kozlov part 1 2002

Analytical modelling of steady-state temperature distributionin thermal microsensors using Fourier method

Part 1. Theory

Alexander G. Kozlov*

Institute of Sensor Microelectronics, Russian Academy of Sciences, Siberian Branch, Pr. Mira 55a, Omsk 644077, Russia

Received 4 December 2001; received in revised form 23 May 2002; accepted 27 June 2002

Abstract

An analytical method is presented that allows one to determine the steady-state temperature distribution in thermal microsensors based on

thermally isolated structures with arbitrary rectangular edges. The structure of thermal microsensors is treated as a 2D structure with a number

of rectangular regions which are classified into some types depending on the boundary conditions at their edges. For each type of the regions,

the equivalent parameters and heat exchange conditions are determined and the expression for temperature distribution in the region is

obtained by means of Fourier method. Heat flux densities between the regions are represented as sums of orthogonal functions with weighting

coefficients. The expressions for temperature distribution in the regions contain unknown weighting coefficients whose values are determined

from adjoint boundary conditions between all the adjacent regions. The system of equations for the weighting coefficients obtained with the

help of the adjoint boundary conditions is that of linear equations. As an example, a system of linear equations for the weighting coefficients of

thermal microsensors based on the membrane thermally isolated structure is presented.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Fourier method; Temperature distribution; Thermal microsensor; Thermally isolated structure

1. Introduction

Thermal microsensors are one class of microsensors [1].

There is a large variety of thermal microsensors:

(1) Direct thermal microsensors converting thermal mea-

surements like temperature or heat to electrical signals,

for example, thermistors, thermocouples, and thermo-

elements based on p-n junctions (diodes and transistors).

(2) Indirect thermal microsensors converting physical

quantities into thermal quantities first and then further

into electrical quantities, for example, thermal radia-

tion microsensors, thermal converters.

(3) Thermal microsensors using thermal actuation effects,

for example, thermal flow microsensors, thermal

pressure microsensors, thermal conductometric gas

microsensors, thermal accelerometer.

(4) Thermal microsensors operating at high temperatures,

for example, gas microsensors based on semiconductor

oxide thin-films and solid-state electrolyte thin-films.

(5) Thermal microsensors using several effects, for exam-

ple, pellistor-type catalytic gas microsensors which

operate at high temperatures and at the same time

convert the concentration of a combustible gas in

ambient air into a thermal quantity and then only into

an electrical quantity.

The main feature of all thermal microsensors is using a

very thin base layer or layers for arrangement of their

elements. These base layers called ‘‘thermally isolated

structures’’ are usually made with an anisotropic etching

of silicon. The thickness of the thermally isolated structure is

commensurable with the thickness of other layers of the

thermal microsensors and equal to some unit micrometers or

less. This feature of thermal microsensors leads to the

decrease of thermal losses and finally allows one to increase

their sensitivity and to decrease their power consumption.

However, the use of very thin base layers in thermal micro-

sensors complicates the design process of these microsen-

sors since, in this case, it is necessary to take into account the

parameters of all the layers. On other hand, the basic part in

the design process of the thermal microsensors is the mod-

elling of the temperature distribution in their structure. In

Sensors and Actuators A 101 (2002) 283–298

* Tel.: þ7-3812-6486-76; fax: þ7-3812-6486-76.

E-mail address: [email protected] (A.G. Kozlov).

0924-4247/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 4 - 4 2 4 7 ( 0 2 ) 0 0 2 0 9 - 1

this connection, it is important that the modelling of the

temperature distribution should be quick and correct.

Presently, for the modelling of the temperature dis-

tribution in thermal microsensors are used the following

methods:

(1) the analytical methods [2–8];

(2) the numerical methods [9–14].

Most of the analytical methods are based on the solution

of 1D heat conduction equations for the simple structures of

thermal microsensors [2–4,6–8]. In these methods, the

properties of all layers of the thermal microsensors are

taken into account by the average or equivalent values of

the corresponding parameters. These values are defined

through the parameters of each layer.

In [6], for the modelling of the temperature distribution in

thermally isolated structures is used the analytical method

that have been suggested in [15] for the thermal analysis of

multilayered parallelepiped structures. This method is based

on the Fourier method. In this method, the properties of all

layers of thermal microsensors are taken into account by

considering their structures as 3D structures and using

adjoint boundary conditions on the boundaries between

the layers. However, this analytical method can be applied

only to multilayered parallelepiped structures in which all

layers have identical dimensions (the length and the width)

and are placed on top of each other. If the thermal micro-

sensor have layers with different dimensions that are situated

in various regions of its thermally isolated structure, this

method can not be used. Furthermore, this method does not

allow one to take into account the convective and radiant

heat transfer and, when it is applied to the modelling of the

temperature distribution in the suspended structure, it is

necessary to determine the thermal impedance of the beams

that support this structure. In addition, the beams must be

narrow enough to be treated as 1D elements from the point of

view of the heat transfer.

Currently, the numerical methods (the finite-element

method [9–12], the boundary-element method [13], and

the finite-difference method [14]) are the most frequently

used to model the temperature distribution in the

thermal microsensors with arbitrary thermally isolated

structures. However, these methods are characterised

by the large amount of computations and the tedious

work of problem-defining. Furthermore, it is difficult

to use these methods in the further analytical proce-

dures, such as analytical determination of the thermal

microsensor parameters and analytical optimisation of its

structure.

The aim of this paper is to present an analytical method

of modelling the steady-state temperature distribution that

is based on the Fourier method and makes it possible to

determine the temperature distribution in 2D structures

like thermally isolated structures with arbitrary rectangular

edges.

2. Problem formulation

The thermal microsensors are usually made on thermally

isolated structures which are a thin base layer or layers and

supported by the silicon bulk substrate. The thermally

isolated structures are classified into the following types:

cantilevers, bridges, membranes, and suspended plates

(floating membranes suspended on thin and narrow beams).

The basic types of thermally isolated structures are shown in

Fig. 1. The thermal microsensors made on these thermally

isolated structures have the following features:

1. The thicknesses of all layers (base layers and element

layers) in the thermal microsensors are commensurable

and much smaller than the other dimensions of these

layers (their length and width), i.e. most of the thermal

microsensors can be considered as 2D structures.

2. The 2D thermally isolated structures on which the

elements of the thermal microsensors are situated are the

structures with arbitrary rectangular edges.

3. The elements of the thermal microsensors consist of the

layers, which are as well the structures with arbitrary

rectangular edges.

4. All the thermal microsensors have heat-generating

elements that are usually placed in the centre of the

membranes and the bridges, or on the tip of cantilevers.

The above-mentioned features of the thermal microsen-

sors allow to be proposed the following algorithm for the

analytical modelling of the temperature distribution in the

structure of these microsensors.

1. The 2D structure of a thermal microsensor is divided

into rectangular regions depending on the composition

of the layers and the heat-generating conditions. Each

region is replaced by an equivalent region with

homogeneous parameters.

2. For each region, are determined the heat exchange

conditions with the environment by means of a heat

transfer through gas medium and a radiant heat transfer,

and with the adjacent regions and the support bulk

silicon substrate by means of a thermal conduction.

3. For each region, a steady-state heat differential equation

is defined and then this equation is solved by Fourier

method. The solution of the equation is presented with

the help of eigenfunctions.

4. Parameters characterising thermal conduction processes

between all adjacent regions are determined using

adjoint boundary conditions.

Consider each step of the present algorithm separately.

3. Division of the thermal microsensors intoregions. Parameters of the regions

As mentioned in Section 2, the first step of the present

modelling of the temperature distribution in a thermal

284 A.G. Kozlov / Sensors and Actuators A 101 (2002) 283–298

microsensor is the division of its 2D structure into rectangular

regions. The necessity of a such division is due to the

possibility of the simplified writing of the analytical expres-

sion of the temperature distribution in each region through

eigenfunctions. In this connection, it is necessary (i) to

formulate requirements which have to be fulfilled for the

division of the 2D structure into rectangular regions, (ii) to

define the substitution conditions of the rectangular regions

consisting of a few layers with various parameters by equiva-

lent regions with homogeneous parameters, and (iii) to state

the analytical expression for the heat generation in the regions.

3.1. Requirements for the division of thermal

microsensors into rectangular regions

The following requirements have to be fulfilled for the

division of the 2D structure of the thermal microsensor into

regions:

(1) the edges of the region have to coincide with the edges

of the rectangular layers of the thermal microsensor;

(2) the edges of the region have to coincide with the

boundaries of the heat-generating area;

(3) on each edge of the rectangular region only one

boundary condition have to be fulfilled, i.e. on each

edge of the region the heterogeneous boundary

conditions must be absent;

(4) each edge of the rectangular region have to border with

only one adjacent region, i.e. the 2D region structure of

the thermal microsensor have to represent a rectangular

mesh, however, this mesh can be non-periodic on both

co-ordinates.

(5) the total thickness of all layers in each region have to

be much smaller than its other dimensions (length and

width), i.e. for any region, the following conditions have

to be fulfilledXn

i¼1

dðjÞi ! lj;

Xn

i¼1

dðjÞi ! bj; (1)

where dðjÞi is the thickness of a layer i in a region j, n the

number of the layers in the region j and lj, bj the length

and width of the region j, respectively.

Fig. 1. Basic types of thermally isolated structures used in thermal microsensors: (a) cantilever; (b) bridge; (c) three-arm bridge; (d) four-arm bridge; (e)

suspended plate; (f) membrane.

A.G. Kozlov / Sensors and Actuators A 101 (2002) 283–298 285

The first and the fifth requirements allow one to sub-

stitute the complicate structure of the thermal microsensor

which consists of a few layers having various dimensions

and parameters with the equivalent structure consisting of a

few rectangular regions each of which has a homogeneous

structure through the thickness and is characterised by

equivalent parameters. The fulfilment of the second

requirement allows one to use the simple analytical expres-

sion for heat generation in the regions since this heat

generation takes place all over their area. The third require-

ment allows one to write the expression for temperature

distribution in each region through the eigenfunctions.

Finally, the fourth requirement allows one to use the

adjoint boundary conditions only between two regions

lying on one boundary. As it will be shown below, the

fulfilment of this requirement simplifies determination of

the temperature distribution in the structure of the thermal

microsensor.

3.2. Equivalent regions and their parameters

The basic condition, which allows us to substitute the

rectangular region consisting of few layers with various

parameters with the equivalent region with homogeneous

parameters, is the fifth requirement (Eq. (1)). In this case, the

temperature gradient along the thickness of the region does

not have to be taken into account and one can assume that a

region has been formed from homogeneous materials with

the following equivalent parameters:

� equivalent thickness of the region j, dðjÞe

dðjÞe ¼

Xn

i¼1

kðjÞi d

ðjÞi ; (2)

� equivalent thermal conductivity of the region j, lðjÞe

lðjÞe ¼Pn

i¼1kðjÞi d

ðjÞi lðjÞi

dðjÞe

; (3)

where kðjÞi is the coefficient equal to the ratio of the total

area of a layer i in a region j to the area of this region, lðjÞi

is the thermal conductivity of the material of the layer i in

the region j. To ensure high accuracy of the modelling,

the values of coefficients kðjÞi must not be less than 0.5.

Otherwise, it is necessary to divide this region into a

number of subregions with different composition of

layers. The better variant is the variant where, for all

layers, kðjÞi is equal to 1.

Each of the regions marked out in a thermal microsensor

has to be considered in its own co-ordinate system. Further-

more, to simplify the analytical modelling of the tem-

perature distribution in the thermal microsensors, the

co-ordinate systems of all the regions must have identical

disposition of the co-ordinate axes. Therefore, we shall

situate the origin of the co-ordinates in the left lower corner

of each rectangular region. The x-axis will pass from left to

right and y-axis will pass upwards. As an example, the co-

ordinates systems for the regions of the thermal micro-

sensor based on the membrane thermally isolated structure

with one rectangular heat-generating region are shown on

Fig. 2. The boundaries of each rectangular region are

parallel to the co-ordinate axes of its own co-ordinate

system. In this connection, one can introduce the following

denotion for the region boundaries: x-boundaries, which

are parallel to x-axis, and y-boundaries, which are parallel

to y-axis. On the boundary between two adjacent regions,

the co-ordinates parallel to this boundary are identical

between themselves.

3.3. Heat generation in the regions

Some of the regions marked out in the thermal micro-

sensors are the heat-generating ones. A heat generation takes

place in an absorbing layer (thermal radiation microsensors),

a catalytic layer (catalytic gas microsensors), or a heater

layer (most thermal microsensors). These heat-generating

layers have a small thickness, which is considerably less

than the other dimensions of the layer (their length and

width), like the other layers of the thermal microsensors. In

this case, one can assume that a heat generation takes place

along the entire thickness of the heat-generating regions and

is independent of their thickness. In order to write heat

differential equations for such regions, it is necessary to state

the analytical dependence of the heat generation on the

region co-ordinates. This analytical dependence must be

adequate to the natural conditions of heat generation. For the

majority of the thermal microsensors, it is enough to use a

linear analytical dependence of the heat generation on the

region co-ordinates. In this case, the dependence of the heat

generation on co-ordinates in region j can be represented as

follows:

qðjÞ ¼ qðjÞ0 þ qðjÞ

x xj þ qðjÞy yj; (4)

where qðjÞ is the power of heat generation per unit area in a

region j, qðjÞ0 , q

ðjÞx , and q

ðjÞy are coefficients, xj and yj are co-

ordinates of the region j.

4. Heat exchange conditions of regions with theenvironment and the other regions

The second step of the present analytical modelling is

determining the heat exchange conditions of the regions

with the environment and the other regions. The basic

ways of the heat exchange for each region are the heat

transfer through the ambient air and the radiant transfer

from the lower and upper surfaces of the region and

the thermal conduction transfer through edges with the

adjacent regions and the support bulk silicon substrate.

Consider these ways of a heat exchange in the regions in

detail.


4.1. Heat transfer from the lower and upper surfaces

of the regions

The heat transfer from the lower and upper surfaces of

regions to an environment is realised by means of the

heat transfer through the ambient air and the radiant heat

transfer. The heat transfer through the ambient air from the

lower and upper surfaces of a region j leads to the heat

losses whose power per unit area (NðjÞa ) can be presented as

follows:

NðjÞa ¼ HðjÞ

a ðTj � TenÞ; (5)

where HðjÞa is the surface heat transfer coefficient, Tj is the

temperature of the region j since based on the conditions (1)

the temperature on the surface of the region is equal to tem-

perature of the region, Ten is the environment temperature.

The value of the surface heat transfer coefficient (HðjÞa ) is

controlled by the design of the thermal microsensor. If the

thermal microsensor is placed in a small package, the value

of the surface heat transfer coefficient is

HðjÞa ¼ la

1

dðjÞl

þ 1

dðjÞu

!; (6)

where la is the thermal conductivity of the ambient air, dðjÞl is

the distance between the lower surface of the region j and the

bottom of the etch pit, for thermal microsensors whose

thermally isolated structure was made with a front aniso-

tropic etching, and is the distance between the lower surface

of the region and the bottom of the package, for thermal

microsensors whose thermally isolated structure was made

with a rear anisotropic etching; dðjÞu is the distance between

the upper surface of the region j and the package cap. If the

thermal microsensor is placed in a package whose dimen-

sions are much larger than those of the thermal microsensor,

the surface heat transfer coefficient is

HðjÞa ¼ hc þ

la

dðjÞl

; (7)

Fig. 2. Membrane thermally isolated structure and its division into regions.


where hc is the convective coefficient for the given structure

of a thermal microsensor.

The radiant heat transfer from the lower and upper surfaces

of the region j leads to the heat losses whose power per

unit area (Nð jÞr ) can be presented as follows:

NðjÞr ¼ sðeðjÞl þ eðjÞu ÞðT4

j � T4enÞ; (8)

where s is Stefan–Boltzmann constant, eðjÞl and eðjÞu are the

emissivities of the lower and upper surfaces of the region j,

respectively. If a difference between the temperature of the

region j and the environment temperature is small, that is

Tj � Ten ! Ten; (9)

expression (8) can be represented as follows:

NðjÞr ¼ 4sðeðjÞl þ eðjÞu ÞT3

enðTj � TenÞ: (10)

In this case, one can combine Eqs. (5) and (10) and write

the expression for the total power of the surface heat losses

per unit area of the region j (NðjÞt ) in the following form

NðjÞt ¼ NðjÞ

a þ NðjÞr ¼ AjðTj � TenÞ; (11)

where Aj is the total surface heat transfer coefficient of the

region j equal to

Aj ¼ HðjÞa þ 4sðeðjÞl þ eðjÞu ÞT3

en: (12)

4.2. Heat transfer through the region edges

The determination of the heat transfer conditions through

the region edges is equivalent to a definition of boundary

conditions on these edges. Depending on a location one can

distinguish the following types of region edges with the

corresponding boundary conditions:

(1) region edges which coincide with the joint line of the

thermally isolated structure and the bulk silicon

substrate; in this case, the bulk silicon substrate is

one of the heat sink whose temperature is equal to the

environment temperature and, therefore, on such edges

the Dirichlet boundary condition must be fulfilled, i.e.

Tjjedge ¼ Ten; (13)

(2) region edges which are contiguous with the ambient

air; for these edges, taking into account the small

thickness of the region in comparison with its length

and width, one can use the homogeneous Neumann

boundary condition

lðjÞe

@Tj

@~n

��edge

¼ 0; (14)

where ~n is the normal to the edge;

(3) region edges which are contiguous with the edges of an

adjacent region; on such edges the adjoint boundary

condition must be fulfilled, i.e. the temperatures and

heat fluxes on the boundary between adjacent regions

(for example, between regions j and s) must be equal

Tjjedge ¼ Tsjedge; (15)

Pjjedge ¼ Psjedge; (16)

where Pj and Ps are the heat fluxes on the contiguous

edges of the regions j and s, respectively.

To write the temperature distribution in any region with

the help of eigenfunctions it is necessary to have the

Dirichlet, Neumann, or Newton boundary conditions on

the region edges. However, for the region edges, which

are contiguous with the edges of the adjacent regions, this

condition is ruled out. Consequently, the adjoint boundary

condition on such edges should be substituted by other

boundary condition, in particular, the inhomogeneous Neu-

mann boundary condition. In an inhomogeneous Neumann

boundary condition is used the value of the heat flux density

on the boundary of a region. Taking into account that the

equivalent thicknesses of adjacent regions are different it is

necessary to establish the relation between heat flux den-

sities on their boundary. For that it should be used the

equality of the heat fluxes on the boundary (second equation

of the adjoint boundary condition (Eq. (16))).

On the boundary between regions j and s (as an example

we consider x-boundary) the heat fluxes in the regions are

defined as follows:

Pðj;sÞj ¼

Z lj

0

qðj;sÞj dðjÞ

e @xj; (17)

Pðj;sÞs ¼

Z ls

0

qðj;sÞs dðsÞ

e @xs; (18)

where qðj;sÞj and q

ðj;sÞs are the heat flux densities on the conti-

guous edges of the regions j and s, respectively, lj ¼ ls is the

length of the boundary between regions j and s.

Using the second equation of the adjoint boundary con-

ditions between regions j and s (Eq. (16)), Eqs. (17) and (18)

one can write the following relation between the heat flux

densities in the regions j and s on the boundary between

these regions

qðj;sÞs ¼ kðj;sÞqðj;sÞ

j ; (19)

where

kðj;sÞ ¼ dðjÞe

dðsÞe

: (20)

The expression (19) connects the heat flux densities on the

boundary of two adjacent regions and allows us to use the

product kðj;sÞqðj;sÞ instead of the heat flux densities qðj;sÞj and

qðj;sÞs . In this case, provided s > j we shall assume that, for

regions with a smaller number (in our case this is region j),

coefficient kðj;sÞ is equal to 1 and, for regions with a bigger

number (in our case this is region s), coefficient kðj;sÞ is equal

to the ratio of the equivalent thickness of a region with a


smaller number to the equivalent thickness of a region with a

bigger number. This allows us to use a common approach to

write the temperature distribution in regions with different

numbers, since each region can be at the same time a region

with a smaller number for one adjacent region and a region

with a bigger number for the other. Furthermore, in this case,

the process of the thermal conduction transfer between two

adjacent regions is characterised by one thermal quantity

qðj;sÞ instead of two quantities qðj;sÞj and q

ðj;sÞs .

Thus, for the region edges, which are contiguous with

the edges of adjacent regions, one can use the Neumann

boundary conditions as in the case of the edges which are

contiguous with the ambient air. However, these boundary

conditions are inhomogeneous and taking into account the

above ones can be written as follows:

�lðjÞe

@Tj

@~n

��edge

¼ kðj;sÞqðj;sÞ: (21)

Any rectangular region marked out in a structure of

thermal microsensors is characterised by four boundary

conditions according to quantity of the edges. Depending

on these boundary conditions all the regions can be divided

into three basic groups (Fig. 3):

(1) A-type regions—these regions have Neumann bound-

ary conditions on all four edges;

Fig. 3. Basic types of regions.


(2) B-type regions—these regions have three Neumann

boundary conditions and one Dirichlet boundary

condition; depending on a location of the edge with

the Dirichlet boundary condition concerning the co-

ordinate system one can distinguish four variants of

B-type regions;

(3) C-type regions—these regions have two Neumann

boundary conditions and two Dirichlet boundary

conditions, moreover, the edges with the same

boundary conditions have the contact between them-

selves in one of the corners of a region; depending on a

location of the edge couple with the same boundary

conditions concerning the co-ordinate system of the

region one can distinguish four variants of C-type

regions.

Regions with other combinations of boundary conditions

(four Dirichlet boundary conditions; one Neumann bound-

ary condition and three Dirichlet boundary conditions; two

Neumann boundary conditions and two Dirichlet boundary

conditions, the boundaries with the same boundary condi-

tions to be located on the opposite edges of a region) appear

as a result of the division of the structure of thermal

microsensors extremely seldom. Therefore, these regions

will not be considered in the present analytical modelling of

the temperature distribution in thermal microsensors. How-

ever, in case of need, the present analytical model can be

used for such regions too. Furthermore, from such regions

one can obtain several regions of types B and/or C by means

of dividing them into subregions and this is the second

reason why such regions are not included into the group of

regions for which we shall determine the temperature dis-

tribution with the help of Fourier method.

Now, as an example, one can take a description of the

regions in the thermal microsensor with membrane ther-

mally isolated structure (Fig. 2). In this structure, region 0 is

A-type region, regions 1–4 are B-type regions, and regions

5–8 are C-type regions.

5. Steady-state heat differential equations for regionsand their solutions

At the third step of the present analytical modelling it is

necessary to write the steady-state heat differential equation

for each region of a thermal microsensor and to solve these

equations by means of Fourier method writing the tempera-

ture distribution in the regions through eigenfunctions.

However, depending on the boundary conditions, in the

structure of a thermal microsensor, the three basic types

of rectangular regions were marked out above. Hence, the

process of the analytical modelling of temperature distribu-

tion is simplified. It is enough to obtain expressions for the

temperature distribution in the three above-mentioned basic

types of regions and then to use the necessary for each region

of any thermal microsensor solutions defining beforehand

the type of this region and its parameters. Thus, there is no

necessity when considering new thermal microsensor to

obtain again the expressions for the temperature distribution

in its regions.

Since the three basic types of regions differ only in their

boundary conditions while they have the same heat

exchange with the environment through the lower and upper

surfaces and are characterised by the equivalent form of a

heat generation, the steady-state heat differential equations

for all the regions must be alike. The typical form of the

steady-state heat differential equation for all the regions can

be represented as follows:

� lðjÞe dðjÞe

@2Tj

@x2j

� lðjÞe dðjÞe

@2Tj

@y2j

þ AjðTj � TenÞ

¼ qðjÞ0 þ qðjÞ

x xj þ qðjÞy yj: (22)

Eq. (22) was obtained by using expression (11) for the heat

exchange with the environment through the lower and upper

surfaces of a region j and expression (4) for the heat

generation in this region. Eq. (22) can be rewritten as

follows:

@2T 0j

@x2j

þ@2T 0

j

@y2j

� p2j T 0

j ¼ jðjÞ0 þ jðjÞ

x xj þ jðjÞy yj; (23)

where

T 0j ¼ Tj � Ten; (24)

pj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi

Aj

lðjÞe dðjÞe

s; (25)

jðjÞ0 ¼ � q

ðjÞ0

lðjÞe dðjÞe

; jðjÞx ¼ � q

ðjÞx

lðjÞe dðjÞe

;

jðjÞy ¼ � q

ðjÞy

lðjÞe dðjÞe

: (26)

The solution of Eq. (23) with using the corresponding

boundary conditions for each type of regions allows us to

determine the temperature distribution in these types of

regions.

5.1. Temperature distribution in A-type regions

The structure of A-type region is shown in Fig. 4. On all

boundaries of this region the Neumann boundary conditions

are fulfilled. These conditions can be written in the following

form

lðjÞe

@T 0j

@xj

��xj¼lj

¼ �kðj;sÞqðj;sÞ; (27)

lðjÞe

@T 0j

@yj

��yj¼bj

¼ �kðj;tÞqðj;tÞ; (28)


lðjÞe

@T 0j

@xj

��xj¼0

¼ kðj;uÞqðj;uÞ; (29)

lðjÞe

@T 0j

@yj

��yj¼0

¼ kðj;vÞqðj;vÞ: (30)

The solution of the Eq. (23) for A-type region with

boundary conditions (27)–(30) can be obtained by Fourier

method and written with the help of the eigenfunctions. In

this case, in general, the steady-state temperature distri-

bution in A-type region can be represented as follows

[16]:

T 0j ðxj; yjÞ ¼ �

X1k¼0

X1m¼0

CðjÞkm

jjCðjÞkmjj

2

1

ðgðjÞkmÞ2

Z lj

0

Z bj

0

CðjÞkmðj

ðjÞ0

þ jðjÞx xj þ jðjÞ

y yjÞ dxj dyj

þX1k¼0

X1m¼0

CðjÞkm

jjCðjÞkmjj

2

1

ðgðjÞkmÞ2

Z

Lj

½YðjÞkmðxj; yjÞ�ðxj;yjÞ2Lj

dxj; (31)

where CðjÞkm are the eigenfunctions of the homogeneous

boundary value problem for A-type region, jjCðjÞkmjj

2is the

square of a norm of the eigenfunctions, ðgðjÞkmÞ2

are the

eigenvalues of the homogeneous boundary value problem

for A-type region, YðjÞkmðxj; yjÞ is the function given on

boundary contour Lj which depends on the boundary con-

ditions and the eigenfunctions of the region, dxj is the

differential of an arc length of boundary contour.

The eigenfunctions of the homogeneous boundary value

problem for A-type region are

CðjÞkm ¼ cos

kpxj

lj

cos

mpyj

bj

: (32)

The corresponding eigenvalues are

ðgðjÞkmÞ2 ¼ kp

lj

2

þ mpbj

2

þp2j : (33)

The square of a norm of the eigenfunctions depends on the

values of indices k and m and is equal to

jjCðjÞkmjj

2 ¼ ljbj; for k ¼ 0 and m ¼ 0; (34)

jjCðjÞkmjj

2 ¼ ljbj

2; for k ¼ 0 and m > 0; (35)

jjCðjÞkmjj

2 ¼ ljbj

2; for k > 0 and m ¼ 0; (36)

jjCðjÞkmjj

2 ¼ ljbj

4; for k > 0 and m > 0: (37)

The function ½YðjÞkmðxj; yjÞ� can be determined as follows:


¼ CðjÞkm

@T 0j

@~nj

��ðxj;yjÞ2Lj

; (38)

where ~n is the normal to the boundary of the region. The

values of @T 0j=@~njjðxj;yjÞ2Lj

in each point of contour Lj can be

defined by using the boundary conditions for the region.

The differential of arc length of the boundary contour is

dxj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdxjÞ2 þ ðdyjÞ2

q: (39)

In order to use expression (31) for calculating the tem-

perature distribution in the A-type region it is necessary to

transform this expression to a form, which contains only

algebraic parts. For this purpose, each component of the

expression (31) should be considered separately. First com-

ponent of expression (31) can be transformed to the follow-

ing form by using Eqs. (32)–(37) and calculating the definite

integral

�X1k¼0

X1m¼0

CðjÞkm

jjCðjÞkmjj

2

1

ðgðjÞkmÞ2

Z lj

0

Z bj

0

CðjÞkmðj

ðjÞ0 þ jðjÞ

x xj þ jðjÞy yjÞ dxj dyj

¼ FðjÞ0 � 2jðjÞ

x lj

p2

X1k¼1

ð�1Þk � 1

k2½ðkp=ljÞ2 þ p2j �

coskpxj

lj

� 2jðjÞy bj

p2

X1m¼1

ð�1Þm � 1

m2½ðmp=bjÞ2 þ p2j �

cosmpyj

bj

; (40)

where

FðjÞ0 ¼ �jðjÞ

0

p2j

� jðjÞx lj

2p2j

� jðjÞy bj

2p2j

: (41)

To transform the second component it is necessary to

determine the function YðjÞkmðxj; yjÞ on all sections of bound-

ary contour Lj. Contour Lj is shown in Fig. 4 and have four

sections (AB, BC, CO, OA) on which the boundary condi-

tions are different. In this connection, the contour integral in

Fig. 4. A-type region and heat flux densities on its edges.


second component of expression (31) must be presented as

the sum of line integrals over sectionsZLj


dxj

¼Z

AB

½YðjÞkmðxj; yjÞ�ðxj;yjÞ2AB dxj

þZ

BC

½YðjÞkmðxj; yjÞ�ðxj;yjÞ2BC dxj

þZ

CO

½YðjÞkmðxj; yjÞ�ðxj;yjÞ2CO dxj

þZ

OA

½YðjÞkmðxj; yjÞ�ðxj;yjÞ2OA dxj: (42)

Using Eq. (38) one can determine the values of the function

YðjÞkmðxj; yjÞ on each section of contour Lj. These values are

� section AB (xj ¼ lj)

½YðjÞkmðxj; yjÞ�ðxj;yjÞ2AB ¼ �ð�1Þk

lðjÞe

kðj;sÞqðj;sÞðyjÞ cosmpyj

bj

;

(43)

� section BC (yj ¼ bj)

½YðjÞkmðxj; yjÞ�ðxj;yjÞ2BC ¼ �ð�1Þm

lðjÞe

kðj;tÞqðj;tÞðxjÞ coskpxj

lj

;

(44)

� section CO (xj ¼ 0)

½YðjÞkmðxj; yjÞ�ðxj;yjÞ2CO ¼ 1

lðjÞe

kðj;uÞqðj;uÞðyjÞ cosmpyj

bj

;

(45)

� section OA (yj ¼ 0)

½YðjÞkmðxj; yjÞ�ðxj;yjÞ2OA ¼ 1

lðjÞe

kðj;vÞqðj;vÞðxjÞ coskpxj

lj

:

(46)

Using Eqs. (39) and (43)–(46) the second component of

Eq. (31) can be written as follows:

X1k¼0

X1m¼0

CðjÞkm

jjCðjÞkmjj

2

1

ðgðjÞkmÞ2

ZLj


dxj

¼ 1

lðjÞe

X1k¼0

X1m¼0

CðjÞkm

jjCðjÞkmjj

2

1

ðgðjÞkmÞ2

"� ð�1Þkkðj;sÞ

Z bj

0

qðj;sÞðyjÞ cosmpyj

bj

dyj

� ð�1Þmkðj;tÞZ lj

0

qðj;tÞðxjÞ cosmpxj

lj

dxj

þ kðj;uÞZ bj

0

qðj;uÞðyjÞ cosmpyj

bj

dyj

þ kðj;vÞZ lj

0

qðj;vÞðxjÞ cosmpxj

lj

dxj

#: (47)

In Eq. (47) unknown quantities are the heat flux densities

on the boundaries of the region with adjacent regions

qðj;sÞðyjÞ, qðj;tÞðxjÞ, qðj;uÞðyjÞ, qðj;vÞðxjÞ. These heat flux den-

sities depend only on one of the co-ordinates xj or yj since the

region has the rectangular shape and its edges are parallel to

the co-ordinates. The dependencies of the heat flux densities

on the corresponding co-ordinate can be determined from

the first expressions of adjoint boundary conditions

(Eq. (15)). However, usage of the expressions for the

temperature where the unknown quantities are parts of

the integrands in the adjoint boundary conditions requires

solution of the integral equations. This approach is very

difficult. Nevertheless, one can avoid this difficulty if the

integrals with the unknown quantities are transformed to

algebraic expressions. As an example, we shall demonstrate

this transformation on the basis of the first definite integral in

Eq. (47) that is

Z bj

0


bj

dyj; m ¼ 0; 1; . . . ;1: (48)

The function qðj;sÞðyjÞ being a part of the integrand in (48)

can be presented as a sum of orthogonal functions Wðj;sÞm0 ðyjÞwith weighting coefficients dðj;sÞm0 , which are independent of

yj-co-ordinate and correspond to the relative contribution of

each orthogonal function

qðj;sÞðyjÞ ¼X1m0¼0

dðj;sÞm0 Wðj;sÞm0 ðyjÞ: (49)

Orthogonal functions Wðj;sÞm0 ðyjÞ must satisfy the following

orthonormalization conditions

Z bj

0

Wðj;sÞm0 ðyjÞ cosmpyj

bj

dyj ¼ 1; for m0 ¼ m; (50)

Z bj

0

Wðj;sÞm0 ðyjÞ cosmpyj

bj

dyj ¼ 0; for m0 6¼ m: (51)

In this case, taking into account Eq. (49) and conditions (50)

and (51) the definite integral (48) can be represent as

follows:

Z bj

0


bj

dyj ¼ dðj;sÞm : (52)

Thus, the present procedure allows us to substitute the

definite integral (48), whose integrand is equal to the product

of the unknown function qðj;sÞðyjÞ and the eigenfunction

cosðmpyj=bjÞ (both functions depend on one co-ordinate

yj), by unknown weighting coefficient dðj;sÞm . In this case, the

unknown function qðj;sÞðyjÞ can not be found. However, if a

need of determining this function arises, it can be found by

using Eq. (49). At this point, orthogonal functions Wðj;sÞm0 ðyjÞin Eq. (49) are determined from conditions (50) and (51) and

weighting coefficients dðj;sÞm0 are determined by a method

which will be considered below.


In the same manner, one can transform the rest of the

integrals in Eq. (47)Z lj

0

qðj;tÞðxjÞ coskpxj

lj

dxj ¼ dðj;tÞk ; (53)

Z bj

0

qðj;uÞðyjÞ cosmpyj

bj

dyj ¼ dðj;uÞm ; (54)

Z lj

0

qðj;vÞðxjÞ coskpxj

lj

dxj ¼ dðj;vÞk : (55)

Finally, using Eqs. (32)–(37), (40), (47) and (52)–(55) in

(31) yields the following expression for the temperature

distribution in A-type regions

In Eq. (56) the unknown quantities are the weighting

coefficients dðj;sÞm , dðj;tÞk , dðj;uÞm , and dðj;vÞk .

5.2. Temperature distribution in B- and C-type

regions

The temperature distribution in B- and C-type regions can

be found in the analogous way as in the case of the

temperature distribution in A-type region. However, for

B- and C-type regions, due to distinctions in boundary

conditions the eigenfunctions and eigenvalues are different

from those for A-type region. The final expressions for the

temperature distributions in B- and C-type regions are given

in the Appendix A.

6. Determination of parameters characterisingprocess of thermal conduction between regions

In the previous section, for each type of regions, the

analytical expressions for the temperature distribution have

been determined. These analytical expressions contain the

unknown weighting coefficients. Therefore, the final step of

the present analytical modelling determines the values of

these coefficients. This problem can be solved by using first

equation of adjoint boundary conditions (Eq. (15)) between

adjacent regions, i.e. the temperature equality conditions on

the boundaries between adjacent regions.

As an example, consider the boundary between regions 0

and 1 in the thermal microsensor with the membrane

thermally isolated structure (Fig. 2). This boundary is y-

boundary. Region 0 is a heat-generating one and region 1 is

not a heat-generating one. The first adjoint boundary con-

dition on the boundary between regions 0 and 1 is

T0jx0¼l0¼ T1jx1¼0: (57)

Using the expressions for the temperature distributions in

region 0 and 1 in Eq. (57) yields to

T 0j ðxj; yjÞ ¼ FðjÞ

0 � 2jðjÞx lj

p2

X1k¼1

ð�1Þk � 1

k2½ðkp=ljÞ2 þ p2j �

coskpxj

lj

� 2jðjÞ

y bj

p2

X1m¼1

ð�1Þm � 1

m2½ðmp=bjÞ2 þ p2j �

cosmpyj

bj

þ�kðj;sÞdðj;sÞ0 � kðj;tÞdðj;tÞ0 þ kðj;uÞdðj;uÞ0 þ kðj;vÞdðj;vÞ0

ljbjlðjÞe p2

j

þ 2

ljbjlðjÞe

X1k¼1

�ð�1Þkkðj;sÞdðj;sÞ0 � kðj;tÞdðj;tÞk þ kðj;uÞdðj;uÞ0 þ kðj;vÞdðj;vÞk

ðkp=ljÞ2 þ p2j

coskpxj

lj

þ 2

ljbjlðjÞe

X1m¼1

�kðj;sÞdðj;sÞm � ð�1Þmkðj;tÞdðj;tÞ0 þ kðj;uÞdðj;uÞm þ kðj;vÞdðj;vÞ0

ðmp=bjÞ2 þ p2j

cosmpyj

bj

þ 4

ljbjlðjÞe

X1k¼1

X1m¼1

�ð�1Þkkðj;sÞdðj;sÞm � ð�1Þmkðj;tÞdðj;tÞk þ kðj;uÞdðj;uÞm þ kðj;vÞdðj;vÞk

ðkp=ljÞ2 þ ðmp=bjÞ2 þ p2j

coskpxj

lj

cos

mpyj

bj

: (56)

�dð0;1Þ0 � dð0;2Þ0 þ dð0;3Þ0 þ dð0;4Þ0

l0b0lð0Þe p2

0

þ 2

l0b0lð0Þe

X1k¼1

ð�1Þk½�ð�1Þkdð0;1Þ0 � dð0;2Þk þ dð0;3Þ0 þ dð0;4Þk �ðkp=l0Þ2 þ p2

0

þ 2

l0b0lð0Þe

X1m¼1

�dð0;1Þm � ð�1Þmdð0;2Þ0 þ dð0;3Þm þ dð0;4Þ0

ðmp=b0Þ2 þ p20

cosmpy0

b0

þ 4

l0b0lð0Þe

X1k¼1

X1m¼1

ð�1Þk½�ð�1Þkdð0;1Þm � �1ð Þmdð0;2Þk þ dð0;3Þm þ dð0;4Þk �ðkp=l0Þ2 þ ðmp=b0Þ2 þ p2

0

cosmpy0

b0

� 2

l1b1lð1Þe

X1k¼1

kð0;1Þdð0;1Þ0 � dð1;5Þk þ dð1;8Þk

½ðð2k � 1Þp=2l1Þ�2 þ p21

� 4

l1b1lð1Þe

X1k¼1

X1m¼1

kð0;1Þdð0;1Þm � �1ð Þmdð1;5Þk þ dð1;8Þk

½ðð2k � 1Þp=2l1Þ�2 þ ðmp=b1Þ2 þ p21

cosmpy1

b1

¼ �FðjÞ0 þ 2jð0Þ

x l0

p2

X1k¼1

1 � ð�1Þk

k2½ðkp=l0Þ2 þ p20�þ 2jð0Þ

y b0

p2

X1m¼1

ð�1Þm � 1

m2½ðmp=b0Þ2 þ p20�

cosmpy0

b0

: (58)


In Eq. (58) the quantities y0 and y1, b0 and b1 are identical,

i.e. y0 � y1 and b0 � b1. Furthermore, in this equation,

coefficients kðj;sÞ are represented in accordance with the

proposition of Section 4 (only for the heat flux densities

in regions with the greater numbers).

To determine the weighting coefficients in Eq. (58) it is

necessary (i) to pass on to finite sums in this equation (to

vary index k from 0 to s and index m from 0 to t, where s

and t are the whole numbers and have high values:

s ! 1; t ! 1), (ii) to represent this equation as a

sum of equations according to index m, and (iii) to divide

this equation into t þ 1 equations for each value of m. In

this case, the following rule should be used. The tem-

perature equality equation for y-boundary is divided into

t þ 1 equations and the temperature equality equation for

x-boundary is divided into s þ 1 equations. Dividing the

Eq. (58) into t þ 1 equations and eliminating the eigen-

functions of y0 and y1 in each obtained equation yield to

the following system of equations concerning weighting

coefficients

�dð0;1Þ0 � dð0;2Þ0 þ dð0;3Þ0 þ dð0;4Þ0

l0b0lð0Þe p2

0

þ 2

l0b0lð0Þe

Xs

k¼1

ð�1Þk½�ð�1Þkdð0;1Þ0 �dð0;2Þk þ dð0;3Þ0 þ dð0;4Þk �ðkp=l0Þ2 þ p2

0

� 2

l1b1lð1Þe

Xs

k¼1

kð0;1Þdð0;1Þ0 � dð1;5Þk þ dð1;8Þk

½ðð2k � 1Þp=2l1Þ�2 þ p21

¼ jð0Þ0

p20

þ jð0Þx l0

2p20

þ jð0Þy b0

2p20

þ 2jð0Þx l0

p2

Xs

k¼1

1 � ð�1Þk

k2½ðkp=l0Þ2 þ p20�; (59)

for m ¼ 0, and

for m ¼ 1; . . . ; t.

This system of equations is that of linear equations. Each

equation of the system contains one unknown weighting

coefficient for each heat flux density flowing through y-

boundaries (weighting coefficients with subscript ‘‘m’’) and

s þ 1 unknown weighting coefficients for each heat flux

density flowing through x-boundaries (weighting coeffi-

cients with subscript ‘‘k’’). On the contrary, for the tem-

perature equality on x-boundary, each equation of the

analogous system will contain one unknown weighting

coefficient for each heat flux density flowing through

x-boundaries (weighting coefficients with subscript ‘‘k’’)

and t þ 1 unknown weighting coefficients for each heat flux

density flowing through y-boundaries (weighting coeffi-

cients with subscript ‘‘m’’).

However, the obtained system of linear equations for one

boundary between regions in the thermal microsensor is

incomplete system concerning the weighting coefficients. To

obtain the complete system for all weighting coefficients it is

necessary, firstly, to write the temperature equality condition

for each boundary, secondly, to obtain the systems of linear

equations for each boundary (analogously for the boundary

between region 0 and 1) and, thirdly, to unite the systems of

linear equations for all boundaries into one generalised

system of linear equations. This generalised system contains

ðs þ 1ÞNx þ ðt þ 1ÞNy equations (Nx and Ny are the quan-

tities of x-boundaries and y-boundaries, respectively) and so

many unknown weighting coefficients. It is convenient to

write this generalised system in a matrix representation

MD ¼ U; (61)

where M is the matrix of the coefficients; D is the vector of

the unknown weighting coefficients; U is the vector of the

right parts.

The matrix M can be represented as the block matrix with

the submatrixes, which characterise each set of coefficients

attached to the weighting coefficients on each boundary. In

this case, one can distinguish the following types of the

submatrixes:

� diagonal submatrixes for coefficients with subscript ‘‘k’’

on x-boundaries and coefficients with subscript ‘‘m’’ on

y-boundaries;

� complete submatrixes for coefficients with subscript ‘‘k’’

on y-boundaries and coefficients with subscript ‘‘m’’ on

x-boundaries;

� zero submatrixes for coefficients attached to weighting

coefficients, which are not members of the temperature

equality equation for the given boundary.

Analogously, one can represent the vectors D and U. The

vector D contains the subvectors, each of which consists

of the weighting coefficients for the heat flux density on the

given boundary. The vector U contains the subvectors,

each of which consists of the right parts of the equations

obtained from the temperature equality equation for the

given boundary.

2

l0b0lð0Þe

�dð0;1Þm � ð�1Þmdð0;2Þ0 þ dð0;3Þm þ dð0;4Þ0

ðmp=b0Þ2 þ p20

þ 4

l0b0lð0Þe

Xs

k¼1

ð�1Þk½�ð�1Þkdð0;1Þm � ð�1Þmdð0;2Þk þ dð0;3Þm þ dð0;4Þk �ðkp=l0Þ2 þ ðmp=b0Þ2 þ p2

0

� 4

l0b0lð0Þe

Xs

k¼1

kð0;1Þdð0;1Þm � ð�1Þmdð1;5Þk þ dð1;8Þk

½ðð2k � 1Þp=2l1Þ�2 þ ðmp=b1Þ2 þ p21

¼ 2jð0Þy b0

p2

ð�1Þm � 1

m2½ðmp=b0Þ2 þ p20�; (60)


For example, for the thermal microsensor based on the

membrane thermally isolated structure shown in Fig. 2, the

components of Eq. (61) can be written using the block matrix

representation as follows:

In the designation of the submatrixes and the subvectors, the

superscript designates that the submatrix or subvector con-

cerns the boundary between the appointed regions (for

example, superscript (0, 1) designates the boundary between

regions 0 and 1). The subscript designates the dependence of

submatrix (or subvector) component on index ‘‘k’’ or ‘‘m’’.

Each row of the block matrix representation in the matrix Mcorresponds to the equation system of the temperature

equality on the appointed boundary whose location is

defined by the superscript index of submatrix being a

member of this row and placed in diagonal of the matrix

M. For example, the third row in matrix M (Eq. (62))

corresponds to the equation system for temperature equality

on the boundary between regions 0 and 3 since submatrix

Cð0;3Þm is situated in the diagonal of matrix M. Subscript ‘‘m’’

indicates that this boundary is y-boundary.

The block matrix representation is convenient to obtain

the complete system of the equations and also it can be used

to determine the temperature distribution in several regions

of the thermal microsensors.

In general, the system of the linear Eq. (61) contains a

great number of unknown weighting coefficients and equa-

tions. However, for practical application, the number of

unknown weighting coefficients and equations can be

restricted in terms of a necessary accuracy of a solution

by choosing the limited values of s and t. In this case, the

system of the linear Eq. (61) is solved by known methods.

The weighting coefficients found as a result of this solution

allow one to determine finally the temperature distribution in

regions of the thermal microsensor.

7. Conclusion

In this paper, the method of modelling the steady-state

temperature distribution in 2D structure like thermal micro-

M ¼

Að0;1Þm A

ð0;2Þk Að0;3Þ

m Að0;4Þk A

ð1;5Þk A

ð1;8Þk 0 0 0 0 0 0

Bð0;1Þm B

ð0;2Þk Bð0;3Þ

m Bð0;4Þk 0 0 Bð2;5Þ

m Bð2;6Þm 0 0 0 0

Cð0;1Þm C

ð0;2Þk Cð0;3Þ

m Cð0;4Þk 0 0 0 0 C

ð3;6Þk C

ð3;7Þk 0 0

Dð0;1Þm D

ð0;2Þk Dð0;3Þ

m Dð0;4Þk 0 0 0 0 0 0 Dð4;7Þ

m Dð4;8Þm

Eð0;1Þm 0 0 0 E

ð1;5Þk E

ð1;8Þk Eð2;5Þ

m 0 0 0 0 0

Fð0;1Þm 0 0 0 F

ð1;5Þk F

ð1;8Þk 0 0 0 0 0 Fð4;8Þ

m

0 Gð0;2Þk 0 0 G

ð1;5Þk 0 Gð2;5Þ

m Gð2;6Þm 0 0 0 0

0 Hð0;2Þk 0 0 0 0 Hð2;5Þ

m Hð2;6Þm H

ð3;6Þk 0 0 0

0 0 Ið0;3Þm 0 0 0 0 Ið2;6Þm Ið3;6Þk I

ð3;7Þk 0 0

0 0 Jð0;3Þm 0 0 0 0 0 J

ð3;6Þk J

ð3;7Þk J

ð4;7Þm 0

0 0 0 Kð0;4Þk 0 0 0 0 0 K

ð3;7Þk Kð4;7Þ

m Kð4;8Þm

0 0 0 Lð0;4Þk 0 L

ð1;8Þk 0 0 0 0 Lð4;7Þ

m Lð4;8Þm

266666666666666666666666666664

377777777777777777777777777775

;

D ¼

Dð0;1Þm

Dð0;2Þk

Dð0;3Þm

Dð0;4Þk

Dð1;5Þk

Dð1;8Þk

Dð2;5Þm

Dð2;6Þm

Dð3;6Þk

Dð3;7Þk

Dð4;7Þm

Dð4;8Þm

26666666666666666666666666664

37777777777777777777777777775

; U ¼

Uð0;1Þm

Uð0;2Þk

Uð0;3Þm

Uð0;4Þk

0

0

0

0

0

0

0

0

266666666666666666666666664

377777777777777777777777775

: (62)


sensors is presented. The method is based upon division of

the realistic 2D structure into rectangular regions whose

structures are substituted by equivalent structures with

homogeneous parameters. For each rectangular region,

the analytical expression of the temperature distribution is

determined using Fourier method. In addition, each heat flux

density between adjacent regions is defined as the sum of

orthogonal functions with unknown weighting coefficients.

In order to find the unknown weighting coefficients are used

the adjoint boundary conditions on the boundaries between

the adjacent regions. In general, the determination of the

weighting coefficients is reduced to solving a system of

linear equations.

In part 2 of this paper, the practical application of the

present method for modelling the temperature distribution in

a number of thermal microsensors will be considered.

However, at this part the following remarks should be made

concerning the present method.

1. Stating this method, we used the linear law for the heat

generation in regions. However, one can use more

complicate laws for the heat generation. In this case, the

basic condition is the ability of determining the definite

integral, which takes into account the heat generation in

an analytical expression for temperature distribution (for

example, the first term in the right part of Eq. (31)).

2. In the present analytical modelling, we considered the

three basic types of regions in 2D structure. However,

this method can be used in 2D structures with other

types of regions.

3. In a number of cases, the present method of modelling

the temperature distribution in 2D structure can

substitute numerical methods. However, to determine

the advantages of this method over numerical methods it

is necessary to carry out comparative investigations on a

number of the parameters (accuracy, rapidity and other).

For the present, the basic advantage of this method is a

possibility to obtain the analytical expressions for the

temperature distribution in thermal microsensors that

can be used in other applications (for example, the

computer aided design and the analytical optimisation of

thermal microsensors).

Appendix A

A.1. Temperature distribution in B-type regions

� B1-type region:

T 0j ðxj; yjÞ ¼ 2

X1k¼1

1

½ðð2k � 1Þp=2ljÞ�2 þ p2j

2jðjÞ0

ð2k � 1Þpþ jðjÞy bj

ð2k � 1Þp

!ð�1Þk þ 2jðjÞ

x lj2

½ð2k � 1Þp�2þ ð�1Þk

ð2k � 1Þp

!" #

cosð2k � 1Þpxj

2lj

� �þ 8jðjÞ

y bj

X1k¼1

X1m¼1

ð�1Þk½ð�1Þm � 1�f½ðð2k � 1Þp=2ljÞ�2 þ ðmp=bjÞ2 þ p2

j gð2k � 1Þm2p3

cosð2k � 1Þpxj

2lj

� �cos

mpyj

bj

þ 2

ljbjlðjÞe

X1k¼1

�kðj;tÞdðj;tÞk þ kðj;uÞdðj;uÞ0 þ kðj;vÞdðj;vÞk

½ðð2k � 1Þp=2ljÞ�2 þ p2j

cosð2k � 1Þpxj

2lj

� �

þ 4

ljbjlðjÞe

X1k¼1

X1m¼1

�ð�1Þmkðj;tÞdðj;tÞk þ kðj;uÞdðj;uÞm þ kðj;vÞdðj;vÞk

½ð2k � 1Þp=2lj�2 þ ðmp=bjÞ2 þ p2j

cosð2k � 1Þpxj

2lj

� �cos

mpyj

bj

:

� B2-type region:

T 0j ðxj; yjÞ ¼ 2

X1m¼1

1

½ð2m � 1Þp=2bj�2 þ p2j

2jðjÞ0

ð2m � 1Þpþ jðjÞx lj

ð2m � 1Þp

!ð�1Þmþ

"2jðjÞ

y bj

2

½ð2m � 1Þp�2þ ð�1Þm

ð2m � 1Þp

!#

cosð2m � 1Þpyj

2bj

� �þ 8jðjÞ

x ljX1k¼1

X1m¼1

ð�1Þm½ð�1Þk � 1�ððkp=ljÞ2 þ ½ð2m � 1Þp=2bj�2 þ p2

j Þð2m � 1Þk2p3

coskpxj

lj

cos

ð2m � 1Þpyj

2bj

� �þ 2

ljbjlðjÞe

X1m¼1

�kðj;sÞdðj;sÞm þ kðj;uÞdðj;uÞm þ kðj;vÞdðj;vÞ0


cosð2m � 1Þpyj

2bj

� �

þ 4

ljbjlðjÞe

X1k¼1

X1m¼1

�ð�1Þkkðj;sÞdðj;sÞm þ kðj;uÞdðj;uÞm þ kðj;vÞdðj;vÞk

ðkp=ljÞ2 þ ½ð2m � 1Þp=2bj�2 þ p2j

coskpxj

lj

cos

ð2m � 1Þpyj

2bj

� �:


� B3-type region:

T 0j ðxj; yjÞ ¼ �2

X1k¼1

1

½ð2k � 1Þp=2lj�2 þ p2j

2jðjÞ0

ð2k � 1Þpþ jðjÞy bj

ð2k � 1Þp� 4ð�1ÞkjðjÞx lj

½ð2k � 1Þp�2

( )sin

ð2k � 1Þpxj

2lj

� �

� 8jðjÞy bj

X1k¼1

X1m¼1

ð�1Þm � 1

f½ð2k � 1Þp=2lj�2 þ ðmp=bjÞ2 þ p2j gð2k � 1Þm2p3

sinð2k � 1Þpxj

2lj

� �cos

mpyj

bj

þ 2

ljbjlðjÞe

X1k¼1

ð�1Þkkðj;sÞdðj;sÞ0 � kðj;tÞdðj;tÞk þ kðj;vÞdðj;vÞk

½ð2k � 1Þp=2lj�2 þ p2j

sinð2k � 1Þpxj

2lj

� �

þ 4

ljbjlðjÞe

X1k¼1

X1m¼1

ð�1Þkkðj;sÞdðj;sÞm � ð�1Þmkðj;tÞdðj;tÞk þ kðj;vÞdðj;vÞk

½ð2k � 1Þp=2lj�2 þ ðmp=bjÞ2 þ p2j

sinð2k � 1Þpxj

2lj

� �cos

mpyj

bj

:

� B4-type region:


X1m¼1

1


2jðjÞ0

ð2m � 1Þpþ jðjÞx lj

ð2m � 1Þp� 4ð�1ÞmjðjÞy bj

½ð2m � 1Þp�2

( )sin

ð2m � 1Þpyj

2bj

� �

� 8jðjÞx lj

X1k¼1

X1m¼1

ð�1Þk � 1

fðkp=ljÞ2 þ ½ð2m � 1Þp=2bj�2 þ p2j gð2m � 1Þk2p3

coskpxj

lj

sin

ð2m � 1Þpyj

2bj

� �

þ 2

ljbjlðjÞe

X1m¼1

�kðj;sÞdðj;sÞm þ ð�1Þmkðj;tÞdðj;tÞ0 þ kðj;uÞdðj;uÞm


sinð2m � 1Þpyj

2bj

� �

þ 4

ljbjlðjÞe

X1k¼1

X1m¼1

�ð�1Þkkðj;sÞdðj;sÞm þ ð�1Þmkðj;tÞdðj;tÞk þ kðj;uÞdðj;uÞm

ðkp=ljÞ2 þ ½ð2m � 1Þp=2bj�2 þ p2j

coskpxj

lj

sin

ð2m � 1Þpyj

2bj

� �:

A.2. Temperature distribution in C-type regions

� C1-type region:


X1k¼1

X1m¼1

jðjÞ0 ð�1Þkð�1Þm þ jðjÞ

x ljð�1Þm½ð�1Þk þ 2=ð2k � 1Þp� þ jðjÞy bjð�1Þk½ð�1Þm þ 2=ð2m � 1Þp�

f½ð2k � 1Þp=2lj�2 þ ½ð2m � 1Þp=2bj�2 þ p2j gð2k � 1Þð2m � 1Þp2

cosð2k � 1Þpxj

2lj

� �cos

ð2m � 1Þpyj

2bj

� �þ 4

ljbjlðjÞe

X1k¼1

X1m¼1

kðj;uÞdðj;uÞm þ kðj;vÞdðj;vÞk

½ð2k � 1Þp=2lj�2 þ ½ð2m � 1Þp=2bj�2 þ p2j

cosð2k � 1Þp

2lj

� �cos

ð2m � 1Þpyj

2bj

� �:

� C2-type region:


X1k¼1

X1m¼1

�jðjÞ0 ð�1Þm þ jðjÞ

x ljð�1Þkð�1Þmð2=ð2k � 1ÞpÞ � jðjÞy bj½ð�1Þm þ ð2=ð2m � 1ÞpÞ�


sinð2k � 1Þpxj

2lj

� �cos

ð2m � 1Þpyj

2bj

� �þ 4

ljbjlðjÞe

X1k¼1

X1m¼1

ð�1Þkkðj;sÞdðj;sÞm þ kðj;vÞdðj;vÞk


sinð2k � 1Þp

2lj

� �cos

ð2m � 1Þpyj

2bj

� �:

� C3-type region:


X1k¼1

X1m¼1

jðjÞ0 � jðjÞ

x ljð�1Þkð2=ð2k � 1ÞpÞ � jðjÞy bjð�1Þmð2=ð2m � 1ÞpÞ


sinð2k � 1Þpxj

2lj

� �

sinð2m � 1Þpyj

2bj

� �þ 4

ljbjlðjÞe

X1k¼1

X1m¼1

ð�1Þkkðj;sÞdðj;sÞm þ ð�1Þmkðj;tÞdðj;tÞk


sinð2k � 1Þp

2lj

� �

sinð2m � 1Þpyj

2bj

� �:


References

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CMOS microsensors, Proc. IEEE 86 (1998) 1660–1678.

[2] T. Elbel, R. Lenggenhager, H. Baltes, Model of thermoelectric

radiation sensors made by CMOS and micromachinihg, Sens.

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(1993) 65–71.

[4] A.U. Dillner, Thermal modeling of multilayer membranes for sensor

applications, Sens. Actuators A 41/42 (1994) 260–267.

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considerations of CMOS compatible IR thermoelectric sensors, Sens.

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tool for ICs, microsystem elements and MCMs: the mS-THERMA-

NAL, Microelectron. J. 29 (1998) 241–255.

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chanical thermoelectric cooler, Sens. Actuators A 75 (1999) 95–

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sensors with comb thermoelectric transducer, Sens. Actuators A 75

(1999) 139–150.

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Actuators 12 (1987) 303–312.

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chem. Soc. 134 (1987) 2937–2940.

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consumption thermal gas-flow sensor based on thermopiles of

highly effective thermoelectric materials, Sens. Actuators A 60

(1997) 1–4.

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a thin-film thermal converter, Microelectron. J. 30 (1999) 1155–

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Biography

Alexander G. Kozlov was born in Omsk, Russia, in 1955. He received the

diploma of an engineer in radioelectronics from the Technical Institute of

Omsk in 1977 and the PhD degree in microelectronics from the

Electrotechnical Institute of St. Petersburg, Russia in 1984. From 1977

to 1981 and from 1985 to 1991 he was with the Technical Institute of

Omsk working in consecutive order assistant lecturer, lecturer. From 1981

to 1984 he was doctoral student in the Electrotechnical Institute of St.

Petersburg, Russia. Since 1992, he has been with the Institute of Sensor

Microelectronics RAS where he is presently a head of laboratory. From

1993 to 1998, he was a visiting associate professor at Omsk State

Technical University. Since 2000, he is a visiting associate professor at

Omsk State University. His current research interests are solid state gas

sensors, simulation and optimisation of semiconductor and thin-film

microsensors.

� C4-type region:


X1k¼1

X1m¼1

�jðjÞ0 ð�1Þk � jðjÞ

x lj½ð�1Þk þ ð2=ð2k � 1ÞpÞ� þ jðjÞy bjð�1Þkð�1Þmð2=ð2m � 1ÞpÞ


cosð2k � 1Þpxj

2lj

� �sin

ð2m � 1Þpyj

2bj

� �þ 4

ljbjlðjÞe

X1k¼1

X1m¼1

ð�1Þmkðj;tÞdðj;tÞk þ kðj;uÞdðj;uÞm


cosð2k � 1Þp

2lj

� �sin

ð2m � 1Þpyj

2bj

� �:


kozlov part 1 2002

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