786
J. Phys. Chem. 1980,84,786-793
Evaluation of Dielectric Behavior by Time Domain Spectroscopy. 3. Precision Difference Methods Robert H. Cole,” Satoru Mashimo,+ and Paul Wlnsor, I V Department of Chemistty, Brown University, Providence, Rhode Island 029 12 (Received September 28, 1979) Publlcation costs assisted by the National Science Foundation
Refinements in time domain total reflection methods are described for precision measurements of dielectric permittivity from 1MHz to several GHz. A single basic cell design with two effective sample lengths can be used for direct and difference measurements of a wide range of permittivities, from strongly polar liquids to dilute, weakly polar solutes in a nonpolar solvent. Loss processes with maximum absorption emf‘ = 0.02 have been measured with satisfactory accuracy as judged by comparison with results from steady state measurements. Methods for precise time referencing and correction of timing differences are described, together with convenient procedures for numerical Fourier transformation of the observed time domain waveforms to obtain the complex permittivity.
I. Introduction In previous papers of this series (I1 and 112),methods for high-frequency measurements of complex permittivity by time domain reflectometry (TDR) were described and illustrated. These methods are based on observation and analysis of the “total”, i.e., multiple, reflection pattern produced by the dielectric sample inserted in or terminating a coaxial line, as shown schematically in Figure 1. In the original arrangements, satisfactory results were obtained for moderately to strongly polar liquids, but the precision in recording amplitudes and times of reflections in relation to the incident tunnel diode pulse was not adequate for measurement of smaller effects, such as relaxation of dilute solutions of polymers in a nonpolar solvent. In this paper, we describe a cell design, time referencing and other circuitry, and methods of data processing which have largely overcome the earlier limitations and have greatly extended the range of usefulness of the basic methods. Examples are given to show that relaxation processes with maximum loss t” of order 0.02 can be satisfactorily measured at frequencies from a few megahertz to several gigahertz. 11. Basic Methods In the arrangement we have found most useful, a dielectric sample is placed in a section of coaxial line of effective length d a t the end of the 50-ohm coaxial line system shown in Figure 2. An incident pulse V&t)at the input to the sample cell produces a reflection R(t)as shown in Figure 1, replicas of which are obtained by probes in the sampler unit and expanded to a longer (millisecond) time scale by the repetitive scanning circuitry of the sampler and oscilloscope. The basis equation for determining relative complex permittivity e* of the sample in this arrangement, derived in I1 from transmission line theory, is conveniently written in the simple form uo-r €*(W) = f(z) d iw(uo + r ) where vo and r are the Laplace transforms of the incident and reflected pulse waveforms, as given by +Department of Physics, Tokai University, Hiratsuka-shi, Kanagawa, Japan. OO22-3654/80/2084-6786$~1.OO/O
uo(w) =
(2)
Jmdt exp(-iot)Vo(t)
with a similar expression for ~ ( w ) .The function f(z) where z = ( w d / ~ ) e * laccounts /~ for propagation and multiple reflections in the sample and for coaxial line geometry is given by f(2)
+
= z cot z = 1 - ‘ / ( w d / c ) % ” - 1/46(0d/c)4€*2 .., (3)
the series expansion being valid for lzl < n. The assumptions of equivalence to a coaxial line of length d may not be sufficiently accurate for some cell designs which can be so devised that f(z) = 1- uz2 - bz4, where u and b are empirical coefficients. In either case, eq 1 and eq 3 or its replacement are readily solved for e* for 1x1 < 1. This condition IzI < 1, corresponding to a maximum frequency w given by w
c c/dlt*’q
(4)
is not a necessary upper limit but is one above which the solution rapidly becomes sensitive to small errors, especially near z = n/2 for which z cot z = 0. We have therefore taken eq 4 as a restriction on cell design and possible range of usefulness. A second upper limit on frequency range is imposed by the frequency spectrum of the incident pulse as modified by the sampler response. For our instrumentation, the pulse can be described approximately by a rise time T,of order 50 ps, for which the transform is zero at 20 GHz, and we have taken the upper limit for adequate frequency content of the pulse to be about 10 GHz. A. Timing Errors. Subject to the limitations just, specified, the accuracy of the basic method is determined by the accuracy with which such transforms as vo - r and iw(u0 r) or iwuo can be obtained from the observed pulses. Since only differences and ratios of uo and r are involved in eq 1 or other working equations based on it, the origin of time in calculating the transforms is arbitrary as long as it precedes the arrival of the pulse, but a difference in the choice for the two transforms does produce errors. The magnitude of the errors which result from a given time difference and an upper limit on the t,ime difference for given acceptable errors are easily estimated. If the origin of time for R ( t ) relative to that for Vo(t) is in error by 6 t , taken as positive for R ( t ) too early, the correct
+
0 1980 American Chemical Society
Tlme Domain Spectroscopy
-~
Flgure 1. Incldont voltage pulse V,(t), as seen In reflectlon from an open clrcult, and muklple reflection R ( t )from a cell of flnke length fllled wlth dlelectrlc. The small Initial peak of both curves Is the tlmlng reference pulse described In the text.
I~SAMPLE: Ill
The Journal of Physlcal Chemistry, Vol. 84, No. 7, 1080 787
minutes of 20 ps or more, resulting in unacceptable errors for many measurements above a few megahertz and only limited possibilities for improvement by signal averaging. We describe below methods for generation and use of timing reference pulses to determine values of 6t which are consistent to within one part in several thousand of‘the observation period for Vo(t) and R(t). B. Recording of the Incident Pulse. The need. for precise relative timing also precludes direct observation of Vo(t) as the incident pulse when it first passes by the sampler probe, as this is located some centimeters and several nanoseconds ahead of the dielectric sample. An ideal open circuit at the position of the front edge of the sample would produce the desired V&t) by reflection in the same range of time as R(t) and could be processed in the same way. An ideal open circuit is difficult to realize exactly, but its equivalent and the desired Vo(t)can be simply obtained by use of a suitable sample cell and reference dielectric with known permittivity e, which is real and independent of frequency or time in the range of interest. From eq 1 for f(z) = 1, Le., (wd/c)2e,
