A phase controlled single phase rectifier, supplied by an AC source, feeds power to an R-L-E load as shown in the figure. The rectifier output voltage has an average value given by \({V_O} = \frac{{\;Vm}}{{2\pi }}\;\left( {3 + cos\;\alpha } \right).\) Where 𝑉_𝑚= 80𝜋 volts and 𝛼 is the firing angle. If the power delivered to the lossless battery is 1600 W, 𝛼 in degree is________ (up to 2 decimal places).

Concept:

The average output voltage of a single-phase semi-converter is given by

\({V_0} = \frac{{{V_m}}}{2\pi }\left( {3 + \cos \alpha } \right)\)

Vm is the maximum value of supply voltage

α is the firing angle

Calculation:

Average output voltage,

\({V_o} = \frac{{{V_m}}}{{2\pi }}\left( {3 + \cos \alpha } \right)\)

Power transfer to the 80 V battery,

P = E I_o = 1600 W

\(\Rightarrow {I_O} = \frac{{1600}}{{80}} = 20A\)

V_m = 80 π

\({V_o} = \frac{{{V_m}}}{{2\pi }}\left( {3 + cos\alpha } \right)\)

\(= \frac{{80\pi }}{{2\pi }}\left( {3 + cos\alpha } \right)\)

The relation between Vo and E is given by,   

V_o = E + I_oR

⇒ 120 + 40 cos α = 80 + 20 (2)