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The Measurement of Dielectrics in the Time Domain by Hugo Fellner-Feldegg Hewlett-Packard Laboratories, Palo Alto, Caltfornia
(Received August 1 5 , 1 9 6 8 )
The time dependence of the reflection of a step pulse from the interface between air and a dielectric medium in a coaxial line has been measured from 30 psec to 200 nsec, corresponding to a frequency range of 1 MHz to 5 GHz. The high-frequency and low-frequency dielectric constant, the relaxation time, and the dielectric loss can be obtained in a fraction of a second. Different alkyl alcohols have been measured over a wide temperature range. The results are essentially the same as obtained in the frequency domain. Introduction Generally, permittivity is measured by placing the substance between two plates of a capacitor (at low frequencies) or into a coaxial line and measuring the complex impedance. A number of measurements over a wide frequency range is required for complete characterization, which is time consuming and demands a considerable investment in instrumentation, particularly in the microwave region. Therefore, and in spite of its usefulness, this method found only rather limited applications in the past. However, one can obtain the same information over a wide frequency range in only a fraction of a second by making the measurement, not in the frequency domain but in the time domain, using a pulse which, simultaneously, contains all the frequencies of interest. This pulse method has been applied sometimes in the past for low-frequency investigations on dielectrics. Modern tunnel diode pulse generators and wide band sampling oscilloscopes allow the extension of this method into the microwave region where savings in time and equipment are most pronounced. Such instruments have been used for years for cable testing and are known as time domain reflectometers (tdr) .
additional signal ( 2 ) which is displayed on the oscilloscope. The time elapsed between the first and second steps is equal to the transit time of the traveling wave from A to B and back to A again. The remainder of the wave, not reflected at B, travels to C. If we terminate the line at C with an open end then all of the wave is reflected back in phase (assuming no losses due to radiation). Part of this pulse is reflected again at B and part of it goes through past A, giving rise to another step (3). The time between steps 2 and 3 is the transit time from B to C and back to B again. The magnitude of the first step, Vop, is
- 20 vop = vo 2z ___ +
2 0
where p is the reflection coefficient and Vo is the pulse height. A coaxial line with the impedance Zo when empty, will have a smaller impedance 2, when filled with a nonconductive dielectric of the permittivity K*
Thus, the reflection coefficient p is a function of
Basic Relations for the Measurement of Dielectric Properties with Tdr The time domain reflectometer consists of a pulse generator which produces a fast rise time step, a sampler which transforms a high-frequency signal into a lower frequency output, and an oscilloscope or any other display or recording device (Figure 1). The pulse from the step generator travels along the coaxial line until it reaches point A. The sampler detects and the oscilloscope displays the voltage step (1) as it travels past point A (Figure 2 ) . The coaxial line which transmits the pulse has a characteristic impedance Zo = 50ohms. Whenever there is a discontinuity in this line, a fraction of the traveling wave will be reflected back into the generator. Therefore, at the interface of the 50-ohm line with any other impedance Z (point B) part of the step pulse will be reflected and will pass point A again, producing an The Journal of Phgsical Chemistry
K
(3) and
(4) For conductive dielectrics jwL
+R
(5)
where R is the series resistance, G is the parallel conductance, u is the low-frequency conductivity in siemens/cm, and G/C = 4 m . I n most applications, with reasonable lengths of transmission lines, R NN 0. Therefore 2 becomes 2 = Z O / ( ~-* j 4 7 r a / ~ ) ' / ~
(6)
MEASUREMENT OF DIELECTRICS IN
TIMEDOMAIN
THE
1-
1
617
,
AMYL ALCOHOL 25'C 2
ZU
OSCl L L O S C O P E
Figure 1. Time domain reflectometer system.
and for
20
lo-% 100 can be measured with tdr, corresponding to u < 0.25 siemens/cm for K = 40. Such a conductivity makes dielectric constant measurements with other methods at high frequencies quite difficult. With tdr, however, materials with ten times higher conductivities could be determined with ease. A remark about the accuracy of the dielectric measurements with tdr. First, it is dependent on the mechanical accuracy of the coaxial line used. Standard 7-mm precision coaxial air lines have an impedance of 50 f 0.1 ohms or a maximum deviation of 0.2%. Since K = (Zo/Z)z,the relative error in K can be f0.4%. This error can be reduced to an insignificant amount by measuring accurately the impedance or the physical dimensions of the coaxial line. A second source of error is any inaccuracy of the measurement of the reflection coefficient p. It may be due to amplifier nonlinearities in the oscilloscope or the recorder, which normally is less than 1%, or due to unwanted reflections from poorly matched transmission lines, including the sampler. Arrangements shown in the next part can suppress the latter and allow, with commercial instruments, an overall measurement accuracy of the reflection coefficient Ap/p I 0.01 and AP I 2 X whichever is greater. The transformation of p into K , however, multiplies this error by a factor
frequency cutoff-of the system. Imagine an ideal dielectric with no relaxation reflecting a &function impulse, which contains all frequencies of equal amplitude. The reflected pulse will be a 6 function again, if the interface air-dielectric is parallel to the electric field. If the interface extends over a certain distance in the direction of the traveling wave, however, then the reflected pulse will be spread out, corresponding to the return trip time of the wave through the interface. This reflected pulse has a limited high-frequency spectrum, The convolution in the time domain of this pulse with the reflection from any real dielectric (with relaxation) having a perfect interface gives the actual reflection obtained from the real dielectric with an extended interface. Errors thus introduced in the measurement of the high-frequency permittivity and relaxation time can be eliminated by terminating the sample cell with plastic or ceramic beads, similar to the ones shown in Figure 9. These beads have an inner and outer conductor of proper dimensions to produce a characteristic impedance of 50 ohms. They are, therefore, electrically in no way different from the remainder of the empty coaxial line but form a perfect physical interface with any liquid in contact with them. Finally, we shall consider the influence of the finite rise time of the step pulse on the measurement of K,. The pulse produced by the tunnel diode generator approximates quite well a linear voltage ramp which rises from zero to Vowithin the time t~ as shown in Figure 10. This ramp is the integral of a step pulse and the response of the dielectric becomes the integral of the step-pulse response divided by t~ 1
f ( t )ramp = tR
f (t) stepdt
Thus, the derivative of the measured time response of the reflection coefficient during the rise time of the pulse times tR gives the response to a step pulse. The linear extrapolation of p to the time t R gives po? and, from eq 4,
AK is shown in Figure 8 for the different sources of error mentioned. Obviously, relative measurements, using a reference sample, can be made with much higher accuracy. A poorly defined interface between air and the dielectric, e.g., a liquid meniscus, produces another error. It reduces in effect the rise time-or high-
/
Km*
Practical Aspects of Tdr Measurements Sample Cells. A simple and versatile cell for measurements of liquids and granular solids is the standard 7-mm diameter precision coaxial line of 10 and 20cm length with Amphenol APC-7 connectors (see Figure 9). As mentioned already, the maximum deviation from the 50-ohm impedance is f 0 . 2 % . The plastic bead on each end is machined accurately enough Volume 76, Number S March 1969
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Figure 10. Reflection coefficient vs, time for step pulse with finite rise time.
t o provide a liquid-tight seal for a few hours. The plastic may be replaced with any ceramic material, such as alumina, for operation at temperatures above 80' or to provide resistance against attack by organic solvents. The assembly on one side, including the center conductor, is held in place by the APC-7 connector, thus allowing one t o fill the line from the other end. The hybrid connector itself is of high quality and does not introduce any significant reflections up to 18 GHz. Solids can be measured in the coaxial cell by filling the line with the molten material, if possible, or by machining the substance to the dimension of the line, or by wrapping it tightly around the center conductor when available as foil, or by shredding or granulating it. Tdr measures an average value of the permittivity if the line is not completely filled with the dielectric. Therefore, for absolute measurements the fill factor has to be known. Relative measurements of the dielectric constant and exact determinations of the relaxation time can be made, however, if the fill factor is unknown but constant. A change of the packing density of the material over the length of the line will cause unwanted reflections. The dielectric properties of large batches of liquids may be measured with a vertical coaxial line immersed into the substance. Since the characteristic impedance, 20,depends only on the ratio of the diameters of inner and outer conductor, the line can be made to any size for mechanical stability as long as no other modes than the TERI mode are excited. A continuous monitoring of chemical reactions by measuring the permittivity in a time interval where it is most sensitive to changes in the chemical composition is entirely feasible. In addition, the position of the surface of the liquid can be measured, thus providing a simple liquid level control. This application has been suggested long ago7 and has found some practical use already. The only limitation of the straight coaxial line is its length. A standard 7-mm line becomes quite unstable, The
JOUTnd
of Physical Chemistry
if more than 30 em long, due to vibrations and sagging of the center conductor. On the other hand, one great advantage of dielectric measurements with tdr is the freedom from multiple reflections if one measures only within the time for the return trip of the wave in the sample cell. A 40 em long line filled with a dielectric of K = 20 limits the range t o
