*2.3. Efficiency Calculation*

It is important to validate the two principal losses that generally occur during the operation of switching devices. These losses are classified as conduction losses and switching losses. Conduction losses (*PConduction*) are defined as the losses of power electronic devices during on-state. In this case, both the diode and the switches are taken into consideration. The instantaneous conduction losses of the IGBT and diodes can be determined using (11) and (12), respectively.

$$P\_{\text{Conduction\\_IGBT}}\left(t\right) = \left[V\_{IGBT} + R\_{IGBT}\left\dot{\imath}(t)\right] \times \dot{\imath}(t)\right] \tag{11}$$

$$P\_{\text{Conduction\\_Diode}}\left(\mathbf{t}\right) = \left[V\_{Diode} + R\_{Diode}\,\dot{\mathbf{r}}(\mathbf{t})\right] \times \dot{\mathbf{r}}(\mathbf{t})\tag{12}$$

Here, *VIGBT* is the voltage of IGBTs while *VDiode* is the voltage of the diodes when these devices are activated. Again, *RIGBT* is the corresponding resistance of IGBTs while *RDiode* represents the corresponding resistance of the diodes. The subsequent results are then summed to calculate the conduction losses of the proposed MLI.

Switching losses (*PSwitching*) are defined as the losses of power electronic modules when they are activated and deactivated. Switching loss is proportional to the switching frequency. The activation (*Won*) and deactivation (*Woff*) energy loss of these modules are obtained by:

$$\mathcal{W}\_{\rm on} = \int\_0^{t\_{\rm on}} \left[ \left( \frac{\upsilon\_{\rm switch}}{t\_{\rm on}} t \right) \left( -\frac{I}{t\_{\rm on}} (t - t\_{\rm on}) \right) \right] dt. \tag{13}$$

$$\mathcal{W}\_{\rm off} = \int\_0^{t\_{\rm off}} \left[ \left( \frac{v\_{\rm switch}}{t\_{\rm off}} t \right) \left( -\frac{I}{t\_{\rm off}} (t - t\_{\rm off}) \right) \right] dt. \tag{14}$$

Here, *ton* and *toff* are the turn-on time and turn-off time of the IGBTs consecutively. *I* represents the current through the IGBT devices before/after they are turned off/on. *v*switch demonstrates the forward voltage drop for the IGBTs. Thus,

$$P\_{switch} = f \left[ \sum\_{i=1}^{N\_{switch}} \left( \sum\_{i=1}^{N\_{out}} \mathcal{W}\_{on,i} + \sum\_{i=1}^{N\_{off}} \mathcal{W}\_{off,i} \right) \right] \tag{15}$$

where *f*, *Non* and *Noff* are the fundamental frequency and the number of times each switch is turned on and off during a time period of *t*, respectively. Again, *Won*,*<sup>i</sup>* and *Woff*,*<sup>i</sup>* are the energy loss of each switch turning on and off for *i*th time. Thus, the total loss (*PTotal*) can be calculated as follows:

$$P\_{\text{Total}} = P\_{\text{conduction}} + P\_{\text{switching}}.\tag{16}$$

The following parameters are considered for calculating the power loss of the proposed module: *VIGBT* = 2.4 V, *RIGBT* = 0.052 Ω, *VDiode* = 2 V and *RDiode* = 0.1 Ω, *f* = 50 Hz, *ton* = *toff* = 1. The proposed inverter is running under the conditions: Total DC-link voltage = 450 V, *Ma* = 1, *f* = 50 Hz, and *P*out = 3.6 kW. Again, a single-phase resistive inductive load (237 Ω-0.53H) is connected with the module as the output. The calculation is conducted for each switch and diode for one period and applying (11) to (16) the total losses (*PTotal*) of the inverter is evaluated to be 88.75 W operating for 1 s. Therefore, the proposed inverter efficiency is 97.59%. This has passed the IEEE 1547 standard for interconnected devices of power grids.
