Single Phase Inverter With AC Motor As Load

Single Phase Inverter With AC Motor As Load

Let us assume that the sinusoidal back emf of ac motor is in synchronism with inverter switching frequency and is purely sine wave without any harmonics.

Let inverter frequency f_p = 1/t_p Hz, e_o = E_o sin(wt + ε_o) = internal e.m.f of motor where Ф_o = Angle of lead with respect to V_o .

For positive cycle of V_o ,

$${{Vs} \over 2} - eo - Roio - Lo{{di} \over {dt}} = 0$$

Solution of equation(i) gives equation for i_o for positive half cycle,

These equation can be easily be solved by superposition theorem.

I_o= I_o1 + I_o2 + I_o3

Where I_o1 = Current due to D.C voltage

I_o2 = Current due to e_o

I_{o3 ­}= Transient Current

1)A steady state part associated with driving function Vs/2 and is given by

I_o1 = V_dc/(2 R_o)

2) I_o2 = A steady part associated wth driving function E_o sin(wt + ε _o) and is given by

$$\eqalign{ & {I_{o2}} = {{Eo} \over {\sqrt {{R_O} + {{(\omega L)}^2}} }}\sin (\omega t + {\varepsilon _o} - {\delta _0}) \cr & where\;{\delta _0}\;is\;given\;by \cr & {\delta _0} = ta{n^{ - 1}}{{\omega L} \over {{R_O}}} \cr}$$

3) I_o3 = Transient part given by solution of following equation $$\eqalign{ & Roio + Lo{{di} \over {dt}} = 0 \cr & Solution\;is\;{i_{03}} = {I_3}{e^{ - t./TO}} \cr & where\;To = LO/RO(time\,\;cons{\mathop{\rm tant}\nolimits} )\;and\;it\;depends\;on\;the\;initial\;condition\;of\;the\;circuit \cr & {i_{03}} = {I_3}{e^{ - \theta /tan{\delta _o}}} \cr}$$

The complete time domain equation is

I_O = I_O1 + I₀₂ + I₀₃

= I₁ - I₂ sin(wt + ε _o- δ_O) + I₃ e^-t/T0 --------(i)

The value of I₃ can be calculated at time t = 0 by assuming I_O = I_a and substituting in the above equation

I₃ = I_a - I₁ + I₂ sin( ε _o- δ_O)

The value of I₃ can be substituted on equation (i) and we get the value of Ib.

Solution for negative half cycle

I_O = -I₁ - I₂ sin(wt + ε _o- δ_O) + I₄ e^-t/T0

To calculate I₄ at pi radian, I_o = I_b

_{$${I_4} = {{Io + I1 + I2Sin(\pi + \varepsilon o - \delta o)} \over {{e^{ - \theta /tan{\delta _o}}}}}$$}