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

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_{L1}^{2}+ I_{L2}^{2 }+ I_{L3}^{2} +…………

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_{L1}^{2 }+ I_{L2}^{2} + I_{L3}^{2} + ……..)

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_{L1}^{2} + _{ }V_{L2}^{2 } +V_{L3}^{2} + ……

The rms value of entire load voltage is given by

#### V = √V_{L(dc)}^{2} + V_{L(ac)}^{2} = √V_{L(dc)} + V_{L1}^{2} + _{ }V_{L2}^{2 } +V_{L3}^{2} + ……

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