Root mean square value

General formula for RMS – root mean square

$$F_{RMS} = \sqrt{\frac{1}{T}\int_{0}^{T}{f^{2}(t)\cdot dt}}$$

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

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