#### Power tube switching delay and turn-on voltage drop

The previous analysis only considers the influence of the artificially added dead time on the output voltage of the bridge arm and the incoming current of the bridge arm to prevent the bridge arm from passing through. However, the turn-on and turn-off of actual power devices is not instantaneous, and there is a turn-on voltage drop during the turn-on process. Figure 1 is a schematic diagram of the output voltage of the bridge arm of the inverter when the non-ideal factors of the switching device are considered in the working stage of the positive half-cycle of the power grid under the condition of unity power factor. Among them, ton is the turn-on time of the power tube (turn-on delay and current rise time), and toff is the turn-off time (storage delay and current fall time), Usat and Ud are the conduction voltage drop of the power tube and its anti-parallel diode respectively, Ton and Toff are the actual turn-on and turn-off time of the power tube S1 respectively, Ts is the switching period, Ts=Ton+Toff. u′inv is the equivalent bridge arm output voltage considering the power tube turn-on time ton and turn-off time toff, u″inv is the equivalent bridge arm output voltage considering both the power tube switching time and the turn-on voltage drop, Δuinv is the difference between the actual output voltage and the ideal output voltage, that is, Δuinv = u″iinv-uinv. The analysis of the negative half cycle of the power grid is similar to that of the positive half cycle of the power grid, and will not be repeated here.

It should be noted that, as mentioned above, at the zero-crossing point of the modulation wave, that is, near the zero-crossing point of the grid voltage, the instantaneous duty cycle of the bridge arm output voltage is very small, and the voltage distortion caused by the non-ideal factors of the device is more obvious than the ideal bridge arm output voltage at this time, which makes the incoming current more prone to distortion. Here, the unit power factor operation of the inverter is analyzed first, and the zero-crossing distortion of the incoming current caused by the dead time and the non-ideal factors of the device is not considered, and it is considered that the current transitions smoothly at the zero-crossing point. According to the approximation theory of equivalent area, the average error voltage waveform of the output voltage of the bridge arm of the inverter in the power frequency modulation period can be obtained, as shown in Figure 2.

In Figure 2, ΔUm is the square wave amplitude of the average error voltage, and its size is shown in formula (1.1). According to the symmetry, the average voltage amplitude of the negative half cycle of the grid is -ΔUm.

In the case of non-unity power factor, the dead time td also affects the output voltage uinv of the bridge arm of the inverter. Assuming that the power factor angle is θ, that is, the phase difference between the voltage and current is θ, the square wave amplitude ΔU’m of the average error voltage at this time is shown in formula (1.2). According to the symmetry, the average voltage amplitude of the negative half cycle of the grid is -ΔU’m.

Fourier decomposition of the average error voltage shown in Figure 2 gives:

In the case of unity power factor, ΔU’m=ΔUm. It can be seen from formula (1.3) that the non-ideal characteristics of the power device not only affect the fundamental wave of the output voltage of the bridge arm, but also bring a large amount of harmonic voltage. Since the positive and negative half cycles of the photovoltaic inverter work symmetrically, the harmonic components only contain odd-numbered terms. In addition, the magnitude of the harmonic voltage is inversely proportional to the harmonic order n, so the low-order harmonics (such as the 3rd, 5th, 7th, etc.) in the output voltage of the bridge arm are relatively large. Since the low-order harmonics are difficult to be filtered out by the filter, these low-order harmonic components become the main factor causing the waveform distortion of the incoming current.