Mathematical Analysis

The analysis is relatively simple when the load is purely resistive. The aims of the analysis are:

i.        To obtain the average output voltage as a function of firing angle,
ii.       To obtain the rms output voltage as a function of firing angle,
iii.      To obtain the ripple factor of output voltage,
iv.      To obtain the rms line current,
v.       To obtain the fundamental component of line current,
vi.      To obtain the Displacement power factor and power factor of line current and
vii.     To obtain the total harmonic distortion(THD) in line current.

The average value of the output voltage is obtained as follows. Let the supply voltage be v_s = E*Sin (q ), where q varies from 0 to 2p radians. Since the output waveform repeats itself for every half-cycle, the average output voltage is expressed as a function of a, the firing angle, as shown in equation (1). The r.m.s. value of output voltage is obtained as shown in equation (2). The ripple factor in output voltage can defined in two ways. The definition followed in this text as follows. The maximum average output voltage occurs at a firing angle of 0^o. Let it be V_om. Then the ripple factor RF(a) is defined as shown in equation (3).

The alternate definition uses V_o,avg(a) as the denominator instead of V_om. If V_o,avg(a) is used as the denominator, then RF(a) can tend to infinity. It is more logical to express the ripple content as a fraction of the maximum average voltage. The variation of average output voltage, rms output voltage and ripple factor with the firing angle have been shown below. The plots shown below have been normalized with respect to V_om. For example, when the firing angle is 90^o, the average output is shown to be 0.5. It means that the actual average output voltage is 0.5V_om. It can also be seen that when the firing angle is 0^o, the r.m.s. output voltage is about 1.1V_om and the ripple factor is about 0.48. The ripple factor increases as the firing angle increases. It increases to 0.658 at a firing angle of 65^o and then it falls as the firing angle increases further. At the firing angle of 65^o, the r.m.s. ripple voltage in the output is 0.658V_om. For a sinusoidal source of 240 V r.m.s., the maximum r.m.s. ripple voltage works out to be 142 V.