβ. The real part α is the attenuation constant, and the imaginary part β is the phase constant.
The transverse wave numbers may also be complex, and are often referred to as eigenvalues, since they are characteristic solutions to the wave equation. The notation e −γzz indicates a wave traveling in the forward, +z, direction, while e γzz indicates reverse travel along −z. If γ z is purely imaginary, the wave is unattenuated.
If γ z is purely real, the wave is evanescent. For complex γ z and Re (γ z ) > 0, the wave Coordinate system used thoughout this thesis. The waveguide axis is z, y is top to bottom, and x is left to right. The waveguide is filled with free space, with parameters (ǫ 0 , µ 0 ) . travels within some envelope of attenuation. Once A z is found, the E and H field components can be determined according to (3.35) and (3.37).

Guided Waves
This section presents the method for solving Maxwell's equations in a rectangular hollow waveguide with (assumed) perfect electric conducting (PEC) walls, and cross section uniform to the direction of propagation. The analysis throughout this section is similar to Chapter 8 of [16]. The geometry of the waveguide is given in Therefore, solutions TM and TE will be investigated.
Transverse magnetic to z modes, as previously mentioned, are generated from A =ẑA z and F = 0. Standing waves will occupy the transverse dimensions, while a traveling wave will exist along z. Therefore, use of the magnetic vector potential in (3.40) is appropriate. Boundary conditions require that the tangential component of the electric field (E z ) vanish at x = 0, x = a, y = 0 and y = b. Therefore, f and g of (3.39) are Each pairing of the integers m and n represent a possible mode. If only forwardtraveling waves are considered, the TM z mode functions are where the constants have been combined. The constraint equation (3.41) then be- The field components are obtained from (3.34) and (3.36). A solution using TE modes can be determined in a similar manner.

Impedance.
The use of the term impedance to describe the complex ratio of voltage and current was applied, in turn, to circuit theory, transmission lines, and electromagnetic fields [24]. Several representations of impedance exist, depending on the field type, direction of travel, and the medium of propagation. The intrinsic impedance η of a material, dependent only the parameters ǫ and µ, is The wave impedance, is the ratio of transverse components of E to transverse components of H. The impedance is dependent on the type of wave: TEM, TE, TM all have characteris-tic wave impedances particular to the type of waveguide, material, and frequency. If a plane wave (TEM, TE, or TM) is normally incident on the material, then the wave impedance Z is reduces to the intrinsic impedance η.

Cutoff Frequency.
Forward wave propagation (without attenuation) in the waveguide occurs when γ z = jβ is purely imaginary. When γ z is purely real, the wave is evanescent and does not propagate. The transition between these two states occurs at the cutoff frequency, when where Ψ is a mode function and ∇ t is the transverse gradient. By Green's first identity, Also, by Green's second identity, If k ci = k cj the integral must vanish, as must the right hand side of (3.49), that is It is equivalent to state that if the inner product of two functions (or vectors) is zero, then they are orthogonal. The proof (from Chapter 8 of [16]) may be extended to the magnetic field vectors h as well. The transverse electromagnetic field at any point can then be expressed as the sum of the mode vectors: where the mode vectors e and h can be either TE or TM.

Power Transmission.
When many modes in a waveguide with PEC walls exist simultaneously, each will propagate energy independently [16]. For the current test setup of a PFW this is especially relevant, since it is expected that an infinite number of TM y modes will exist on both sides of the discontinuity. In the fully filled waveguide, all the energy is transmitted by the dominant TE 10 mode; this allows the use of closed form algorithms, such as NRW, which consider single-mode propagation. When multiple modes exist, it is necessary to take into account the complex power transmitted by both the propagating and evanescent modes. For N modes, the total z-directed complex power in a section of waveguide is the sum of the power contained in each mode, such that where V i is the mode voltage and I i is the mode current of the ith mode. It is apparent that this is the expected result, since the complex power of any system is

Summary
A cursory review of electromagnetics was presented in this chapter, including the use of vector potentials and the modal expansion of fields in a waveguide. These concepts form the basis of the modal analysis found in Chapter IV.

IV. Mode-Matching Analysis
The characterization of electromagnetic materials by transmission and reflection methods in a rectangular waveguide assumes the sample to be homogeneous and precisely machined, i.e. it completely fills the inner dimensions of the waveguide and is perpendicularly planar to the waveguide walls, as seen in Figure 4.1. At high temperatures, thermal expansion distorts the geometry of both the sample and the waveguide itself [21]. The metal waveguide tends to expand at a greater rate than the sample under test, resulting in air gaps between the sample and the waveguide walls. The boundary conditions at the gap require the excitation of higher order modes which are not accounted for in closed form solutions such as the NRW algorithm, which uses the dominant (TE 10 ) mode only [17,21].
Development of the solution for a single top gap will follow that suggested by Collin [10] and Harrington [16], using TM y modes to construct the electric and magnetic fields in the unobstructed and partially filled waveguide regions. By applying appropriate boundary conditions, and ensuring that tangential components of E and H are matched, a system of equations that accurately describes the structure can be developed. In turn, the forward reflection (S thy 11 ) and transmission (S thy 21 ) coefficients can be extracted. A minimization of the difference between the theoretical and experimental S-parameters is calculated according to the minimization equations A Newton-Raphson root search of the complex parameters ǫ and µ is performed until an acceptable minimum tolerance is reached. In addition, it is necessary to have available a reference plane independent measurement scheme to compensate for error in sample placement or if alignment with the calibration plane cannot be guaranteed.
This method is presented in the final section of the chapter.

Newton-Raphson Root Search
The minimization equations used in this thesis can be thought of as two functions, dependent on values of ǫ r and µ r that force the differences in (4.1) below the required tolerance. The complexity of the desired root search depends on the parameters of the material. If a material has both electric and magnetic loss properties, then a 2-dimensional root search, using two equations and two unknowns, must be used.
However, if a material is a non-magnetic dielectric, only ǫ r is the unknown (µ r = 1), and a complex 1-dimensional root search of one equation and one unknown can be applied. This technique will be shown first.
One of the most common techniques in numerical analysis used to determine roots of equations is the Newton-Raphson method [5]. If p is an unknown root, and a function f is differentiable on the interval of all approximations to p, then f (p) = 0. where ξ is on the range x 0 to x 0 + ∆x. If the approximation x 0 + ∆x is set equal to p, then he first derivative of f can be calculated numerically, using forward, central, or backward differences. If ∆x is small, then the third and higher order terms can be neglected. This leads to an estimate for ∆x, The next approximation x 1 to the root is obtained by adding ∆x to the previous estimate, x 0 , In general, then, the nth approximation to the root p is Convergence is dependent on a "good" initial guess for the root p. If the root is known to be complex, the initial guess must also be complex; the method will not converge to a complex root if a purely real initial guess is supplied [1].
If roots to more than one equation are to be found simultaneously, Newton-Raphson's method can easily be expanded to higher dimensions. Consider two func- with roots u and v. It is assumed both functions are differentiable on the interval, or surface, of approximations to u and v. If x = x 0 and y = y 0 are supplied as estimates to the roots, then the Taylor series expansion around the approximations is where the higher order derivatives of the series have been neglected. Letting x 0 +∆x = u and y 0 + ∆y = v, (4.8) reduces to which can be recast in matrix notation as The matrix can be inverted to obtain expressions for ∆x and ∆y, Proceeding as before, the next set of estimates to the roots are found by adding ∆x and ∆y to the previous estimates, x 0 and y 0 .
The nth root approximation is therefore x n = x n−1 + ∆x n−1 y n = y n−1 + ∆y n−1 (4.13) Depending on the slope of each function f and g, the initial guesses usually must be good approximations to the root. If the third term of the Taylor series is too large, then it may be necessary to include it in order to achieve root convergence [5].

Partially Filled
The wave impedance, by definition, is the ratio of transverse e and h vector field components of the dominant mode and is therefore equal to Since this is equal to the wave impedance of a TM y 10 mode, and because all the scattered modes will be in the set of TM y 1n , the complete mode set can be constructed using TM y modes. The three computational regions used in field construction: Region I: empty; Region II: PFW; Region III: empty.

Field Construction. It is sufficient to use the magnetic vector potential
A to generate the necessary electromagnetic field components required for satisfying boundary conditions in all the regions of the waveguide. where k x , k y , and γ 0 are the transverse wave numbers and complex propagation wave number, respectively, that satisfy the constraint equation in free space, The use of e −γ 0 z denotes a forward propagating wave in theẑ direction (i.e. through the waveguide), and γ 0 is, in general, a complex quantity. The respective E and H-fields can be generated by Substituting (4.15) into (4.16) and simplifying, the E components may be written as and the H components as Satisfaction of electric field boundary conditions at the waveguide walls requires B 2 and C 1 to vanish, leading to the definition of the wave numbers k x and k y : where a is the long dimension of the waveguide cross section, and b is the short dimension,x andŷ respectively (refer to Figure 4.2 (a)). By the Uniqueness Theorem, it is sufficient to satisfy only the electric field condition [4].
The complex mode propagation wave number γ 0 of the empty region of the waveguide can then be written as By applying the aforementioned boundary conditions, the Region I electric and magnetic field components of (4.17) can simplified as Using PFW theory in Region II, spatial shift factors are used to create an alternative magnetic vector potential A, which is then applied to construct the fields in each subregion (material and free space), namely where the subscripts 1 and 2 represent each subregion.
In Region II, the electric field boundary condition inx is identical to that of Region I, namely, the tangential fields vanish at x = 0 and x = a. Therefore, a standing sine wave exists in both subregions of Region II and throughout Region I; this is manifested in the choice of harmonic functions in (4.19). From (4.16) the fields in each region can be calculated. If the sample has height h, then fields in the first subregion (0 < y < h) exist within the material: Fields in the second subregion (h < y < b) exist in free space: Boundary conditions require that the tangential electric field must vanish at y = 0 and y = b, forcing the constants C 1 and C 2 to zero. Continuity of tangential E and H at y = h requires that k x and γ zn be the same in each subregion [16]. In addition, the tangential components e x1 , e x2 , e z1 , e z2 , e x1 , h x2 , h z1 and h z2 must be continuous at the boundary between the subregions, y = h. Equating e z1 and e z2 of (4.20) and (4.21), the requirement is obtained. Similarly, the continuity between h z1 and h z2 requires Using either (4.22) or (4.23), the propagation coefficient of subregion 1, B ± n , can be expressed in terms of the propagation coefficient of subregion 2, C ± n , that is Division of (4.22) by (4.23) gives or, equivalently such that the wave numbers in each subregion satisfy the constraint equation of (4.26b). Since both k y1 and k y2 are dependent on γ z , (4.26a) represents a transcendental eigenvalue equation for possible values of the PFW mode-propagation constant γ z . Therefore, the equation must be solved numerically. When the correct value of γ is found, the ratio B ± n C ± n is given by (4.24). Incorporating this ratio into (4.20), the field 35 components of subregion 1 are (4.27) and the field components of subregion 2 are The wave number γ z is determined using the height iteration method discussed in Section 4.2.6. The wave numbers k y1 and k y2 , which represent variation in theŷ direction in each subregion, are not expected to be equal to each other. The fields in each of the three regions having now been constructed, it becomes possible to define the system modes.

Modal Analysis.
In an empty waveguide, only the dominant mode propagates completely through the waveguide. All other higher-order modes are The TM y 10 mode is incident on the sample, and scatters the mode set TM ±y 1n . Hybrid TM y modes exist in Region II, between z = 0 and z = d. In Region III (z > d), forward propagating TM y modes exist. All modes for n > 0 are evanescent. evanescent and rapidly decay before reaching the network analyzer ports. For the geometry described, which is discontinuous inŷ, the dominant mode is TM y 10 and the scattered mode set is TM y 1n , where the index n indicates the mode number. In any region of the system, the total transverse electric and magnetic field can be represented as the superposition of the forward and reverse propagating modes: where the coefficients a ± n represent the complex weighting constants of each mode. The field vectors e n and h n describe the vector components of the electric and magnetic fields of each mode, respectively [13]. Depending on the value of z, either the complex propagation wave number of free space γ 0 , or that of the PFW region, γ z should be used.
Using (4.29) as a guide, the total transverse fields in each region of the system can be constructed. In Region I (z < 0), a TM y 10 mode propagates in the forward direction, and an infinite number of TM y 1n modes scatter in the reverse direction due to scattering from the sample (see Although an infinite number of reverse scattered modes exist, only the n = 0 mode will propagate completely through the waveguide to the network analyzer. In Region II (0 < z < d), the existence of an infinite number of forward and reverse traveling modes are necessary to satisfy the boundary conditions at the discontinuity in the geometry of the sample. The transverse fields can be written as cutoff exists and all modes will propagate, although the modes that exist above the operation frequency will rapidly decay [17].
In Region III (z > d), the total fields can be represented by the forward propagating mode set of TM y 1n ; no reverse traveling waves are present due to the absence 38 of reflecting obstacles in this region. The final mode set can be described by where γ 0 is the propagation wave number of free space. A phase shift has been introduced into the complex exponential to facilitate matching of boundary conditions between regions.

Boundary Conditions.
It is necessary to maintain continuity of the transverse E and H field components at the interfaces between each region, namely at z = 0 and z = d, which implies the satisfaction of the conditions Since the scattered mode set is TM y , the field components h y do not exist. It will be shown that this allows e x to be written as a function of the field components e y and h x , thereby eliminating the need to satisfy e x explicitly.

From (3.1) and (3.2), Maxwell's curl equations (Faraday's Law and Ampere's
Law) in a source free region are Performing the curl in rectangular coordinates, and letting h y = 0, this can be expanded into six equations, Performing the partial derivative on e x in (4.34b) and using the expression for e x from (4.17) yields where the sign is chosen with respect to either forward or reverse traveling waves.
Taking the two terms on the right hand side of (4.39), substituting ∂ 2 ex ∂y 2 = −k 2 y e x , multiplying both sides by ω 2 ǫµ, and rearranging terms yields The total transverse field consists of both forward and reverse propagating modes, and the (±) is applied to each type of mode respectively. At material boundaries, regardless of the presence of discontinuities, tangential E and H and their derivatives must be continuous [10,16]. Having shown that e x is linearly dependent on the second derivatives of e y and h x , the interfacial boundary conditions on e y and h x will be satisfied analytically. The resultant continuity of e x can be shown numerically given a successful solution of the mode matching matrix of (4.56).
Region II is divided into two subregions, each with different material parameters. Therefore, continuity of e y and h x , and also (4.40), must satisfy the boundary conditions (4.33) piecewise, that is at the z = 0 plane. Additionally, at the z = d plane, the piecewise equalities must hold, where e II x , e II y and h II x are functions of position in y, corresponding to the vector field components of each respective subregion. The total fields are matched at the boundary are the superposition of an infinite number of higher order modes, which must be truncated through practically to N modes for computational purposes. Now that the required transverse field components, namely e y and h x , have been identified, the boundary conditions of (4.33) can be applied to the mode sets of The constants of (4.48) correspond to the interfacial reflection and transmission coefficients of the sample, at the front (Γ n , t n ) and back (r n , T n ) sample interfaces, respectively. The required S-parameters of the system can be determined from the mode coefficients, as they are simply If the sample is a simple material, the reverse S-parameters can be accurately equated to the forward parameters, that is to be applied to each mode vector, which is the inner product of a mode vector f and the testing function ψ. This choice of operation, when used on the fields of Regions I and III, will cause many terms to vanish, since sinusoidal functions of different mode indices are mutually orthogonal. The linear dependence of e x on the field components e y and h x removes the requirement to integrate the testing function over the waveguide cross section, as is often done in other methods [7,13,17,18]. The testing operation is applied term by term to the linear system of equations in (4.47), which can then be written as To further reduce the required number of calculations, it is useful to observe that the transverse e and h field components have similar spatial variation, containing harmonic functions of identical arguments. This means that the testing operation integrals need only be calculated once, and then can be multiplied by an appropriate scaling factor to correspond to the necessary field component. To facilitate anticipated matrix algebra, the testing operations of (4.53) corresponding to the electric field mode vectors can be represented by submatrices, such that where ζ is the scaling factor from (4.24). Submatrices U np and V np correspond to the testing operation applied to the electric field vector of subregions 1 and 2, respectively.
When the inner product is performed on the magnetic field vectors, the resultant submatrices are where Z is the z-directed wave impedance in each region, and the notation is that of (4.54). The respective integrals of (4.54) and (4.55) are presented in Appendix B Using (4.54) and (4.55), the system of (4.53) can be represented in matrix form As has already been indicated, the S-parameters S thy 11 and S thy 21 are identically Γ 1 and T 1 of the solution set. The minimization equation (4.1) can now be used to search for the correct values of ǫ and µ.

Height Iteration Method.
It has been stated that the eigenvalue equation of (4.26a) is a transcendental equation for possible values of the complex propagation wave number γ z,P F W and must be solved numerically [1]. A good initial guess is imperative to successfully finding the root of the equation. The value of γ z,F F W of the fully fillwed waveguide (FFW) is used as the initial, unperturbed guess for γ z,P F W in (4.26a) with an initial guess as h = b − δ, where δ is a small value.
A Newton-Raphson root search algorithm, using central-difference derivatives, uses these values to converge on a new γ, which becomes the initial guess in the next iteration. The height h is decreased again by δ, and the process repeats until the input height h is the actual value. If the final iteration converges, then γ is accepted as γ P F W , which is used to fill the mode matrix of (4.56). The top-level root search, using the minimization equation of (4.1), continues the iteration for parameters ǫ and µ.

Reference Plane Independent Measurement
The two measurements discussed in the previous section, S 11 and S 21 , are the reflection and transmission S-parameters, respectively. The reflection measurement is highly dependent on the sample location relative to the calibration plane along the z-axis (see Figure 4.1), since a change in position incurs a two-way phase delay (or advance) in the measurement. The transmission measurement is not sensitive to the placement of the sample.
To perform calculations, it is necessary to extract the true S-parameters (denoted with superscript s ) from the measured S-parameters (denoted with superscript ms ) obtained from the network analyzer. This section will assume that the sample has thickness d, and is in a waveguide sample holder of width w. Also, the notation k z , instead of γ z , is used for the propagation wave number. Using Figure 4.5 (a) as a guide, and using complex exponential notation to denote phase shifts, the measured S-parameters are:  It can be seen that the forward and reverse transmission measurements of (4.60) and (4.59) are the same. However, the forward and reverse reflection measurements of (4.60) incorporate the two-way phase delay/advance caused by the shift. Even a small value of δ will cause a significant phase shift of the S-parameters.
It is quite possible that, in the course of handling the material sample and the waveguide system, the exact distance between the sample and the calibration plane may not be known. During high-temperature measurements, this problem can be exacerbated by waveguide expansion in extreme heat. To perform accurate material characterization in these situations, it is necessary to eliminate the dependence on the reference plane. By multiplying the forward and reverse reflection measurements In a simple material, the respective forward and reverse reflection and transmission measurements are equal, i.e. S 11 = S 22 and S 21 = S 12 . This allows Γ 1 and T 1 of the solution set of (4.56) to be used for both the forward and reverse coefficients, saving significant computational effort.

Summary
A method for characterizing an electromagnetic material that partially fills a rectangular metal waveguide in one dimension was presented using TM y modal analysis. The matrix A contains the region-to-region mode coupling information and the solution vector x relates A to the field excitation vector B. The theoretical reflection and transmission coefficients of the dominant mode, S thy 11 and S thy 21 , are extracted from x and compared to the experimental S-parameters. A 2-D Newton-Raphson root search iterates the parameters ǫ r and µ r until the absolute difference between the theoretical data and experimental data is within a specified tolerance.
In addition, a method was developed for removing measurement dependence on a sample's axial position in the waveguide with respect to the reference plane. This was accomplished by multiplying the forward and reverse reflection measurements

V. Results
The PFW measurement correction and the reference plane independence analysis were developed in the previous chapter for the case of a single air gap between the top of the material sample and the waveguide. The analysis uses mode matching to calculate the full set of theoretical scattering parameters and compares them to the experimentally measured scattering parameters. A two-dimensional Newton-Raphson root search is used to minimize the difference between the theoretical and experimental S-parameters using the equations where it is accepted that the actual values of permittivity ǫ and permeability µ will drive the two functions of (5.1) below the specified tolerance.

Test Procedure
Microwave measurements were performed at room temperature in two frequency bands, S-band (2.6 − 3.95 GHz) and X-band (8.

51
The results presented are the characterization of two materials, acrylic and FGM-125. The acrylic material is a commercially available, ideally lossless dielectric.
FGM-125 is a commercially available rubberized magnetic shielding material, having both electric and magnetic losses, manufactured by Emerson & Cuming. Measurements of the acrylic were only done at S-band, while FGM-125 was measured at S-band and X-band. This research is intended specifically for magnetic materials, but it can be applied generally, hence the inclusion of the acrylic measurements.
Parameter data is presented visually using the convention ǫ r = ǫ ′ − jǫ ′′ and µ r = µ ′ − jµ ′′ . Truth data was calculated using NRW on a measurement of a sample that completely filled the waveguide. RPI was not used to calculate the truth data, since it is not a standard measurement technique. The uncorrected data is the raw data of the PFW measurement, and NRW is used to extract the effective permittivity and permeability of this measurement. Additionally, the minimization equations of (4.63) are used in conjunction with PFW theory to characterize samples using the reference plane independent formulation. The corrected data corresponding to this method is labeled RPI. The RPI correction is always calculated with the same number of modes as the standard mode-matching correction.   It has been suggested that the real and imaginary components of permittivity and permeability be described as functions of h and d [12,13]. However, this choice of functions does not reflect the mechanics of the algorithm itself, since the actual solution set is theoretical S-parameters. An error analysis of the S-parameters themselves would not be particularly useful, since the purpose of the research is to calculate ǫ and µ. Therefore, the error analysis of ǫ and µ will consider propagation of uncertainty in the calculated S-parameters due to the measurement uncertainty in sample height and thickness.
Let κ be either relative permittivity or permeability, such that The real part of the calculated material parameter due to a height h and thickness d is expanded in a Taylor series around the point h 0 and d 0 , where δh and δd are the measurement uncertainties (±2 mils) in height and thickness, respectively. The higher order terms of the Taylor series can be neglected since the uncertainty is so small. The total uncertainty in κ ′ can now be approximated by which represents the uncertainty contribution of both measurements. Since an analytic expression for the partial derivatives does not exist, they must be calculated via a numerical approximation. Using forward differences, the two partial derivatives of (5.3) are The quantities δh and δd can in general be positive or negative, but this analysis will proceed considering the compounded uncertainty of overmeasuring both sample dimensions. While all the errors, assuming they are independent and random, could be combined in quadrature, a worst-case approximation is The Triangle Inequality shows that the absolute value of the sum of the errors is always less than the sum of the absolute values [25]. Substituting the real and imaginary parts of ǫ and µ in for κ completes the error analysis.
The height iteration method used to solve the eigenvalue equation for the correct value of the complex propagation constant γ z in the PFW region is, unfortunately, not consistently stable. Providing an initial guess of γ F F W for a particular mode does not guarantee that the root search for γ P F W will converge on the next consecutive root, if it converges at all. It is also possible that the root search will find the same root twice, or skip a root. If this happens, the subsequent filling of the mode matrix in (4.56) will be inaccurate, due to the use of incorrect values of γ z . Verification of sample homogeneity is also beyond the scope of this research, so it is assumed that material samples are homogeneous.

Acrylic.
Although this mode-matching technique is meant to be applied to magnetic materials, a correction can be applied to a dielectric material in order to verify that the algorithm is working. Therefore, acrylic samples were ob-tained from the AFIT machine shop, manufactured to the dimensions shown in Table   5.1. Acrylic is non-magnetic, so it was assumed that the sample permeability is that the results for real permittivity displayed in Figure 5.3. Acrylic is essentially lossless (ǫ ′′ ≈ 0), so the imaginary component is not presented graphically. The corrected permittivity is within 2% of the true value for the entire band, and provides a noticeable improvement over the uncorrected data. When combined with the RPI formulation (also using 5modes), the correction is nearly perfect.
Having shown the performance of the correction for this simple case, it is now appropriate to consider the more complex situation of magnetically lossy material.

FGM-125.
Samples of FGM-125 were obtained and machined to the dimensions shown in Table 5.2, to be used in tests at S-band (2.6 -3.95 GHz). In addition, samples were machined to the dimensions shown in Table 5.3 to be used in tests at X-band . The S-band results will be discussed first.   This is an expected property of magnetic materials, that both µ ′ and µ ′′ tend to decrease quickly for increasingly high frequencies [21].
From Figure 5.4 (a), it is apparent that the ǫ ′ extraction is highly sensitive to gap size, since the uncorrected measurement is consistently 8% below the true data.
The modal correction, both with and without the RPI formulation, converges to a slightly higher value of real permittivity, just outside the range of the error bars, within 5% of the truth data. The uncorrected data for ǫ ′′ is sufficiently close to the true value of zero, so a modal correction is not actually necessary. However, it can be seen that inclusion of RPI yields a superior result than when it is neglected.
Examining the plots of real and imaginary permeability in Figure 5.5 it can be seen that these measurements are not as sensitive to top/bottom air gap, given the proximity of the uncorrected data to the truth data. The 10 mode correction does not much improve the extraction for either real or imaginary parts, although using RPI has a considerable effect, especially for the imaginary component.
In a lab environment, it is expected that the waveguide samples will not be deliberately machined with large gaps. Even in a high temperature measurement, unless the sample falls over onto its side, extremely large gaps are not expected.
However, in order to illustrate a more drastic case of a PFW, and to test the accuracy of the modal method for large gaps, Sample 2, with a gap of 81 mils was tested. As the largest gap size under test, it will be easier to see the benefit of including successive higher-order in the mode-matching correction. Therefore, a 10 mode correction is presented in Figure 5.6 and Figure 5.7.     The correction of ǫ ′ in Figure 5.6 (a) yields excellent results, as the truth data is within the uncertainty of the modal correction, both with and without the RPI formulation. Considering the correction to ǫ ′′ in Figure 5.6 (b), once again the uncorrected data yields an acceptable value. Of the modal correction methods, however, RPI is the preferred solution , since it provides the most improvement. Likewise, in Figure 5.7, the use of RPI is preferred as a correction to µ ′ , and µ ′′ .
The difference between the uncorrected permittivity data of the 27 mil gap (Figure 5.4 (a)) and the 81 mil gap, and the relative stability of the other three complex parameters, is a further indication of the sensitivity of permittivity extractions to this particular PFW geometry. This is due to the field pattern in the PFW region.
The electric field vector e y is discontinuous inŷ between the material and free space, creating a capacitive charge distribution [22]. The measurement of electric permittivity, therefore, is very sensitive to this geometry. However, the magnetic field vector h x , for a given value of y, is continuous inx. Accordingly, the measurement of magnetic permeability is much more stable than the permittivity measurement for top air gaps. If, for example, the air gap was located in the other dimension (i.e. left/right), an inductive charge distribution would be created. It can reasonably be assumed that in this scenario, the permeability measurement would suffer more than permittivity, using similar reasoning.
The second round of room temperature measurements were performed at Xband, using the samples described by Table 5 The real permittivity extraction of Figure 5.8 (a) represents an unsuccessful attempt to improve ǫ ′ , as both corrections are less than the truth data. This may be due to a strong capacitive effect at the measurement frequency. The corrected and uncorrected data sets corresponding to ǫ ′′ in Figure 5.8 (b) are nearly identical throughout the entire band, which is consistent with previous measurements. The 10 mode correction, when applied to magnetic permeability, is very accurate, having better performance than the RPI formulation, which can be verified in Figure 5.9.
It must also be pointed out that the 2 mil uncertainty in height and thickness has almost no effect on extraction of permeability. This is in contrast to permittivity, which is clearly seen to be affected by a small amount of uncertainty in the specified sample dimensions. The standard mode correction would be preferred in this case, since it has a better average value and the average RPI value is too low.
The extraction of ǫ ′′ in Figure 5.10 (b) is the first instance when the uncorrected data did not accurately approximate the truth data. In addition, the two modal corrections show significant variation with frequency. This may be due in part to the instability of the Newton-Raphson Root search when used in parameter extractions on lossless materials. It is well known that "good" guesses are critical when using this algorithm [1,13]. Of the corrections to permeability, shown in Figure 5.11, the mode-matching correction using RPI was superior for both µ ′ and µ ′′ .

Complex Propagation Wave Number.
It has been mentioned that a valid PFW correction is contingent upon satisfaction of the transcendental equation   The RPI correction is the best of the 3 data sets, but still varies significantly from the truth data. which, through finding γ z , enables the determination of the wave numbers k y1 and k y2 . Using the procedure outlined in Section 4.2.6, an initial guess of γ F F W , the propagation wave number of a fully filled waveguide, is provided to the root search.
It is trivial to calculate γ F F W for any number of desired modes, since it is based on the waveguide geometry. However, using a numerical method to determine γ P F W may lead to erroneous results, due to instability in the iteration.
Therefore, it is useful to observe the response of the mode-matching correction as successive modes are included in the analysis. This will be done with FGM-125 Sample 3 (S-band), having an air gap of 81 miles. Corrections to ǫ r and µ r using 1, 2, 3, 6 and 10 modes are illustrated in Figures 5.12 and 5.13. In Figure 5.12 (a), it is clearly seen that the uncorrected permittivity obtained using NRW is far below the true value rendering it completely inaccurate. Using the mode-matching technique with 1 mode (i.e. no higher-order modes) yields a small improvement. Adding 1 higher order mode improves the measurement further. Including 2 higher order modes (i.e. 3 modes total) results in an overcorrection, but as more modes are considered, the modal correction moves closer to the true value. Similar observations can be made about the ǫ ′′ correction in Figure 5.12 (b), although in that case the uncorrected data is closest to the truth data. This is also true for the corrections to µ ′ and µ ′′ in Figure   5 The ratio of α to β, or the propagation -attenuation ratio, for each root of γ from Figure 5.14 is given in Table 5  is an "overcorrection", but as more modes are included the correction rights itself. (b) Similar to (a), it is better to use more modes than too few. Uncorrected data gives the best result.  It is expected that the the dominant mode in a PFW system will have a small propagation-attenuation ratio, and this can be verified in Figure 5.14 (a), where the dominant mode can clearly be identified. The fact that the higher order mode roots in (a) are close to zero indicates minimal propagation. This is similar to the behavior of modes in a fully filled guide, in which the higher order modes (if they even exist) attenuate rapidly.
However, for the 81 mil gap case in Figure 5.14 (b), the situation is very different.
Both the real and imaginary components of the roots are much larger than in (a). This causes the α-β ratio at every mode, except the first, to be much smaller than the 27 mil gap case. The first mode has 7 times the attenuation of the 27 mil gap dominant mode, so it can immediately be recognized that each γ, especially in the large gap scenario, is both highly propagating and highly attenuated. The combination of these effects implies that many higher order modes are necessary for a proper correction using the PFW mode-matching technique. This is consistent with the observations made of the successive mode corrections of Figures 5.12 and 5.13. Including several higher order modes balances the total field, which leads to a more accurate correction. The potential exists for instability in the numerical root search algorithm used to converge to roots of the eigenvalue equation. If the solution space has a shallow gradient, such in some regions of Figure 5.14 (b), the Newton-Raphson method (which 72 uses a tangent line to approximate the next guess) may obtain a poor approximation.
It also is helpful if the roots are clearly distinguishable from one another, as in Figure   5.14 (a). In cases where the roots are very close together, or do not stand out greatly from the surrounding space, it may be appropriate for a different root search, such as the Muller Method, to be used [1,13], although this step was not taken here.

VI. Conclusions and Recommendations
This thesis has demonstrated the feasibility of using modal analysis to accurately perform electromagnetic material characterization in a partially filled waveguide. Waveguide expansion is a common problem in high-temperature material measurements, since gaps form between the sample and waveguide walls. This discontinuous geometry excites higher order which are not accounted for in algorithms such as Nicolson-Ross-Weir. A correction was accomplished by considering a single air gap between the top of the sample and the waveguide. Because the waveguide is partially filled in the y-axis, the scattered mode set can be completely described using TM y modes. Measurements of acrylic and magnetic shielding material in a PFW geometry were performed at room temperature in S-band and X-band. For most air gaps, acceptable corrections, i.e. within 10% of the true value, could usually be achieved using 15 modes or less. At S-band, modal corrections for air gaps greater than 100 mils did not converge. Modal corrections on the 45 mil gap, the largest gap at X-band, did not converge using more than 10 modes.
It was difficult to obtain useful modal corrections of the large gap PFW acrylic samples. Even when using a single transmission measurement to extract permittivity, i.e. a 1-D search, the near-lossless property of the sample made the root searching unstable, and no amount of modes used in the correction would converge. Corrections to air gaps in magnetic shielding material measurements were usually more successful, unless a significant gap (60-100 mils) was present. In most cases, the algorithm could be expected to compute the entire data set (201 points) in less than 20 minutes using a 15 mode correction.
The calculation of real permittivity (regardless of material) is more sensitive to a top air gap than either imaginary permeability or real and imaginary permeability.
This is due to the electric field vector discontinuity that exists across the material boundary, which does not exist in the two other transverse field vectors. Therefore, it is reasonable that the mode-matching technique will be most effective at correcting the parameter most corrupted by the presence of an air gap. If the gap was in the other dimension, i.e. left-right, it is expected that the permeability measurement would suffer. The modal correction was usually able to obtain an acceptable correction for ǫ ′ when ǫ ′′ was very small. The success of modal corrections to µ r depended on the gap size and frequency range. It was not unusual for the "uncorrected" PFW data to be the most acceptable data set.
The addition of a reference plane independent formulation to the mode-matching technique contributed significant improvements to the correction. Foremost, this indicates that samples were not precisely aligned with the reference plane. Secondly, by using both forward and reverse scattering parameters, the effects of sample inhomogeneity on permittivity and permeability extraction is reduced. Based on these observations, it is recommended that future material characterizations include a reference plane independent solution method.
The experiment was not performed at high temperature, but is a presented as a proof of concept technique for a partially filled waveguide environment.
This research presents a novel method for characterization improvements in high temperature rectangular waveguide measurements. The foundation of the single air gap correction is a mode-matching technique of the dominant and higher order scattered TM y modes. The mode-matching is then combined with reference plane independence to eliminate possible phase shift errors. Finally, the method is applied to measurements involving a magnetic material. The combination of these components has not, to the best knowledge of the author, been presented before.

Future Work
It has been observed that a vital part of obtaining a correct solution using modal analysis is the accurate determination of the complex propagation constant γ P F W . If an iterative search is used, care must be taken to ensure that "good" initial guesses are supplied to the algorithm. As an alternative, however, a root-search method which can identify and sort a desired number of mode propagation constants could be sought. Use of finite element analysis, genetic algorithms, or a less-sensitive iterative root search are possible options for improvement in this area. With more accurate values of γ P F W , corrections for larger gaps can be calculated.
The top air gap PFW analysis should be combined with side air gaps to better simulate a high-temperature situation, since it is expected that all dimensions of the waveguide will expand. The room temperature tests had a priori knowledge of the dimensions, specifically the height, of the sample under test. This observation, however, cannot be made at high-temperature. It is recommended that known thermal expansion coefficients be combined with the PFW analysis to obtain a reasonably accurate guess for the air gap between the sample and waveguide while at hightemperature.

Appendix A. Nicolson-Ross-Weir Algorithm
Nicolson, Ross [23] and Weir [27] took expressions for the S 11 reflection coefficient and S 21 transmission coefficient and derived explicit formulas for the calculation of the material parameters permittivity and permeability (ǫ r , µ r ). The derivation presented here pertains to use in a rectangular waveguide where a ≈ 2b.
Given the scattering parameters S 11 and S 21 , the reflection coefficient R is given by The choice of sign in (A.1) which forces |R| < 1 is taken. Now, the transmission coefficient P is given by The impedance z can also be found, as it is Z 1 e 1,y1n , ψ y = 1 Z 2 e 2,y1n , ψ y = 1