General formula for RMS – root mean square

The RMS value interpretation based on the electrical current root mean square value.

W = \int{p \cdot dt}

W = \int{u \cdot i \cdot dt}

W = R\cdot \int{ i^{2} \cdot dt}

where:

W – work

p – temporal electric power

p = u \cdot i

u – temporal electric voltage

i – temporal electric current

If the electrical current is periodical then

W_{T} = R\cdot \int_{0}^{T}{ i^{2} \cdot dt}Root mean square value of the alternate electric current is an equivalent direct electric current which will produce exactly same amount of heat.

R\cdot \int_{0}^{T}{ i^{2} \cdot dt} = R \cdot I^{2} \cdot T

Dividing both sides of the above equation by resistance R

\int_{0}^{T}{ i^{2} \cdot dt} = I^{2} \cdot T

Next swapping sides in order to find relation for the DC electrical current I

I^{2} \cdot T = \int_{0}^{T}{ i^{2} \cdot dt}

Finally following is received

Root mean square value for the alternate electric current

I_{RMS} = \sqrt{\frac{1}{T}\cdot \int_{0}^{T}{ i^{2} \cdot dt}}

Root mean square value for the alternate electric voltage

U_{RMS} = \sqrt{\frac{1}{T}\cdot \int_{0}^{T}{ u^{2} \cdot dt}}