## 2 Transmission Line Modeling Method (TLM)  
TLM is a form of finite difference numerical method whose foundations are based on electromagnetic theory. TLM can be used in general to solve the following problems: Diffusion problem [29], Vibration [30], Heat transfer [31], Radar [32], Electromagnetic compatibility [33]. Transmission Line modeling is a space and time discrete method that solves field problems using their electrical circuit equivalent. TLM represents the model as a matrix of transmission lines with voltage pulses traveling through them, which is analogous to flow of heat through a device's material. For propagation in loss medium a pulse wave obeys Maxwell's curl equation on a transmission-line, which had a form to the telegrapher's equation [34]:  
\[\
abla^{2}\varnothing=2\alpha\cdot R_{d}C_{d}\frac{\partial\varnothing}{ \partial t}+\alpha\cdot L_{d}C_{d}\frac{\partial^{2}\varnothing}{\partial t^{ 2}} \tag{1}\]  
where:  
\(\Phi\): is the potential depends on (\(x\),\(y\),\(z\),\(t\)). \(R_{d}\), \(C_{d}\), \(L_{d}\): are distributed line elements; the Resistance, the Capacitance and the Inductance per unit length respectively. \(\alpha\): is the constant, may take \(\alpha=1\), 2, 3 for 1D, 2D, and 3D. Steady state and transient thermal responses of integrated circuit (IC) devices are determined by solving of the heat flow nonlinear basic equation given by [31]:  
\[\
abla^{2}T=\frac{\rho\cdot C_{p}}{K_{t}(T)}\cdot\left(\frac{\partial T}{ \partial t}\right) \tag{2}\]  
where:  
\(K_{t}(T)\): is the temperature-dependent thermal conductivity. \(C_{p}\): is a specific heat of the material. \(\rho\): is the material density. In Equation (1), when \(R_{d}=0\) (medium without loss), this equation becomes identical to a wave one. If \(R_{d}\
eq 0\) (lossy) the equation describes an attenuated wave, and therefore, Equation (2) establishes an equivalence with thermal diffusion process, if the first time derivate term in the right-hand side of the Equation (1) dominates the second time derivate term. By using TLM network to model the thermal heat flow in terms of lumped RLC network, the combinations of capacitor and inductance per unit length (\(C_{d}\), \(L_{d}\)) will be replaced by loss-free transmissions-lines, involving impedance \(Z\), which connect each node to its neighbors and carry voltage pulses between the nodes during a time step \(\Delta t\). Hence, it may be possible to model the heat transfer processes in terms of a lumped elements \(R_{d}\), \(C_{d}\) network where temperature is represented by voltage and heat by current [35]. According to the fundamental transmission line matrix theory, impedance \(Z\) is given by the following equation:  
\[Z=\sqrt{\frac{L}{C}}=\frac{3\cdot\Delta t}{C}=\frac{L}{3\cdot\Delta t} \tag{3}\]  
In TLM 3D modeling, the system is divided in small volumes elements. Each element is represented by a node that represents a unit of a line transmission. Figure 1 represents a single volume three dimensional TLM node (\(x\), \(y\), \(z\)). Each element of length \(\Delta x\), \(\Delta y\), \(\Delta z\), represents spatial discretization. Using analogy between Equations (1) and (2), the values of three-dimensional \(R\) and a common \(C\) at each node are defined by:  
\[R_{x}=\frac{\Delta x}{2\cdot K_{t}\cdot\Delta_{y}\cdot\Delta z},\ R_{y}=\frac{ \Delta y}{2\cdot K_{t}\cdot\Delta_{x}\cdot\Delta_{z}},\ R_{z}=\frac{\Delta z}{ 2\cdot K_{t}\cdot\Delta_{x}\cdot\Delta y},\ C=\rho\cdot C_{p}\cdot\Delta x \cdot\Delta y\cdot\Delta z \tag{4}\]  
As we have a cubic elemental volume, the three local distances between nodes are equidistant, that means; \(\Delta x=\Delta y=\Delta z\). The TLM is an iterative method. Injection of delta temperature pulses is to be incident simultaneously after each time step \(\Delta t\) on all parts of all nodes of the network. During \(\Delta t\), the incident temperature pulses are scattered instantaneously into reflected pulses which travel along link transmission and become new incidents upon neighboring nodes.