The load current I_L consist of a dc component I_L(dc) and an ac component I_L(ac). The Fourier series of the Half-Wave rectifier rectified current following through the load is found to be

i_L = I_LM(1/ π + sinωt/2 – 2 cos 2 ωt/3π – 2 cos 4 ωt /15 π + ……………)

As seen the Half-Wave rectifier current consists of a large number of ac components (which constitute the ripple) in addition to the dc component. The first term is I_LM/2 which represents the dc component I_L(dc). Sin ωt has a peak value of (I_LM/2). It is called the fundamental or first harmonic component and it’s rms value is

I_L1 = I_LM/2√2

The third term represents the second harmonic component whose frequency is doubled that of the supply frequency. The rms value is

I_L2 = peak value/√2 = 2I_LM/3π× √2 = √2 I_LM /3π.

The fourth terms represents the third harmonics component whose frequency is four times the supply frequency. It’s rms value is

2 I_LM/15π × √2 = √2× I_LM / 15 π.

The rms values of other components can be similarly calculated, However, they found to be of continuously diminishing value.

The rectified output (or load ) current consist of

(i)                  dc component, I_L(dc) = I_LM/π

(ii)                ac component of rms value I_L1, I_L2 and I_L3 etc. Their combined rms is given by

I_L(ac) = √I_L1­­­²+ I_L2­² + I_L3² +…………

The rms (or effective) value of the total load current is given by

I_L = √I_L(dc)² + I_L(ac)²

    = √I_L(dc)² + (I_L1² + I_L2² + I_L3² + ……..)

Similarly, the Fourier series of the load voltage is given by

U_L = V_LM (1/π + sin ωt/2 – 2cos 2 ωt/3 π – 2 cos 4 ωt/15 π……..)

It also consist of

(i)                 a dc component, V_L(dc) = V_LM/π

(ii)               ac component of rms value V_L1, V_L2 and V_L3 etc. which are given by

V_L1 = V_LM/2√2,

V_L2 = √2.V_LM/3 π,

V_L3 = √2V_LM/15 π etc.

Again,

V_L(ac) = √V_L1² +  V_L2²  +V_L3² + ……

The rms value of entire load voltage is given by

V = √V_L(dc)² + V­_L(ac)²  =  √V_L(dc) + V_L1² +  V_L2²  +V_L3² + ……

Related Topic

1. Single-phase half-wave rectifier
2. AVERAGE AND RMS VALUE OF SINGLE-PHASE HALF-WAVE RECTIFIER
3. Efficiency of single-phase half-wave rectifier
4. Ripple Factor of single phase Half-Wave rectifier
5. Peak Inverse Voltage (PIV) of single phase half wave rectifier
6. Peak current of single phase half wave rectifier
7. Transformer Utilization Factor (TUF) of single phase half wave rectifier 
8. Advantage and Disadvantage of single-phase half-wave rectifier