Frequency Component of single-phase Half-Wave Rectifier Voltage and Current

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

iL = ILM(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 ILM/2 which represents the dc component IL(dc). Sin ωt has a peak value of (ILM/2). It is called the fundamental or first harmonic component and it’s rms value is

IL1 = ILM/2√2

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

IL2 = peak value/√2 = 2ILM/3π× √2 = √2 ILM /3π.

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

2 ILM/15π × √2 = √2× ILM / 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, IL(dc) = ILM

(ii)                ac component of rms value IL1, IL2 and IL3 etc. Their combined rms is given by

IL(ac) = √IL1­­­2+ IL2­2 + IL32 +…………

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

IL = √IL(dc)2 + IL(ac)2

    = √IL(dc)2 + (IL12 + IL22 + IL32 + ……..)

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

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

It also consist of

(i)                 a dc component, VL(dc) = VLM

(ii)               ac component of rms value VL1, VL2 and VL3 etc. which are given by

VL1 = VLM/2√2,

VL2 = √2.VLM/3 π,

VL3 = √2VLM/15 π etc.


VL(ac) = √VL12 +  VL22  +VL32 + ……

The rms value of entire load voltage is given by

V = √VL(dc)2 + V­L(ac)2  =  √VL(dc) + VL12 +  VL22  +VL32 + ……




Related Topic

  1. 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

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