(Solved): The following figure shows an HB inverter used to inject power to the grid, which is tied to the gr ...

The following figure shows an $\mathrm{HB}$ inverter used to inject power to the grid, which is tied to the grid through an $L$ filter. The grid $v_G$ has a peak magnitude $v_{G\text{,peak }}=400\mathrm{V}$, at $60\mathrm{Hz}$, and the inductor has a magnitude $L=2\mathrm{mH}$. Assume that the inverter already operates in the steady state, and that a controller has been properly designed such that, in the steady state, the injected current $i=i^{?}$, where $i^{?}$ is the desired current reference. Obtain: a. The exact dynamical model of the inverter. Also, include the expression of $e_{AB}$ in terms of the switch positions. b. The power $P_0$ effectively injected to the g? assuming that the $DC$ source generates a pov $P_{DC}=2000\mathrm{W}$, and that the efficiency of the inverter is $97\%$. c. The expression of the current reference $i^{?}$ to inject such a $P_0$ and guaranteeing a $PF=1$. Hint: For this, consider that the current reference is defined as $i^{?}=\left(\frac{P_0}{v_{G,RMS}^2}\right)v_G$, where $P_0$ is the power to be injected to the grid, and value of the grid voltage. d. The steady state expression, as a function of time, of the average voltage drop $v_L$ in the terminals of the inductor $L$ due to the current $i=i^{?}$. Hint: use the constitutive equation of the inductor. e. The steady state expression, as a function of time, of the injected voltage $e_{AB}^{?}$ in average. Hint: notice that $e_{AB}^{?}$ is the sum of the voltage drop $v_L$ and the grid voltage $v_G$. Hint: better use the phasor concept to avoid the trigonometric identities. f. Modulation index $m_a$ considering a $v_{DC}=500\mathrm{V}$.