Mathematical Analysis

I. BASED ON THE ASSUMPTION THAT LOAD INDUCTANCE IS INFINITE

When the load inductance is infinite, we can assume that the load current is continuous and steady without ripple. Let the firing angle be a. Let the commutation overlap period last from wt = a till wt = b.

During the period, a < wt < b, the output voltage is zero because all the SCRs are in conduction. If the SCRs are ideal, the drop across an SCR in conduction is zero and hence the output voltage is zero. During b < wt < ( p + a), the output voltage equals E*Sin (wt) and then the average output voltage can be obtained as shown in equation (1).

This expression is not very useful because the value of b is required to be known. The value of a is known, since it is the firing angle, whereas the value of b is not likely to be known. Here b is the angle at which process of commutation overlap ends and the duration of commutation depends on the firing angle, the value of source reactance and the load impedance and value of b is variable and unknown. Hence it is preferable to derive an alternate expression. It is possible to derive an alternate expression for the case when the load reactance is large enough to ensure that the load current remains steady without ripple at a given firing angle. Let us assume that the load current be I and the firing angle be a. Let the line current change from - I to + I during commutation overlap when wt changes from a to b. From the waveforms shown above, the area of volt-seconds lost to output due to commutation overlap is computed as shown in equation (2).

Since this area is lost over p radians, the average value of output voltage lost due to commutation is calculated as shown in equation (3). It is known that the average output voltage with no commutation overlap is (2E/p)*Cos (a). By subtracting the voltage lost due to commutation, we can get the average output voltage taking into account the effect of commutation overlap, as shown in equation (4).

The waveforms appear as shown below. Key-in a firing angle less than 90^o and then press the Start button.

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II. ANALYSIS WITH A FINITE LOAD INDUCTANCE

With a finite load inductance, the conduction through the load can be either continuous or discontinuous. Let angle j be defined as shown in equation (5). If the firing angle a is greater than j, then the current through the load is discontinuous and the analysis is similar to that used for the circuit without a source inductance. When a < j , then the conduction is continuous. The analysis is carried out as follows. In these equations, a is the firing angle, b is the angle at which commutation overlap ends and j has been defined above. Let i_L be the load current and i_s is the source current. Let the source voltage v_s = E *Sin (q), where q = wt and 0 < q < 2p . Then the equation that is applicable during b < q < (p + a) can be expressed as shown in equation (6). During this period, the line current is equal to load current as defined by equation (7).

During b < q < (p + a), the voltage that appears as output is almost the negative of the source voltage and hence equation (8) is used and the line current has the same magnitude as the load current, but its polarity is opposite to that of load current as indicated by equation (9).

There are two instances of switching in one input cycle and commutation overlap occurs immediately after triggering either pair of SCRs. The pair consisting of S₁ and S₃ is triggered at wt = a and commuation overlap lasts from wt = a till wt =b and the output voltage is zero during this period as indicated by equation (10). Similarly after S₂ and S₄ are triggered at wt = (p + a), the output voltage is zero from wt = (p + a) till wt = (p + b), as indicated by equation (11). During the commuation overlap, the entire input voltage is applied across the source inductance, as indicated by equations (12) and (13).

The average and rms output voltage can be obtained as shown by equations (14) and (15). The maximum output voltage that can occur is indicated by equation (16) neglecting the loss in output that may occur due to commutation overlap. Then the ripple factor of output voltage can be expressed as shown by equation (17). If the ripple factor is multiplied by V_om, the rms value of ripple content in output voltage is obtained.

The fundamental component of the source current can be determined and then the THD, DPF and apparent power factor can be determined. The programs for simulation have been based on the equations displayed above.