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A new transmission method in dielectric time domain spectroscopy
Vohxme 47, number 2
CHEMICAL
PHYSICS LETTERS
14 ApriI 1977
A NEW TRANSMISSION METHOD IN DIELECTRIC TIME DOMAIN SPECTROSCOPY B. GESTBLOM and E. NORELAND Institute of Physics. University of Uppsala, S-75 I21 Uppsala, Sweden Received 26 January 1977
A new transmission TDS method for permittivity measurements using the linear dependence of the attenuation coeffrcient and the phase coefficient with sample length is descriied. Experimental resuks for methanol in carbontetrachloride at room temperature are reported.
1. Introduction
coefficient is given by
The advent of pulse generators with a puke rise time
in the picosecond range rendered possible the developtime domain methods (TDS) for the study of the permittivity of dielectrics [ l] . Methods now exi?t which cover a frequency range from a few MHz up tp the order of 10 GHz J2J _In the method most frequently used so far the reflection of a fast rising pulse from a long dielectric sample in a coaxial line is monitored. From the Fourier transforms of the incident and reflected pulses the permittivity of the sample can be calculated at any chosen frequency. ment of
2. Theory In a recent paper it was shown that transmission methods offer an attractive alternative to the reflection methods in TDS 1-31.In the total transmission method the pulse r(f) transmitted through a short sample is monitored and compared to the incident pulse u(f) transmitted though the same length of air filled coaxial line. The two pulses are Fourier transformed, F(o)
= S f(r) ewiwt dt -0D
(I)
into the frequency domain in which the transmission
T(w) = fTlei@
= (1-$)exp {+wZ/c) [(c*)rl2 - I J 3 * I--$- exp(-(2iwl/c)(@2 J Here w is the angular frequency, i is the sampie length. c is the speed of light in vacuum and E* is the compiex permittivity e’ - k”. The reflection coefficient p is given by p = [l - (e*)‘/*J/[l
+(?)‘/2]
.
(3)
The solution of the transcendental equation (21 by a search procedure in the complex E’, i’ plane gives the permittivity for the chosen frequency_ A study of eq. (2) shows that the dominant term in the transmission coefficient is the exponentiat term exp {-(k&/c) [(e*)lj2 - I ] ). This can be rewritten as e--v and it is seen that the attenuation coefficient 01= Re fiWrf(e * ) U2 - I j/c) and the phase coefficient B = Im {iwZJ(e*) If2 - I f jc) are both Iinearly dependent on the sample length I. This suggests that a fogarithmic plot of the amplitude I Tl and a linear plot of the phase 40as a function of sample length wauld give straight lines, the-slopes of which give the pmmittivity E* at the chosen frequency. A requirement for this analysis is that the other factors in eq. (2) do not vary significantly with sampk Iength. For not too high perrruttivities the term p2 will not be very large as can be judged from the reflec349
CHEMICAL PHYSICS LETTERS
Volume 41, number 2
tion coefficient diagram in ref. [3] or from eq. (3). For instance, if we assume E* = 6 we find p* = 0.18 and 1~21 diminishes with lower Ic*l. In the denominator the term p*e-*ae-i(*flf2wllc) will be oscillating with the sample length I, with its maximum = p* for I = 0 when Q = f3= 0. This term will be damped with growing length of the sampIe, i.e. growing value of 01. This implies that for drelectrics with low permittivity the pre-exponential factor C= (I -p*)/{l--p*
exp[-2iwZ(e*)l/*/c]
)
in the transmission coefficient in eq. (2) will differ from unity by a correction term of the order of 10%. This correction term wiII vary in a damped sinusoidal manner with increasing sample length. The above analysis suggests the following TDS method for permittivity measurements. The transmitted pulse through a sample is recorded for different sample lengths. From the Fourier transforms of these pulses and the reference pulse the transmission coefficients T(w) = ITI eiq are calculated. Plots of log ITI and p as functions of sample length I are made. These will in a first approximation be straight lines with superimposed oscillatory deviations. From the slopes of the straight lines (Yand fl are determined and first approximations to E* are calculated for the chosen frequencies. Using these values of E* the preexponential factor C is evaluated. This factor is now multiplied with the experimentally determined T,i.e. we make plots of TC as functions of 1. These plots will now produce straight lines which pass through the origin, and from which the oscillatory perturbations are removed. The slopes of these lines give the permittivity E*. If considered necessary this correction procedure can be iterated several times.
3. Experimental Fig. 1 shows the experimental setup used for the TDS transmission measurements. A dual channel sampling oscilloscope, HP 1811 A, was used. Two step pulse generators are triggered from the same triggering pulse taken from the oscilloscope. The pulse in channel B gives the time reference point necessary in the Fourier transform procedures. The oscilloscope is modified to allow the external sweep of the time base from a ramp volt generator_ The output from channel B is 350
15
April 1977
Fig. 1. Experimental setup for transmission measurements using a dual channel sampling oscilloscope. derivated in a differentiating circuit, which causes the reference pulse to appear as a narrow spike. This spike is added to the pulse to be studied from channel A in a summing amplifser and displayed on an X-Y recorder. In the measurements the pulse transmitted through the empty coaxial line is first recorded. The studied liquid is then injected by a syringe into the coaxial line through a 1 mm hole in the wall and the transmitted pulse recorded under otherwise unchanged conditions for successively increasing length of the sample. Since no digital data acquisition system was available, the recorded curves were digitized manually. No averaging over several scans could consequently be performed. The Fourier transformations were done by the Samulon formula [4]. The phase in the Fourier transform is critically dependent on the stability of the time reference. The relative stability of the two pulses in the two channels indicates that the accuracy of the time referencing procedure is better than 10 ps. This would give an error in the phase of the Fourier transform of IO-l1 w rad for a chosen frequency. As a test case a sample of methanol in carbontetrachloride (17% by volume) was studied at room temperature. The transmitted pulse was recorded at sample lengths from 32 mm to 222 mm in steps of 32 mm, corresponding to the addition of 0.5 ml into the coaxial line. After Fourier transformations the transmission coefficients were evaluated and the iteration procedure applied as described above. The resulting straight lines are shown in fig. 2 for a number of frequencies from 100 MHz to 10 GHz. Transmission coefficients < 10% have been rejected. The permittivities evaluated from the slopes of the lines in fig. 2 have been plotted in the form of a ColeCole diagram in fig. 3. The permittivities seem to foilow a skewed arc dependence. Other studies of alcohols have shown the existence of several dispersion regions
T
CHEMICAL
Volume 47. number 2
0
100
200
PHYSICS LETTERS
Ilmmi
15 April 1977
100
0
Fig. 2. Phase upand transmission coefficient lrl as a function of sample length I for methanol tion procedures as described in the text. Frequency range 100 MHz to 10 GHz.
--+B 200
in carbontetrachloride
IlrrWl
after two iters-
eters r = 0.26 ns and fl= 0.52. The correspcnding arc has been drawn in fig. 3. The root mean square deviatiori is O.!%.
Acknowledgement
Fig. 3. Cole-Cole diagram for methanol. The permittivities in fig 2. The full were evaluated from the slopes of iii6- .‘:no@ . ..__ _.. line denotes :: skewed arc from a least squares tit to the experimental data.
with different reiaxation times [5,6!. Pince high frequency data are missing in this study, no attempt has been made to resolve the dispersion curve in two or more regions. Instead the datd have been fitted to the Cole-Davidson skewed arc funcllon [7] E* = E, + (es
. )/(I + iw7)P .
A least squares fit gives for the permittivities es = 5.4, La = 2.5 and for the relaxation and distribution param-
Fiuancia: support from the Swedish Natural Science Research Council is gratefully acknowledged.
References [ 11 [2] [3] [4] [5]
H. Fellner-Feldegg, J. Phys. Chem. 73 (1969) 616. M.J.C. van Gemert, Philip? Res. Rep!. 28 (1973) 53L:. B. Gestblom and E. Noreland, to be publishes. H.A. Samulon, Proc. IRE 39 (1951) 175. G.P. Johari and C.P. Smyth, J. Am. Cbem. Sot. 91 (I 969) 6215. 161 J.A. Saxton, R.A. Bond, G T. Coats and R.M. IJiciith~~L. J. Chem. Phys 37 I1Sc.L) 2:;2. 171 D.W. Davidsonand R.H. Cole. J. Chen:. Phqs. 18 (1951) 1417.
,?5 !
CHEMICAL
PHYSICS LETTERS
14 ApriI 1977
A NEW TRANSMISSION METHOD IN DIELECTRIC TIME DOMAIN SPECTROSCOPY B. GESTBLOM and E. NORELAND Institute of Physics. University of Uppsala, S-75 I21 Uppsala, Sweden Received 26 January 1977
A new transmission TDS method for permittivity measurements using the linear dependence of the attenuation coeffrcient and the phase coefficient with sample length is descriied. Experimental resuks for methanol in carbontetrachloride at room temperature are reported.
1. Introduction
coefficient is given by
The advent of pulse generators with a puke rise time
in the picosecond range rendered possible the developtime domain methods (TDS) for the study of the permittivity of dielectrics [ l] . Methods now exi?t which cover a frequency range from a few MHz up tp the order of 10 GHz J2J _In the method most frequently used so far the reflection of a fast rising pulse from a long dielectric sample in a coaxial line is monitored. From the Fourier transforms of the incident and reflected pulses the permittivity of the sample can be calculated at any chosen frequency. ment of
2. Theory In a recent paper it was shown that transmission methods offer an attractive alternative to the reflection methods in TDS 1-31.In the total transmission method the pulse r(f) transmitted through a short sample is monitored and compared to the incident pulse u(f) transmitted though the same length of air filled coaxial line. The two pulses are Fourier transformed, F(o)
= S f(r) ewiwt dt -0D
(I)
into the frequency domain in which the transmission
T(w) = fTlei@
= (1-$)exp {+wZ/c) [(c*)rl2 - I J 3 * I--$- exp(-(2iwl/c)(@2 J Here w is the angular frequency, i is the sampie length. c is the speed of light in vacuum and E* is the compiex permittivity e’ - k”. The reflection coefficient p is given by p = [l - (e*)‘/*J/[l
+(?)‘/2]
.
(3)
The solution of the transcendental equation (21 by a search procedure in the complex E’, i’ plane gives the permittivity for the chosen frequency_ A study of eq. (2) shows that the dominant term in the transmission coefficient is the exponentiat term exp {-(k&/c) [(e*)lj2 - I ] ). This can be rewritten as e--v and it is seen that the attenuation coefficient 01= Re fiWrf(e * ) U2 - I j/c) and the phase coefficient B = Im {iwZJ(e*) If2 - I f jc) are both Iinearly dependent on the sample length I. This suggests that a fogarithmic plot of the amplitude I Tl and a linear plot of the phase 40as a function of sample length wauld give straight lines, the-slopes of which give the pmmittivity E* at the chosen frequency. A requirement for this analysis is that the other factors in eq. (2) do not vary significantly with sampk Iength. For not too high perrruttivities the term p2 will not be very large as can be judged from the reflec349
CHEMICAL PHYSICS LETTERS
Volume 41, number 2
tion coefficient diagram in ref. [3] or from eq. (3). For instance, if we assume E* = 6 we find p* = 0.18 and 1~21 diminishes with lower Ic*l. In the denominator the term p*e-*ae-i(*flf2wllc) will be oscillating with the sample length I, with its maximum = p* for I = 0 when Q = f3= 0. This term will be damped with growing length of the sampIe, i.e. growing value of 01. This implies that for drelectrics with low permittivity the pre-exponential factor C= (I -p*)/{l--p*
exp[-2iwZ(e*)l/*/c]
)
in the transmission coefficient in eq. (2) will differ from unity by a correction term of the order of 10%. This correction term wiII vary in a damped sinusoidal manner with increasing sample length. The above analysis suggests the following TDS method for permittivity measurements. The transmitted pulse through a sample is recorded for different sample lengths. From the Fourier transforms of these pulses and the reference pulse the transmission coefficients T(w) = ITI eiq are calculated. Plots of log ITI and p as functions of sample length I are made. These will in a first approximation be straight lines with superimposed oscillatory deviations. From the slopes of the straight lines (Yand fl are determined and first approximations to E* are calculated for the chosen frequencies. Using these values of E* the preexponential factor C is evaluated. This factor is now multiplied with the experimentally determined T,i.e. we make plots of TC as functions of 1. These plots will now produce straight lines which pass through the origin, and from which the oscillatory perturbations are removed. The slopes of these lines give the permittivity E*. If considered necessary this correction procedure can be iterated several times.
3. Experimental Fig. 1 shows the experimental setup used for the TDS transmission measurements. A dual channel sampling oscilloscope, HP 1811 A, was used. Two step pulse generators are triggered from the same triggering pulse taken from the oscilloscope. The pulse in channel B gives the time reference point necessary in the Fourier transform procedures. The oscilloscope is modified to allow the external sweep of the time base from a ramp volt generator_ The output from channel B is 350
15
April 1977
Fig. 1. Experimental setup for transmission measurements using a dual channel sampling oscilloscope. derivated in a differentiating circuit, which causes the reference pulse to appear as a narrow spike. This spike is added to the pulse to be studied from channel A in a summing amplifser and displayed on an X-Y recorder. In the measurements the pulse transmitted through the empty coaxial line is first recorded. The studied liquid is then injected by a syringe into the coaxial line through a 1 mm hole in the wall and the transmitted pulse recorded under otherwise unchanged conditions for successively increasing length of the sample. Since no digital data acquisition system was available, the recorded curves were digitized manually. No averaging over several scans could consequently be performed. The Fourier transformations were done by the Samulon formula [4]. The phase in the Fourier transform is critically dependent on the stability of the time reference. The relative stability of the two pulses in the two channels indicates that the accuracy of the time referencing procedure is better than 10 ps. This would give an error in the phase of the Fourier transform of IO-l1 w rad for a chosen frequency. As a test case a sample of methanol in carbontetrachloride (17% by volume) was studied at room temperature. The transmitted pulse was recorded at sample lengths from 32 mm to 222 mm in steps of 32 mm, corresponding to the addition of 0.5 ml into the coaxial line. After Fourier transformations the transmission coefficients were evaluated and the iteration procedure applied as described above. The resulting straight lines are shown in fig. 2 for a number of frequencies from 100 MHz to 10 GHz. Transmission coefficients < 10% have been rejected. The permittivities evaluated from the slopes of the lines in fig. 2 have been plotted in the form of a ColeCole diagram in fig. 3. The permittivities seem to foilow a skewed arc dependence. Other studies of alcohols have shown the existence of several dispersion regions
T
CHEMICAL
Volume 47. number 2
0
100
200
PHYSICS LETTERS
Ilmmi
15 April 1977
100
0
Fig. 2. Phase upand transmission coefficient lrl as a function of sample length I for methanol tion procedures as described in the text. Frequency range 100 MHz to 10 GHz.
--+B 200
in carbontetrachloride
IlrrWl
after two iters-
eters r = 0.26 ns and fl= 0.52. The correspcnding arc has been drawn in fig. 3. The root mean square deviatiori is O.!%.
Acknowledgement
Fig. 3. Cole-Cole diagram for methanol. The permittivities in fig 2. The full were evaluated from the slopes of iii6- .‘:no@ . ..__ _.. line denotes :: skewed arc from a least squares tit to the experimental data.
with different reiaxation times [5,6!. Pince high frequency data are missing in this study, no attempt has been made to resolve the dispersion curve in two or more regions. Instead the datd have been fitted to the Cole-Davidson skewed arc funcllon [7] E* = E, + (es
. )/(I + iw7)P .
A least squares fit gives for the permittivities es = 5.4, La = 2.5 and for the relaxation and distribution param-
Fiuancia: support from the Swedish Natural Science Research Council is gratefully acknowledged.
References [ 11 [2] [3] [4] [5]
H. Fellner-Feldegg, J. Phys. Chem. 73 (1969) 616. M.J.C. van Gemert, Philip? Res. Rep!. 28 (1973) 53L:. B. Gestblom and E. Noreland, to be publishes. H.A. Samulon, Proc. IRE 39 (1951) 175. G.P. Johari and C.P. Smyth, J. Am. Cbem. Sot. 91 (I 969) 6215. 161 J.A. Saxton, R.A. Bond, G T. Coats and R.M. IJiciith~~L. J. Chem. Phys 37 I1Sc.L) 2:;2. 171 D.W. Davidsonand R.H. Cole. J. Chen:. Phqs. 18 (1951) 1417.
,?5 !