# Important values of Alternating Quantities

## Important values of Alternating Quantities

1) Peak Value$$\left( {{i}_{0}}\,or\ {{V}_{0}} \right)$$: The maximum value of alternating quantity (i or V) is defined as peak value or amplitude.

2) Mean square value$$\left( \overline{{{V}^{2}}}\,or\,\overline{\,{{i}^{2}}} \right)$$: The average of square of instantaneous values in one cycle is called mean square value. It is always positive for one complete cycle. E.g. $$\overline{{{V}^{2}}}=\frac{1}{T}\int\limits_{0}^{T}{{{V}^{2}}dt}=\frac{{{V}_{0}}^{2}}{2}$$ (or) $$\overline{{{i}_{0}}^{2}}=\frac{{{i}_{0}}^{2}}{2}$$.

3) Root mean square value: Root of mean of square of voltage or current in an ac circuit for one complete cycle is called root mean square value. It is denoted by $${{V}_{rms}}$$ (or)$${{i}_{rms}}$$ .

$${{i}_{rms}}=\sqrt{\frac{{{i}_{1}}^{2}+{{i}_{2}}^{2}+……}{n}}=\sqrt{\overline{{{i}^{2}}}}=\sqrt{\frac{\int\limits_{0}^{T}{{{i}^{2}}dt}}{\int\limits_{0}^{T}{dt}}}=\frac{{{i}_{0}}}{\sqrt{2}}=0.707{{i}_{0}}=70.7%{{i}_{0}}$$  Similarly,

$${{V}_{rms}}=\frac{{{V}_{0}}}{\sqrt{2}}=0.707{{V}_{0}}=70.7%\,of\,{{V}_{0}}$$

r.m.s value of ac is equal to that value of dc, which when passed through a resistance for a given time will produce the same amount of heat as produced by the alternating current when passed through the same resistance for same time.

4) Mean of Average value$$\left( {{i}_{avg}}\,or\,{{V}_{avg}}\, \right)$$: The average value of alternating quantity for one complete cycle is zero. The average value of ac over half cycle$$\left( t=0\,\,to\,\,\frac{T}{2} \right)$$$${{i}_{avg}}=\frac{\int_{0}^{T/2}{i\,dt}}{\int_{0}^{T/2}{dt}}=\frac{2{{i}_{0}}}{\pi }=0.637{{i}_{0}}=63.7%\,\,of\,{{i}_{0}}\,$$ similarly, $${{V}_{avg}}=\frac{2{{V}_{0}}}{\pi }=0.637{{V}_{0}}=63.7%\,\,of\,\,{{V}_{0}}$$.

5) Peak to Peak value: It is equal to the sum of the magnitudes of positive and negative peak values. Therefore, Peak to peak value = $${{V}_{0}}+{{V}_{0}}=2{{V}_{0}}=2\sqrt{2}{{V}_{rms}}=2.828{{V}_{rms}}$$.

6) Form factor and Peak factor: The ratio of r.m.s value of ac to its average during half cycle is defined as form factor. The ratio of peak value and r.m.s value is called peak factor.