# Model-Based Analysis and Improvement of Vehicle Radiation Emissions at Low Frequency
## 3 Topological Modelling Process
### Global Network Model  
Based on the topological model introduced in Section 3.1, the algebraic function is established to calculate interferences induced on the sensitive equipment by combining all sub-systems into one global network as shown in Figure 3.  
In Figure 3, the dotted box represents the global multi-port network containing \(h\) sub-networks indexed by \(1,\cdots,h\) one by one. It should be noted that here \"Interference sources\" refers to the parts whose equivalent voltage is not zero and \"Sensitive equipment\" refers to the parts with zero equivalent voltage. The direct connection ports among different  
Figure 3: Global multi-port network. sub-networks are denoted by \(J_{\mathrm{q},i}\) and \(J_{\mathrm{p},i}\) (\(i=1,\cdots,j\)). For each multi-port network, it satisfies [10]:  
\[U_{i}=Z_{i}\cdot I_{i} \tag{1}\]  
where \(Z_{i}\) is the input impedance of the network, and \(U_{i}=\left[ \begin{array}{c}U_{i,1}\\ U_{i,2}\\ \vdots\end{array} \right]\) and \(I_{i}=\left[ \begin{array}{c}I_{i,1}\\ I_{i,2}\\ \vdots\end{array} \right]\) are the port voltage and port current, respectively. Here the Z-parameter of multi-port network is selected because it is independent of the port impedance. For each circular node, we have:  
\[V_{i}-U_{i}=Z_{\mathrm{in},i}\cdot I_{i} \tag{2}\]  
where \(V_{i}\) is the equivalent voltage of the circuit node and \(Z_{\mathrm{in},i}\) is a diagonal matrix whose diagonal elements are the port impedances. We have the following equations for each multi-port network by combing Equations (1) and (2) together:  
\[I_{i}=(Z_{i}+Z_{\mathrm{in},i})^{-1}\cdot V_{i},\quad U_{i}=Z_{i}\cdot(Z_{i}+ Z_{\mathrm{in},i})^{-1}\cdot V_{i}. \tag{3}\]  
To construct the equation of the global network, the variables in Equation (3) is rearranged according to their types, i.e., \"Interference sources\", \"Sensitive equipment\" or direct connections. The following vectors are defined to describe the global network:  
\[U=\left[ \begin{array}{c}U_{\mathrm{sen}}\\ U_{\mathrm{exc}}\\ U_{\mathrm{q}}\\ U_{\mathrm{p}}\end{array} \right],\quad\quad I=\left[ \begin{array}{c}I_{ \mathrm{sen}}\\ I_{\mathrm{exc}}\\ I_{\mathrm{q}}\\ I_{\mathrm{p}}\end{array} \right],\quad\quad V=\left[ \begin{array}{c}V_{ \mathrm{sen}}\\ V_{\mathrm{exc}}\\ V_{\mathrm{q}}\\ V_{\mathrm{p}}\end{array} \right], \tag{4}\]  
where the subscript denotes the types of ports with \"sen\" for \"Sensitive equipments\", \"exc\" for \"Interference sources\", \"q\" and \"p\" for the junction nodes \(J_{\mathrm{q}}\) and \(J_{\mathrm{p}}\), respectively. The vectors, \(U_{\mathrm{sen}},\ I_{\mathrm{sen}}\) and \(V_{\mathrm{sen}}\), are composed of \(U_{\mathrm{sen},i},\ I_{\mathrm{sen},i}\) and \(V_{\mathrm{sen},i},\ i=1,\cdots,n\) with the same order. Other vectors in Equation (4) are defined similarly. Then being similar to Equation (4), the parameter of the global network is also re-arranged and defined as:  
\[Z=\left[ \begin{array}{c}Z_{\mathrm{sen},\mathrm{sen}}\\ Z_{\mathrm{exc},\mathrm{sen}}\\ Z_{\mathrm{exc},\mathrm{exc}}\\ Z_{\mathrm{q},\mathrm{sen}}\\ Z_{\mathrm{p},\mathrm{sen}}\\ Z_{\mathrm{p},\mathrm{esc}}\\ \end{array} \right],\ Z_{\mathrm{in}}=\mathrm{diag}\big{(}\quad Z_{\mathrm{ sen},\mathrm{in}},\quad Z_{\mathrm{exc},\mathrm{in}},\quad\quad 0_{j},\quad 0_{j}\quad\big{)} \tag{5}\]  
where \(0_{j}\) denotes the \(j\)-dimensional zero matrix, \(\mathrm{diag}(. )\) denotes the diagonal matrix taking the parameter as its diagonal elements, \(Z\) and \(Z_{\mathrm{in}}\) are the input impedance and port impedance of the global network, respectively. With the parameters and variables defined by Equations (4) and (5), the following function can be obtained according to the multi-port network theory of Equation (10):  
\[U=Z\cdot\left[ \begin{array}{c}G_{\mathrm{sen}}\cdot Z+G_{\mathrm{sen}}\cdot Z _{\mathrm{in}}\\ G_{\mathrm{exc}}\cdot Z+G_{\mathrm{exc}}\cdot Z_{\mathrm{in}}\\ G_{U}\cdot Z\\ G_{I}\cdot Z\\ G_{I}\end{array} \right]^{-1}\cdot V \tag{6}\]  
where \(G_{U}=\left[