022486-2-T INTERPRETATION OF FREE-SPACE DIELECTRIC MEASUREMENTS M. Hallikainen RP T. Ulaby Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 May 1986 Prepared for: NASA Headquarters Code HWD, Room 726 300 7th Street, S.W. Washington, DC 20546 Contract NAGW 733

TABLE OF CONTENTS Page LIST OF FIGURES....................... ii LIST OF TABLES........................ v LIST OF SYMBOLS......................... vi ABSTRACT............................ viii 1.0 INTRODUCTION....................... 1 2.0 BASIC EQUATIONS FOR THE FREE-SPACE SYSTEM......... 2 3.0 EFFECT OF OSCILLATIONS................... 5 3.1 Oscillation of Loss... ~........ 5 3.2 Oscillation of Phase Shift.............. 9 4.0 INTERPRETATION OF FREE-SPACE MEASUREMENTS FOR PERPENDICULAR 13 INCIDENCE....................... 4.1 Interpretation Procedures.............. 14 4.1.1 Correction for Mismatch Loss Only....... 15 4.1.2 Correction for Loss Oscillation........ 16 4.1.3 Correction for Loss and Phase Shift.... 18 4.2 Comparison of Procedures............... 18 4.2.1 No Measurement Error............. 19 4.2.2 Assuming a Measurement Error......... 20 5.0 INTERPRETATION OF FREE-SPACE MEASUREMENTS FOR OBLIQUE 22 ANGLES OF INCIDENCE................... 6.0 CONCLUSIONS........................ 23 REFERENCES........................... 25 APPENDIX i

LIST OF FIGURES Figure 1. Theoretical loss at 4 GHz as a function of sample thickness for various dielectrics. Figure 2. Theoretical loss at 18 GHz as a function of sample thickness for various dielectrics. Figure 3. Theoretical phase shift as a function of sample thickness for various dielectrics. Figure 4. Definition of terms related to sample loss. Figure 5. Theoretical mismatch loss as a function of sample c'. Figure 6. Theoretical maximum deviation from the average loss as a function of average loss, with sample e' as a parameter. Figure 7. Theoretical minimum required average loss to ensure a desired accuracy for the measured loss as a function of sample c', when the oscillation of loss is neglected. Figure 8. Definition of terms related to sample phase shift. Figure 9. Graphical solution to maximum deviation from the average phase shift (see text). Figure 10. Theoretical maximum deviation from the average phase shift as a function of average loss, with sample e' as a parameter. Figure 11. Theoretical minimum required average loss to ensure a desired accuracy for the phase shift as a function of average phase shift with sample &' as a parameter, when the oscillation of phase shift is neglected. Figure 12. Block diagram of Method 1 that accounts for the mismatch loss only. Figure 13. Block diagram of Method 2 that accounts for the loss oscillation. Figure 14. Block diagram of Method 3 that accounts for the oscillation in the loss and phase shift. Figure 15. Comparison of Method 2 (phase not corrected) and Method 3 (phase corrected) for e' as a function of sample thickness (e = 3- j0.2). No measurement error is assumed.

Figure 16. Comparison of Method 2 (phase not corrected) and Method 3 (phase corrected) for e' as a function of-sample thickness (E = 10 - j1.5). No measurement error is assumed. Figure 17. Comparison of Method 2 (phase not corrected) and Method 3 (phase corrected) for e' as a function of sample thickness (E = 25 - j5). No measurement error is assumed. Figure 18. Comparison of interpretation methods for e" at 4 GHz as a function of sample thickness (e = 3 - jO.2). No measurement error is assumed. Figure 19. Comparison of interpretation methods for e" at 4 GHz as a function of sample thickness (e = 10 - j1.5). No measurement error is assumed. Figure 20. Comparison of interpretation methods for e" at 4 GHz as a function of sample thickness (E = 25 - j5). No measurement error is assumed. Figure 21. Comparison of interpretation methods for e" at 18 GHz as a function of sample thickness (E = 3 - jO.2). No measurement error is assumed. Figure 22. Comparison of interpretation methods for e" at 18 GHz as a function of sample thickness (e = 10 - j1.5). No measurement error is assumed. Figure 23. Comparison of interpretation methods for ~" at 18 GHz as a function of sample thickness (e = 25 - j5). No measurement error is assumed. Figure 24. Comparison of Method 2 (phase not corrected) and Method 3 (phase corrected) for E' as a function of sample thickness (e = 3 - j0.2). Assumed measurement errors as shown. Figure 25. Comparison of Method 2 (phase not corrected) and Method 3 (phase corrected) for e' as a function of sample thickness (~ = 25 - j5). Assumed measurement errors as shown. Figure 26. Comparison of interpretation methods for E" at 4 GHz as a function of sample thickness (e = 3 - jO.2). Assumed measurement errors as shown. Figure 27. Comparison of interpretation methods for ~" at 4 GHz as a function of sample thickness (~ = 25 - j5). Assumed measurement errors as shown. iii

Figure 28. Comparison of interpretation methods for E" at 18 GHz as a function of sample thickness (~ = 3 - jO.2). Assumed measurement errors as shown. Figure 29. Comparison of interpretation methods for e" at 18 GHz as a function of sample thickness (e = 25 - j5). Assumed measurement errors as shown. Figure 30. Comparison of Methods 2 (phase not corrected) and 3 (phase corrected) for ~' at 4 GHz as a function of sample thickness (a = 3 - jO.2). Assumed measurement errors as shown. Figure 31. Comparison of Methods 2 (phase not corrected) and 3 (phase corrected) for c' at 4 GHz as a function of sample thickness (e = 25 - j5). Assumed measurement errors as shown. Figure 32. Effect of assumed measurement error to a" from Method 2 at 4 GHz (E = 3 - jO.2). Figure 33. Effect of assumed measurement error to ~" from Method 3 at 4 GHz (e = 3 - jO.2). Figure 34. Effect of assumed measurement error to E" from Method 2 at 4 GHz (e = 25 - j5). Figure 35. Effect of assumed measurement error to a" from Method 3 at 4 GHz (e = 25 - j5). Figure 36. Effect of measured phase constant (with respect to air) to total phase constant as a function of sample a'. Figure 37. Propagation path with and without the sample. iv

LIST OF TABLES Table 1. Assumed measurement errors for comparing the interpretation methods.

LIST OF SYMBOLS d thickness of sample deff effective thickness of sample f frequency k constant L loss of sample Lav average loss of sample (assuming no multiple reflections within the sample) LaVmin minimum average loss of sample Ld dielectric loss of sample LM mismatch loss of sample Lmeas measured loss of sample x path-length difference in air due to reflection T complex transmission coefficient attenuation constant phase constant SO phase constant for free space sm phase constant with respect to air Y complex propagation constant 5 phase angle of reflection coefficient tan6 loss tangent (c"/l ) AL deviation from average loss ALmax maximum deviation from average loss ALmaxl ALmax when L > Lav ALmax2 ALmax when L < Lav ALmx peak-to-peak oscillation of loss a change in a A3 change in n vi

phase shift Sl av average phase shift (assuming no multiple reflections within the sample) ARM mismatch phase shift Ummeas measured phase shift Aomm measured phase shift with respect to air Ao deviation from average phase shift Aemax maximum deviation from average phase shift Aemaxpp peak-to-peak maximum oscillation of phase shift C complex relative dielectric constant co dielectric constant for free space e' real part of e e:" imaginary part of ~ 9! angle of incidence e2 propagation angle in sample Xo rwavelength in free space permeability p reflection coefficient X angular frequency vii1

INTERPRETATION OF FREE-SPACE DIELECTRIC MEASUREMENTS Martti Hallikainen and Fawwaz T. Ulaby Remote Sensing Laboratory University of Kansas Center for Research, Inc. Lawrence, Kansas 66045-2969 -ABSTRACT This report presents an analysis of the multiple reflections associated with free-space measurements of dielectric slabs. Expressions are provided relating the dielectric loss and phase shift to the thickness of the sample and its dielectric properties. Methods for interpreting the measured loss and phase shift in terms of the dielectric properties of the sample are examined and compared. This work was started at the University of Kansas and completed at the University of Michigan. V11 1

1.0 INTRODUCTION In the free-space transmission technique, a sample is placed between the transmitting and receiving antennas, and its loss and phase shift are measured. The interpretation of the measured quantities, in terms of the complex dielectric constant ~ of the sample, is straightforward if the sample is thick and its measured loss is high. For low-loss materials, the multiple reflections within the sample make the loss and phase shift oscillate around their average values as a function of the sample thickness. The usual procedure for removing the uncertainty in the results is to measure several identical samples of different thicknesses [1,2]. The attenuation constant~ a and the phase constant B are then obtained by fitting the data points for each to a straight line and taking the slope of the line. This method is timeconsuming and is not accurate for low-loss samples unless numerous sample thicknesses are used. In single-frequency measurements the use of multiple thicknesses can often be avoided by simply making the sample thickness (and the loss) large enough. The multiple reflections within the sample can then be neglected; however, this is often not possible at low microwave frequencies. In wide-band measurements the maximum sample thickness that the system can handle is usually determined by the loss the sample exhibits at the highest measurement frequency. For example, in recent soil measurements using a 3-18 GHz free-space system, the measured loss of an 18-mm thick sample was about 41 dB at 18 GHz, but was only 8 dB at 4 GHz [3]. The measured phase shift (with respect to air) varied from 800 degrees at 18 GHz to 200 degrees at 4 GHz. 1

Clearly, the multiple reflections can be neglected at 18 GHz, but they must be considered at 4 GHz. In this report the multiple reflections within the sample are analyzed for the free-space measurement technique. Equations for the deviation from the average loss and phase shift are derived as a function of the properties of the sample. Three different procedures are presented to interpret the measured loss and phase shift in terms of the dielectric properties of the sample. Finally, the effect of measurement errors on the accuracy of these procedures is discussed. Since the measurements in the present experimental program were made using perpendicular incidence only, the effect of oscillation and the interpretation procedure is discussed in a detailed manner for that case only [4]. In Section 5 the interpretation procedure for oblique angles of incidence is briefly discussed. A brief bibliography of reported free-space measurements is given in [3]. 2.0 BASIC EQUATIONS FOR THE FREE-SPACE SYSTEM When an electromagnetic plane wave is transmitted through a homogeneous dielectric slab, the complex transmission coefficient T for viewing angle 01 = 00 can be expressed as 2

(1 - p2) e'd T(e1 = 0~) = - (1) 1 - p2e -2yd where p = reflection coefficient at the slab-air interface y = complex propagation constant d = thickness of the slab. The reflection coefficient at the slab-air interface is p= -, (2) 1 + JF where E is the complex dielectric constant, E = E' - j ". (3) The complex propagation constant y consists of the attenuation constant a and the phase constant A: y = a + jp3. (4)

2 + r2 (5) 2 f( = E( ) + (6) in Equations (5.) and (6) X0 is the wavelength in free space. The loss L and the phase shift Ap of the sample are obtained from the transmission coefficient T: 2 L = iTI; L > 1 (7) Ap = -tan(Im —- ( ) + 2r n; 0A > O (8) Re{T) In Equation (8), n is zero or a positive integer. The ambiguity in the phase shift, as described by Equation (8) is a property of all transmission dielectric measurement methods. Both the loss and the phase shift are taken to be positive quantities in this report. In general, the loss and the phase shift do not increase linearly with increasing sample thickness; rather, they oscillate around the average value. The oscillations slowly damp out when the sample loss becomes large enough to suppress the multiple reflections within the sample. As illustrated in Figures 1 to3, the oscillations for higher frequencies appear at smaller sample thicknesses than they do for lower frequencies. In addition, the amplitude of the oscillations is much larger for the loss than for the phase shift.

3.0 EFFECT OF OSCILLATIONS If the multiple reflections within the sample are not accounted for when interpreting the measured loss and phase shift, they can result in errors in the estimated values of ~' and s". The reflections can be neglected only when the measured loss is high enough or e' of the sample is low enough. The deviation from the average loss, and its maximum value as a function of the properties of the sample, are discussed in Section 3.1. The oscillation of the phase shift is discussed in Section 3.2. 3.1 Oscillation of Loss The quantities related to the oscillation of the loss L are illustrated in Figure 4. The loss of the dielectric sample consists of both mismatch loss and dielectric loss. The mismatch loss in dB is given by LM = -20 log1011 - 21 (dB). (9) The mi smatch loss increases with increasing', as illustrated in Figure 5. For oblique angles of incidence the mismatch loss may be either higher (horizontal polarization) or lower (vertical polarization) than for perpendicular incidence. The average loss, Lav, is defined by 1 2 L -yd (10) Lav (1 - p2) e (10) 5

and is the loss without any oscillations present. The deviation from the average loss, AL, is defined by _L AL k' (11) av where L is the actual loss. This form is convenient for expressing AL in dB. The deviation AL, when expressed in dB can be either positive or negative. The maximum deviation from the average loss is defined by ALmax (Lav) maximum or minimum (12) In Equation (12) the maximum value of L/Lav is positive and the minimum value is negative, when expressed in dB. From Equations (7) and (10), AL is found to be 2 -2yd 2 AL = 11 - p e (13) In order to develop AL into a more convenient form, the following notation is used for the reflection coefficient p: P = IPI ej6. (14) Equation (13) can now be expressed as -4td -2cd AL: 1 + p!4 e - 21pl2 e cos[26 - 2 Ed]. (15) 6

Equation (15) gives the deviation from the average loss as a function of the properties of the sample. The maximum deviation is obtained by setting the cosine term equal to + 1: -2cad ALmaxl = (1 + PI2 e ) (16) -2ad ALtmax2 = (1 - I2 e )2(17) In Equation (16), ALmaxl is the maximum deviation when the actual loss is higher than the average loss. In Equation (17), aLmax2 is the corresponding value when the actual loss is lower than the average loss. Equations (16) and (17) give the envelopes for the maximum deviation from the average loss (see Figure 4). The peakto-peak oscillation of the loss is ALmaxpp =ALmaxl + L max2 dB. (18) The disadvantage in Equations (16) and (17) is that a is needed to calculate the maximum deviation. Hence it is recalled from Equation (10) that the sample thickness d can be expressed using the average loss: d = log10(11 - P I La) (19) 10o av2' Using Equation (19) aL can be written in a more practical form: 7

AL = 1 + I4 1 21P12 (Ii - P22 Lav ) * cos[26 - 2 Ed]. (20) 1 - p212 La The maximum deviation from the average loss can be expressed as aLmaxl = (1 + Lv I i2 ) (21) ALmax2 ( 1 Lv 211-p ) (22) -av 1 - p Equations (20) to (22) give the oscillation characteristics of the loss as a function of the average loss. Since the ~" dependence of p is weak for " << ~', it can often be neglected. Hence a good estimate of the uncertainty in the measured loss is obtained from Lav and ~'. The maximum deviation from the average loss increases with increasing E' and decreasing average loss, as illustrated in Figure 6. Since the average loss, as defined by Equation (10), includes the mismatch loss, the oscillation chart in Figure 6 also includes it. The curve for each value of s' starts below the average loss from a point at which the dielectric loss is zero. The accuracy of'" can be increased by using correction procedures that take the oscillation into consideration. 8

Equations (21) and (22) can be used to calculate the minimum average loss that is required to obtain a certain accuracy for the measured loss, when the oscillations are not accounted for. As expected, the minimum average loss increases with increasing a' and required accuracy, as illustrated in Figure 7. For a' = 10, an accuracy of 1% for the measured loss is guaranteed by an average loss of about 14 dB. However, this does not guarantee an accuracy of 1% to s", because the mismatch loss must be subtracted from the measured loss when calculating a. Because of the standing waves between the antennas of the free-space system, a higher average loss than those mentioned above is required to obtain an accuracy of 1% [3]. Power reflections between the antennas may change the reference reading (with no sample) considerably, unless the distance between the antennas is large enough. The reflections also occur through a low-loss sample. Methods for eliminating the standing waves are discussed elsewhere [3]. 3.2 Oscillation of Phase Shift The quantities related to the oscillation of phase shift are illustrated in Figure 8. The mismatch phase shift is defined as -1 Im( - (23) A~taM n Re(1 - p2)

Usually Im(1 - p2) << Re(1 - p2) and ALM can be neglected. For example, for e = 25 - j5 at 01 = 0~ ARM = 3.80. The average phase shift is defined as 2 -yd = -1 Im[(1 - p )e ] + 2IIn, av -Yd Re[1 - p )e ] (24) and is the phase shift without any oscillations present. The deviation from the average phase shift, Ae, is defined by A8 = Adp - @Aav (25) and can be either positive or negative. The maximum deviation from the average phase shift is defined as Aemax = (A* - A av) maximum or minimum (26) In Equation (26) the maximum value of Af - Aav is positive and the minimum value is negative. From Equations (1) and (24) Ae is found to be -2yd Ae = -tan-1 Im[l - p e ] (27) 2 - 2 d Re[1 - p2e ] and can also be expressed as 10

2 2 2ad e -tan-1 IP2 e-2 sin[2=6 -2d] (28) -2ad 1 - Ip1 e cos[26 - 2sd] The condition for the maximum deviation from the average phase shift is that the phase of the denominator in Equation (1) has either a local maximum (positive) or a local minimum (negative) value. By assuming that the attenuation constant of the sample, a, is small, the trace of the rotating vector -p2 e-2d in Figure 9 can be considered as a circle. The maximum deviation from the average phase is obtained when the angle between the 2 -2yd 2 -2yd vectors 1 - 2 e and -p e in the complex plane is + t/2. This condition gives Aemax -2:d ACmax =+ sin-I (IPl2e ) (29) Using Equation (10) aemax can be expressed as max = sin-1 [ I 21) (30) Lav - p av Equation (30) is similar to Equations (21) and (22) which give the maximum deviation from the average loss. For the phase shift, however, the maximum deviation above the average phase shift is equal to that below it. Because of the assumption that a is small, Equation (30) tends to slightly overestimate Aemax for high-loss samples. Comparison between the values for aemax from 11

Equations (30) and (28) was madeat 18GHz and 3GHZ using e = 25 - j5 and e = 25- jO.5. For the two values of e the maximum difference was less than 0.2% at both frequencies. The largest observed absolute difference between the values for Aemax from Equations (30) and (28) was 0.13%. As in the case of the loss oscillation, e" has only a small effect on the term Ip/(1 - p2)12 in Equation (30). The average loss is, of course, dependent on e" of the sample. Because of the symmetry, the peak-to-peak oscillation of the phase shift is AOmaxpp = 21Aemax. (31) The maximum deviation from the average phase shift increases with increasing E' and decreasing average loss, as illustrated in Figure 10. The small mismatch phase shift is included in the results. The curve for each value of e' starts from a point where Aemax has its largest value: (Aomax)maximum = sin-1 (1pl2) (32) Equation (32) assumes that the thickness of the sample approaches zero. By taking the oscillation of the phase shift into account with a 12

correction procedure, accuracy of the interpretation can be increased. Equation (30) can be used to calculate the minimum average loss that is required to obtain a certain accuracy for the measured phase shift, when the oscillations are not accounted for. The required average loss increases with decreasing average phase shift and with increasing required accuracy and e' of the sample, as illustrated in Figure 11. For ~' = 10 the required average loss for an accuracy of 1% for the phase shift is 12 dB when the average phase shift is 200~. In practice, higher average losses than any of those mentioned above are required to obtain an accuracy of 1%. This is because of the standing waves between the antennas when no sample is inserted (reference measurement). Unless the measurement frequency is properly selected (a frequency corresponding to the maximum or minimum loss value without a sample), the reference phase shift may deviate from the average phase shift.This results in a lower accuracy [3]. 4.0 INTERPRETATION OF FREE-SPACE MEASUREMENTS FOR PERPENDICULAR INCIDENCE As discussed in Section 3, the oscillation in both the loss and phase shift and phase shift may result in a considerable error for' - jE", unless it is accounted for. In previous measurements reported in the literature [1,2], the usual method to avoid dealing with the oscillation has been to make measurements on identical samples of 13

different thicknesses. The attenuation constant a and the phase constant Bm, have been found from the average slopes of loss and phase shift versus sample thickness. From a and Sm, a' - jE" can be calculated: = 82 a2 (33) 2 2 2~ll (34) where = + 30 (35) and to,,, co, and am are the angular frequency, permeability of the sample, the permittivity of free space, and the phase constant with respect to air, respectively. Repeating the measurements for different sample thickness is time-consuming, for example in the measurement of wet soil, where careful sample preparation is required. 4.1 Interpretation Procedures Three different procedures for interpreting the measured phase shift and loss in terms of ~' and I" are discussed. The 14

simplest way is to neglect the oscillation of the loss and the phase shift and to account only for the mismatch loss, as will be discussed in Section 4.1.1. A better approach is given in Section 4.1.2 which takes the oscillation of the loss into account (in addition to compensating for the mismatch loss) but neglects the oscillation of the phase shift. For dry and wet snow, this method is adequate, because Aemax is very small for low values of c'. The third method takes advantage of Equation (1) using an iteration process, as will be discussed in Section 4.1.3. This method accounts for the oscillation in both the loss and phase shift. The computer program for the three procedures is given in the Appendix. 4.1.1 Correction for Mismatch Loss Only A block diagram of the method is shown in Figure 12. The phase shift from the sample Ap is assumed to depend linearly on the sample thickness d. The phase constant with respect to air Bm is calculated from = Ap/d. (36) Then a zero-order estimate of ~', denoted by ~'(0), can be obtained from Equation (33) by assuming a = 0. Since in most cases d >> a, this gives a fair estimate of c'. Using'(O0), the mismatch loss is then obtained from Equation (9), where again it is assumed that a = O. The dielectric loss is L(0): L - L(0) (dB), (37) d meas LM 15

where Lmeas is the measured loss. An estimate of the attenuation coefficient a, denoted by a(0), is (0) Ld (0) d (38) Estimates of e' and ~" can now be obtained from Equations (33) and (34). They are denoted by e'(1) and c"(1) and are the starting values for the iteration process. New values to LM, and a are calculated from Equations (9), (37), and (38). From s, LM and (1), new estimates E'(2) and E"(2) are obtained using Equations (33) and (34). The iteration process will be continued until the desired accuracy is obtained:'(i) - E'(i 1)- < k1 (39) I:"( i) - "(i - 1)i < k2 (40) In Equations (39) and (40), k1 and k2 are constants. Since, in most cases, the mismatch loss depends very little on c", few iteration steps are required to obtain the final values of a' and "'. This procedure results in a negative value for E", if the measured loss is smaller than the mismatch loss. This is possible in low-loss samples. 4.1.2 Correction for Loss Oscillation A block diagram of the procedure is shown in Figure 13. As in the procedure in Section 4.1.1, the oscillations of the loss 16

and phase shift are at first neglected. The first estimates of E' and s" are obtained by using the procedure outlined in Section 4.1.1 as the first step. Since the mismatch loss depends little on e", even the first estimates of c' and c", denoted by e'(') and e"(1) are often accurate enough to start the iteration process in Figure 13. Due to the oscillation of the loss, a(1) may be slightly negative. In that case a small positive value is assigned to a(1). The iteration is started by obtaining the first estimate of the deviation from the average loss, AL(1), from Equation (15). Then the first estimate of the average loss is obtained: L(1) = L - AL(1) (41) av meas New estimates of ac, c, ~1', p, LM, and AL are obtained from Equations (5), (33), (34), (2), (9), and (15). They are denoted by a(2),,(2) E11(2), p(2), LM(2), and AL(2), respectively. The iteration is continued until the criterion in Equation (42) is valid: L(i) +av eas < k (42) In Equation (42), a value that is smaller than the observed standard deviation for the measured loss is assigned to k3. The value for c' derived from this method is usually very close to that from the method discussed in Section 4.4.1. The value for e" derived from this method may be negative for low-loss samples with a large value for C'. 17

4.1.3 Correction for Loss and Phase Shift A block diagram of this method is shown in Figure 14. Again, the procedure in Section 4.1.1 is used as the first step in obtaining the estimates for E' and s" for the iteration process. They are denoted F,(1) and e"(1). From Equations (5), (6), and (2), a(1), (1), and p(1) are calculated and then the loss and phase shift, denoted by L(1) and aM(1) are obtained from Equation (1). L(1) and Ac(1) are compared with the measured values Lmeas and 4ameas. The iteration is continued, until IL(i) L_ measl < k4 (43) |ap(i) - YAmeasl < k< (44) For k4 and k5, values that are smaller than the observed standard deviation for the measured loss and the phase shift, correspondingly, are selected. 4.2 Comparison of Procedures The results from the three procedures are compared for three different dielectric materials, c = 3 - jO.2, C = 10 - jl.5 and F = 25 - j5. Sample thicknesses between 4 mm and 50 mm were used at 4 GHz, and thicknesses between 2 mm and 20 mm at 18 GHz. In Section 4.2.1 a comparison is made, assuming no measurement error. The effect of measurement errors in the loss and phase shift on the accuracy of the interpretation procedures is discussed in Section 4.2.2. 18

4.2.1 No Measurement Error The results of interpreting s' at 4 GHz and 18 GHz for the three dielectric materials are shown in Figures 15 to 17. Methods 1 and 2, which do not correct the phase, give practically identical values that oscillate with the sample thickness. Method 3, which accounts for the oscillation both in the loss and the phase shift, gives the correct value at both frequencies. For Methods 1 and 2, the maximum errors of the obtained C' in Figures 15 to 17 are the same at 4 GHz and 18 GHz, but for 18 GHz they occur at smaller thicknesses. The interpretation error for s' from Methods 1 and 2 is smaller than 5% at 4 GHz for sample thicknesses over 17 mm, and for thicknesses over 4 Ima at 18 GHz. The error is smaller than 2% for thicknesses above 20 mm at 4 GHz, and for those above 7 mm at 18 GHz. In order to make the interpretation error for Methods 1 and 2 negligible at 4 GHz, the sample thickness should be of the order of 40 mn. The interpretation error is much more serious for s", as illustrated in Figures 18 to 20 for 4 GHz and Figures 21 to 23 for 18 GHz. Method 1, which accounts for the mismatch loss only, may produce for low-loss cases an error that is over 100%. Therefore the use of Method 1 should be limited to high-loss cases where the measured loss is at least 15 dB. Method 2 gives substantially better results. For ~ = 3 - jO.2 the error is negligible for sample thicknesses above 40 mn. For higher values of s' - jc", smaller thicknesses are acceptable. Method 3 gives a correct 19

value for E" at all thicknesses. At 18 GHz the thicknesses required to obtain a reasonable accuracy are again smaller. For s = 3 - jO.2, Method 2 gives a good accuracy for thicknesses over 6 mm. 4.2.2 Assuming a Measurement Error A realistic estimate Of the accuracy of the interpretation methods is obtained by assuming that a measurement error is involved. The errors used in the calculations were (a) the standard errors observed for the 3-18 GHz free-space system [3], and (b) larger errors for 4 GHz to evaluate more closely the effect on s' and s" of increasing error. In Table 1, the sign of the errors is chosen so that the errors add up when calculating c' - jE". Therefore, the errors obtained in this way cannot exist simultaneously for a' and ~". The calculations are made for C = 3 - jO.2 and e = 25 - j5 only. The results for the case where the errors were assumed to be equal to the standard deviation of the observed errors for the 3-18 GHz free-space system are shown in Figures 24 and 25 for s' and Figures 26 to 29 for s". With the assumed errors, none of the interpretation methods gives the correct value. From Equation (33) it is expected that the error for s' is IE-I 21- i + 21ca (45) where A/S are the relative errors for B and a, respectively. Since S >> a in all cases, the error for s' is mostly due to that in S. Indeed, the additional error for 20

Methods 1 and 2 (phase not corrected) is 2.1% when AB/S = 1%. Method 3 produced no error for ~' when the assumed measurement error was zero. With the assumed error in the phase shift and loss, it showed an average error of 2.1% for e'. For e", Methods 1 and 2 give results the overall behavior of which is similar to those obtained with no error assumed, as illustrated in Figures 26 to 29. From Equation (34) the error for s" is found to be IAI = IaI + Il (46) Since the error for the loss was assumed to be ~5% of the total loss (including the mismatch loss), ha/a > 5% in all cases. At 4 GHz the average error for s" is about 8% for all methods and all dielectric materials. At 18 GHz it is 2.1%, which is a result of a smaller assumed error of Ah/a = 1% for the loss. Calculations for Methods 2 and 3 assuming larger errors were made at 4 GHz for ~ = 3 - jO.2 and e = 25 - j5. Figures 30 and 31 show the results for a' and Figures 32 to 35 for s". Comparing Figures 30 and 31 indicates that with increasing error in the phase shift, the behavior of ~' from Method 3 begins to approach that of ~' from Method 2. This can be expected, because Method 3 relies on correct phase-shift information. However, the oscillation in e' from Method 3 is substantially smaller than that from Method 2. The importance of correct phase-shift information for Method 3 is even more pronounced for s". With increasing error, Method 2 provides better accuracy for thicknesses above 12 mmn than 21

Method 3, as illustrated in Figures 32 to 35. In all the above calculations of the effect of phase shift and loss errors on c'- je", the phase shift was used in the same sense as it is in Equation (1). Usually the phase shift is measured with respect to a reference material (air). In order to relate the measured value to- that from Equation (1), the phase shift due to air must be added, as pointed out in Equation (35). Hence the ratio of the measured phase coefficient (with respect to air) to the total phase coefficient varies with ~' of the sample: Sm Y- /C- 1 - (47) As illustrated in Figure 36, 5m/B is small for samples with a low ~'; therefore it. is important to separate the two quantities. 5.0 INTERPRETATION OF FREE-SPACE MEASUREMENTS FOR OBLIQUE ANGLES OF INCIDENCE For an oblique angle of incidence, the propagation angle in the sample depends on the dielectric properties of the sample. Therefore, the effective thickness of the sample length of the sample length of the propagation path depends on ~ as well: d (48) deff cos 02( (48) Additionally, the length of the reference path is no longer equal to that of the sample. This is illustrated in Figure 37, where AB (sample) - AC (air) gives the difference. Hence, in order to obtain from the measured value with respect to air (Arm) the'total 22

measured phase shift in the sense of Equation (1), Equation (49) should be used: = A B + aod- os~e - 6 (49) AYmeas:im + od cos(1 _ 02) Methods 1, 2, and 3 can be applied to the interpretation of the measured loss and phase shift for oblique angles of incidence. Since o2 depends on s', values for "', s", and 02 must be solved simultaneously. Every time a new value for a' and c" is calculated, new values for 62, deff, a, a, and LM must be obtained in Method 1 as well. In Method 2, AL also depends on 92, and in Method 3, L depends on it. Since all the dielectric measurements of soils using the 3-18 GHz free-space system were made for perpendicular incidence [4], a computer program for interpreting measurements for oblique angles of incidence is not currently available. 6.0 CONCLUSIONS Analysis of the multiple reflections within the sample for the free-space transmission method was represented. As a result of these reflections, the measured loss and the phase shift oscillate around their average values as a function of the thickness and the dielectric properties of the sample. Three different methods for interpreting the measured results in terms of e' - js" were discussed. Method 1, which accounts only for the mismatch loss, can be used for high-loss samples. This is because the multiple reflections are suppressed by the high loss and can be 23

neglected. Method 2, which takes the oscillation of the loss into account, gives a reasonably good accuracy for s"; however, the value for e' is practically identical to that derived from Method 1. Method 2 can be used, in addition to high-loss and medium-loss samples, for cases where c' is reasonably low. Method 3 accounts for the oscillation both in the loss and the phase shift. If no measurement error is assumed, it gives the correct value for a' - jE". The three methods were also analyzed in the presence of an error in the measured loss and phase shift. The errors increase the inaccuracy of Methods 1 and 2 by the amount of the error. The average error for a' as derived from Method 3 is equal to that from Method 2, but the amplitude of oscillation as a function of sample thickness is still substantially smaller. However, E" from Method 3 is affected by the lack of correct phase information, and with increasing error in the measured loss and phase shift, its accuracy becomes worse than that of Method 2. 24

REFERENCES [1] Wiebe, M. L., "Various Techniques of Dielectric Constant Measurements as Applied to the Relative Dielectric Constant of Sand as a Function of Moisture Content," Technical Memorandum RSC-22, Remote Sensing Center, Texas A M University, College Station, Texas, April 1971. [2] Perry, J. W., and A. W. Straiton, "Dielectric Constant of Ice at 35.3 GHz and 94.5 GHz," Journal of Applied Physics, Vol. 43, No. 2, p. 731, 1972. [3] Hallikainen, M., and F. T. Ulaby, "A Free-Space System for Dielectric Measurements in the 3-18 GHz Frequency Range," RSL Technical Report 545-3, University of Kansas Center for Research, Inc., Lawrence, Kansas 66045-2969, 1983. [4] Hallikainen, M., F. T. Ulaby, M. El-Rayes, and M. C. Dobson, "Microwave Dielectric Behavior of Wet Soil, Part III: Effect of Frequency, Soil Texture and Temperature," RSL Technical Report 545-5, University of Kansas Center for Resea rch, Inc., Lawrence, Kansas 66045-2969, 1983. 25

Table 1. Assumed measurement errors for comparing the interpretation methods. Error (%) Dielectric Constant 4 GHz 18 GHz (a) (b) (a) Phase Shift -1 -1, -3 -1 Loss (dB) +5 +5, +10 +1 Dielectric Loss Factor Phase Shift +1 +1, +3 +1 Loss (dB) +5 +5, +10 +1 26

APPENDIX Computer program to calculate s'-js" from free-space transmission measurements.

0150 INPUT U1,W2 0161 PRINT 0162 PRINT "TOTAL NUMBER OF MEASUREMENTS?" 0163 INPUT IO 0164 FOR 1=1 TO 10 0165 PRINT 0170 PRINT"FREf (6HZ), AMPL(SAMPLE) (DB), AMPL(REF) (B), PHASE(SAMPLE) (PEG), PHASE(REF) (PEG), 0200 INPUT F(I),A1(I),A2(1),P1 (I),P2(I) 0210 NEXT I 0212 PRINT 0221 PRINT "VANT TO CORRECT THE VALUES AT SOME FREQUENCY?" 0222 PRINT "IF YES PRINT YH 0223 INPUT Y$ 0224 IF YS="Y" THEN 226 0225 GOTO 230 0226 PRINT "MEASUREMENT NUMBER, F(GHZ), A10(B), A2(DD), P1(DEG), P2(DE6) 7" 0227 INPUT I,F(I),A1 (I),A2(I),P l(I),'P2(I) 0228 GOTO 221 0230 PRINT 0231 PRINT "VANT RESULTS FOR NOMINAL THICKNESS PLUS/MINUS DEVIATION?"1 0232 PRINT "IF YES PRINT Y" 0233 INPUT Y$ 0234 IF Y$="Y' THEN 236 0235 GOTO 239 0 236 PRINT 0237 PRINT "DEVIATION FROM NOMINAL THICKNESS (MM) 7"1 0238 INPUT TO 0239 REM GRAVIMETRIC WETNESS, DRY BASIS 0240 M1=(1-U2)/U2 0250 REM GRAVIMETRIC WETNESS, WET BASIS 0260 M2=(U1-W2)/Ul 027`0 REM AMOUNT OF WATER IN THE SAMPLE (6) 0280 U3=M2*U 0290 REM AMOUNT OF SOIL IN THE SAMPLE (6) 0300 W4=(1-M2)*U 0310 REM BULI< DENSITY 0320 R=U4/V 0330 REM VOLUMETRIC WETNESS 0340 M3:=R*M1 0350 PRINT

0020 REM THE PROGRAM COMPUTES THE DIELECTRIC PROPERTIES OF 0030 REM SOIL FROM FREE-SPACE MEASUREMENTS 0031 REM USING THREE DIFFERENT METHODS. 0032 REM THE FIRST METHOD ACCOUNTS ONLY FOR THE MISMATCH LOSS, AND 0033 REM THE SECOND ASSUMES THE LOSS TO BE A NONLINEAR FUNCTION 0034 REM OF SAMPLE THICKNESS AND THE PHASE SHIFT TO DEPEND LINEARLY 0035 REM ON SAMPLE THICKNESS. THE THIRD METHOD ASSUMES BOTH THE LOSS 0036 REM AND THE PHASE SHIFT 10 BE NONLINEAR FUNCTIONS OF 0037 REM SAMPLE THICKNESS. 0038 REM THE ANGLE OF INCIDENCE IS 0 DEGREES (PERPENDICULAR). 0050 REM 0060 REM 0070 DIM F(20),A1(20),A2(20),Pl(20),P2(20) 0071 TO=O 0072 PRINT 0074 PRINT "INPUT FOR SOIL MEASUREMENT INTERPRETATION PROGRAM" 0O75 PRINT 0076 PRINT 0078 PRINT 0080 PRINT "MONTH, DAY, YEAR?" 0090 INPUT DID2,D3 0095 PRINT 0100 PRINT "SOIL TYPE?u 0110 INPUT S$ 0115 PRINT 0117 PRINT "MAXIMUM VALUE FOR THE REAL PART?" 0119 INPUT K 0 120 PRINT 0122 PRINT "MINIMUM VALUE FOR IHE REAL PART Y' 0124 INPUT RI 0126 PRINT 0 12' i#PRINT 0128 PRINT "MASS (G), VOLUME (CM3) AND THICKNESS (MM) OF SAMPLE?" 0130 INPUT W,V,T1 0132 VO=V/T1 0135 PRINT 0140 PRINT "MASS (G) OF TEST SAMPLE AS WET AND D1Y?'

0352 PRINT 0354 PRINT 0356 PRINT 0358 PRINT 0360 PRINT 0370 PRINT "RESULTS OF SOIL MEASUREMENTS" 03 72 PRINT"- -- - -------- 0380 PRINT 0390 PRINT "DATE "Dl"-"D2"-"D3 0400 PRINT "SOIL TYPE "5$ 0421 PRINT 0422 FOR T=TW-TO TO T1+TO STEP TO 0423 PRINT "THICKNESS ="T"NM"rl 0425 R=14/(VO*T) 0426 M3=R*Ml 0427 PRINT "BULK DENSITY ="R1" G/CM3" 0428 PRINT "VOLUMETRIC WETNESS:"M3" CN3/CN3" 0430 PRINT 0440 PRINT 0450 P7'RINT " F(G1-HZ) RE E IN E A(DB/M)'TAN D A(DB) PH.SHLFT(DEB)" 0455 PRINT " NO. OF ITERATIONS" 0460 PRINT 0470 FOR 1=1 TO 10 0480 PO=1.2*F(I) 0490 IF P2(I)(0 THEN 515 0500 P3=P2(I)-P1(I) 505 IF P3>0 THEN 540 507 P3=P3+360 0510 GOTO 540 0515 IF P1(I)K:0 THEN 500 0520 P3=P2(I)-P1(I)+360 0530 REM P IS TOTAL PHASE SHIFT INCLUDING THAI DUE TO FREE SPACE 0540 P=P3+PO*T 0550 REH B IS PHASE COEFFICIENT 0560 B=(1000*P/T)*3.1415926/180 0570 REM E IS DIELECTRIC CONSIANT FOR LOSSLESS CASE 0580 E=(F/(T*F0))**2 0581 REN 0 82 IEM

0583 REM THE FIRST METHOD 0584 REM 0585 REN 0590 REM DIELECTRIC PROPERTIES OF SAMPLE ARE CALCULATED ONLY 0600 REH UHEN THE REAL PART I5 BETWEEN R1 (MINIMUM) AND K (MAXIMUM) 0670 IF E>R1 THEN 716 0680 P=P+360 0690 B-(1000*P/T)3. 1415926/180 0700 E=(P/(T*PO))**2 0710 IF E(R1 THEN 680 0712 60TO 716 0716 A5-((1-SQR(E))/(I 1SQR(E)))),t2 0718 A6=20*LOG(1-A5)/2.3026 0720 A3=A1(I) —A2(I) 0722 A4A=3+A6 0730 REM A IS ATTENUATION COEFFICIENT 0740 A=(1000*A4/T)/8.686 0744 IF A:::-O0 THEN 750 0747 A=0.001 0750 Y=439.2473*F( I)**2 0760 REM COMPLEX DIELECIRIC CONSTANT OF SOIL IS E1.- J-2 0770 E1=(B**2-A**2)/Y 0780 E2=2A*A*B/Y 0784 IF E1>K' THEN 1530 0786 D4=E2/E 1 0787 IF D4<0.03 THEN 807 0788 REM CORRECTION OF A A*D B TO 0789 REM ACCOUNT FOR HIGO LOSS. 0790 M4=SQR ( SQR (E 1 t**2+E2 e ) )*COS (0. 5*ATN (-E2/E1 ) ) 0791 M5=SQR(SQR(E1**2+E2** 2) ) SIN(O.5,ATN( —E2/E1- ) ) 079? H6=SQR( ( 1-M4)**s2HM5:t*.2)/SOR ( ( 1 +M4) st+M:.2) 0793 C6=M6**2 0794 M7=ATN(-M5/(1-M4))-ATN(M5/(1tM4)) 0795 8=H6*COS(M7 ) 0796 M9=M6*SIN(M7 ) 0797 HO= ( 1 -H M8* 2+H 9 **) **2 + ( 2* M8H? ) *9 2 0798 C9=10*LO(GMO)/2.3026 0799 A4-3 +C9 0800 Al('00IA4/' ),/8.636

0801 06 ATN( -42*M8fl9(S/ *:2+9*:.2) )* 1/ 3.84/3. 92 654 0802 P4=P+D6 0803 B=(10O0*P4/T)*3.141592654/180 0804 E1=(B**2-A**2)/Y 0805 E2=2*A*B/Y 0806 D4=E2/EI 0807 A:=A*8.686 0808 PRINT 810 PRINT F(I);E1;E2;A/7;D4;A3;P-PO:*r 0811 REM 0812 REM 0813 REM THE SECOND METHOD 0814 REM 0815 REM 0817 H=O 0818 C9=A6 0840 BO=2*3.1415926*F( I)*T/300 0865 REM DEVIATION FROM THE AVERAGE LOSS 0874 A9=1+A5**2*EXP(-4*A*T/1000) 0876 A9=A9-2*A5*EXP(-2*A*T/1000)*COS( —2*10*SQR(E1)) 0880 A9=10*LOG(A9)/2.3026 0885 REM NEU VALUE FOR THE AVERAGE LOSS 0890 C4:A3-A9 0895 REM NEU VALUE FOR THE AIIENUA1l.[ON CI4EF[ICIIEN'T 0900 U5=(1000*(C4+C9)/T)/8.686 0905 REM NEW VALUE 1::Qj% THE COMPLEX DIELiECTRIC CONSTANT 0910 E3=(B**2-C5**2)/Y 0920 E4=2*C5*B/Y 0925 REM NEU REFLECTION COEFFICIENT IS M8 +JM9 0926 M4:SOR(SOR(E3.i **2 +E4:* *2))*COS(0.5*ATN( —E4/E3)) 0927 M5=S0R(SOR(E3**2+E4:*2))*SIN(0.5*ATN(-E4I43)) 0928 M6=SQR((1-M4)** n, N2+5:MS*2)/SQR((1+M4):te**2 tNM5:**2) 0929 C6=M6**2 0930 M7=ATN( —05/(1-M4))-ATN(MS/(l+M4)) 0931 M8=M6*COS(M?) 0932 M9=M6sSIN(M7) 0933 MO=( 1 -M8**. 2+M9:*:*2 ) *:*2 ( 2 *M8n *M9):fc*2 0934 REM C9 IS THE MISMATCH LOSS

0937 C9=10*LOG(MO)112.302 0968 REN NEU VALUE FOR DEVIATION FROM THE AVERAGE LOSS (A9) 0995 A9=1+C6**2*EXP(-4*C5*'T/100) 0997 A 9= A9-2*C6* EX P( -2 It C5*T/ 1000) zCOS ( -2 *B0SOR(E 3)) 1000 A9=10*LOG(A9)/2.3026 1005 REM D5 IS THE TOTAL LOSS 1010 D5=C4+A9 1020 H=H+1 1030 IF H>20 THEN 1050 1035 REM TEST FOR ACCURACY (LOSS) 1040 IF ABS(D5-A3)::0.001 THEN 890 1050 CBC5*8.686 1055 D4=E4/E3 1060 PRINT TAB(6);E3;E4;C8;D4;H 1061 REM 1062 REM 1063 REM THE THIRD METHOD 1064 REM 1065 REM 1090 Hl=l 1091 REM NEW VALUE FOR THE REFLECTION COEFFICIENT 1100 NO=SOR(SQR(E**2tE2:**2))SCOS(0.5*ATN(-E2/El)) 1110 Nh=SQR(SOR(E1*:*2+fE24:ft**2?)):)SIN(O.5*ATN( —E2/E)) 1 1210 N2&=SQR( ( I1-NO)**2t N 1;f.**2 )/SQR ( ( 1 +NO)*42 +N142) 1130 N3=ATN(-N1/(1-NO))-ATN(N1/(1I-NO)) 1140 N4=N2*C0S(3) 1150 N5=N2*SIN(N3) 1160 REM N4 + JN5 IS THE COMPLEX REFLECTION COEFFICIENT 1170 N6=1-N4**2+NS**2 1180 N7:-2tN4*N5 1190 REM N6 + JN7 IS THE TRANSMISSION COEFFICIENT 1 —R**'2 1194 D6=ATN(N7/N6)*180/3.14159265 1200 REM N8 + JN9 IS THE POUER REFLECTION COEFFICIENT R**12 1210 N8=N4**2-N5**2 1220 N9=2*N4:tN5 1230 REM NEXT COMPUTATION OF COMPLEX TRANSMISSION 1240 REM COEFFICIENT THROUGH A DIELECTRIC SLAB 1244 L=COS(-D*T/10*0)-*EXP(-A.T/1*00) 1246 L2zSJN(-3*T/1*O0)*EXP( —A*T/1000)

1 250 L3=COSI(-2E*T/iO00)*X(-' X At"I /100) 1260 L4=SIN(-2 00Tf O *)1XXFP(-2l*A*/lo 0) 1270 L5=N6*LI-N7*L2 1280 L6=N6*L2+N7/*L 1290 L7=N8*L3 —-N9*L4 1300 L8=N8*L4-+N9:L3 1310 L9=(L5**2+L6**l2)(1-L.7)**2+L8**2) 1315 REM LO IS THE COMPUTED LOSS IN BD 1320 LO=-10*LOG(L9)/2.3026 1325 REM D4 IS THE COMPUTED PHASE ANGLE IN DEGREES 1330 D4=(ATN(L6/L5)-ATN(-L8/(1-L) )))*1l8/3.141592654 1335 D5=D4 1340 IF D4('0 THEN 1360 1350 D4=D4-180 1360 IF ADS(P+t4)(45 THEN 1390 1370 IF ABS(D14)>'25000 THEN 1522 1380 GOTO 1350 1390 REM MAKE PHASE SHIFT POSITIVE 1392 [4=-D14'1393 REM TEST FOR ACCURACY (PHASE SHIF'I) 1394 IF (AbS(F-D4)) j0.5 THEN 1398 1396 GOTO 1400 1397 REM TEST FOR ACCURACY (LOSS) 1398 IF ABS(A3-LO)O0.05 THEN 1513 1400 IF D4>P THEN 1430 1410 El=El*(1+0.9*(P-D4)/P) 1420 GOTO 1440 1430 EI=El*(1-0.9*(D4-P)/D4) 1440 IF LO>A3 THEN 1460 1450 E2=E2:*((EXP(0.23*(A3-LO))-1)/3+1) 1455 GOTO 1470 1460 E2:=E2/((EXP(0.23*(LO-A3))-1)/3t1) 1461 REM IF THE MEASURED LOSS IS SO SMALL THAT THE ITERATION 1462 REM PROCESS CANNOT FIND A VALUE FOR E2, AN ESTIMATE IS 1463 REM OBTAINED BY USING DIFFERENT VA-UlUES FOR E2.: AND 1464 REM COMPARING THE CONPUTED LOSS UITH THE MEASIJREiD VALUE. 1465 REM THE BEST ESTIMATE IS ["RINTE.D. 1470 IF E2K1.E-03 THEN 1475 1471 471B= 2*3.141592654 / (0. 31F (I) sSUR (0. 5*E 1: (SUR + (E4E21/E'i ) *:$:2 ~)+1)

1472 A=2*3.141592654/(0.3/F(I)): SQR(O.5*I:1(SOR( t(E'2/E 1)*2)-1 )) 1473 H1=H1+1 1474 GOTO 1100 1475 Y I= —( OOO:tA6/T)/8.686 1476 Y=22*Y1*B/Y 1477 Y3=Y2/200 1478 Y6=100 1480 YO=El 1484 FOR Y4:=Y3 TO Y2 STEP Y3 1486 B=2*t3.141592654/(O.3/F ( I) )SQR(O.5YO(SQR( ( Y4/YO)*:t ) +1 )) 1487 =23. 141592654/(. 3/F ( I) ) SQR ( O.5YO ( SQ ( + (Y4/ );t ) — 1 ) ) 1488 L 1 =COS(-B*T/1000)EXP (-A: T. 1 000) 1489 L2=SIN(-B*T/ 1000 )EXP(-'I-AT/1000) 1490 L3=COS(-2*B T/ OOO) tEXf ( -2*l'T /1000 ) 1491 L4=SIN (-2*Bf'T/1000)*EXP (-24*AT/1000 ) 1492 L5=N6*L1-N7*L2 1493 L6=N6*L2+N7?L1 1494 L7=N8*L3 —N9*L4 1495 L8=N8*L4t+N9*L3 1496 L9=(L5S*2+L6**2 ) / ( ( 11-7 )**2tL8 2) 1497 REM LO IS THE COMPUTED LOSS IN DB 1498 LO=- 10*LOG(L9)/2.3026 1499 YS=ABS (A3-LO) 1500 IF Y5>Y6 THEN 1508 1501 Y6bY5 1 502 E2=Y4 1503 El=YO 1504 L=LO 1508 NEXT Y4 1510 H1=0 1511 LO=L 1513 C8=A*;8.686'1514 D4=E2/E1 1515 PRINT TAB(6);E1;E2;C8;D4;H1 1516 IF H1::0.5 THEN 1518 1517 PRINT TAB(15)" COMPUTED LOSS:"LO"DB" 1518 F=P+360 1519 GOTO 560 152 1 GOTO 1530

1522 PRINI "COI'%"RLC f ["HASE ANGLE NOT FOUND, PHASE:".I5" DEGREES" 1524 GOTO 1514 1530 NEXT 1 1540 PRINT 1550 PRINT 1560 IF TO)0.01 THEN 1580 1570 STOP 1580 NEXT T 1590 PRINT 1600 PRINT H 1610 END

20 Frequency: 4 GHz S = 25 - j 5 15 0 9,j 10 up I n -1 0 - jil. 5 E=3-j 0.2 0 10 20 30 40 50 Thickness d (mm) Figure 1. Theoretical loss at 4 GHz as a function of sample thickness for various dielectrics.

40 ~-Frequency: 18 GHz 35 E = 25 - j 5 30 25 15 10 5 0 5 10 15 20 25 Thickness d (mm) Figure 2. Theoretical loss at 18 GHz as a function of sample thickness for various dielectrics.

2000 18 GHz I 1500,'1000 ZOOO4 GHz 0 10 20 30 40 50 0 10 20 30 40 50 Thickness d (mm) Figure 3. Theoretical phase shift as a function of sample thickness for various dielectrics.

Actual Loss C~7~3 1@ \-Average Loss 0 Envelope of Maximum Mismatch 0 / Deviation from Average Loss Loss Sample Thickness Figure 4. Definition of terms related to sample loss.

14 12 tan 6 ~< 1 10 Ln 4-0 -J -c E In 4 2 0 0 10 20 30 40 50 60 70 80 100 Dielectric Constant E' Figure 5. Theoretical mismatch loss as a function of sample -'.

6o 4 der' tar ~ d e c "~''-.?er~9/ /~ / /perB~di(U~aT ~nddcu -- - -, tanD<<! ~s,~~~~ 26 0 ~ ~ ~ ~ ~ ~~~.2 Nt smatch Loss: ~ 12 16 d-10 0 2 4 6 $ 10 I eaerageOSS t ( 4 x2verage o s o { loss as ~ deviation Irom the r. -[heotet~Ca~ _.,4m sample ~ as a parameter Fvgure 6. average toss,-....

30 Angle of Incidence: 00 ~ 25 Accuracy: 1% C 2 OE I ~ 20 -J 0~~~~~~~~~~~~~~~0 ~15 E 10 E 0 1 2 3 5 10 20 30 50 100 Dielectric Constant E' Figure 7. Theoretical minimum required average loss to ensure a desired accuracy for the measured loss as a function of sample e', when the oscillation of loss is neglected.

Envelope of Maximum Deviation from Average (I, \\ Phase Shift a) Actual Phase Shift Average Phase Shift Sample Thickness Figure 8. Definition of terms related to sample phase shift.

Im(z) S- I2erp2e-2Yd Re(z) Figure 9. Graphical solution to maximum deviation from the average phase shift (see text).

45 A' 40,/\E' 100 -c / Xm 35 / Angle of Incidence: 00 i,: / tanb << 1 e/303 E 25 a e90X 20?0 /10 15 10 /5 5E /3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Average Loss Lc,(dB) Figure 10. Theoretical maximum deviation from the average phase shift as a function of averaae loss. with samnip. s:':: n nrnmPtPr

Angle of I ncidence: 0~ tan 6 << 1 -— 30 LAccuracy: 1% 30 --— | Accuracy: 2% O 15 10 ~ t, \\ ~E'' 120 10 0 100 200 300 400 500 Average Phase Shift A4av (degrees) Figure 11. Theoretical minimum required average loss to ensure a desired accuracy for the phase shift as a function of average phase shift with sample e' as a parameter, when the oscillation of phase shift is neglected.

Calculate 1 and E' for Lossless Case Ca Icu late LM and a Calculate E' and " Calculate New Values 1 forLManda Calculate New Values j for E' and E" El"'i - E"i-l < k2 No Yes Print Results Figure 12. Block diagram of Method 1 that accounts for the mismatch loss only.

Assume: Ls= L, lt meas: av Calculate E' and E" as in P rocedure 1 If a< O, make a=O. 1 N/m Calculate E' and E" Calculate AL Calculate New Value for La, from Lmeasand AL Calculate New Values for a,', ", LM, AL Is AL + La - Lmeas < k3 No Yes Print Resu Its 1 Figure 13. Block diagram of Method 2 that accounts for the loss oscillation.

Assume: Lmeas L, A'Dmeas =A(av Calculate E' and E" as in Procedure 1 If a< O, Make a = 0.1 N/m Calculate E' and E" Calculate L and Aav, from E', E", a,, Is IL - LmeasI< k4 lID - ADmeas| < k5 No Yes I I Change Print E' and E" Results P roperly Calculate aand j Figure 14. Block diagram of Method 3 that accounts for the oscillation in the loss and phase shift.

4.0 - -30 =441...ap o e 2.5 4 GHz PhaseNotCorrected 20 -2 - 4 GHz and 18 GHz, Phase 2.0 I I I'' I 5 10 15 20 25 30 35 40' 45 50 Thickness d (mm) Figure 15. Comparison of Method 2 (phase not corrected) and Method 3 (phase corrected) for E' as a function of sample thickness (e = 3 - jO.2). No measurement error is assumed.

18 E1 = 10- j 1.5 ---- 18 GHz, Phase Not Corrected 16 -~ 4 GHz, Phase Not Corrected 4 GHz and 18 GHz, Phase and w14 Amplitude Corrected 1I r ~~~~~~~~~~~~~30 12 20 G3 I I I w; 10 0 10 8 -20 6 L. I I I I I I i | 0 5 10 15 20 25 30 35 40 45 50 Thickness d (mm) Figure 16. Comparison of Method 2 (phase not corrected) and Method 3 (phase

20 29.- ~ i A l1 — 8,Phs o orce 10l t~~~~~~~~~~~~i |1oHz, Phase ~ot Corrected -1.@~~~ In ~ 4 4GHz and18 GHz, Phasel 17 - - 0 5 10 15 20 25 30 35 45 50 -30 Thickness d (mam) Figure 17. Gomparison of Method 2 (phase not corrected) and Method 3 (phase corrected) for a' as a function of sample thickness (E_= 25 -iS). No meHz Phasurement eCorretor is assumed.21 14 GHz and 18 GHz, Phase ii ~~~~~~~~~~~and Amplitude Corrected 2 19 17 0 5 10 15 20 25.30 35 40 45 5 Thickness d (m m) Figure 17. Comparison of Method 2 (phase not corrected) and Method 3 (phase corrected) for C as a function of sample thickness (v, = 25 - j5). No measurement error is assumed.

0.5 E = 3 - j 0.2 140 Frequency: 4 GHz 120 0.4/ - Phase and Amplitude Corrected I ---- Amplitude Corrected = l — m- Mismatch Loss Accounted for 80 w I 60 p0.3 40 U) I I u0_ |4w \ / 20 0 U / -20 I -_ -80 0.0 0 100 0 5 10 15 20 25 30 35 40 45 50 Thickness d (mm) Finl irp 1R A (.nmnnricnn nf intfrnrotntinn mathnr4e fnr C" At 1 rAL' —e' fl ani ftin ~t

5 = 10-j1.5 200 Frequency: 4GHz Phase and Amplitude Corrected 150 -- - Amplitude Corrected,Um - Mismatch Loss Accounted for o M 3 U. L, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Li I-) 00 %%~~ 00"* Awy U I ( a) L A ~ -~ 0 5 10 15 20 25 30 35 40 45 50 Thickness d (mm) Figure 19. Comparison of interpretation methods for c" at 4 GHz as a function of sample thickness (e = 10 - ji.5). No measurement error is assumed.

12 E = 25 - j5 Frequency: 4GHz 10 1 Phase and Amplitude Corrected 100 I\ mm ---- Amplitude Corrected 80 w- - - Mismatch Loss Accounted for -60 -05 10 15 20 25 30 35 40 45540 4(u4 ~ ~ ~ ~~Tikesd{4 2 I -60 0 L~:~-~~~~ ~-80 0 I I I I I I I I -100 0 5 10 15 20 25 30 35 40 45 50 Thickness d (mm)

0.5 E = 3 - j 0.2 Frequency: 18GHz 0.4 Phase and Amplitude Corrected 100 I —- Amplitude Corrected Iw l l lMismatch Loss Accounted for 1 80 I - 60 0.3 0. () 00 0 2 4 6 8 10 12 14 16 18 20 Thickness d (mm) Figure 21. Comparison of interpretation methods for E" at 18 GHz as a function of sample thickness (c = 3 - jO.2). No measurement error is assumed.

2. 5 60 2.0 C 40 2.0 2 1.5 I... I,J E =10-j1.5 -20' 1.0- I Frequency: 18 GHz I --- -- Phase and Amplitude Corrected -40 -\ I -— m - Amplitude Corrected 0.5 -- Mismatch Loss Accounted for -60 -80 0. 0 0.0 I''''I''' — 0 0 2 4 6 8 10 12 14 16 18 20 Thickness d (mm)

6.5 E= 25-j5 Frequency: 18GHz |I ----— Phase and Amplitude Corrected 6.0 ---- Amplitude Corrected 20 I%~ -- 5 0 -—' Mismatch Loss Accounted for 45.0 moo% z\\~- X10, 5.5 4.0 4I 5 -\ 0 2 4 6 8 10 12 14 16 18 20 Thickness d (mm) Figure 23. Comparison of interpretation methods for &" at 18 GHz as a function of sample thickness (~ = 25 - j5). No measurement error is assumed.

4.0 30 E 3 -j 2 020 i3 5 Assumed Error: Phase Shift: 41% cM Loss: + 1% at 18 GHz 10 6 tA +5% at4GHz 3.o 00' 18 GHz, Phase Corrected 2.5 ---- 18 GHz, Phase Not Corrected -4 GHz Phase Corrected ---- 4 GHz, Phase Not Corrected 2.0 0 5 10 15 20 25 30 35 40 45 50 Thickness d (mm) Figure 24. Comparison of Method 2 (phase not ~orrected) and Method 3 (phase

20 29 27 10 23 ii D 4 7 Q 9 E= 23 5E-2j5 Assumed Error: Phase Shift: -1% -10 |'I ~ Loss: +1% at 18 GHz 21 II +5% at4GHz 18 GHz, Phase Corrected -20.. —-- 18 GHz, Phase Not Corrected 19 -0 4GHz, Phase Corrected — ~ —- 4 GHz, Phase NotCorrected 1 7 L' I I I I m I-30 0 5 10 15 20 25 30 35 40 45 50 Thickness d (mm) Figure 25. Comparison of Method 2 (phasenot corrected) and Method 3 (phase corrected) for c' as a function of sample thickness (~ = 25 - j5). Assumed measurement errors as shown.

0. 5 140 I, \ o120 0. 4 I -100 0.4 E = 3 -j 0. 2 I | We Frequency: 4 GHz " | |I Assumed Error: Phase Shift: +1% 60 A u0. 3 Loss: +5% o.! I,,,, | | | | I. 1 1-100 0 5 - 10 15 20 25 30 35 40 45540 I aO 0. zU 0., I. 0 5 10 15 20 25 30 35 40 45 Thickness d (mm) Figure 26. Comparison of interpretation Mlethogls for E" at 4 GHz as a function of

12 120 10 - - 100 II 80 8 E =25- j5 60 I% | \ PFrequency: 4GHz Assumed Error:,Phase Shift:+l% 4-% 4 Loss: +5% 2 0 2L I IPhase and Amplitude Corrected -60 IJ --— Amplitude Corrected — Mismatch Loss Accounted for -80 0 I A' 0'' o 5 10 15 20 25 30 35 40 45 50 Thickness d (mm) Figure 27. Comparison of interpretation methods for e" at 4 GHz as a function of sample thickness (c = 25 - j5). Assumed measurement errors as shown.

0. 5 120 0.410 = 3-j0.2 80 Frequency: 18 GHZ.2 Assumed Error: Phase Shift: +1% 60 ~0.3 Loss: +1% 40 20~ U" CAC o. 2 Q) II ~~~~~~~~~~~~~~~~~~~~~-20 0 -40 0.1 Phase and Amplitude Corrected ---- Amplitude Corrected --- Mismatch Loss Accounted -80 0.0 I I0 2 4 6 8 10 12 14 16 18 Thickness d (mm) Figure 28. Comparison of interpretation methods for c" at 18 0Hz as a function of

6.5 1 m30 6.0 - 20 E!25-j 5 5 |5 I t Frequency: 18 GHz 5. 5 Assumed Error: Phase Shift: +1-A I ---- AmpLoss: +1%'0- - N or, 5. 0 I I - 4.5 5 Phase and Amplitude Corrected - -10 I —...Amplitude Corrected ---— /Viismatch Loss Accounted for 4.0 -20 0 2 4 6 8 10 12 14 16 18 20 Thickness d (mm) Figure 29. Comparison of interpretation methods for ~" at 18 GHz as a function of sample thickness (e = 25 - j5). Assumed measurement errors as shown.

Frequency: 4 GHz 4.0 E = 3 - j 0. 2 - Phase Not Corrected ~- - Phase Corrected Assumed Error: 0.20 " 3. 5 / ~ I Phase Shift: -1%, Loss: +5% 2.5 Phase Shift: -3%, Loss: + 5%'.10 20 30 40 50 Thickness d (mm) Figure 30. Comparison of Methods 2 (phase not corrected) and 3 (phase

Frequency: 4 6Hz 31 E 0 25-j5 -P ~~Phase Not Corrected 20 -9 -~m~Phase Corrected 29 Assumed Error: 0 IbnC / ~ ~~~~~~ Phase Shift: -11o, Loss: +57o11 r'27 Phase Shift: -3%, Loss: +571 ( 25 o 0 a -t-L 00e~~~~~~~~~~~~~~~ 23 w o I ~~~~~~~~~~~/ -1~~~~~~~-10 21 -20 19 5 10 15 20 2 Thickness d (mm) Figure 31. Comparison of Methods 2 (phase not corrected) and 3 (phase corrected) for v' at 4 GHz as a function of sample thickness (C = 25 - j5). Assumed measurement errors as shown.

0.3 Amplitude Corrected Frequency: 4 GHz 0. 0.2; 0.1 -I Assumed Error: -al-Phase Shift: +1%, Loss: +5% -60 -I -Phase Shift: +3%, Loss: +5% Phase Shift: +3%, Loss: +10% 0.0.I-1 0 10 20 30 40 Thickness d (mm) Fin irl -3 Fffprt nf +i imrl m i rnm3nrt rrnr 9- 4r %" 1A^-kr~e+k,4, A -%IA +I

0.3 Phase and Amplitude Corrected Frequency: 4 GHz E=3-j0.2.~~~~~ 0.2~~~~~ffr W I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I 0.2 0 c.) 0. 1 Assumed Error: - Phase Shift: +1%, Loss: +5% -60 Phase Shift: +3o%, Loss: +5% ---- Phase Shift: +3%, Loss: +10% 0.0 I 0 I I.-100 0 10 20 30 40 50 Thickness d (mm) Figure 33. Effect of assumed measurement error to e" from Method 3 at 4 GHz (e = 3 - jO.2).

8 Amplitude Corrected 6 Frequency: 4 GHz 7 E -25-j5 40 ~~~~~4 20._LL I k \ / ~~~~Assumed Error: 1#~~~~ \L /..0 Phase Shift: +1%, Loss: +5%' - 0 Phase Shift: +3%, Loss: +5% r —-- Phase Shift: +3%, Loss: +10% 2 — 60 5 10 15 20 25 Thickness d (mm) Finl irn WA Pffor't n f mcclim r m nc lrnmgnr r -rrr%~ " Ad { r,-ors Jl A 0 f+ A BU-S

Phase and Amplitude Corrected Frequency: 4 GHz 7 E =25 J5 540 7 LA) 0 04O Y, O ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~tet" g L)~ U -20 4) Assumed Error: Phase Shift: +1%, Loss: +55% --- — ~Phase Shift: +3%, Loss: +5% rn —rn Phase Shift: +3%, Loss: +10% 2 -60 5 10 15 20 25 Thickness d (mm) Figure 35. Effect of assumed measurement error to c" from Method 3 at 4 GHz (c = 25 - j).

1.0 0 0. 8 - 0.6.k-a 0.4 0 a.) 0.0 1 2 3 5 10 20 30 50 100 Dielectric Constant E' Figure 36. Effect of measured phase constant (with respect to air) to total phase

Sample Reference Path (No Sample) Path with Sample I n/serted Ce ~/ A Figure 37. Propagation path with and without the sample.

UNIVERSITY OF MICHIGAN 3 9015 03027 5203