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HUGOFELLNER-FELDEGG
been proposed by Zundel.'O It is based on the idea of fluctuations of protonic energy levels, resulting from the splitting of vibrational states. These fluctuations are caused by the environmental field, inducing a continuous level distribution. This effect may be especially pronounced in cases when the two potential energy minima are nearly equivalent, i.e., in H bonds with large A; values (intermediate between AZO and Ajil), which is observed in our case. Thus, the Aji values can be described by means of a formal proton-transfer model, and the ir data seem to
support the conclusion that this effect is of great importance in the investigated systems. It should be remembered, however, that Aji and XPT were calculated based on assumptions which may be to some extent questionable. Also, the contribution of the electron delocalization cannot be neglected. Further studies are needed to clarify the role of these two effects in H bonds of intermediate strength. (10) G. Zundel, Allg. Prakt. Chem., 21, 329 (1970); see also E. G . Weidemann and G. Zundel, Z. Phys., 198,288 (1967).
A Thin-Sample Method for the Measurement of Permeability, Permittivity,
and Conductivity in the Frequency and Time Domain1 by Hugo Fellner-Feldegg Hewlett-Packard Laboratories, Palo Alto, California 94304
(Received January 4, 1972)
Publication costs assisted by the Hewlett-Packard Company
A method for measuring the permeability, permittivity, and conductivity of materials using thin samples in coaxial lines is presented. I n the time domain, one obtains directly the impulse response of the permeability and permittivity and the step response of the conductivity in a single measurement over a frequency range which is limited only by the rise time of the oscilloscope-pulse generator system and the duration of the applied pulse. There is a linear superposition of these material parameters which facilitates the transformation between time and frequency domain. The only condition is the maximum thickness of the sample, which depends on the material parameters and the frequency range used. The required sample volume is approximately 1 pl.
Introduction Some time ago time domain reflectometry was introduced for the measurement of dielectric relaxation properties in the range of 30 psec to about 100 nseca2 It allowed the determination of high- and low-frequency permittivity and the relaxation time from a single time domain measurement over a range which required a fairly substantial instrument investment in the past. The method used only the first reflection from the interface air-dielectric, which made it easy to interpret. It required, on the other hand, rather long coaxial sample cells for the measurement of long relaxation times and also required the Fourier transformation into the frequency domain for nonideal dielectric^.^ This single-reflection time domain reflectometry has been shown to produce surprisingly precise data.4 Recently Nicolson and Ross6have reported a method which uses a combination of both the reflected and transmitted wave from samples of arbitrary thickness The Journal of Physical Chemistry, Vol. 76, No. 16, 1972
to measure the complex permittivity and permeability, using Fourier transformation. We have used a similar approach of applying transmission line theory for the measurement of material properties and found that reducing the sample thickness below a certain value and terminating the line with a matched load simplify the result considerably and give directly the impulse response of the permeability and permittivity, and the step response of the conductivity. Thus, it is possible to derive the transient behavior of the material from the time domain or from the frequency domain without the (1) This method was first reported a t the meeting of the Dielectrics Discussion Group, Bedford College, University of London, April 5-7,
1971. (2) H. Fellner-Feldegg, J. Phys. Chem., 73, 616 (1969). (3) (a) T. A. Whittingham, ibid., 74, 1824 (1970); (b) H. FellnerFeldegg and E. F. Barnett, ibid., 74, 1962 (1970). (4) A. Suggett, P. A. Mackness, and M. J. Tait, Nature (London), 228, 456 (1970). (5) A. M. Nicolson and G. F. Ross, IEEE Trans. Instrum. Meas., 19, 377 (1970).
MEASUREMENT OF PERMEABILITY, PERMITTIVITY, AND CONDUCTIVITY requirement of long sample cells for the measurement of long relaxation times. Furthermore, the sample volume is in the microliter range, which is of advantage in many applications. The equipment required is the same as described in ref 2. There, a sampling oscilloscope and a tunnel diode pulse generator were used for measurements in the microwave region. Since the thin-sample method, reported here, is not limited in the lower frequency range by the sample length, real time oscilloscopes and suitable pulse generators may be used for measurements below a few hundred megahertz.
2117
[Szl - exp( - r01)I = lexp(-rO - 11[1 p2 exp(-$)I 1 - p2 exp(-2yl)
+
+
[1 - exp(-roOI
(6)
Expanding the exponential term into a Taylor series and using only the linear term gives the reflection coefficient
and the difference in the transmission coefficient
Mathematical Treatment We will first derive the equations for the frequency domain using a well-known transmission line equation, and then transform the result into the time domain. The input impedance 2 of a transmission line of an impedance 2 1 and length 1, terminated with 20 (see Figure 1) is given by
where y is the propagation constant and p is the reflection coefficient of a line of the impedance 20,terminated with 21. The line with the impedance 21is obtained by filling a coaxial line of the characteristic impedance Zo with the sample of a length 1. The sample shall have a relative permittivity K* = K' - j u " , a conductivity u, and a relative permeability p* = p' - j p " . We are neglecting any series resistance in this line. Then the impedance is
702
- -(P* 2
-t K* -k J w - 2)
(8)
Equations 7 and 8 can be used for the measurement in the frequency domain. The response to a step pulse is in the frequency domain
and likewise
It is the advantage of this approach that one obtains a linear superposition of the frequency response of and the propagation constant is permeability, permittivity, and conductivity (the latter divided by j w ) . This not only makes it easy to = z/jW~p*(jwcK* G) = yov'p(3) Tu interpret the results in the frequency domain, but also allows making the transformation into the time domain with separately for each term. G jw The Laplace transformation into the time domain - = 4nu; yo = j w z / L c = c then gives and
+
zo=g,
We are using the cgs system and are, therefore, including the factor 4 n in the conductivity term. The reflection coefficient from the sample, expressed as a scattering coefficient S, is then SI1
=
[1 - exp(-22rOIp 1 - p2 exp(-2yZ)
and the change in the transmission coefficient due to the sample, referenced to the output port of the sample, is given by
- 42c [~(t)
- 1
+ ~ ( t -) 1 4-4a
L'
u(l)dt]
(12)
where [ S ( l ) l l is a unit st.ep function at the time t = Z/c and p ( t ) , ~ ( t ) u, ( t ) are the impulse response of the permeability, permittivity, *and conductivity, respecThe Journal of Phusical Chemistry, Vol. 76, N o . 16, 197.8
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HUGOFELLNER-FELDEGQ
and
Figure 1. Coaxial line with sample and 50-ohm termination.
Figure 5 shows the time response schematically. area of the 6 function is proportional to (1/2c)(l and the area under the curve is sll(t)dt = -(1 1 2c
o
i
i
i
i n s
Figure 2. Reflection and transmission from a thin sample of two ferrites: (1) ferramic Q g , ( 2 ) ferramic Qa.
tively. The integral of the impulse response u ( t ) is the step response of the conductivity. The only difference between the reflected and transmitted waves, both in the frequency and time domain, is the sign of the permeability term. This allows one to separate it from the permittivity and conductivity terms by measuring the transmission and the reflection. However, the reflection measurement will, in general, be more accurate, since the signal is directly proportional to the material constants and the sample length and can be well separated from the incident step pulse, whereas in the transmission measurement one has to take the difference between exp(-yy,l) or S(1) and the relatively small value of (821)step and (SZi)step, respectively. l / c = 3.3 X lo-" cm-' sec can also be expressed as 30 ohms, since 1 ohm = 1.1 X 10-l2 cm-' sec. This may be used for the conductivity term in eq 9-12 when expressing a in ohms-' cm-'. Figures 2, 3, and 4 give some typical time responses obtained with thin-sample time domain spectroscopy for magnetic, dielectric, and conductive materials. The results obtained in the time domain can either be transformed into the frequency domain by numerical transformation or evaluated directly. We shall explain the latter using the example of an ideal and a lossy dielectric, but will keep in mind that it is equally applicable for any dielectric or magnetic sample. Consider a thin sample of a nonmagnetic ( p = 1) Debye dielectric with no conductivity. The frequency and time response of the reflection coefficient to a step pulse is then The Journal of Physical Chemistry, Val. 76, N o . 16,1972
The
-
K')
(15)
KO)
Figure 6 shows the close match between theoretical and experimental values obtained from a number of dielectrics. The low value of si1 for water may be due to incomplete filling of the sample cell because of the strongly hydrophobic Teflon beads. The exponential part of the time response curve gives the relaxation time, which can be obtained simply from the slope of the log 811 vs. time plot. For nonideal dielectrics one obtains ~1 and KO as described above; however, the log $11 os. time plot will deviate from a straight line, depending on the relaxation properties of the dielectric. The finite response of the measuring system degrades
t
d2,-S(lr/ -0.05
-0.1
0.5
Lon5
Figure 3. Reflection and transmission from a thin sample of dielectric; glycerine, 25".
Figure 4. Reflection from a thin sample of a dielectric with conductivity; methanol, saturated with potassium chloride, 25".
2119
MEASUREMENT OF PERMEABILITY, PERMITTIVITY, AND CONDUCTIVITY
TIME
Figure 5. Schematic time response of the reflection from a thin sample of a dielectric.
’““1.
I
/’
9. ,/
20
0
40
80
60
100
Korp
Figure 7 . Reflection and transmission signal from a thin sample us. relative permeability or permittivity for different values of rol.
(Ko-l)
c
IO
/
/ /
/
/
/
/
/ 1 Trichloraethylene
2 3 4 5 6 7 8 9
Pyridlne Ethanol Acefone n-Propanol Ethanol Methanol Dimethylformamide Water
PEAK AREA A,, t .
sec
Figure 6. Experimental (points) and theoretical (dashed line) values of the static relative permittivity obtained with the thin-sample reflection method.
the 6 function in eq 14 to a peak of finite width whose area is proportional to ( l / 2 c ) (1 - KI). Electrical conductivity, present in the sample, is added to the frequency or time dependence of the reflection coefficient. If the conductivity is independent of frequency, it will produce an offset of the base line of ( ~ l l ) ~=~ ~~ Td U ~ /=C
188.5~1
(16)
with u expressed in ohms-l centimeter-’. Finally, we have to specify the error introduced by using only the first term of the Taylor expansion of eq 5 and 6. We will solve the problem for the frequency domain first. Figure 7 shows a computer plot of the exact value of SIIor [Szi - exp( - -,do)] vs. K or p for different values of 701,according to eq 5 and 6. The dashed lines give the approximate values, derived from eq 7 or 8. The difference between eq 5 and 6 (reflection and transmission, respectively) contains only the third and higher order terms of the Taylor expansion and is about for the range of values plotted.
,005
,002
0
io
40
$0
io Ibo K or c
c
Figure 8. Error, f i x 1 - A, made using the signal amplitude, fitl, to correct for the thin-sample approximation in the frequency domain.
For a given value of 811,the relative difference A of K , derived from the exact and the approximate equations, A = ( K - K . ~ ~ ~ ~ ~is) very / K , closely equal to the value of 811. The difference &I - A vs. K is plotted in Figure 8. For small values of rol and for K >> 1, this difference approaches r01/4. It is therefore quite convenient to use this relation between A and 811not only to estimate the error made in using the thin-sample approximation but also to correct for it, which in turn permits one to employ relatively thick samples, producing sufficiently strong signals for exact measurements in the frequency domain. It is difficult to transform these relations into the time domain. Therefore, we will derive eq 11 (without conductivity term) in the time domain to obtain A. Consider again a section of a transmission line of the length I , filled with a material with the permeability fi The Journal of Physical Chemistrg, Vol. 76, No. 16, 1978
HUGOFELLNER-FELDEGG
2120
and the permittivity K inserted into a line of the impedance 20 and terminated with the same impedance. We further stipulate that p and K do not vary during the time interval tr, considered in the following. Then the reflection coefficient for the first reflection will be dP/K = 4pJK
n=l
n=l
-1
+1
n.2
The return time within the sample is
n=2 n.3
214&
to
= C
Consecutive multiple reflections can easily be analyzeda and are shown in Figure 9. They produce a reflected signal which is shown schematically in Figure 10. The step height, referenced to Vo,is
I
4-P F
Figure 9. Multiple reflections from a transmission line with the impedance 21and length 1.
and the area of step 1t o n is n 1
Rc
I
p
m
A,, = to
n=3
n94
p2"
-1 -1
= top ___ p2
(20)
Consequently, the total area is given by
Using eq 17 and 18 gives Figure 10. Schematic time response of a section of transmission with impedance 21 and length E.
which is the same as eq 11 without a conductivity term. We now specify a time interval t, = ntO,which is defined either by the rise time of the measuring system or by the condition that p or K do not vary during this time. We will make an error A = (A, - A,,)/A, of the measurement of p or K which is A =
p2n
(23)
or
This function is shown in Figure 11. For t r = 33 psec, a typical rise time of a sampling oscilloscope system, trc is 1cm and eq 24 can be approximated by
I
=
0 . 5 ( ~- 1) for A = 2%
1 = 0 . 6 7 ( ~- 1) for A = 5%
(25)
for p = 1. This can be used to conservatively estimate the maximum permissible sample length. The error A affects also the shape of the measured response curve. Let us approximate the time response by a step function, similar to Figure 10, however now with a step time tr = nto as used in eq 24. Then, The Journal of Physical Chemistry, Vol. 76, N o . 16, 1978
K0,p
Figure 11. t,c/21 us. relative permeability or permittivity for different values of the relative measurement error A = A K / Kin the time domain.
each step with a theoretical step height f(t/T) will have an actual value fi(t/Ti)
=
(1 - A)f(t/T) -t- Af((t -
(26)
where TI is a relaxation time derived from the experimental curve and T is the true relaxation time. (6)
B.M. Oliver, Hewlett-Packard J . , 15 ( 6 ) (1964).
MEASUREMENT OF PERMEABILITY, PERMITTIVITY, AND CONDUCTIVITY
2121
The second term in eq 26 comes from the contribution of the preceding time element and can be approximated with
Since Atr/T
