## 3 Mathematical Model and Numerical Simulation
### Mathematical Model of the Thermal Control System  
In order to estimate the performance of the thermal control system, a mathematical model of the system was established to calculate the cooling capacity. The thermal resistance network diagram is shown in Figure 7.  
R\({}_{1}\) is the thermal resistance of the thermally conductive grease between the substrate of the heat sink and the cold side of the TEMs. \(R_{2}\) is the thermal resistance of the thermally conductive grease between the hot side of the TEMs and the bottom inside of the cabinet. \(R_{1}\) is approximately equal to \(R_{2}\). \(R_{3}\) is the thermal resistance from the bottom inside of the cabinet to the ambient environment. \(T_{1}\) is the upper surface temperature of heat sink, since the 4 LD modules and fibers are closely fixed on the heat sink and they are the main heat sources. Thus, it is important to maintain \(T_{1}\) at a suitable and stable temperature to ensure the pump laser module works reliably. \(T_{c}\) is the cold side temperature of the TEMs, \(T_{h}\) is the hot side temperature of TEMs and \(T_{a}\) is the temperature of the ambient environment. \(Q_{pump}\) is the heat generated by the pump laser module and \(Q_{c}\) is the cooling capacity of the TEM, since there are 16 pieces of TECs used in the system, and assuming each TEC haste same \(Q_{c}\), it can be deduced that the equation at the steady heat transfer state is  
\[Q_{pump}=16\times Q_{c}. \tag{8}\]  
Figure 7: The thermal resistance network diagram.  
The one-dimensional steady heat transfer equation of the thermal resistance, \(R_{1}\), can be carried out by thermal analysis as shown below:  
\[Q_{c}=\frac{T_{1}-T_{c}}{R_{1}}. \tag{9}\]  
\(Q_{h}\) is the heat release output at the hot side of TEM. Thus, we can determine the one-dimensional steady heat transfer equation:  
\[Q_{h}=\frac{T_{h}-T_{a}}{R_{2}+R_{3}}. \tag{10}\]  
From thermodynamic Equations (3), (4) and (8)-(10), the TEC cooling capacity (\(Q_{c}\)) can be written as the following expression:  
\[Q_{c}(T_{1},T_{a},I)=\frac{[{S_{m}}^{2}l^{2}(R_{2}+R_{3})-S_{m}I-K_{m}]T_{1}+K _{m}T_{a}+\frac{1}{2}l^{2}R_{m}[(2K_{m}-S_{m}I)(R_{2}+R_{3})+1]}{[S_{m}I(R_{2}+ R_{3})-1](S_{m}IR_{1}+1)-K_{m}(R_{1}+R_{2}+R_{3})}. \tag{11}\]  
From Equation (11), it can be deduced that \(Q_{c}\) mainly depends on the TEM performance parameters: \(S_{m}\), \(R_{m}\), \(K_{m}\), TEC drive current I, the system thermal resistances (\(R_{1}\), \(R_{2}\) and\(R_{3}\)), the upper surface temperature of the heat sink (\(T_{1}\)) and the ambient environment temperature (\(T_{a}\)). If it is assumed that \(T_{1}\) is 293.15 K at which the pump laser module could work steadily and normally, \(I\) and \(T_{a}\) are set to regular values, and \(S_{m}\), \(R_{m}\) and \(K_{m}\) are calculated according to the TEM parameters of \(I_{\max}\), \(\Delta T_{\max}\), \(T_{h}\) and \(V_{\max}\) (which are given by the manufacturer), then we just need to know the system's thermal resistances (\(R_{1}\), \(R_{2}\) and \(R_{3}\)) to calculate \(Q_{c}\). If \(Q_{c}\) multiplied by 16 is greater than \(Q_{pump}\), we can make the conclusion that the pump laser module can work at a suitable and stable temperature. The temperature of heat sink \(T_{1}\) can also be written as the following expression:  
\[T_{1}(Q_{c},T_{a},I)=\frac{\{[S_{m}I(R_{2}+R_{3})-1](S_{m}IR_{1}+1)-K_{m}(R_{ 1}+R_{2}+R_{3})\}Q_{c}-K_{m}T_{a}-\frac{1}{2}l^{2}R_{m}[(2K_{m}-S_{m}I)(R_{2} +R_{3})+1]}{S_{m}I^{2}(R_{2}+R_{3})-S_{m}I-K_{m}}. \tag{12}\]  
If assuming Equation (8) is established and \(Q_{pump}\) is at its maximum value, then we can obtain \(Q_{c}\) as a fixed constant, setting \(I\) and \(T_{a}\) at regular values, and calculate \(S_{m}\), \(R_{m}\) and \(K_{m}\), allowing the temperature (\(T_{1}\)) to be calculated out by the expression. Finally, it can be concluded whether the pump laser module is working at a suitable temperature by comparing \(T_{1}\) with the normal operating temperature range.