#### DC component of the half rectified sine wave

This is given by

####
U_{RMS} of the half rectified sine wave

As seen in part I, this is

#### RMS value of the alternating component

Now, the next question is: how "strong" is the total alternating component in the half rectified sine wave?

In part I we established that URMS is splitted into two parts: a continuous component and the sum of sines. The latter, the sum, is the total alternating component of the signal. The formula is

where C=U

_{DC}. Substituting, we arrive at the RMS value for the alternating part#### Tying this with Fourier

This gives the RMS value for the total alternating component, but what if we are interested in the RMS value of a specific harmonic?

The Fourier Transform will be our tool: it transforms a time-domain signal to a frequency-domain one, and the Fourier Series gives us the amplitude of each harmonic.

For a half rectified sine wave of period

Here is a table of the first 5 harmonics amplitudes relative to the peak amplitude of the original signal.

For a half rectified sine wave of period

*T*, the amplitudes are given by:Here is a table of the first 5 harmonics amplitudes relative to the peak amplitude of the original signal.

Harmonic | Amplitude |
---|---|

1 | 0.5 |

2 | 0.21221 |

4 | 0.042441 |

6 | 0.018189 |

8 | 0.010105 |

The following graph shows the harmonic amplitudes.

Remembering that the RMS value for a sine signal is the peak intensity divided by the square root of two, we have

Harmonic | RMS Amplitude | Cumulative RMS |
---|---|---|

1 | 0.35355 | 0.35355 |

2 | 0.15005 | 0.38408 |

4 | 0.030011 | 0.38525 |

6 | 0.012862 | 0.38546 |

8 | 0.0071454 | 0.38553 |

After just a few harmonics, we are almost at the calculated total value already.

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