Usually, in a reflection method, only one complex reflection coefficient is measured, from which only one complex materials property parameter, either permittivity or permeability, can be derived. To obtain both the complex permittivity and complex permeability, at least two independent reflection measurements are required. Here, we discuss six examples for the measurement of both permittivity and permeability using reflection method.
3.4.1 Two-thickness method
As shown in Figure 3.33, in a two-thickness method, the two independent reflection measure- ments are made by measuring the reflections from two samples made of the same material under test but with different thickness values. We assume that
L1
L
Sample 2 Port 1
Sample 1
Short circuit L2
Figure 3.33 Two samples in a short-circuited trans- mission line. Source: Baker-Jarvis, J. Domich, M. D.
and Geyer, R. G. (1993), Transmission/reflection and short-circuit line methods for measuring permittivity and permeability, NIST Technical Note 1355 (revised), National Institute of Standards and Technology, U.S.
Department of Commerce
the thickness values of the two samples areLand αL respectively, and their corresponding reflec- tions measurements are S11(1) and S11(2) respec- tively.
According to the transmission theory, the two independent reflections are given by
S11(1)= Ŵ−Z2
1−ŴZ2 (3.90)
S11(2)= Ŵ−Z2α
1−ŴZ2α (3.91) with the reflection coefficientŴ given by
Ŵ= (à/γ )−(à0/γ0)
(à/γ )+(à0/γ0) (3.92) and the transmission coefficientZgiven by
Z=exp(−γ L) (3.93) From Eq. (3.90), we have
Z2= S11(1)−Ŵ
S11(1)Ŵ−1 (3.94)
Substituting Eq. (3.94) into Eq. (3.61) we get
S11(2)= Ŵ−
S11(1)−Ŵ S11(1)Ŵ−1
α
1−Ŵ
S11(1)−Ŵ S11(1)Ŵ−1
α (3.95)
Eq. (3.95) can be solved iteratively forŴ.
With Ŵ given by Eq. (3.95) and Z given by Eq. (3.93), we can calculate the properties of the material using the following equations (Baker- Jarviset al. 1993):
εr= λ20 àr
1 λ2c − 1
2
(3.96)
àr= 1+Ŵ
(1−Ŵ)
1/λ20−1/λ2c
(3.97) with
1 2 = −
1 2π Lln
1 Z
2
(3.98)
where λ0 is the free-space wavelength and λc
is the cutoff wavelength of the transmission structure. It should be noted that Eq. (3.98) has an infinite number of roots because the logarithm of a complex number is multivalued. In order to choose the correct root, it is necessary to compare the measured group delay to the calculated group delay.
3.4.2 Different-position method
The two independent reflections for the determi- nation of permittivity and permeability can be obtained when the sample is placed at different positions of a transmission line (Baker-Jarviset al.
1993). As shown in Figure 3.34, if measurements are made when the sample is placed at two dif- ferent positions at a short-circuited transmission line, explicit solution to Eq. (3.6) can be obtained by solving the Eq. (3.6) at a given short-circuit position (position 1) for tanhγ L and then sub- stituting this expression into Eq. (3.6) at another short-circuit position (position 2).
The reflection coefficients corresponding to the two different positions shown in Figure 3.34 are given by
Ŵ1= 2βδ1−[(δ1+1)+(δ1−1)β2] tanhγ L
−2β+[(δ1−1)β2−(δ1+1)] tanhγ L (3.99) Ŵ2= 2βδ2−[(δ2+1)+(δ2−1)β2] tanhγ L
−2β+[(δ2−1)β2−(δ2+1)] tanhγ L (3.100)
L1
L2 L ∆L2
∆L1 L
Position 1 Port 1
Short circuit Position 2
Figure 3.34 Two-position measurements in a short-circuited transmission line. Source: Baker-Jarvis, J. Janezic, M. D. Grosvenor, J. H. Jr. and Geyer, R. G. (1993), Transmission/reflection and short-circuit line methods for measuring permittivity and permeability, NIST Technical Note 1355 (revised), National Institute of Standards and Technology, U.S. Department of Commerce
whereδ1 andδ2 denote the phases calculated from Eq. (3.8) for L1 and L2 respectively. From Eqs. (3.99) and (3.100), we can get
tanhγ L
= 2β(δ1+Ŵ1)
β2(Ŵ1+1)(δ1−1)+(1−Ŵ1)(δ1+1) (3.101) γ = 1
L
tanh−1
2β(δ1+Ŵ1)
β2(Ŵ1+1)(Ŵ1−1)+(1−Ŵ1)(δ1+1)
+2nπj
(3.102) where nis an integer, whose correct value can be determined from the group delay.
Meanwhile, we have
β2 =
δ1[δ2(Ŵ1−Ŵ2)+Ŵ1Ŵ2+1−Ŵ2]
−{δ2[Ŵ1(Ŵ2−2)+1]+Ŵ2−Ŵ1} δ1[δ2(Ŵ1−Ŵ2)+Ŵ1Ŵ2+1+2Ŵ2]
−{δ2[Ŵ1(Ŵ2+2)+1]+Ŵ2−Ŵ1} (3.103) Once the value of β is obtained, the permittivity and permeability of the material can be calculated according to Eqs. (3.102) and (3.103).
3.4.3 Combination method
The two independent complex parameters can be obtained by the method shown in Figure 3.35.
First, measure the complex reflection Ŵ1 by directly terminating the probe with the sample shorted by a metal plate. Second, measure the complex reflectionŴ2 by inserting a material with known electromagnetic properties, such as Teflon, between the probe and the sample under test. From the two complex reflection coefficientsŴ1 andŴ2, the materials complex parameters εr and àr can be obtained.
In the calculation of the permittivity and per- meability of the sample, we need to know the
Metal plate
Metal plate
d
(a) (b)
er
mr
d d1
er er1
mr1 mr
Figure 3.35 Two independent reflection mea- surements. (a) Sample directly contacts the probe and (b) a layer of material with known permittivity and per- meability is inserted between the probe and the sample
admittance values at the two conditions shown in Figure 3.35. The admittance of the probe con- nected with the layered media can be analyzed using the method discussed in Section 3.2.2.
3.4.4 Different backing method
As shown in Figure 3.36, the two independent reflection measurements can be made by back- ing the sample with free space and metal plate respectively. We directly terminate probe with the sample under test and measure the reflec- tion coefficients when the sample is backed by free space and metal plate respectively. From the two complex reflection coefficients, the two com- plex material property parameters εr and àr can be obtained.
Similar to the combination method discussed above, in the calculation of materials properties, we need to know the admittance values when the sample is backed by free space and metal plate respectively. The admittance of the probe at these two conditions can be done using the method discussed in Section 3.2.2.
3.4.5 Frequency-variation method
The frequency-variation method was proposed by Wang et al. for the measurement of both permittivity and permeability using a reflection probe (Wang et al. 1998). This method takes fre- quency as an independent variable, and only needs one frequency-sweeping measurement. This
(a) (b)
Free space
d er
mr
Metal plate
d er
mr
Figure 3.36 Two independent reflection measure- ments. (a) Sample backed by free space and (b) sample backed by a metal plate
method has some similarity with the two-thickness method discussed in Section 3.4.1. For two mea- surements made at two different frequencies, though the physical thickness of the sample does not change, the sample has different values of elec- tric thickness at different frequencies. If we assume that the sample has the same permittivity and per- meability at different frequencies, the two reflec- tions measured at two frequencies, corresponding to two values of electric thickness of the sample, can be used to determine the permittivity and per- meability of the sample.
In this method, the open-ended probe may be made from a coaxial line, rectangular or circular waveguide. The material under test may be semi- infinite, or it can be a single layer or multiple-layer sample backed by free space or metal plate. As an example, Figure 3.37 shows a flanged open-ended coaxial probe terminated by a single-layer sample backed by a metal plate. In this configuration, the reflection coefficientŴ(a, b, d, εr, àr, f) is a func- tion of the coaxial dimensions a and b, sample thickness d, electromagnetic property parameters εr and àr, and measurement frequency f. From the reflection coefficients Ŵ1 and Ŵ2 measured at frequency f1 andf2 respectively, the permittivity and permeability of the sample can be calculated.
The obtained permittivity and permeability can be taken as the averages of the values at frequencies f1 andf2.
d er mr
Metal plate
Sample Transmission line
Flange
Figure 3.37 Configuration for a typical reflection measurement using a coaxial probe
In some cases, we need to consider the frequency dependence of the electromagnetic properties of materials. In these cases, the permit- tivity εr(f ) and permeability àr(f ) are functions of frequency, and so interpolation techniques are needed for extracting permittivity and permeabil- ity from the frequency-sweeping reflection data.
Linear interpolation is the simplest interpolation:
εr(f )=af +b (3.104) àr(f )=cf +d (3.105) By using the linear interpolation in Eqs. (3.104) and (3.105), the determination of permittivity and permeability becomes the determination of four complex parameters a, b, c, and d, and so the reflection coefficients at four frequency points are needed. If we use the parabolic interpolation, reflection coefficients at six frequency points are needed to determine the six complex parameters involved. As at microwave frequencies, most of the polarizations are Debye type, Debye equations are often used in interpolation.
In the application of the frequency-variation method, after the reflection coefficients are mea- sured over a wide frequency range, the complex permittivity and permeability are reconstructed from the reflection coefficients at frequency points in a certain frequency band according to the adopted interpolation technique. This process con- tinues throughout the entire measurement fre- quency range.
The choice of the interpolation and frequency interval f depends on many factors such as the natural characteristics of the electromagnetic prop- erties of the materials and the requirements on measurement speed and accuracy. In general, for highly dispersive media and for critical accuracy requirements, higher-order interpolation should be used. At the same time, the inverse problem will become more difficult and time consuming. A trade-off between accuracy and speed must be made in practical real-time and on-site measure- ments. The frequency interval should guarantee that the reflection coefficient is changed enough to be distinguished by the network analyzer. Mean- while, to improve the measurement resolution, the frequency interval selected should be as small
Incident wave
Region 1 (air)
Γ1 Γ2
Region 2 (sample)
Figure 3.38 A short-ended coaxial line with a sample loaded at the end. The time histories of the incident wave and the first and second reflections (Ŵ1andŴ2)are measured at a location in front of the interface between the air and the sample
as possible, especially for the highly dispersive materials whose permittivity and permeability vary rapidly with frequency.
3.4.6 Time-domain method
Besides the five methods discussed below, Court- ney and Motil proposed a time-domain measure- ment in a short-circuited coaxial geometry and data-reduction procedure to experimentally deter- mine the complex permittivity and permeabil- ity of materials (Courtney and Motil 1999). As shown in Figure 3.38, using time-domain tech- niques, this method measures the first and second reflection waves (partial components of the com- plete reflection) in the short-circuited coaxial trans- mission line that holds the material under test. In this method, the two independent parameters for the determination of permittivity and permeability are the first and second reflections (Ŵ1 and Ŵ2).
Detailed discussions on this method can be found in (Courtney and Motil 1999; Courtney 1998).