Separation of Variables Method

The heat equation is a partial differential equation (PDE) which can be solved by the classical method called the separation of variables. The separation of variables method represents one of the most powerful and most used analytical techniques for solving a variety of PDEs. The following approach was first proposed by Fourier in his classical work Thorie analytique de la chaleur (1822; The Analytical Theory of Heat) [81].

Assume the solution can be separated, i.e.:

T(x, t) =X(x)T(t). (3.17)

Begin by substituting (3.17) into (3.14), giving:

∂

∂t[X(x)T(t)] =α ∂2

∂x2 [X(x)T(t)]. (3.18)

Separate the equation so that the one side depends only on t, while the other depends only on x. Both sides must be equal to a constant (λ) since one side depends only on t

and the other only on x, so the equation becomes:

T0_(t)

αT(t) =

X00_(x)

X(x) =−λ. (3.19)

The minus sign appears for convenience. The equation in (3.19) contains a pair of separate ordinary differential equations (ODEs). The first ODE to solve is:

3.4 Derivation of the Heat Conduction Equation

This is straightforward to solve, rearranging and integrating leads to:

Z 1 T(t)dT(t) = Z −λαdt. (3.21) Therefore: T(t) = CCe−λαt, (3.22)

and using the law of indices:

T(t) =CCeα_e−λt_{. (3.23)}

As a constant multiplied by a constant generates another constant, the expression in Equation (3.23) can simply be written as:

T(t) =CCe−λt_{. (3.24)}

Eigenvalues and eigenfunctions must be found for the second ODE which is:

X00_{(x) +λX(x) = 0. (3.25)}

When λ <0, there exists real numbers D, E such that:

X0_{(x) =}√_−λDe√−λx₋√_−λEe−√−λx_{. (3.26)}

3.4 Derivation of the Heat Conduction Equation

is identically 0. Also, suppose thatλ = 0, then there exists real numbersD,E such that:

X(x) =Dx+E. (3.27)

From (3.15), in the same manner as in the previous case, it can be concluded that that

D= 0. So every constant function X0(x) = E is an eigenvalue λ0 = 0.

When λ >0 there exists real numbers D,E such that:

X0(x) = −D√λsin(√λx) +E√λcos(√λx). (3.28)

The boundary condition X0_{(0) = 0 means E = 0, and the boundary condition X}0_{(l) = 0}

provides the expression:

Dsin(√λl) = 0 (3.29)

To avoid having a trivial solution, √λl must be equal to nπ. Therefore, the eigenvalues and eigenfunctions are:

λn= ³_nπ l ´₂ n = 0,1,2...., (3.30) and Xn(x) = cos ³_nπx l ´ n = 0,1,2... (3.31)

Notice the result fromλ= 0 case is incorporated into this solution. The resulting functions associated with (3.24) are:

3.4 Derivation of the Heat Conduction Equation

The linear combination of (3.31) and (3.32), written as the formal infinite sum for any integer N and constants {CCn}Nn=0, is:

Tn(x, t) = 1 2CC0+ ∞ X n=1 CCne( nπ l ) 2 t_cos³nπx l ´ , n = 0,1,2... (3.33)

The initial condition (3.16) means the following must be satisfied:

T(x,0) =f(x) = 1 2CC0+ ∞ X n=1 CCncos ³_nπx l ´ , n = 0,1,2... (3.34)

This is nothing more than a Fourier cosine series expansion of the function f(x) over the interval (0, l). Fourier’s successful approach was to substitute the expression for {CCn}Nn=0, from the Fourier series definition, into (3.33). This gave rise to a solution to

1D heat equation in terms ofT(x, t). Prior to Fourier’s work there was no known solution to the heat equation.

The problem with the customary separation of variables technique is that it can only provide a solution to the 1-D heat equation, in terms of T(x, t), for known analytic time-varying boundary conditions. For non-analytic time-varying boundary conditions the traditional separation of variables approach is inappropriate for the task. It merely reduces the 1D heat equation into a set of two ODEs, one ODE in terms of T(x) and the other in terms of T(t), which can not be solved. Therefore it is not worth incorporating the customary separation of variable technique into a thermal model to attempt to fulfil the requirements of this work. Non-analytic time-varying boundary conditions, in the form of heat fluxes, must exist to accurately simulate the material interfaces which are present in the layered structure of a power module.

Other existing thermal models should be examined to see if there are any that can fulfill all of aims set out in the motivation section of this thesis.

Chapter

4

Thermal Models for Power Converters

This chapter introduces existing thermal models which are frequently employed to cal- culate the temperature of power semiconductor devices used in power converters. The models are divided into separate categories; compact thermal models, dynamic compact thermal models, numerical methods and analytic solutions. The theory behind each of the thermal models and their mode of operation is considered. At the end of the chapter is a detailed discussion which examines the advantages and disadvantages of existing models. A suitable way to proceed, considering the motivation for this thesis, is then stated.

4.1

Steady State Compact Thermal Models

Compact thermal models (CTMs) provide a simple quantitative description of a modelled packaging structure. CTMs provide an abstract description of power module packaging behaviour when construction details are too detailed to be of use at the desired level of analysis. CTMs simulate quickly when compared with detailed thermal models. A requirement for a CTM is to be reasonably boundary condition independent so that the variation of the environment does not affect the compact thermal model [82].

4.1 Steady State Compact Thermal Models