The root mean square value of an alternating current is defined as the square root of the mean of the squares of all currents over one cycle.

**RMS value of AC**

The term RMS refers to
time-varying sinusoidal currents and voltages and not used in DC systems.

**The root mean square
value of an alternating current is defined as the square root of the mean of
the squares of all currents over one cycle. **It is denoted by I_{RMS} . For alternating voltages, the RMS value is given by V_{RMS}.

The alternating current
i = I _{m} sin Ï‰ t or i = I_{m} sinÎ¸ , is represented
graphically in Figure 4.42. The corresponding squared current wave is also
shown by the dotted lines.

The sum of the squares
of all currents over one cycle is given by the area of one cycle of squared
wave. Therefore,

An elementary area of
thickness dÎ¸ is considered in the first half-cycle of the squared current wave
as shown in Figure 4.42. Let *i*^{2} be the mid-ordinate of the
element.

Area of the element = *i*
2*d*Î¸

Thus we find that for a symmetrical sinusoidal current rms value of current is 70.7 % of its peak value.

Similarly for alternating voltage, it can be shown that

**EXAMPLE 4.18**

Write down the equation
for a sinusoidal voltage of 50 Hz and its peak value is 20 V. Draw the
corresponding voltage versus time graph.

**Solution**

**EXAMPLE 4.19**

The equation for an
alternating current is given by *i* =
77 sin 314*t*. Find the peak value,
frequency, time period and instantaneous value at t = 2 ms.

**Solution**

i = 77 sin 314t ; t = 2
ms = 2Ã—10^{-3} s

The general equation of
an alternating current is i = I_{m} sin Ï‰t . On comparsion,

(i) Peak value, I_{m}
= 77 A

(ii) Frequency, *f* = Ï‰/2Ï€ = 314 / 2 Ã—3.14 = 50 Hz

Time period, T = 1/*f* = 150 = 0 .02 *s*

(iv) At t = 2 m s,

Instantaneous value,

*i* = 77sin(314Ã—2Ã—10^{âˆ’3} )

i = 45.24 A

A sinusoidal alternating
voltage (or current) can be represented by a vector which rotates about the
origin in anti-clockwise direction at a constant angular velocity Ï‰. Such a
rotating vector is called a phasor. A phasor is drawn in such a way that

Â·
the length of the line segment equals the peak value *V _{m}*
(or

Â·
its angular velocity Ï‰ is equal to the angular frequency of the
alternating voltage (or current)

Â·
the projection of phasor on any vertical axis gives the
instantaneous value of the alternating voltage (or current)

Â·
the angle between the phasor and the axis of reference (positive
x-axis) indicates the phase of the alternating voltage (or current).

The notion of phasors is
introduced to analyse phase relationship between voltage and current in
different AC circuits.

The diagram which shows
various phasors and their phase relations is called phasor diagram. Consider a
sinusoidal alternating voltage *Ï…* = *V _{m}* sin Ï‰t applied
to a circuit. This voltage can be represented by a phasor, namely as shown in Figure 4.43.

Here the length of equals the peak value (*V _{m}*), the angle it makes with x-axis
gives the phase (Ï‰t) of the applied voltage. Its projection on y-axis provides
the instantaneous value (

When rotates about *O* with angular velocity Ï‰ in anti-clockwise direction, the waveform of the voltage
is generated. For one full rotation of , one cycle of
voltage is produced.

The alternating current
in the same circuit may be given by the relation *i*â€†=â€†*I _{m}*sin
(Ï‰t + Ï•) which is represented by

Tags : Phasor and phasor diagram, Definition, Explanation, Formulas, Solved Example Problems | Alternating Current (AC) , 12th Physics : Electromagnetic Induction and Alternating Current

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12th Physics : Electromagnetic Induction and Alternating Current : RMS value of AC | Phasor and phasor diagram, Definition, Explanation, Formulas, Solved Example Problems | Alternating Current (AC)

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