Deficiency in the Error Propagation Method for Sensitivity Analysis of Free Space Material Characterization Raenita A. Fenner
Edward J. Rothwell
Department of Engineering Loyola University Maryland Baltimore, MD 21210 Email:
[email protected]
Department of Electrical and Computer Engineering Michigan State University East Lansing, MI 48824 Email:
[email protected]
Abstract—An important aspect in the development of any material characterization method is error analysis. As the necessity for non-destructive evaluation and quality assurance applications increase, free space material characterization methods along with error analysis of these methods will become imperative. A common approach for performing error analysis is the error propagation method. The error propagation method is a standard method to predict the amount of propagated uncertainty in an experimentally determined quantity. This paper demonstrates a deficiency in the error propagation method in predicting propagated error in the extracted permeability and permittivity of free space material characterization methods. The paper provides an overview of the error propagation method and demonstrates the deficiency through error analysis of a free space method entitled the dual polarization method.
methods become an essential method in quality control and non-destructive evaluation, error analysis of these methods will simultaneously become essential. A common method for performing error analysis is the error propagation method. This paper presents a deficiency in the error propagation method in predicting the amount of propagated error in free space methods. An overview of this deficiency is presented with instances where the error propagation fails and links the deficiency to physical phenomena within a particular free space method called the dual polarization method. Comparisons between the propagated error calculated by the error propagation method and Monte Carlo simulations are also shown. II. T HE E RROR P ROPAGATION M ETHOD
I. I NTRODUCTION Microwave materials are critical to a number of applications such as, but not limited to, high-speed, high-frequency circuits, communication devices, and military satellites. An important aspect of the design and maintenance of applications which depend upon microwave materials is knowledge of the electric and magnetic properties of materials. The process of determining the relative electric permittivity, r , and the relative magnetic permeability, µr , of a material is termed material characterization. Material characterization is not only important to design applications, but also to quality control and non-destructive evaluation applications. Within this subset of applications, material characterization methods which allow minimal contact to the material and reduce compromising the material sample are highly desirable. Material characterization methods can broadly be classified into two categories - resonant and non-resonant methods. Resonant methods are used to accurately determine r and µr at a single frequency or several frequencies. Non-resonant methods are used to determine r and µr over a band of frequencies. Non-resonant methods can be implemented via waveguides, probes, or free space methods [3]. Of these non-resonant methods, only free space methods can be implemented in a contactless manner. As free space material characterization 978-1-4673-0292-0/12/$31.00 ©2012 IEEE
The error propagation method determines the measurement uncertainty (denoted σ) in a quantity which is computed from direct measurement of other quantities. For example, suppose calculation of a hypothetical quantity A is desired. The calculation of A is dependent on the measurement of quantities x, y, and z. However, quantities x, y, and z have their own uncertainties σx , σy , and σz . Therefore, A will have inherent uncertainty due to the uncertainties in x, y, and z. The measurement uncertainty is an expression for the random error which propagates into the final computed quantity. The measurement uncertainty also provides the range where approximately 68.0% of all the measured quantities will lie. For a quantity dependent on other measured quantities, the final solution is presented in the format best estimate ± measurement uncertainty. The formula for the standard deviation of A due to the uncertainties in independent quantities x, y, and z is the quadrature formula
σA =
s
σx2
∂A ∂x
2
+ σy2
∂A ∂y
2
+ σz2
∂A ∂z
2
.
(1)
A complete explanation of the derivation of the error propagation method is found in [4]. Key assumptions for the
error propagation method to be valid are that the uncertainties in the independent variables are small and the independent variables are independent from one another. III. T HE D UAL P OLARIZATION M ETHOD The dual polarization method is a free space material characterization method. The dual polarization method uses the measured reflection coefficients from parallel and perpendicular polarized incident plane waves. In this manner, closed form expressions can be found for µr and r . The dual polarization method is especially useful for characterizing conductor backed media as it only requires reflection data to characterize the material under test (MUT). Fig. 1 illustrates the implementation of the dual polarization method.
θ
Measurement II (Perpendicular Polarization)
k E
Conductor
k H
Material Under Test
θ
Measurement I (Parallel Polarization)
the material under test (MUT) the impedance is kη/kz , ⊥-polarization Z= kz η/k, k-polarization,
while for the overlay the impedance is k¯η¯/k¯z = η¯/ cos θ, ⊥-polarization Z¯ = (6) k¯z η¯/k¯ = η¯ cos θ, k-polarization. p √ µ , η = η µr /r , and kz = Here k = ω/c, k = k r r 0 0 0 p ¯ η¯, and k¯z follow, k0 µr r − µ¯r ¯r sin2 θ. The definitions of k, with (µ¯r , ¯r ) in place of (µr , r ). The dual polarization method has closed form extraction equations for µr and r . The extraction equations for µr and r are in (7) and (8) respectively. The terms X, Y , and Q are defined in (9)-(11). s sin2 θ X (7) µr = ±µ¯r Q2 1 − Y cos2 θ s Q2 sin2 θ r = ±¯r . (8) X 1 − Y cos2 θ 1 + Γk 1 + Γ⊥ (9) X= 1 − Γ⊥ 1 − Γk 1 + Γk 1 − Γ⊥ Y = (10) 1 + Γ⊥ 1 − Γk
IV. E RROR
Diagram illustrating the dual polarization method
Consider a conductor-backed planar material layer with thickness ∆, complex permeability µ(ω) = µ0 µr (ω) and complex permittivity (ω) = 0 r (ω), as shown in Fig. 1. The layer is illuminated by a uniform plane wave of frequency ω, originating in an overlay region with complex permeability µ¯r (ω) = µ0 µ¯r (ω) and complex permittivity ¯r (ω) = 0 ¯r (ω). If illumination occurs at an incidence angle θ from the normal to the surface, then the electric field reflected by the layer is determined by the global reflection coefficient [1] Γ(ω) =
R(ω) − P 2 (ω) . 1 − R(ω)P 2 (ω)
(2)
(3)
and R is the interfacial (Fresnel) reflection coefficient R(ω) =
¯ Z(ω) − Z(ω) . ¯ Z(ω) + Z(ω)
Based upon the extraction equations for µr and r shown in (7) - (8), measurement of the incidence angle, θ, MUT thickness, ∆, and the global reflection coefficients, Γ⊥ and Γk , are necessary to perform the dual polarization method. Each one of these independent variables has their own associated measurement uncertainty and will thus introduce uncertainty into the extracted µr and r . The uncertainty introduced into the extracted µr and r by θ, ∆, Γ⊥ , and Γk can be analyzed separately or simultaneously. The purpose of this paper is to demonstrate instances where the error propagation method will yield inconsistencies when compared to other error analysis methods such as Monte Carlo analysis. Therefore, for simplicity error analysis due to the uncertainty of θ is only presented. V. D EFICIENCY
Here P is the propagation factor P (ω) = e−jkz (ω)∆
1 − P2 = j tan(kz ∆) (11) 1 + P2 A NALYSIS OF THE D UAL P OLARIZATION M ETHOD Q=
d Fig. 1.
(5)
(4)
The wave impedances are dependent on the polarization of the illuminating field with respect to the plane of incidence. For
IN THE
E RROR P ROPAGATION M ETHOD
To demonstrate the deficiency in the error propagation method, consider error analysis of the dual polarization method for uncertainty in θ. Take the MUT to be a 40 mil thick slab of FGM40. FGM40 is a type of magnetic radar absorbing material (MagRAM) with r = 21.864 − j.390 and µr = 2.088 − j2.538 at f = 8.20 GHz. Assume an uncertainty in θ of σθ = 0.5◦ with θ = 45◦ . Implementation of the error propagation method to determine σµ0r , σµ00r , σ0r , and σ00r yields the results shown in Table 1.
Error Propagation 5.9868 4.8131 17.6710 53.0800
σµ0r σµ00 r σ0r σ00 r
Monte Carlo 0.8029 3.8937 7.0998 8.2487
σµ0r σµ00 r σ0r σ00 r
TABLE I AND r DUE TO σθ PREDICTED WITH THE ERROR PROPAGATION METHOD AND M ONTE C ARLO SIMULATIONS
Additionally, σµ0r , σµ00r , σ0r , and σ00r predicted by running 100,000 Monte Carlo simulations are shown in Table 1. Observation of Table 1 shows a percent difference ranging from 23.6% for σµ00r to 543.50% for σ00r . There are two interpretations to explain the discrepancies between the error propagation method and Monte Carlo simulation. One interpretation is that σθ = 0.5◦ is too large in order to accurately predict σµr and σr . One major requirement for the error propagation method is that the uncertainty in the independent variable (σθ in this case) is small. Thus, sufficient decrease in σθ should result in both the error propagation method and Monte Carlo simulations yielding the same σµr and σr . In this instance, reduction of σθ ≈ 0.04 is necessary to achieve more comparable results between the error propagation method and Monte Carlo simulations. For σθ ≈ 0.04, percent differences ranging from 5.4% for σµ0r to 9.0% for σ0r are achieved. Further decrease in σθ will yield even better agreement between the error propagation method and Monte Carlo simulation. A second interpretation for the discrepancy between the error propagation method and Monto Carlo simulations is linked to the physical phenomena which occur during implementation of the dual polarization method. In the first interpretation, σθ was considered too large. A contrary explanation would be that the first derivatives of µr and r with respect to θ are too large. Consequently it can be concluded that there is rapid variation in µr and r with respect to θ. Rapid variation of µr and r with respect to θ can be linked to the reflection coefficient, wave impedance in the MUT, propagation factor, etc. In this particular case, the rapid variation is due to the mismatch between the MUT material parameters and the material parameters of the media where the MUT is immersed or the overlay region (denoted µ ¯ r and ¯r ). This effect can be seen by examining the first derivatives of µr and r with respect to θ, also termed amplification factors. Computing the first derivatives of (7)-(8) and simplifying the results gives
where Fθ = Gθ = kz0
=
∂r = ¯r (Fθ − Gθ ) ∂θ
cot θ (1 − Y ) 1 − Y cos2 θ
(14)
2
Y k12 sin2 θ cos 2θ − Y cos4 θ 2kz (1 − Y cos2 θ)
(12)
(13)
−4jdP 2 kz0
Q (1 + P 2 )
Monte Carlo 0.2021 0.1856 0.7464 2.3996
TABLE II
U NCERTAINTY IN µr
∂µr =µ ¯ r (Fθ + Gθ ), ∂θ
Error Propagation 0.1916 0.1886 0.8800 2.3203
2
(15)
U NCERTAINTY IN µr
AND r DUE TO σθ PREDICTED WITH THE ERROR PROPAGATION METHOD AND M ONTE C ARLO SIMULATIONS FOR ¯ r = 20
Observation of the denominator in (15) shows that if the term Yr cos2 θ is close to unity, (15) can become large due to cancellation effects. Recalling the fact that the term Y (defined in (10)) is essentially a product of Γ⊥ and Γk , Y can be controlled by the level of impedance mismatch (or media contrast) of the overlay and the MUT. In this manner, a better impedance match between the MUT and overlay region leads to reasonably valued amplification factors and accurate use of the error propagation method. Table 2 gives affirmation of this effect as it shows good agreement between the error propagation method and Monte Carlo simulations for σθ = 0.5◦ when ¯r is increased to 20 − j0. VI. C ONCLUSION A deficiency in the error propagation method to predict accurate uncertainty in r and µr for the dual polarization method has been demonstrated. Although the deficiency has only been demonstrated for the dual polarization method in this paper, it has also been evidenced in other free space material characterization methods such as the layer shift method, two-thickness, etc. described in [2]. The deficiency has been linked to large amplification factors which are linked to physical phenomena that occur during the implementation of the free space methods. The deficiency can be avoided with careful study of the extraction equations and amplification factors for possible instances where the error propagation method will not give accurate predictions of measurement uncertainty. R EFERENCES [1] M.J. Cloud E.J. Rothwell. Electromagnetics. CRC Press, Boca Raton, FL, 2nd edition, 2009. [2] E. J. Rothwell Fenner, R. A. and L. L. Frasch. A comprehensive analysis of free-space and guided-wave techniques for extracting the permeability and permittivity of materials using reflection-only measurements. Radio Science, 2012. [3] C.P. Neo V.V. Varadan V.K. Varadan L.F. Chen, C.K. Ong. Microwave Electronics. John Wiley & Sons Ltd, 2004. [4] John R. Taylor. An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books, 1997.
