RMS or virtual or Effective value of AC

It is that steady DC which would produce same amount of heat as is produced by AC when both are allowed to flow in dependently through the same circuit for same time ‘t’.

Relation between  RMS and Peak value of AC

P = I²R

dH/dt = I²R

dH =I²R.dt

dH = (I₀sinwt)²R.dt

dH = I₀² sin²wt.R.dt

dH = I₀² R((1 – cos2wt)/2).dt

H = (I₀²R/2) ∫ (1 – cos2wt).dt

H = (I₀²R/2) [T – 1/2w(sin2wt)^T₀]

H = (I₀²R/2) [T – 0]

H = (I₀²R/2) T

H_ac = (T₀²R/2)T                    Heat produced by AC

For DC

Heat Produced:-

H = I²_dc.Rt

H_dc = I²_dc Rt   Hdc = I²_dc RT

If H_ac = H_ac

Then, I_dc  I_r.m.s

I²_r.m.s R.t = (I₀²R/2)T

I_rms = I₀/(2)^1/2

I_rms = 0.707 I₀

The above relations for average or rms values are also applicable for Alternating voltage

V_avg = 0.636 V₀

V_avg = (2/π).V₀

V_rms = V₀/√2

V_rms = 0.707 V₀

For household

V_rms = V_o/√2

V₀ = √2 V_rms

V₀ = 1.41 x 220 = 311 volts.