Half Wave Rectifier : Part1

  In half wave rectifier , rectifying element conducts only during positive half cycle of input a.c. supply. The negative half cycles of a.c. supply are eliminated from the output.

      This rectifier circuit consists of resistive load, rectifying element, i.e. p-n junction diode, and the source of a.c. voltage, all connected in series. The circuit diagram is shown in the Fig. 1. Usually, the rectifier circuits are operated from a.c. mains supply. To obtain the desired d.c. voltage across the load, the a.c. voltage is applied to rectifier circuit using suitable step-up or step-down transformer, mostly a step-down one, with necessary turns ratio.

Fig. 1  Half wave rectifier

       The input voltage to the half-wave rectifier circuit shown in the Fig. 1 is a sinusoidal a.c. voltage, having a frequency which is the supply frequency, 50 Hz.

       The transformer decides the peak value of the secondary voltage. If  N₁ are the primary number of turns and N₂ are the secondary number of turns and E_pm is the peak value of the primary voltage then,

       where         E_sm   = The peak value of the secondary a.c. voltage.

       As the return of E_sm is sinusoidal the instantaneous value will be,
                 e_s = E_sm  sinωt
                 ω = 2πf
                 f = Supply frequency

       Let R_f represents the forward resistance of the diode. Assume that, under reverse biased condition, the diode acts almost as an open circuit, conducting no current.

1.1 Operation of the Circuit

       During the positive half cycle of secondary a.c. voltage, terminal (A) becomes positive with resoect to terminal (B). The diode is forward biased and the current flows in the circuit in the clockwise direction, as shown in the Fig. 2(a). The current will flow for almost full positive half cycle. This current is also flowing through load resistance R_L hence denoted as I_L, the load current.

       During negative half cycle when terminal (A) is negative with respect to terminal (B), diode becomes reverse biased. Hence no current flows in the circuit as shown in the Fig. 2(b). Thus the circuit current, which is also the load current, is in the form of half sinusoidal pulses.

Fig. 2

       The load voltage, being the product of load current and load resistance, will also be in the form of half sinusoidal pulses. The different waveforms are illustrated in Fig. 3.

Fig. 3  Load current and load voltage waveforms for half wave rectifier

       The d.c. output waveform is expected to be a straight line but the half wave rectifier gives output in the form of positive sinusoidal pulses. Hence the output is called pulsating d.c. It is discontinuous in nature. Hence it is necessary to calculate the average value of load current and average value of output voltage.

1.2 Average DC Load Current ()

       The average or dc value of alternating current is obtained by integration.

       For finding out the average value of an alternating waveform, we have to determine the area under the curve over one complete cycle i.e. from 0 to 2π and then dividing it by the base i.e. 2π.

       Mathematically, current waveform can be described as,
                        I_L = I_m sin t                  for 0 ≤ ωt ≤ π
                        I_L = 0                        for π ≤ ωt ≤ 2π
       Where  I_m = Peak value of load current

       As no current flows during negative half cycle of a.c. input voltage, i.e. between ωt = π to ωt = 2π, we change the limits of integration.

       Applying Kirchhoff's voltage law we can write,

      Where R_s = Resistance of secondary winding of transformer. If R_s is not given it should be neglected while calculating I_m.

1.3 Average DC Load Voltage (E_DC)

It is the product of average D.C. load current and the load resistance R_L.

       The winding resistance R_s and forward diode resistance R_f are practically very small compared to R_L.

       But as R_f and R_s are small compared to R_L, (R_f +R_s )/R_L is negligibly small compared to 1. So neglecting it we get,

Note : When R_f and R_s are finite, calculate I_m, then I_DC and from that calculate as .
Note : Do not calculate E_DC as E_sm/π directly for finite R_f and R_s. Calculate I_m1 then and then E_DC = I_DC R_L .

1.4 R.M.S. Value of Load Current (I_RMS)

       The R.M.S. means squaring, finding mean and then finding square root. Hence R.M.S. value of load current can be obtained as,

Note : remember that this R.M.S. value is for half wave rectified waveform hence it is I_m/2. For full sine wave it is I_m/√2, which is derived later.

1.5 R.M.S. Value of the Load Voltage

       The r.m.s. value of the load voltage is the r.m.s. value of the total output voltage which includes d.c. output and the a.c. ripples. As the load is resistive, the r.m.s. value of the load voltage is given by,