# Thermal Mapping of Power Semiconductors in H-Bridge Circuit +
## 3 Loss Dissipation and Junction Temperature of Power Devices
### Loss Dissipation  
As the power loss is symmetrically distributed among the upper and lower switches in the two-level power converter, the upper IGBT \(T_{1}\) and lower diode \(D_{4}\) are selected as an example. Under the steady-state operation, the loading current flowing through the upper IGBT \(T_{1}\) and lower diode \(D_{4}\) is described in Figure 3. It is noted that \(I_{lo\_T}\) and \(I_{lo\_D}\) indicate the loading current through the IGBT and diode, \(P_{con\_T}\) and \(P_{con\_D}\) indicate the power loss of the IGBT and diode, and \(E_{sw\_T}\) and \(E_{sw\_D}\) indicate the switching energy for the IGBT and diode, respectively. Due to the direction of the loading current, both the IGBT and the diode conducts only half of the fundamental period \(T_{f}\). Within a switching period \(T_{sw}\), as the loading current commutes from \(D_{4}\) to \(T_{1}\) once, the conduction losses and switching losses are generated in both the IGBT and the diode. Due to the fact that the sinusoidal loading current induces different dissipations in every switching period, the average power loss can be calculated, which is accumulated from each switching pattern during a fundamental period. The conduction loss of the IGBT \(P_{con\_T}\) can be calculated as,  
\[P_{con\_T}=\frac{1}{T_{f}}\int_{0}^{\frac{T_{f}}{2}}V_{CE}(\mathbf{t})\cdot i _{lo}(\mathbf{t})\cdot d(\mathbf{t})dt \tag{2}\]  
where \(T_{f}\) denotes the fundamental period, \(V_{CE}\) denotes the on-state voltage of the IGBT, \(i_{lo}\) denotes the loading current, and \(d\) denotes the duty cycle. Considering the displacement angle between the loading current and modulated voltage \(\varphi_{u}\), and making a linear approximation of the IGBT forward characteristics (initial on-state voltage drop \(V_{CE0}\) and equivalent resistor \(r_{CE}\)), the conduction loss of the IGBT can be deduced by [19],  
\[P_{con\_T}=\frac{1}{2}\bigg{(}\frac{1}{\pi}V_{CE0}I_{m}+\frac{1}{4}r_{CE}\cdot I _{m}{}^{2}\bigg{)}+M\cos\varphi_{u}\bigg{(}\frac{1}{8}V_{CE0}I_{m}+\frac{1}{3 \pi}r_{CE}\cdot I_{m}{}^{2}\bigg{)} \tag{3}\]  
where \(I_{m}\) denotes the amplitude of the loading current, and \(M\) denotes the modulation index, which equals the converter output voltage over half of the dc-bus voltage. Similarly, the conduction loss of the diode \(P_{con\_D}\) can be calculated as,  
\[P_{con\_D}=\frac{1}{T_{f}}\int_{0}^{\frac{T_{f}}{2}}V_{F}(\mathbf{t})\cdot i _{lo}(\mathbf{t})\cdot(1-d(\mathbf{t}))dt \tag{4}\]  
where \(V_{F}\) denotes the on-state voltage of the freewheeling diode. It can be noted that the lower diode conducts if its driving signal is high. Assuming the linear approximation of the diode forward characteristics (initial on-state voltage drop \(V_{F0}\) and equivalent resistor \(r_{F}\)), the conduction loss of the diode can be deduced by,  
\[P_{con\_D}=\frac{1}{2}\bigg{(}\frac{1}{\pi}V_{F0}I_{m}+\frac{1}{4}r_{F}\cdot I _{m}{}^{2}\bigg{)}-M\cos\varphi_{u}\bigg{(}\frac{1}{8}V_{F0}I_{m}+\frac{1}{3 \pi}r_{F}\cdot I_{m}{}^{2}\bigg{)} \tag{5}\]As the turn-on energies \(E_{on}\) and turn-off energies \(E_{off}\) of the IGBT can be described as a polynomial function [20], where \(a_{T}\) denotes the zero-polynomial term, \(b_{T}\) denotes the linear term, and \(c_{T}\) denotes the quadratic term,  
\[E_{sw\_T}(\mathbf{t})=E_{on}(\mathbf{t})+E_{off}(\mathbf{t})=\frac{V_{dc}}{{V_ {dc}}^{*}}(a_{T}+b_{T}I_{m}(\mathbf{t})+c_{T}{I_{m}}^{2}(\mathbf{t})) \tag{6}\]  
the switching loss \(P_{swT}\) can be calculated within a fundamental period,  
\[P_{sw\_T}=\frac{1}{T_{f}}\sum E_{swT}(\mathbf{i})=f_{sw}\frac{V_{dc}}{V_{dc} ^{*}}\bigg{(}\frac{a_{T}}{2}+\frac{b_{T}}{\pi}I_{m}+\frac{c_{T}}{4}{I_{m}}^{2} \bigg{)} \tag{7}\]  
where \(f_{sw}\) denotes the switching frequency, \(V_{dc}\) denotes dc-bus voltage, and \(V_{dc}^{*}\) denotes the dc-bus voltage under the testing condition.