Mathematical Analysis

An expression for the current through the SCR can be obtained as shown below. It is assumed that the current flows for a < wt < d, where d > p . When the SCR conducts, the driving function for the differential equation is the sinusoidal function defining the source voltage. Outside this period, the SCR blocks current and acts as an open switch. For this period, there is no equation defining the behaviour of the circuit. For a < wt < d , equation (1) applies. Given a linear differential equation, the solution is found out in two parts. The homogeneous equation is given by equation (2), where a is the firing angle. The value of constant A in the complimentary solution is to be evaluated later. The particular solution is the steady-state response and is diplayed as equation (3). The total solution is the sum of both the complimentary and the particular solution and is presented as equation (4). The value of A is obtained using the initial condition. Since the SCR starts conducting at wt = a and the current starts building up from zero, i(a) = 0. In the expression above t = wL/R. Then A can be expressed as in equation (5).

Once the value of A is known, the expression for current is known. When the firing angle a and the extinction angle d are known, the average output voltage at the cathode of the SCR can be evaluated as shown in equation (6).

The average load current can be obtained by dividing the average load voltage by the load resistance, since the average voltage across the inductor is zero.

 
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