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Harmonic Formulas for Filtering Applications

Trigonometric Series and Harmonic Analysis:

The history of the trigonometric series, for all practical purposes, came of age in 1822 with Joseph De Fourier’s book that developed the concept. The basic idea of trigonometric series is to represent periodic functions by series of particular (trigonometric) period functions.

Series of functions gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia in which the general term is gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia, with constant coefficients of an and bn, are called trigonometric series.

Trigonometric series will converge in an interval of length, 2gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia, since this is true the functions are periodic and will converge for all x and represents a periodic function gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia. This function is not necessary continuous however. We only consider series that converges uniformly, and then its sum gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia is continuous. This is what we have for harmonic analysis in electrical systems.

In Our Case a Connection Can be Established Between Coefficients an and bn and the Sum Function of gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia. The Multiplication of the Series Looks Something Like this

gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia

Factors Bounded by cos(px) or sin(px), WHere "P" is a Positive Integer Allows us to Calculate

gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia and gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia

Integration of the Series gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia or gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia for the Integrals Over the Interval gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia

When gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia"P" these Integrals have a Value of 0. For p=n They are

gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia for n>0

gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia for n=0

Because n=0 Behaves Like it Does it is Now Conventional to Write the Trigonometric Series as

gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia

The Euler-Fourier Formula for ngilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia0 looks Like This

gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia


Harmonic Analysis and Formulas

Let look back at basic circuits for a moment. We learned from Ohms law V = IR and P=VI and P=I2R. So the power delivered by an a-c circuit at any given time is:

 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia
(1)

Now We Need the Trigonometric Identity We Learned Years Ago

 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia
(2)

But the average power delivered by a a-c source is the first term only, since the average value of a cosine wave is zero. It may have twice the frequency of the original input current waveform. Equating the average power of an a-c generator to that delivered by a d-c source:

Pav(a-c) = Pd-c

 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia leaves us with gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia
(3)

So the Effective Values Give Us

 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia and gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia
(4)

This Can Be Obtained with the Following

 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia or gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia
(5)

This finally takes us to where we always wanted to go.

By Applying this Equation to the Following Fourier Series We Get


 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia
(6)

Then Form (5) and After Performing the Indicated Operations We Have the Following

 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia
(7)

But, Since

 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia

 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia
(8)

Similarly, For

 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia
(9)
 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia (10)
 gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia (11)

Looking What is Called in Mathematics as Odd Functions, This Means x, x3, x5, x7 and the Like for Which We Have

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Understand that even functions have nothing to do with even harmonics, nor odd functions with odd harmonics. An odd function is simply a function with odd powers of x and an odd harmonic has an odd multiple of the fundamental frequency.

This being state it is obvious that the sine wave is odd and the cosine wave is even. Therefore, the sum of sine waves will be odd, and a Fourier series containing only sine components represents an odd function.

Just an aside when even and odd functions are added the sum will be neither odd nor even. We will not get into that in this paper.

Plotting three sine waves, 60 hertz as fundamental, and the 5th and 7th harmonic of 60 hertz. Using 360 degrees for the fundamental, we get 1800 degrees for the 5th and 2520 degrees for the 7th. The individual sine waves look like this.

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We allowed the amplitude to 20 for the fundamental, 10 for the 5th harmonic and 5 for the 7th harmonic. Adding these simultaneously we get the following distorted wave.

gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia

Assuming These are Currents From (8), We Get

gilbert electrical systems, document library, basic electrical formulas, ungrounded, unbalance, calculation, capacitor, beaver, west virginia = 22.913 amps

This is known as the Total Harmonic Distortion Current or THDI.
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