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Mathematical Simulation for Dielectric Spectrometry
Copyright@ IFAC Control Applications of Optimization. St. Petersburg. Russia. 2000
MATHEMATICAL SIMULATION FOR DIELECTRIC SPECTROMETRY
Andrey G. Karpov, Oleg O. Olekhnovich
Faculty for Applied Mathematics and Control Processes. St-Petersburg University. Bibliotechnaya pi.2, St-Petersburg, 198904, Russia e-mail: nve@ Jtserv.apmath.spbu.ru
Abstract: A mathematical simulation for dielectric spectrometry is described. A consideration of the dielectrometrical response shape made it possible to sufficiently reduce the measurement time. The principle of this measurement and processing method consist in recording and analysis the time moments of crossing equidistant 7 voltage tresholds with a 1O. s resolution . The treshold voltages are set by digital-toanalog converter . The total measurement and processing error is < I %. Copyright © 2000IFAC Keywords: automatic operation, algebraic approaches, computational methods, computer-aided diagnosis, data processing, spectroscopy, spectrum analysis .
I. INTRODUCTION
41C17a3/(k7), where 17 - interior friction coefficient, a - radius of the ball as a model of rotating molecule (for water r = 25ps). If r large then l/j iffrequency of exciting electric field) , the dipole have no time to do a complete turning, and E decreases. 3. The volume charges making - ion moving in viscous environment; relaxation time up to hours. This mechanism is very interesting for modem industry of composite and synthetic materials.
One of the methods for investigation and diagnosis of materials, especially of dielectric materials, is the investigation of dielectric permittivity over a wide range of frequency of exciting (or diagnosis) signal. The correspondence between specimen response and exciting signal is determined by polarization phenomenon, which is numerically characterized by dielectric permittivity magnitude c. The value of dielectric permittivity is connected with polarization mechanisma, which take place in specimen under investigation. We can point a few polarization mechanisma; every one of which is characterized with relaxation time r : I. The forming of dipoles - displacement of atom electrons, relaxation time is near to speed of light. 2. The orientation of dipoles, which depend on temperature, environment viscousness, size and structure of molecules; relaxation time is in range from 0.1ns to IOms. For example, the relaxation time for simple liquid is determinating as r =
Because of depending on viscousness c contains the information about material structure, that is about material quality. Therefore measurement and analysis of c and r may be a method of diagnosis . For this reason, there is currently great interest in the of techniques for dielectric development spectrometry; such systems will permit the rapid evaluation and display of complex permittivity over a wide range of frequency .
173
For measuring and data processing we use the stepresponse method, where the current (or time) response to a voltage step is measured, combined with Fourier transformation to obtain the required a.c. parameters.
as reaction on alone step voltage stimulus, when each successive measuring is accompained only with decrementation of V,-{;ode . Such decomposition of recording algorithm into two parts permits us to decrease the recording time, for example, by 64 times for 8-bits length of V"
2. DATA RECORDING
To present we developed the automated unit for electric parameters determination of dielectric by pulse polarization method (Karpov and Egorov, 1999). Unit works with computer by on-line mode (Fig. 2). The information interchange is performed along universal serial line. On the command from computer unit gives the stream of time moments of achieving the successive decreasing levels of voltage
A relaxation system can be defined as one whose response to a step-function stimulus ie a monotonic decrease towards zero. In the dielectric case, a voltage step applied to the specimen results in a monotonically decaying current i(t) ; the voltage across the measurement resistor also behaves monotonically. During the developing the hardware of system (see Fig. I) we took into account the following conditions: I. During the begin period the decaying speed of response is too large to record the data with highest spectrometer resolution because of computer speed; 2. At the end period of the response decaying the decreasing speed is low; and computer speed permits the data recording with spectrometer resolution ; 3. It can be measured only such number of response values, which hardware lenght of response digital code permits, for instance, if response value is transformed into I2-bits code, we can get 4096 different values of response.
control unit
spectrometer " head" standard serial line Real part of c· (w)
r---..,~
voltage t - -... commutator supply
Imaginary part 0 f c' (w)
start/sto V, code (to be measured) comparison timer
time Fig. I. Measuring unit of system. SiO after 450'C heating
Thus there is not necessity to measure absolute values of the monotonic signal u(t), but time intervals, in which u(t) achieves every possible value, may be recorded. In begin and end periods of u(t) decaying the registration algorithms are different. In the begin period the registration of each successive value is the separate measurement. It consist of reset the timer, switching off the commutator, transfer the V,-{;ode to digital-analog converter (DAC); and then, after discharging the measurement capacitor, switching on the timer and the commutator, and recording the timer state into computer memory after transition of the comparison signal into high level. In the end period successive measurements are obtained
Si02 after 1200' C heating Fig. 2. The spectromety system and results of measuring and processing.
174
by dielectric response on step voltage stimulus. The measuring error is lower than 1%.
Values .1, ak, and b* are preliminary calculated, therefore processing itself includes only simple operations (addition, multiplication, substraction). It permits to accelerate and simplify the processing. The iterative calculation technique give us the processing in real time scale and permits to interrupt the processing if necessary or if required precision of result have been achieved. The functions of frequency become more precise with recording and processing of the next Ti , and become fixed in their high frequency part. And if only high frequency part is of interest, investigation may be interrupt in desired moment.
In our case time quantum for timer is equal to W- 7 s, therefore we can investigate parameters of materials with 10 MHz resolution in frequency.
3. DA TA PROCESSING FOR ANALYSIS OF DIELECTRlCAL COMPLEX PERMITTIVITY We worked out the processing technique for dielectric permittivity. According to measured current/time response the imaginary and real parts of complex permittivity are calculated in real-time scale. The result displayed in diagram form is renewed by acquisition and processing of i(t}--data.
4. DATA PROCESSING FOR ANALYSIS OF RELAXATION TIMES As it is described above, each polarization mechanism is characterized by its own relaxation time. And different mechanisma of polarization may take place in material under investigation. Thus in general case the response of material is the sum of exponents:
For a capacitor having unity vacuum capacitance, filled with the material as dielectric, the current response i(t} to a unit voltage step is connected with complex permittivity by the Fourier transform: c· (a»
= c'( a»
= c'" +
- jc"( a»
E!( a» = c'" + E!'( a»
r
o
i(t}exp(-Jrd}dt,
(I)
j(t)
f i(t}cos(rd}dt ,
=
o
h,exp(-tl r)dr,
(3)
hr, exp( -t 1r, ) ,
(4)
o
'"
=
f
'"
or in discrete case
f i(t}sin(rd}dt. o
f(t}
In real-time scale the real and imaginary part of Fourier-image are calculated with r values of i k, and we use the iterative technique:
=
I
where Tt - relaxation time; or in another statement
f
i(t) =
g(r)r-1exp(-tlr)dr,
(5)
o where g( r) - relaxation-time distribution function :
r
+L ik[bk_i_1- (.1+ I )bk., + .1bk_i+ I]);
f
1<=2
g( r)dr
=
1.
o Then complex dielectric permittivity
r
+L h[ak-,-I - (.1+ I )ak_, + .1ak_i+I]),
co
1<=2
/(a»
where (4 = w./.1'-1 (.1 > I), IV) - maximum angular frequency, tk - time moment for ik determination (tk = tl.1*-I, to = 0, tl = I1 w.), ak = (.1- I rl(l - sin(.1k) .1-*), bk = (.1- Ir\1 - cos(.1 k» .1-*, k = ... , -1, 0, 1,2,3,
calculation of
g(r)(l+ja>rr1dr+ &(00),
(6)
If the system can be described by a unique relaxation time ro, i.e. g( r) = 0 ( r - '0), the above reduce to the Debye case: i(t)
= ro-Iexp(-tlro),
/(a»-c(oo)=(I+ja>rorl. (7)
The computer processing is based on the mathematical model, which takes into account the discrete technique of measuring. The specimen voltage response can be expressed as co
u(t)
= Vo -
I
Uk
exp( -t Ird,
(8)
k=O
where Vo - exciting step amplitude. In initial moment u(O) = 0 and there is maximum current through discharged condeser, which contains the dielectrical spesimen as filler.
T(tk), and changing i(tk) in
quadrature formula by i (t k )
f
o c(oo) = lim Re[c*(a».
For using of this formula it is necessary to know values of it = i(t) in fixed time moments tt. but i(t) is measured in arbitrary time moments Tn. Therefore the data preprocessing is necessary. The calculation of i(t*) is performed as follows : a) determination of time interval [Tn, Tn+.J, which contains tt; b) approximation of function i(t) in this interval by line segment T(t) = i(Tn) + (t - Tn)(i(Tn+1) - i(Tn»/(Tn+1 - Tn); c)
=
•
175
The value U(/) is known in each measured time Equation (8) can be rewritten as
device time resolution and precision of voltage level putting by DAC. However we have the essential advantage in recording and processing time.
I;.
N
u(/;) = Uo -
I
Uk exp( -1,I'k),
(9)
REFERENCES
k=O
because of discreteness of time intervals I; and finity of measurings number. Here i = I , ... , N - number of measuring and N - number of performed measurings.
Karpov, A.G. and Egorov, N.V. (1999). An automated dielectrometer. Instruments and Experimental Techniques, 42, No. 6, pp. 790793.
The discrete recording technique gives data precision not grater than data discreteness. We nothing know about response signal value in intervals between successive achievings the digitally set voltage levels. The voltage resolution is equal OU = Urnax1n,
(10)
where Urna• - largest output signal of DAC, n - total number of levels. The system time resolution is equal to 1O-7s, as it was said above. Therefore we can detennine relaxation times of polarization phenomena 'k only as approximation to measured times I, of successive coincidences between specimen response signal U and digitally decrement voltage level. The decisions Uk of system N
U(I;} = Uo -
L Uk exp(-I, llk} , i = I, ... , N,
(11)
k ~O
are the weights of polarization phenomena with relaxation times 'k ('k = IJ in total dielectric specimen response . The numbers of I; and Ik are equal, and I, = Ik if i = k. If some Uk < c (where c beforehand set positive value) then according polarization phenomenon with relaxation time is absent.
'k
Consider the detenninant of matrix {exp(-t;ltk)}. Because of discreteness of data recording t; = 1;0 t and tk = mko t, where I; and mk are integer. Then exp(-t;ltk) = exp(-l;Otlmkot) = exp(-l/mk)' (12) i = I, .. . , N, k = I, . .. , N. The rows and columns of such elements will be linear independent if all I; are different, and henceforth our matrix will be not special. Therefore system solution is unique, there is a unique set of relaxation times in another words . The values exp(-I;lmk) can be calculated before measuring that allows to speed up the calculation of relaxation times.
CONCLUSION Data recording technique and mathematical processing allow to determine relaxation times only as approximation to measured times t;. The precision of relaxation times detennination depends on device properties and may not be higher than
'k
176
MATHEMATICAL SIMULATION FOR DIELECTRIC SPECTROMETRY
Andrey G. Karpov, Oleg O. Olekhnovich
Faculty for Applied Mathematics and Control Processes. St-Petersburg University. Bibliotechnaya pi.2, St-Petersburg, 198904, Russia e-mail: nve@ Jtserv.apmath.spbu.ru
Abstract: A mathematical simulation for dielectric spectrometry is described. A consideration of the dielectrometrical response shape made it possible to sufficiently reduce the measurement time. The principle of this measurement and processing method consist in recording and analysis the time moments of crossing equidistant 7 voltage tresholds with a 1O. s resolution . The treshold voltages are set by digital-toanalog converter . The total measurement and processing error is < I %. Copyright © 2000IFAC Keywords: automatic operation, algebraic approaches, computational methods, computer-aided diagnosis, data processing, spectroscopy, spectrum analysis .
I. INTRODUCTION
41C17a3/(k7), where 17 - interior friction coefficient, a - radius of the ball as a model of rotating molecule (for water r = 25ps). If r large then l/j iffrequency of exciting electric field) , the dipole have no time to do a complete turning, and E decreases. 3. The volume charges making - ion moving in viscous environment; relaxation time up to hours. This mechanism is very interesting for modem industry of composite and synthetic materials.
One of the methods for investigation and diagnosis of materials, especially of dielectric materials, is the investigation of dielectric permittivity over a wide range of frequency of exciting (or diagnosis) signal. The correspondence between specimen response and exciting signal is determined by polarization phenomenon, which is numerically characterized by dielectric permittivity magnitude c. The value of dielectric permittivity is connected with polarization mechanisma, which take place in specimen under investigation. We can point a few polarization mechanisma; every one of which is characterized with relaxation time r : I. The forming of dipoles - displacement of atom electrons, relaxation time is near to speed of light. 2. The orientation of dipoles, which depend on temperature, environment viscousness, size and structure of molecules; relaxation time is in range from 0.1ns to IOms. For example, the relaxation time for simple liquid is determinating as r =
Because of depending on viscousness c contains the information about material structure, that is about material quality. Therefore measurement and analysis of c and r may be a method of diagnosis . For this reason, there is currently great interest in the of techniques for dielectric development spectrometry; such systems will permit the rapid evaluation and display of complex permittivity over a wide range of frequency .
173
For measuring and data processing we use the stepresponse method, where the current (or time) response to a voltage step is measured, combined with Fourier transformation to obtain the required a.c. parameters.
as reaction on alone step voltage stimulus, when each successive measuring is accompained only with decrementation of V,-{;ode . Such decomposition of recording algorithm into two parts permits us to decrease the recording time, for example, by 64 times for 8-bits length of V"
2. DATA RECORDING
To present we developed the automated unit for electric parameters determination of dielectric by pulse polarization method (Karpov and Egorov, 1999). Unit works with computer by on-line mode (Fig. 2). The information interchange is performed along universal serial line. On the command from computer unit gives the stream of time moments of achieving the successive decreasing levels of voltage
A relaxation system can be defined as one whose response to a step-function stimulus ie a monotonic decrease towards zero. In the dielectric case, a voltage step applied to the specimen results in a monotonically decaying current i(t) ; the voltage across the measurement resistor also behaves monotonically. During the developing the hardware of system (see Fig. I) we took into account the following conditions: I. During the begin period the decaying speed of response is too large to record the data with highest spectrometer resolution because of computer speed; 2. At the end period of the response decaying the decreasing speed is low; and computer speed permits the data recording with spectrometer resolution ; 3. It can be measured only such number of response values, which hardware lenght of response digital code permits, for instance, if response value is transformed into I2-bits code, we can get 4096 different values of response.
control unit
spectrometer " head" standard serial line Real part of c· (w)
r---..,~
voltage t - -... commutator supply
Imaginary part 0 f c' (w)
start/sto V, code (to be measured) comparison timer
time Fig. I. Measuring unit of system. SiO after 450'C heating
Thus there is not necessity to measure absolute values of the monotonic signal u(t), but time intervals, in which u(t) achieves every possible value, may be recorded. In begin and end periods of u(t) decaying the registration algorithms are different. In the begin period the registration of each successive value is the separate measurement. It consist of reset the timer, switching off the commutator, transfer the V,-{;ode to digital-analog converter (DAC); and then, after discharging the measurement capacitor, switching on the timer and the commutator, and recording the timer state into computer memory after transition of the comparison signal into high level. In the end period successive measurements are obtained
Si02 after 1200' C heating Fig. 2. The spectromety system and results of measuring and processing.
174
by dielectric response on step voltage stimulus. The measuring error is lower than 1%.
Values .1, ak, and b* are preliminary calculated, therefore processing itself includes only simple operations (addition, multiplication, substraction). It permits to accelerate and simplify the processing. The iterative calculation technique give us the processing in real time scale and permits to interrupt the processing if necessary or if required precision of result have been achieved. The functions of frequency become more precise with recording and processing of the next Ti , and become fixed in their high frequency part. And if only high frequency part is of interest, investigation may be interrupt in desired moment.
In our case time quantum for timer is equal to W- 7 s, therefore we can investigate parameters of materials with 10 MHz resolution in frequency.
3. DA TA PROCESSING FOR ANALYSIS OF DIELECTRlCAL COMPLEX PERMITTIVITY We worked out the processing technique for dielectric permittivity. According to measured current/time response the imaginary and real parts of complex permittivity are calculated in real-time scale. The result displayed in diagram form is renewed by acquisition and processing of i(t}--data.
4. DATA PROCESSING FOR ANALYSIS OF RELAXATION TIMES As it is described above, each polarization mechanism is characterized by its own relaxation time. And different mechanisma of polarization may take place in material under investigation. Thus in general case the response of material is the sum of exponents:
For a capacitor having unity vacuum capacitance, filled with the material as dielectric, the current response i(t} to a unit voltage step is connected with complex permittivity by the Fourier transform: c· (a»
= c'( a»
= c'" +
- jc"( a»
E!( a» = c'" + E!'( a»
r
o
i(t}exp(-Jrd}dt,
(I)
j(t)
f i(t}cos(rd}dt ,
=
o
h,exp(-tl r)dr,
(3)
hr, exp( -t 1r, ) ,
(4)
o
'"
=
f
'"
or in discrete case
f i(t}sin(rd}dt. o
f(t}
In real-time scale the real and imaginary part of Fourier-image are calculated with r values of i k, and we use the iterative technique:
=
I
where Tt - relaxation time; or in another statement
f
i(t) =
g(r)r-1exp(-tlr)dr,
(5)
o where g( r) - relaxation-time distribution function :
r
+L ik[bk_i_1- (.1+ I )bk., + .1bk_i+ I]);
f
1<=2
g( r)dr
=
1.
o Then complex dielectric permittivity
r
+L h[ak-,-I - (.1+ I )ak_, + .1ak_i+I]),
co
1<=2
/(a»
where (4 = w./.1'-1 (.1 > I), IV) - maximum angular frequency, tk - time moment for ik determination (tk = tl.1*-I, to = 0, tl = I1 w.), ak = (.1- I rl(l - sin(.1k) .1-*), bk = (.1- Ir\1 - cos(.1 k» .1-*, k = ... , -1, 0, 1,2,3,
calculation of
g(r)(l+ja>rr1dr+ &(00),
(6)
If the system can be described by a unique relaxation time ro, i.e. g( r) = 0 ( r - '0), the above reduce to the Debye case: i(t)
= ro-Iexp(-tlro),
/(a»-c(oo)=(I+ja>rorl. (7)
The computer processing is based on the mathematical model, which takes into account the discrete technique of measuring. The specimen voltage response can be expressed as co
u(t)
= Vo -
I
Uk
exp( -t Ird,
(8)
k=O
where Vo - exciting step amplitude. In initial moment u(O) = 0 and there is maximum current through discharged condeser, which contains the dielectrical spesimen as filler.
T(tk), and changing i(tk) in
quadrature formula by i (t k )
f
o c(oo) = lim Re[c*(a».
For using of this formula it is necessary to know values of it = i(t) in fixed time moments tt. but i(t) is measured in arbitrary time moments Tn. Therefore the data preprocessing is necessary. The calculation of i(t*) is performed as follows : a) determination of time interval [Tn, Tn+.J, which contains tt; b) approximation of function i(t) in this interval by line segment T(t) = i(Tn) + (t - Tn)(i(Tn+1) - i(Tn»/(Tn+1 - Tn); c)
=
•
175
The value U(/) is known in each measured time Equation (8) can be rewritten as
device time resolution and precision of voltage level putting by DAC. However we have the essential advantage in recording and processing time.
I;.
N
u(/;) = Uo -
I
Uk exp( -1,I'k),
(9)
REFERENCES
k=O
because of discreteness of time intervals I; and finity of measurings number. Here i = I , ... , N - number of measuring and N - number of performed measurings.
Karpov, A.G. and Egorov, N.V. (1999). An automated dielectrometer. Instruments and Experimental Techniques, 42, No. 6, pp. 790793.
The discrete recording technique gives data precision not grater than data discreteness. We nothing know about response signal value in intervals between successive achievings the digitally set voltage levels. The voltage resolution is equal OU = Urnax1n,
(10)
where Urna• - largest output signal of DAC, n - total number of levels. The system time resolution is equal to 1O-7s, as it was said above. Therefore we can detennine relaxation times of polarization phenomena 'k only as approximation to measured times I, of successive coincidences between specimen response signal U and digitally decrement voltage level. The decisions Uk of system N
U(I;} = Uo -
L Uk exp(-I, llk} , i = I, ... , N,
(11)
k ~O
are the weights of polarization phenomena with relaxation times 'k ('k = IJ in total dielectric specimen response . The numbers of I; and Ik are equal, and I, = Ik if i = k. If some Uk < c (where c beforehand set positive value) then according polarization phenomenon with relaxation time is absent.
'k
Consider the detenninant of matrix {exp(-t;ltk)}. Because of discreteness of data recording t; = 1;0 t and tk = mko t, where I; and mk are integer. Then exp(-t;ltk) = exp(-l;Otlmkot) = exp(-l/mk)' (12) i = I, .. . , N, k = I, . .. , N. The rows and columns of such elements will be linear independent if all I; are different, and henceforth our matrix will be not special. Therefore system solution is unique, there is a unique set of relaxation times in another words . The values exp(-I;lmk) can be calculated before measuring that allows to speed up the calculation of relaxation times.
CONCLUSION Data recording technique and mathematical processing allow to determine relaxation times only as approximation to measured times t;. The precision of relaxation times detennination depends on device properties and may not be higher than
'k
176