Mathematical Analysis

An expression for the current through the diode can be obtained as shown below. It is assumed that the current flows for 0 < wt < b, where b > p . When the diode conducts, the driving function for the differential equation is the sinusoidal function defining the source voltage. During the period defined by b < wt < 2p, the diode blocks current and acts as an open switch. For this period, there is no equation defining the behaviour of the circuit. For 0 < wt < b, the equation (1) defined below applies.

Given a linear differential equation, the solution is found out in two parts. The homogeneous equation is defined by equation (2). It is preferable to express the equation in terms of the angle q instead of 't'. Since q = wt, we get that dq = w.dt. Then equation (2) then gets converted to equation (3). Equation (4) shown above is the solution to this homogeneous equation and is called the complementary integral.

The value of constant A in the complimentary solution is to be evaluated later.

The particular solution is the steady-state response and equation (5) expresses the particular solution. The steady-state response is the current that would flow in steady-state in a circuit that contains only the source, the resistor and the inductor shown in the circuit above, the only element missing being the diode. This response can be obtained using the differential equation or the Laplace transform or the ac sinusoidal circuit analysis. The total solution is the sum of both the complimentary and the particular solution and it is shown as equation (6). The value of A is obtained using the initial condition. Since the diode starts conducting at wt = 0 and the current starts building up from zero, i(0) = 0. The value of A is expressed by equation (7).

Once the value of A is known, the expression for current is known. After evaluating A, current can be evaluated at different values of wt, starting from wt = p. As wt increases, the current would keep decreasing. For some value of wt, say b , the current would be zero. If wt > b , the current would evaluate to a negative value. Since the diode blocks current in the reverse direction, the diode stops conducting when wt reaches b. Then an expression for the average output voltage can be obtained. Since the average voltage across the inductor has to be zero, the average voltage across the resistor and the average voltage at the cathode of the diode are the same. This average value can be obtained as shown in equation (8).