Root Mean Square Voltage
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Exercise:
Show that the root mean square value of a sinusoidal voltage signal vt V_ sinomega t is sscVrms fracV_sqrt
Solution:
The rms value corresponds to the square root of the average value of the square of the voltage signal over one period. The average over one period is given by sscV^rms fracT_^T v^t textdt fracT_^T V_^ sin^omega t textdt labeleq There are several ways to find a value for the egral in refeq: abcliste abc Using the identiy sin^omega t frac-cos omega t we find _^T sin^omega t textdt frac_^T left - cos omega tright textdt fracleft_^T textdt - _^T cos omega t textdt right frac left leftt right_^T - leftfracsin omega t omega right_^T right frac left T - fracsin omega T - sin omega right With omega T pi and sin sinpi this leads to _^T sin^omega t textdt frac left T - fracsinpi - sin omega right frac left T - right fracT labeleqa abc Using egration by parts _^T ut dotvt textdt leftut vtright_^T - _^T dotut vt textdt with ut sinomega t Longrightarrow dotut omega cosomega t textand nonumber dotvt sinomega t Longrightarrow vt -fraccosomega tomega we find _^T sin^omega t textdt _^T sinomega t sinomega t textdt left-fracsinomega t cosomega tomegaright_^T + _^T fracomega cosomega t cosomega tomega textdt + _^T cos^omega t textdt Adding sin^omega t on both sides of the yields _^T sin^omega t textdt _^T sin^omega t textdt + _^T cos^omega t textdt _^T leftsin^omega t + cos^omega tright textdt _^T textdt lefttright_^T T and hence _^T sin^omega t textdt fracT which agrees with refeqa. abcliste It follows with refeq sscV^rms fracV_^T fracT fracV_^ Longrightarrow sscVrms fracV_sqrt
Show that the root mean square value of a sinusoidal voltage signal vt V_ sinomega t is sscVrms fracV_sqrt
Solution:
The rms value corresponds to the square root of the average value of the square of the voltage signal over one period. The average over one period is given by sscV^rms fracT_^T v^t textdt fracT_^T V_^ sin^omega t textdt labeleq There are several ways to find a value for the egral in refeq: abcliste abc Using the identiy sin^omega t frac-cos omega t we find _^T sin^omega t textdt frac_^T left - cos omega tright textdt fracleft_^T textdt - _^T cos omega t textdt right frac left leftt right_^T - leftfracsin omega t omega right_^T right frac left T - fracsin omega T - sin omega right With omega T pi and sin sinpi this leads to _^T sin^omega t textdt frac left T - fracsinpi - sin omega right frac left T - right fracT labeleqa abc Using egration by parts _^T ut dotvt textdt leftut vtright_^T - _^T dotut vt textdt with ut sinomega t Longrightarrow dotut omega cosomega t textand nonumber dotvt sinomega t Longrightarrow vt -fraccosomega tomega we find _^T sin^omega t textdt _^T sinomega t sinomega t textdt left-fracsinomega t cosomega tomegaright_^T + _^T fracomega cosomega t cosomega tomega textdt + _^T cos^omega t textdt Adding sin^omega t on both sides of the yields _^T sin^omega t textdt _^T sin^omega t textdt + _^T cos^omega t textdt _^T leftsin^omega t + cos^omega tright textdt _^T textdt lefttright_^T T and hence _^T sin^omega t textdt fracT which agrees with refeqa. abcliste It follows with refeq sscV^rms fracV_^T fracT fracV_^ Longrightarrow sscVrms fracV_sqrt