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Solution To
"Harmonics And True RMS"

Solution To Question #1:
The total RMS value of the load current is 11.18 amps RMS. A True RMS meter will read this 11.18A value

Solution To Question #2:
The THD of the load current is 50%

Solution To Question #3:
The total RMS value of the new load current is still 11.18A. It does not matter what the frequency is of the harmonic. If you have 10A of fundamental current and a 5A harmonic, the True RMS meter still reads 11.18A. Remember, the RMS value is independent of frequency.

Solution To Question #4:
The THD of the new load current remains the same. THD is simply a measurement of how much RMS harmonic current is present in comparison to the amount of fundamental current. THD does NOT take into account frequency.

EXPLANATION

The RMS value of a sine wave regardless of frequency is simply the peak value of the sine wave divided by the square root of 2. (V means voltage)

Linear loads on a power system draw a load current that is a pure sine wave at the power frequency of 60Hz. Most everyone has seen the typical sine wave of 60Hz voltage that can be displayed on an oscilloscope. A linear load draws a current waveform of the same shape.

When a non linear load draws current from the power system (i.e. from the transformer), the current waveform is distorted from the mathematically perfect shape of a sine wave. A square wave or saw tooth waveform is an example of a distorted waveform.

The distorted waveform is actually a summation of the fundamental frequency sine wave and a variety of harmonics. Harmonics are actually pure sine waves themselves but each has a frequency that oscillates at a multiple of 60Hz (i.e. 3rd harmonic = 3 x 60 = 180Hz, 5th harmonic = 5 x 60 = 300Hz).

To find the total RMS value of any distorted wave, you have to take "the square root of the sum of the squares" of the RMS value of the fundamental and the series of harmonics. (admittedly it's easier to see this demonstrated than to explain it!!)

Here's the equation (The letter "I" stands for current):

Note that the RMS value of each sine wave of current I1 through I4 and beyond is squared. Then these squared values are added together and the square root of the total addition is taken.

I1 represents the RMS value of the fundamental component. The other I's are simply the RMS values of any harmonics which are present in the distorted waveform.

Irms is the total RMS value that would be read on the True RMS meter.

For our non linear load, the 60Hz current fundamental is 10A RMS. For the harmonic (and it's frequency does not matter) we have a 5A RMS value. Therefore the total RMS is

(10 x 10 + 5 x 5) = (100 + 25) = 125

The square root of 125 is 11.18A as the meter would show.

Total Harmonic Distortion is a measurement of the RMS value of ONLY the harmonics divided by the RMS value of the fundamental and is expressed in percent.

In this example the fundamental current is 10 amps. The harmonic RMS value is 5 amps.

5 / 10 x 100% = 0.5 x 100% = 50%

Therefore the THD is 50% regardless of the harmonic frequency.

In truth, THD is limited in its application. It is really only good for expressing harmonic limits on a feeder or that is produced by a load in a general way. It doesn't tell you which harmonics are present and how much of each harmonic (i.e. their RMS values) is present.

For that you need to perform a Harmonic Spectrum Analysis which is easily done with clamp on meters that are readily available. (Fluke has some hand held models that cost around $1500) These meters will tell you which harmonics are present, their respective RMS values, and even the phase angle between the harmonic and the fundamental. (The phase angle is seldom used in any calculation or harmonic study.)

You must have a Spectrum Analysis performed at different points on your power system (such as at a non linear load, at the transformer, at capacitor banks, at the panelboard, etc.) before any solution to any harmonic problem can be undertaken.

Once the analysis is complete, and if the harmonics are causing you trouble (such as overheating transformers, blowing cap bank fuses, tripping breakers, causing telephone interference, etc.), the information can be used to apply harmonic filters, to size K rated transformers, or to install harmonic cancellation equipment.

The theory of harmonics is explored in "The Electric Power Forum's" seminars. The possible problems and the practical solutions are explained. After completing the course, you won't be an expert on them, but you will be well versed in what they are, why they exist, what problems they cause, and how one mitigates them.

You'll also find out exactly what all the buzzword terminology means such as K Factor, THD, delta winding trapping third order harmonics, etc. If you've ever heard the phrase "third order harmonics add in the neutral" and wondered what it meant and why it occurs - we'll show you (We are very proud of this particular demonstration!) Many people find this part of the course the most worthwhile...and it is (if we do say so ourselves)!

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