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Root Mean Square (RMS) and Overall Level

Siemens Experimenter

09-12-2017
09:37 AM

There are several different ways that the term Root Mean Square (RMS) is used.

This article will go over the following:

**RMS Amplitude Format:**How the equivalent energy of a sine wave is expressed in a spectrum**RMS of a Spectrum:**A representation of the overall energy in a spectrum – often called the overall level**Tracked RMS / Tracked Overall Level:**Tracking the RMS of each average against a tracking parameter to see how the energy content changes**RMS in Simcenter Testlab**

**RMS Amplitude Format:**

Amplitude formats describe how the amplitude of a sine wave is expressed in a spectrum.

The RMS amplitude format is valuable because it indicates the equivalent steady state energy value of an oscillating signal.

For example, a sine wave (like a pressure wave) oscillates above and below zero Pascal. The average value of the wave is zero. This is not a good representation of the effective energy of the wave. To get a better understanding of the equivalent steady state value, the RMS is used.

The *RMS amplitude format *is calculated by squaring the peak amplitude (A) of the sine wave, diving it by two, and then taking the square root of that quantity. For a single sine wave, the RMS amplitude can be represented as 0.707*A.

Look at the spectrum below. The curves are the *exact same data* but expressed with different amplitude formats. The black curve is expressed in peak format. The pink curve is expressed in RMS format. Notice that all amplitude levels of the pink curve are below the black curve (Figure 2).

In the zoomed in section, each spectral line is represented by a cross. When converting to RMS amplitude format, each spectral line is multiplied by ~0.707. For example, the spectral line at 694Hz has a peak amplitude of 1.000g. This corresponds to a RMS amplitude of 0.707g.

The RMS amplitude format is used to represent the equivalent steady state value of a sine wave at each spectral line.

**RMS of a Spectrum:**

Often, an RMS value of a spectrum is desired to be calculated. The RMS of a spectrum is a single number that represents the overall level of energy across a frequency range.

In the graphic below, the RMS of the spectrum is 54.08g.

To calculate the RMS of a spectrum, the root sum square of all the spectral lines within the frequency range of interest must be calculated.

The “A” term in the above equation represents the amplitude of a spectral line. It is important that this amplitude (A) value is in RMS format and has window correction factors taken into consideration (more about this below).

A_0 is the first spectral line under consideration, A_k is the last spectral line under consideration.

There are some important considerations that must be taken to ensure the calculated RMS value is correct:

1. The individual spectral line values (A) must be in *Linear* *format*. If the units were squared or power units, they must be squared rooted before performing the calculation. For example, an autopower of sound pressure with units of Pa2 must be square rooted to units of Pa.

2. Each individual spectral line must be in *RMS format* for the calculation. No matter what format is used for *displaying* the spectrum, RMS amplitude format is used when *calculating* the RMS of the spectrum. The RMS calculation produces the same result no matter how the spectrum is displayed.

3. Window correction factors may need to be applied to the spectrum. If a window was used during acquisition, energy correction must be applied to the spectrum. The usage of windows to compensate for leakage can cause spectral lines to appear lower in amplitude / energy content than they really are. To get appropriate values for RMS spectral energy, energy correction must be applied.

The RMS calculation has a pre-requisite of linear, RMS, energy corrected values.

The RMS value calculated in Simcenter Testlab will automatically convert data to energy, RMS, and linear values "behind the scenes", regardless of how it is displayed in the software interface. This ensures consistent values for RMS calculations.

Users who want to manually calculate RMS must take these elements into consideration.

**Tracked Overall Level:**

The RMS value of a spectrum is often called the overall level. The overall level can be tracked versus speed or time to see how the amount of energy in the signal changes.

When recording an event that is changing, it is often preferred to track the event on time or RPM. This means that for every increment of time or RPM a spectrum is calculated.

In the below example, a spectrum is calculated every 25 rpm between 3500 RPM and 4000 RPM. The overall level (RMS) for each of these spectrums is calculated.

Then, these RMS values are plotted vs. RPM. This way, it is possible to see which RPM values have especially high (or higher than expected) energy content.

**RMS in Simcenter Testlab:**

This section will cover:

- Changing Amplitude Format in Simcenter Testlab
- Calculating the RMS of a frequency spectrum
- Calculating the RMS over a specific frequency range
- Calculating Tracked Overall Level in Testlab

*Changing Amplitude Format in Simcenter Testlab:*

To change the amplitude format in Simcenter Testlab, right lick on the y-axis of the plot, select “Processing” and then choose an amplitude format under “Spectrum & Section Scaling”.

*Calculating the RMS of a frequency spectrum:*

To calculate the RMS of a spectrum in Simcenter Testlab, right click on the legend and choose “Options”.

Next, calculate the RMS of the spectrum by selecting RMS in the “Calculated Content” tab, and the clicking the “Add to selection” arrow. Additionally, the Unit Label can be turned on in the lower right.

The RMS value of the entire spectrum will now be displayed in the legend. Amplitude format and energy corrections are applied in the background as needed.

*Calculating the RMS over a specific frequency range:*

Alternatively, the RMS value over a smaller frequency range can also be calculated.

Add a double cursor to the plot and position the cursor over a range of values.

Then, right click on the cursor, and select Calculations -> RMS.

The RMS value of the content between the cursors will be calculated and displayed in the cursor legend.

*Calculating Tracked Overall Level in Testlab:*

To calculate a tracked overall level in Simcenter Testlab, ensure the measurement mode is set to “Tracked”.

In the Section Settings dialog, make sure that “Overall level” is checked on under the “Overall Level” tab.

The data will be saved in the Overall Level folder within the Sections folder.

**Conclusion: **

It is important to understand the difference between RMS amplitude format, the RMS of a spectrum, and the tracked RMS.

RMS amplitude format represents the equivalent steady state value of a sine wave in a spectrum.

The RMS value of an entire spectrum represents the overall energy level. This value can be tracked to better understand how it changes.

**Questions? **Post to this article, or email jacklyn.kinsler@siemens.com, or contact Siemens PLM GTAC support.

**Related Links**

- Free On-Demand Webinar: Digital Signal Processing Fundamentals
- Digital Signal Processing: Sampling Rates, Bandwidth, Spectral Lines, and more...
- Gain, Range, Quantization
- Aliasing
- Overloads
- Averaging Types: What's the difference?
- Spectrum versus Autopower
- Autopower Function...Demystified!
- Power Spectral Density
- Windows and Leakage
- Window Types
- Window correction factors
- RMS Calculations
- The Gibbs Phenomenon
- Introduction to Filters: FIR and IIR
- Simcenter Testlab: The "Double X cursor"

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RoyDuan

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01-02-2018
09:55 PM

1. In equation 2 about calculating RMS of a spectrum, how to understand the first and the last spectrum line amplitude square should be devided by two (A0^2/2, Ak^2/2)? May be the start and end spectrum range not coincide with spectrum line.

2. How to calculate RMS of a 1/3 CPB spectrum?

Kevin_Grenier

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01-03-2018
08:02 AM

Hello Roy,

The RMS calculation is defined pretty well by an FAQ on the Global Technical Assistance Center (GTAC). If you log into GTAC (www.siemens.com/gtac) then the FAQ will be located under the Solution center at: https://solutions.industrysoftware.automation.siemens.com/view.php?sort=desc&q=RMS&dt=faq&pd=lms-tes...

This is from the Functions.pdf theory document provided with LMS Test.Lab under the Documentation\ Theory Documents folder (for exampe, look under C:\Program Files (x86)\LMS\LMS Test.Lab {revision}\central\Help\Theory documents PDF).

RoyDuan

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01-04-2018
06:30 AM

Hi Kevin,

Thank you for your reply.

1. I cannot have access to the Solution center of GTAC as you mentioned. I don't know the reason. Could you show me the related information?

2. The spectrum RMS calculation equation in the Functions.pdf you showed was different from equaciton 2 in this page. Why?

3. RMS is a measure of energy in a signal. How to understand why the lines in the beginning and end should be half?

4. How to calculate RMS from a 1/3 Octave spectrum?

Roy

Kevin_Grenier

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01-04-2018
08:25 AM

Hello Roy,

1. All customers on maintenance have access to the Global Technical Assistance Center. Using the sold-to and Web Access Code (WAC) located at the top of each license file generated in the last 3 years will be this information. You can use this information to create a personal Webkey (login) to the GTAC site. Please reach out to your local customers services team for assistance. Or call the support line shown at www.siemens.com/gtac for your region. I cannot share information from the GTAC site in this forum.

2. The RMS calculation depends on many things which the FAQ addresses: What function was used (spectrum, Autopower linear, Autopower Power, Autopower PSD, etc). Is the data Peak Amplitude or RMS Amplitude? What window and which window correction was used? As the functions, amplitude scaling and window correction modes change so will the formula to calculate RMS.

3. I believe it's related to using 1/2 the douple sided frequency spectrum. We only use the positive frequencies and not the negative ones. As stated, when 0 Hz is the first spectral line it is not halved.

4. When displaying an octave spectrum in an octave display the Overall Level can be shown on the right side. I show RMS calculated 3 ways below: 1) Showing the Overall Level, 2) Adding the RMS level to the Legend Calculated Content and 3) using a double X cursor to calculate the RMS value across the entire octave range. As you can see, they all match:

RoyDuan

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01-04-2018
09:08 AM

Hi Kevin,

I still have some doubts about this three questions.

1. The two equations is very similar but not the same. And from the statements ,both for the spectrum RMS.

2. As you said only use the positive frequencies and not the negative ones, I think the all spectral lines should be halved. However, from the equation, only the beginning and end have been halved. How to understand?

3. I mean the 1/3 Octave spectrum RMS equation or formula which can be used by using Excel. I want to know the formula like this fft spectrum.

Roy

Kevin_Grenier

Siemens Phenom

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01-05-2018
02:25 PM

Hello Roy, I believe your questions are getting too specific for this Knowledge Base or Forum.

These questions are best left to your local cusomer services team because you can provide your data, they can explain the FAQ and then show the steps in Excel. As I stated there are many things that concern how the RMS calculation is done.

Please reach out to them as they are really the best resource to assist.

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