Difference between revisions of "RMS amplitude"

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<math>\sqrt{2}/2</math><math>x_RMS = \sqrt{\tfrac{1}{N} \sum_{i=1}^n x_i^2}                        </math><math>f_RMS = \sqrt{\tfrac{1}{T_2-T_1} \int_{T_1}^{T_2} [f(t)]^2 dt }</math><math>f_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [f(t)]^2 dt }</math><math>y(t) = Asin (2\pift+\varphi) = Asin(\omegat+\varphi)</math><math>Y_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omegat)]^2 dt}</math><math>      = A\sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omegat)}{2} dt}</math><math>      = A\sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omegat)}{4\omega}]_0^T}</math><math>      = \tfrac{A}{\sqrt{2}}</math><math>=\sqrt{\frac{1}{N}\sum_{i=1}^nx_i^2}</math><math>x_RMS=\sqrt{\tfrac{1}{n}\sum_{i=1}^nx_i^2}                                                                    (1)
 
The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is
 
f_{RMS}=\sqrt{\tfrac{T_2}{T_1}\textstyle \int_{T_1}^{T_2} \displaystyle [f(t)]^2}dt            (2)
 
and the RMS for a function over all time is
 
f_{RMS}=\sqrt{\tfrac{1}{T}\int_{0}^{T} [f(t)]^2}dt                                                                (3)
 
For a sine wave
 
y(t)=A sin⁡(2πft+φ)=Asin(ωt+φ)                                                                                          (4)
 
where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero)
 
Y_{RMS}=\sqrt{\tfrac{1}{T}\int_{0}^{T} [Asin(\omegat)]^2dt}
 
              =A\sqrt{\tfrac{1}{T}\int\limits_{0}^{T} \tfrac{1-cos(2\omegat)}{2}dt}
 
              =A\sqrt{\tfrac{1}{T}[\tfrac{T}{2}-\tfrac{sin(2\omegat)}{4\omega}]_0^T}
 
              =\tfrac{A}{\sqrt{2}}                                                                                                (5)
 
  
</math><math>fRMS=\sqrt{\frac{1}{T_2-T_1}\int_{T_1}^{T_2} [f(t)]^2}dt</math><math>f_RMS = \sqrt{\tfrac{1}{T}\int_{0}^{T}[f(2)]^2dt }</math><math>=\sqrt{\frac{1}{N}\sum_{i=1}^nx_i^2}</math><math>x_{RMS} = \sqrt{\tfrac{1}{n} \sum_{i=1}^n x_i^2 }</math>
 

Revision as of 10:15, 21 October 2019