Mathematical Analysis

An expression for the current through the load can be obtained as shown below. It can be assumed that the load current flows all the time. In other words, the load current is continuous. When the SCR conducts, the driving function for the differential equation is the sinusoidal function defining the source voltage. During the period defined by p < wt < (2p + a), the SCR blocks current and acts as an open switch. On the other hand, the free wheeling diode conducts during this period, and the driving function can be set to be zero volts. For a < wt < p, equation (1) applies whereas equation (2) applies for the rest of the cycle.

As in the previous cases, the solution is obtained in two parts. The expressions for the complementary integral and the particular integral are the same. The expression for the complementary integral is presented as equation (3). The particular solution is the steady-state response and is presented as equation (4).

The total solution is the sum of both the complimentary and the particular solution for a < wt < p.

The difference in solution is in how the constant A in complementary integral is evaluated. In the case of the circuit without free-wheeling diode, i(a) = 0, since the current starts building up from zero when the SCR is triggered during the positive half-cycle. On the other hand, the current-flow is continuous, when there is a free wheeling diode. Since the input to the RL circuit is a periodic function, we expect that the response of the circuit should also be periodic. That means, the current through the load is periodic. It means that

i(a) = i(2p + a).

Since the current through the load free-wheels during p < q < (2p + a) , we get that,

i(q) = i(p) *exp[-(q - p)/t] , where t = wL/R.

We use ( q - p ) for the elapsed period in radians instead of q itself, since the free-wheeling action starts at q = p . From the total solution, we get i(p).

i(p) = A*exp(- p/t) + (E/Z)*sin (p - a).

To obtain A, the following steps are necessary. From the total solution, obtain an expression for i(a) by substituting a for q. From the expression for the free-wheeling period, obtain i(2p + a) by letting q = (2p + a) . Since i(a) = i(2p + a), we can obtain A to be: