dynamic non-linear electro-thermal simulation of a thin-film thermal converter

Dynamic non-linear electro-thermal simulation of a thin-film thermalconverter

H. Laiza, M. Klonzb,*aInstituto Nacional de Tecnologı´a Industrial, CC 157, 1650 San Martı´n, Argentina

bPhysikalisch-Technische Bundesanstalt, Bundesalle 100, 38114 Braunschweig, Germany

Abstract

This article describes the electro-thermal simulation of the ac–dc transfer differences at low frequencies of a thin-film thermal converter.The dynamic non-linear model includes the temperature dependence of all the material parameters, and the radiation losses. It is used tooptimise the performance of the device at low frequencies, where temperature oscillations are present due to the lack of integration of theoscillating Joule heat. The results of the simulation are compared with those of the measurements using a digital method.q 1999 ElsevierScience Ltd. All rights reserved.

Keywords:Electro-thermal simulation; Thin-film thermal converter; Low frequency ac–dc transfer

1. Introduction

In metrology laboratories, achieving the highest accuracyreference standards for the electrical units are reproducedand maintained by means of two quantitative experiments,the Josephson and the Quantum Hall effects [1]. Theseexperiments were, however, done using dc. To relate therms value of an ac signal to these dc units, a transfer methodhas to be used. Nowadays the highest accuracy in suchtransfers is achieved with thermal converters. In thesedevices one or more thermocouples measure the tempera-ture produced by the Joule heat in a resistor by a known dcvoltage (or current) and an unknown ac. Ideally thetemperature should be equal for the two signals of thesame rms value. Thus, the unknown rms ac value can berelated to the equivalent dc. The ac–dc voltagetransferdifferenced of a thermal converter is defined as

d � Uiac 2 Uidc

UidcuUoac� Uodc; �1�

where Uiac is the ac input voltage.Uidc is the dc inputvoltage, which when reversed produces the same meanoutput voltage asUiac. Uoac andUodc are the output voltages

of the converter with ac and dc input voltages, respectively.For an ideal thermal converterd is equal to zero.

1.1. Thin-film multijunction thermal converter

Originally, a thermal converter consisted of a thinresistive wire of a stable alloy, like Evanohm, with a ther-mocouple thermally attached to, but electrically insulatedfrom the resistor by a small glass bead, all inside an evac-uated ampoule. The highest accuracy was obtained withmultijunction thermal converters (MJTC) in which up to120 thermocouples were attached to a bifilar twisted wire.These devices are difficult to fabricate and thus costly [2].

Modern technologies, like thin-film, photolithographictechniques and micromechanics have provided the oppor-tunity to redesign the thermal transfer devices. Thin-filmtechnology allows mass fabrication and thus a dramaticreduction of the cost [3]. In the thin-film or planar multi-junction thermal converter (PMJTC), the bifilar heater andup to 100 thermocouples are sputtered on a Si3N4/SiO2/Si3N4

sandwich membrane, covering a window which is etchedinto a silicon wafer (Fig. 1).

The hot junctions of the thermocouple array are locatedalong the heater and the cold junctions are arranged sym-metrical on the silicon that acts as a heat sink. The thindielectric membrane and the thermocouple system offerlow thermal conductance, thus providing a high sensitivityof the device.

In the audio frequency range, the ac–dc transfer differ-ence is caused by the Thomson effect in the heater and

Microelectronics Journal 30 (1999) 1155–1162

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* Corresponding author. PTB Laboratory 2.32, Bundesalle 100, 38116Braunschweig, Germany. Tel.:1 49-531-592-2320; fax:1 49-531-592-2345.

E-mail addresses:[email protected] (H. Laiz), [email protected](M. Klonz)

Peltier effect in the connection pads. In the actual design, thetemperature is constant along the heater leading to a reduc-tion of the Thomson heat. Both connection pads of theheater are placed at a short distance on the silicon frameelectrically insulated by the sandwich membrane, but ther-mally short-circuited by the high thermal conductance of thesilicon underneath, leading to a reduction of the temperaturedifference due to the Peltier effect in the pads.

All types of the thermal converters show an increasingac–dc transfer differences at low frequencies. Some modelswere proposed to study the relation between the ac–dctransfer difference, lack of integration and non-linearitieslike the temperature dependence of material parametersand radiation losses in single junctions units [4,5]. We intro-duce in this article a model to study this problem in thin filmunits.

2. Electro-thermal model

Modeling of a thermal converter provides a useful tool tostudy its behavior and test potential changes in the design. Inthe development of the PTB thin-film MJTC, the analogmodels and the finite element method (FEM) have beenapplied to optimize the converter geometry, with the aimof improving its sensitivity and temperature profile to mini-mise the Thomson heat along the heater [6]. The CMOSthermal converters were modelled by Jaeggi also using theFEM [7]. In this model, Jaeggi included the Peltier effectand anisotropic characteristics in the multilayer structures.These models are, however, not capable of calculating theac–dc transfer difference of the device, because they cannot

deal with the ac inputs and do not include non-linearities inthe material parameters, which are crucial for the lowfrequency performance.

Measurement of the ac quantities with the thermalconverters implies the transformation of the electricalenergy into thermal energy by means of the Joule heatPJ

dissipated in the heater resistor. The Joule heat flowsthrough the membrane with the thermal conductanceG(sum of the conduction and radiation losses) and a thermalcapacitanceC, producing a temperature riseDT in the hotjunctions of the thermal element. By means of the Seebeckeffecta , this temperature rise is transformed into an outputvoltageUo. Thus, in steady state

Ui ! PJ � U2i

RH�T� ! DT � PJ

G�T� ! Uo � a�T�DT: �2�

This can be depicted in a simple lumped temperature-dependent parameter model (Fig. 2).

The Joule heat oscillates with a frequency that is twice thefrequency of the input voltage. At frequencies well abovethe inverse of the thermal time-constant of the converter, i.e.the time needed by the output voltage to reach 0.63 of itsfinal value after a step input voltage is applied, the heat isintegrated and the temperature of the device, and thus, theoutput voltage are constant, like in the dc.

On the contrary, if the period of the input voltage is largein comparison to the thermal time-constant, the temperatureof the whole device follows the Joule heat and the outputvoltage shows a double-frequency ripple on its dc voltage.

Due to the dependence of the electrical resistance, ther-mal conductance, and specific heat on temperature and thehigh non-linear characteristic of radiation losses, an oscil-lation in the temperature will result in the mean value of theheater temperature being different at ac and dc.

H. Laiz, M. Klonz / Microelectronics Journal 30 (1999) 1155–11621156

Fig. 1. PTB planar multijunction thermal converter.

Fig. 2. Lumped parameter model of a thermal converter.

Fig. 3. Measured ac–dc transfer differencesd at low frequenciesf of thetwo PMJTC, with the silicon obelisk underneath the heater (t � 1.3 s) andwithout obelisk (t � 0.032 s).

The temperature rise of the heater is converted into avoltage by means of the Seebeck effect which in turn isalso temperature-dependant. Consequently, the mean outputvoltage will, generally, differ when applying the dc andlow frequency ac inputs, leading to an ac–dc transferdifference.

Due to this fact, the PMJTC shows increasing ac–dctransfer differences to lower frequencies. The increase ofthe time-constant by adding some thermal mass underneaththe heater (the so-called silicon obelisk) has improved theperformance of the device at low frequencies [8]. Fig. 3shows the ac–dc transfer differences at low frequencies ofthe two PMJTC with different thermal time-constantst . Itcan be seen that increasingt shifts the high values ofd tolower frequencies.

As the main aim of this project is to improve theperformance of the device at low frequencies, wehave to find out the relation between the ac–dc transferdifferences at these frequencies and the temperaturedependence of the material parameters and heat trans-port mechanisms. To find out this relation, when thetemperature varies with the oscillating input power,one has to consider the dynamic behavior of the device.Hence, to determine the temperature distribution in thePMJTC with time, thedynamic heat diffusion equationhas to be solved under the conditions imposed on its

boundaries

7�k7T�1 g�x; y; z; t� � rc2T2t

; �3�

where the tensork is the thermal conductivityof themedium in W m21 K 21, g�x; y; z; t� the heat generationrate per unit of volume in the medium in W m23, r themedium density in kg m23 and c its specific heatin J kg21 K 21. The radiation losses from the surfacesare evaluated according to the Stefan–Boltzmann law as

q� 1msA�T41 2 T4

a�; �4�where 1m denotes the emissivity coefficient of thedifferent materials of the PMJTC,s the Stefan–Boltz-mann constant,A the surface area,T1 the temperature ofthe surface andTa the external temperature. Simulationswere performed in vacuum and in air, where heat lossesby conduction in air are included as a boundary condition.

2.1. Heat generation

In Eq. (3), g�x; y; z; t� denotes the Jouleheat generationrate per unit of volume in the heater. Due to its particularbifilar geometry, the electric current densityj is notuniformly distributed. Furthermore, the dependence oftheelectrical resistivity re with the temperature and the

H. Laiz, M. Klonz / Microelectronics Journal 30 (1999) 1155–1162 1157

Fig. 4. Geometry of the model. Due to its symmetry, only the solution in half the membrane is needed.

expected temperature differences along the heater claims forthe knowledge of the electric field distribution in order tocalculate the generated Joule heat. Consequently, theJouleheat per unit of volume in (x,y,z)should be calculated as

PJ�x; y; z� � E2

re�x; y; z;T� ; �5�

whereE is theelectric field intensitywhich is calculated asthe gradient of theelectric potential V, which in turn is thesolution of the Laplace equation under appropriate bound-ary conditions, i.e.

71

re�x; y; z;T� 7V� �

� 0: �6�

Hence, the determination of the output voltage of thePMJTK for a particulartime instant tinvolves the iterativesolution of Eqs. (3) and (6). Afterwards, the output voltagecan be calculated with theSeebeck coefficienta (T) of eachthermocouple for its particular temperature.

2.2. Calculation of the ac–dc transfer differences

For small voltage excursions, thermal converters have apotential relationship between the inputUi and outputUo

voltages of the type

Uo � kUni ; �7�

where, for ideal devices,n� 2. In the case of the PMJTC,nis between 1.97 and 2.

When we differentiate Eq. (7), and divide it byUo

dUo

Uo� n

dUi

Ui�8�

and regarding Eq. (1), we can calculate the ac–dc transferdifference for smallD � Uodc 2 Uoac as

d � Uodc 2 Uoac

nUodcuUiac � Uidc: �9�

The main purpose of the electro-thermal model is to studythe influence of the non-linear material parameters and theradiation losses in the ac–dc transfer difference. Conse-quently, no thermoelectric effects, such as Thomson andPeltier effects, were included in the model. This meansthat the output voltage will be the same with the dc inputsof both polarities. Thus, to calculated with Eq. (9), we needthe solution of the thermoelectrically coupled problem withthe ac and dc excitations as electrical boundary conditions inorder to calculateUoac andUodc.

2.3. Boundary conditions

Due to the particular symmetry of the device we can solvethe coupled problem only in half of the membrane.

Hence, we have asthermal boundary conditions(see Fig.4):

(a) Dirichlet’s bc or bc of the first kind:The external

boundaries of the membrane are

T�x;0; z; t� � Ta �10�

T�x; y5; z; t� � Ta

T�x5; y; z; t� � Ta

(b) Neumann’s bcs or bcs of the second kind:The plane ofsymmetry is

2T�x; y; z�2x

� 0 ux� 0: �11�

(c) Newton’ bcs or bcs of the third kind:The lowersurface of the membrane

2T�x; y; z�2z

� 2h�T�x; y; z; t�2 Ta� uz� 0: �12�

The upper surface of the membrane is

2T�x; y; z�2z

� 2h�T�x; y; z; t�2 Ta� �13�

uz� z1 on the membrane; z� z2 on the heater;

z� z3 on the thermopile;

where the heattransfer coefficient haccounts for theconduction through air. No convection exists due to thesmall distances between the PMJTC and its cover andbetween the PMJTC and its carrier [9]. When we studythe thermal converter in vacuum,h is taken as zero.

The electrical boundary conditions differ when we dealwith the dc or ac input voltages.

2.3.1. Solution with dc input voltageIn this case, the boundary conditions of the first kind for

the electric field problem are (see Fig. 4)

input terminal of the heater

U�x;0; z� � Udc=2; x1 , x , x2; �14�

plane of symmetry U�0; y; z� � 0; 0 , y , y5 �15�and a steady-state solution is performed for the electric (Eq.(6)) and thermal (Eq. (3)) field problems, that is the deriva-tive regarding time is taken as zero. Then,Uo is determinedfrom the temperature distribution by means of theSeebeckcoefficienta�T�.

2.3.2. Solution with ac input voltageIn this case the boundary conditions of the first kind for

the electric field problem must be time-dependent. A discre-tization of the sinusoidal input voltage is performed and asolution for the coupled electrical and thermal problem isfound in each time step. If the input voltageui is sinusoidal:

ui ��2p

U sin�vt� �16�


the voltage at time stepk is

Uik ��2p

U sin�kvDt�; �17�whereDt is the chosen time step. Consequently the bound-ary conditions of the first kind for the electrical problem forthe time instanttk � kDt, will be

input terminal of the heater

U�x;0; z� � Uik=2; x1 , x , x2; �18�

plane of symmetry U�0; y; z� � 0; 0 , y , y5:

�19�Although we are dealing with the ac input quantities, a

steady-state solution is performed for the electric field, andconsequently for the Joule heat, in each step. This meansthat no capacitive or inductive effects are included, which ismost reasonable when we limit our study to the highestfrequency of 20 Hz. For the solution of the thermal problem,a transient solution is performed for each step, this meansthe time diffusion term in Eq. (3) is included. The thermalboundary conditions remain constant for all the time steps,but the internal heat generation of the time stepk is the Jouleheat calculated in the same step and the initial temperaturedistribution of the time stepk is the temperature distribution

obtained as a solution for the stepk 2 1.

Initial bc Tibc�x; y; z; tk� � T�x; y; z; tk21�: �20�The ANSYS was used to solve the coupled problem. The

first step of this iterative process provides a solution for Eq.(6). From this, in the second step,the electric field intensityE is derived and theJoule heatdistributionPJ is calculatedusing Eq. (5). In the third step Eq. (3) is solved to obtain thetemperature distribution withPJ as theinternal heat gener-ation. Fig. 5 depicts the solutions of the three steps. Due tothe temperature dependence ofre (Eq. (6), first step), and ofc, k and theheat transfer coefficient in air h, and the non-linear characteristic of the radiation lasses (Eq. (3), thirdstep), the three steps should be iterated until an equilibriumsolution is reached.

As previously shown, the PMJTC under study consists ofa heater and a multilevel thermopile deposited on a dielec-tric sandwich membrane. We modelled the entire three-dimensional structure, but to keep the number of elementswithin reasonable limits, the thermopile and the membranewere modelled as two separate homogeneous layers with anequivalent thermal conductivity tensor, i.e. direction depen-dent [7]. Although, due to its small thickness, no significanttemperature variations are expected in thez-directions [6],the membrane and the thermopile are kept as separate layersbecause this simplifies the analysis of the influence of thematerial and dimensional changes in the design.

The Newton–Raphson procedure is used to solve the non-linear set of equations. Convergence is obtained when thesize of the Euclidean norm of the residual is less than 1023

times a reference value. Each time step usually requiresbetween two and four iterations to converge. The conver-gence criterion is based on the electrical current and the heatflow for the electrical and thermal problems, respectively.Simulations were carried out with different number of stepsper period. A good compromise between the accuracy andthe speed was found with 20 steps. An increment does notproduce a significant change in the results, but it increasesthe time required for a solution.

In Fig. 6(a) the simulated output voltage with an inputfrequency of 0.1 Hz and with dc is plotted as a function oftime for a PMJTC with a thermal time-constantt of about0.2 s. The mean value of the last cycle of the ac voltage isalso plotted. In Fig. 6(b) the same results are plotted with aninput frequency of 10 Hz, in which the effect of the thermalintegration can be observed. With ac input voltages thetimeindex kis incremented up to the mean value of theoutputvoltage Uo, for cyclen equals its mean value for cyclen 2 1.

3. Parameter adjustment and comparison of results withmeasurements

As the thermal and electrical parameters of the differentPMJTCs can differ due to tolerance and fabrication pro-cedures, an adjustment of the parameters for a particular


Fig. 5. Electric potential distribution (a), Joule heat (b), and temperaturedistribution (c). Due to its symmetry, only the solution in half themembrane is needed.

thermal converter has to be carried out in order to comparethe results of the simulations with those of the measure-ments. The sensitivity and the time-constant are used forthe adjusting procedure.

The sensitivity of the deviceSis defined as the ratio of theoutput voltageUo and its input powerPJ.

S� Uo

PJ: �21�

The temperature coefficient of the sensitivity is defined as[4]

bS� 1S

dSdT

: �22�

A low bS is important for obtaining a low dependence ofthe output voltage on the ambient temperature. This con-dition improves both the settling time and the standarddeviation during measurement. It leads to the correspondingimprovement of the standard measurement uncertainty. A

power coefficient indicates that a change in the input poweris not followed by a proportional change in the outputvoltage. As the output voltage is proportional to the inputpower, the power coefficient of the sensitivity can be definedas

wSUo � 1S

dSdUo

: �23�

To adjust the temperature coefficients of the thermalconductivities and the emissivity coefficients, we performedsteady-state simulations for the sensitivities for differentambient temperatures and input voltages in vacuum. Theywere adjusted in order to match the simulated and measured


Fig. 6. Output voltage with the dc and ac input voltages (t � 0.2 s). (a)f�0.1 Hz (b)f � 10 Hz. The differenceD between the output voltage with thedc and the mean output voltage of the last cycle in the ac is used to calculatethe ac–dc transfer difference (see Eq. (9)).

Fig. 7. MeasuredS�m� and calculated sensitivitiesS�c� as a function of thetemperature riseDT of the hot junctions at different ambient temperatures.

Fig. 8. Measured and calculated ratio of the ac voltage (peak-to-peak)Uoac

superimposed to the dc output voltageUodc at different frequenciesf.

temperature and power coefficients. In Fig. 7 comparisonsof this simulation with measurements are presented for aPMJTC with a heater resistor of 184V and a thermaltime-constant of about 7.5 s (in vacuum). These simulationswere performed using dc, this means, with steady-state sol-utions in which the right term of Eq. (3) is zero. Afterwards,the heat transfer coefficient in air was adjusted in order tomatch the simulated and measured vacuum factor and itspower dependence.

Afterwards theheat capacity�rc� of the device wasadjusted so as to approximate the amount of the ac super-imposed on the dc output voltage for a particular frequency.Fig. 8 depicts the measured and calculated percentages ofthe ac ripple.

Using these adjusted parameters for a particular PMJTC,

we performed comparisons of the calculated and measuredac–dc transfer differences for different types of the thermalconverters, in order to test the accuracy of the model. In Fig.9 the results for the two PMJTC with different time-constantst are presented.

4. The reasons for the ac–dc transfer differences at lowfrequencies

The developed model is used to investigate the causes andto search a minimization ofd at low frequencies. A goodexample for the usefulness of this method is to study theinfluence of the radiation losses. The calculations performedwith the model show that in air for a PMJTC with Si obelisk


Fig. 9. Comparison of the measured and calculated ac–dc voltage transferdifferencesd at frequenciesf. (a) PMJTC witht � 27 ms (without Siobelisk in air) (b) PMJTC witht � 7.5 s (with Si obelisk in vacuum).

Fig. 10. Influence of the radiation losses ond at low frequencies in air andin vacuum.

radiation losses represent 3.9% of the total thermal conduc-tance and, consequently, 96.1% corresponds to the conduc-tion through air and the layer materials. In vacuum, theparticipation of radiation losses increases to 21.2 and78.8% corresponds to the conduction through the layermaterials. Moreover, their influence is even greater in thetemperature coefficientbS and power coefficientqSUo of thedevice. In air the absolute value ofbS is reduced by 16.6%when radiation is avoided, and in vacuum it can even changesign. The highly non-linear characteristic of the radiationlosses explains the fact that, although they represent about4% of the heat flow in air, they have a considerable influ-ence on thebS andqSUo.

Fig. 10 depicts the calculatedd at low frequencies withand without radiation losses for a PMJTC with Si obelisk inair and in vacuum. In aird decreases about 400× 1026 at0.01 Hz. The effect is still clearer in vacuum.d is reduced tovery low values and can even reverse the sign, when radi-ation losses are eliminated. Therefore, an attempt is made toreduce the radiation losses by means of the deposition of areflecting aluminum thin film over the silicon obelisk.

5. Conclusions

We made a dynamic non-linear thermo-electrical modelfor a planar MJTC to simulate its input–output relation. Themodel has the accuracy necessary for the calculation of theac–dc transfer differences of thin-film thermal converters atlow frequencies. It is used to investigate the causes andsearch for a reduction ofd at low frequencies. This reduc-tion can be achieved by increasing the thermal time-constant or by compensation of the power dependence of

S. Some possibilities of compensating this power depen-dence are under investigation:

• reduction of radiation losses,• use of thermo-electric materials with appropriate

temperature coefficient of the Seebeck effect,• depositing an extra layer with appropriate temperature

coefficient of the thermal conductivity.

References

[1] T.J. Quinn, Base units of the systeme international d’unites, their accu-racy, dissemination and international traceability, Metrologia 31(1995) 515–527.

[2] B.D. Inglis, Standards for AC–DC transfer, Metrologia 29 (1991) 191–199.

[3] M. Klonz, T. Weimann, Accurate thin-film multijunction thermalconverter on a silicon chip, IEEE Trans. Instrum. Meas. 38 (1989)335–337.

[4] T. Takeishi, Characteristics of vacuo-thermo junctions at ultra-lowfrequency, IEEE Trans. Instrum. Meas. 34 (1) (1985) 34–41.

[5] F.L. Hermach, Thermal converters as ac–dc transfer standards forcurrent and voltage measurements at audio frequencies, J. Res. NBS48 (1952) 121–138.

[6] U. Dillner, E. Kessler, M. Klonz, C. Thomes, Micro-machined planarac–dc multijunction thermal converter: aspects of design and layermaterials, in: H. Reichl, A. Heuberger (Eds.), MICRO SYSTEM Tech-nologies 1994, VDE-Verlag GmbH, Berlin, 1994, pp. 773–782.

[7] D. Jaeggi, Thermal converters by CMOS technology, PhD Thesis,Physical Electronics Laboratory, ETH Zurich.

[8] M. Klonz, T. Weimann, Increasing the time-constant of a thin filmmultijunction thermal converter for low frequency application, IEEETrans. Instrum. Meas. 40 (2) (1991) 350–351.

[9] T. Elbel, Miniaturised thermoelectric radiation sensors covering a widerange with respect to sensitivity or time constant, Sensor Actuators A25–27 (1991) 653–656.


dynamic non-linear electro-thermal simulation of a thin-film thermal converter

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