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

# Analysis of an AC electrical circuit

A considered below AC electric circuit is composed of an AC voltage source, a resistor, a capacitor, an AC current source and an inductive coil. The nodal analysis method is being applied to calculate currents in the circuit’s branches. It is recalled that in the nodal analysis method it is always assumed that an electrical potential of one of nodes is equal to 0.

Since the considered electric circuit is an AC circuit following formulas are to be applied:

$$\underline{Z} = \frac{1}{\underline{Y}}$$ $$\underline{Y} = \frac{1}{\underline{Z}}$$ $$\underline{I}_1$$ $$\sum{(\underline{I}_s)_a} = \underline{Y}_{R1C1} \cdot \underline{V}_{s1} + \underline{I}_{s2} = \underline{V}_a \cdot ( \underline{Y}_{R1C1} + \underline{Y}_{L1} ) - \underline{V}_b \cdot ( \underline{Y}_{R1C1} + \underline{Y}_{L1} )$$ $$\underline{Y}_{R1C1} = \frac{1}{R1 - j \cdot \frac{1}{\omega \cdot C1}}$$ $$\underline{Y}_{L1} = \frac{1}{\omega \cdot L1}$$

The further calculations related to the present example are available – Node voltage method example 2.

# Nodal analysis

Nodal analysis is one of methods used for electrical networks analysis. Nodal analysis is based on Kirchhoff’s current law. Main idea of this method is to calculate electrical potentials of every node. This will allow to calculate voltages in branches since voltage is a difference of potentials. This approach has one rule which requires to assume that potential of one chosen node has be equal zero volts. Symbolically this chosen node is connected to the ground on electrical diagram.

Node voltage method