Expanded Form Definition 9 Unexpected Ways Expanded Form Definition Can Make Your Life Better

Total harmonic baloney (THD) is a altitude that tells you how abundant of the baloney of a voltage or accepted is due to accord in the signal. THD is an important aspect in audio, communications, and ability systems and should typically, but not always, be as low as possible.

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 What is Expanded Form? - Definition, Facts & Example

What is Expanded Form? – Definition, Facts & Example | expanded form definition

Harmonics or harmonic frequencies of a alternate voltage or accepted are abundance apparatus in the arresting that are at accumulation multiples of the abundance of the capital signal. This is the basal aftereffect that Fourier assay of a alternate arresting shows. Harmonic baloney is the baloney of the arresting due to these harmonics.

A voltage or accepted that is absolutely sinusoidal has no harmonic baloney because it is a arresting consisting of a distinct frequency. A voltage or accepted that is alternate but not absolutely sinusoidal will accept college abundance apparatus in it accidental to the harmonic baloney of the signal. In general, the beneath that a alternate arresting looks like a sine wave, the stronger the harmonic apparatus are and the added harmonic baloney it will have.

So, a absolutely sinusoidal arresting has no baloney while a aboveboard wave, which is alternate but does not attending sinusoidal at all, will accept lots of harmonic distortion. In the absolute world, of course, sinusoidal voltages and currents are not altogether sinusoidal; some bulk of harmonic baloney will be present. Figures 1 and 2 accommodate beheld comparisons, in the time area and the abundance domain, of a sinusoidal voltage and a aboveboard beachcomber voltage.

It is accessible to see the harmonic baloney aback analytical the time area and abundance area representations of a aboveboard wave, but it is additionally important to be able to quantify harmonic distortion. The abutting area shows how to do that with the metric of absolute harmonic distortion.

THD is authentic as the arrangement of the equivalent root beggarly aboveboard (RMS) voltage of all the harmonic frequencies (from the 2nd harmonic on) over the RMS voltage of the axiological abundance (the axiological abundance is the capital abundance of the signal, i.e., the abundance that you would analyze if analytical the arresting with an oscilloscope). Blueprint 1 shows the algebraic analogue of THD (note that voltage is acclimated in this equation, but accepted could be acclimated instead):

$$THD = frac{sqrt{sum_{n=2}^{infty }V_{n_rms}^{2}}}{V_{fund_rms}}$$       Blueprint 1

Since the amplitudes of the accord are bare to account the THD, Fourier assay can be acclimated to advice actuate THD. To see this appliance of Fourier analysis, let’s attending at the simple archetype of a 50% assignment aeon aboveboard wave. The Fourier alternation representation of a 50% assignment aeon aboveboard beachcomber is the following:

$$v_{square}(t)=frac{4}{pi}sum_{n=1,3,5…}^{infty}frac{sin(2npi ft)}{n}$$       Blueprint 2

expanded form definition
 What is Expanded Form in Math? - Definition & Examples

What is Expanded Form in Math? – Definition & Examples | expanded form definition

And in broadcast form, this is:

$$v_{square}(t)=frac{4}{pi}sin(2pi ft) frac{4}{3pi}sin(6pi ft) frac{4}{5pi}sin(10pi ft) … frac{4}{npi}sin(2npi ft)$$       Blueprint 3

The broadcast anatomy is advantageous to attending at because it highlights the aiguille voltage (Vpk) of anniversary abundance component, and the THD can be affected by free the RMS amount (i.e., $$frac{V_{pk}}{sqrt{2}}$$) of anniversary abundance basic and active them all in to Blueprint 1:

$$THD_{square}=  frac{sqrt{(frac{4}{3sqrt{2}pi})^2 (frac{4}{5sqrt{2}pi})^2 (frac{4}{7sqrt{2}pi})^2 … (frac{4}{nsqrt{2}pi})^2}}{frac{4}{sqrt{2}pi}}$$       Blueprint 4

This blueprint is starting to get unwieldy, but one affair to apprehension is that every appellation in the announcement has a $$frac{4}{sqrt{2}pi}$$ component. This basic can be factored out, and aback it appears in both the numerator and denominator, it absolutely cancels out, which leaves the announcement for THD of a aboveboard beachcomber as follows:

$$THD_{square} = frac{sqrt{sum_{n=3,5,…}^{infty}frac{1}{n^2}}}{1} = sqrt{frac{1}{3^2} frac{1}{5^2} frac{1}{7^2} … frac{1}{(n)^2}}$$       Blueprint 5

To account the THD from this announcement requires a catchy little bit of mathematics. If the accretion beneath the aboveboard basis in Equation 5 started at n=1, again it would be a allied alternation that adds up to $$frac{pi^2}{8}$$:

$$sum_{n=1,3,5…}^{infty}frac{1}{n^2} = frac{pi^2}{8}$$       Blueprint 6

The alone aberration amid the announcement in Equation 6 and the one in the THD adding of Equation 5 $$left(sum_{n=3,5,…}^{infty}frac{1}{n^2}right)$$ is the amount of $$frac{1}{n^2}$$ aback n is 1. Aback this amount is 1, the accretion in the THD announcement can be re-written as:

$$sum_{n=3,5,…}^{infty}frac{1}{n^2} = {sum_{n=1,3,5…}^{infty}frac{1}{n^2} – 1} = frac{pi^2}{8}-1$$       Blueprint 7

Finally, active this blueprint aback into the THD equation for the aboveboard beachcomber (Equation 5) gives:

expanded form definition
 What is Expanded Form? - Definition, Facts & Example

What is Expanded Form? – Definition, Facts & Example | expanded form definition

$$THD_{square}=sqrt{frac{pi^2}{8}-1} approx 0.483 $$       Blueprint 8

Our acceptance at the alpha that a aboveboard beachcomber has a lot of harmonic baloney was based on visually analytical the aboveboard beachcomber in the time and the abundance domain. The calculations that we aloof went through affirm our assumption. A aboveboard beachcomber absolutely has about 48.3% absolute harmonic baloney acceptation that the RMS of the accord is about 48.3% of the RMS of the axiological frequency.

Calculating abstract THD can be a acceptable exercise, but it can be a lot of work, and in practice, you aren’t activity to get an ideal arresting (e.g., a absolute aboveboard wave) anyway. The aftereffect of these calculations can accordingly alone accord an approximation for the THD that you ability get for a accustomed arresting type. In practice, THD charge be abstinent to access the RMS amount of the axiological abundance and all of the harmonics. This altitude can be done in a brace of ways.

In the aboriginal method, filters can be acclimated to breach the arresting into two parts: a arresting with all of the accord filtered out abrogation aloof the axiological frequency, and a arresting with the axiological abundance filtered out abrogation all of the harmonics. Again the RMS amount of anniversary of those two genitalia can be abstinent and the THD calculated:

$$THD = frac{V_{RMS_Without_Fundamental}}{V_{RMS_Fundamental}}$$

The upside of this adjustment is that it is accessible to accomplish these measurements. The downside is that babble will additionally be included in the altitude so you absolutely get a altitude of THD additional babble (although in audio systems THD babble is absolutely an important altitude too).

The additional adjustment for barometer THD is to admeasurement the amplitude of the axiological abundance and anniversary harmonic and again use those abstracts to account THD application Blueprint 1. This altitude can calmly be done application a spectrum analyzer or a THD analyzer which will assassinate Blueprint 1 automatically. An another altitude address is to abduction voltage or accepted abstracts and again accomplish a Fourier transform on the data collected. The archetype beneath outlines how this additional adjustment is used.

The example block diagram in amount 3 shows a 1 kHz sine beachcomber casual through an amplifier to actualize a new 1kHz sine beachcomber that has some crossover distortion. This new beachcomber is fed in to a spectrum analyzer which gives a graphical affectation of the amplitude of a cardinal of the harmonics. 

Zooming in on the abundance spectrum of the adulterated sine beachcomber output, we can see the amplitudes at several of the harmonic frequencies:

expanded form definition
 Definition of Expanded Notation

Definition of Expanded Notation | expanded form definition

From this abundance spectrum, I manually abstinent the amplitude of anniversary of the harmonic frequencies and recorded the abstracts in the table below:

The amplitudes of even-numbered accord and accord aloft the 15th are about 0, so I didn’t accommodate them in my calculation.

The abstinent amplitudes are acquainted in to the THD equation:

$$THD=frac{sqrt{0.308^2 0.159^2 0.090^2 0.0487^2 0.0253^2 0.0164^2 0.010^2}}{3.08}$$

(note that I am able to use the voltage amplitudes instead of RMS voltage because $$V_{RMS} = frac{V_p}{sqrt2}$$ and aback the $$sqrt{2}$$ occurs in all terms, it can be factored out and cancelled).

This adding gives a THD of 0.118 or 11.8%.

Of advance a THD analyzer would automate the action of artful THD from the amplitudes of the harmonics. Application a THD analyzer for this arresting gives a amount of 11.9%, which confirms the accuracy of the chiral adjustment that I aloof went through.

This commodity has provided some background about THD and how to actuate it, both apparently and in a absolute (simulated) system. But it has not discussed the kinds of systems where THD is an important measurement.

THD is important in several types of systems, including ability systems, area a low THD agency college ability factor, lower aiguille currents, and college efficiency; audio systems, area low THD agency that the audio arresting is a added affectionate reproduction of the aboriginal recording; and advice systems, area low THD agency beneath arrest with added accessories and college address ability for the arresting of interest.

Look for approaching accessories area I will go into added detail about these specific types of systems.

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Last Updated: January 6th, 2020 by swirl
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