What is RMS Value of AC?

In this article, we shall discuss the RMS (Root Mean Square) value of AC (Alternating Current). So let us start with the basic introduction of RMS value.

RMS Value of AC

The RMS (Root Mean Square) value, also called the Effective Value of AC, of alternating current is the value of AC defined in terms of its equivalent direct current.

The RMS value of AC can be defined as under:

When an alternating current is flowing through a resistor for a time interval, produces a certain amount of heat. Now, a direct current is made to flow through the same resistor for the same period of time, so that it produces the same amount of heat as produced by the alternating current. Then, this value of direct current (DC) that produces the equivalent heat as produced by the AC is called the RMS value of AC.

The power transfer capability of an alternating current depends upon the RMS value of the current. Hence, the RMS value of AC is of considerable importance in electrical circuits. Also, RMS value plays a vital role in all calculations of AC analysis.

The RMS value of alternating current is generally denoted by I or I_RMS.

Determine the RMS Value of AC

The RMS value of an AC wave can be determined by the following two methods:

• Mid-Ordinate Method
• Analytical Method

Let us discuss each method of determining RMS value in detail.

(1). Mid-Ordinate Method of RMS Value:

In the mid-ordinate method of determining the RMS value of AC, the base of the current wave is divided into n number of equal intervals, each of duration t/n seconds as shown in figure-1.

Here, i₁, i₂, …i_n are the mid-ordinate magnitudes of the wave in the successive time interval.

Now, let us consider this current wave is passed through a resistor of resistance R, for a time duration of t seconds. Then, the heat produced by this current wave in the n^th time interval is given by,

`\H_n=i_n^2×R×t/n" Joules"`

Therefore, the total heat produced by the current wave in t seconds is given by,

`\H=((i_1^2 Rt)/n+(i_2^2 Rt)/n+⋯+(i_n^2 Rt)/n)`

`\⇒H=Rt((i_1^2+i_2^2+⋯+i_n^2)/n)…(1)`

Now, let I be the direct current (DC) that produces the same amount of heat in the resistance R during the same time interval t. Therefore, the total heat produced by the direct current is given by,

`\H=I^2 Rt…(2)`

Hence, from equations (1) and (2), we can write,

`\I^2 Rt=Rt((i_1^2+i_2^2+⋯+i_n^2)/n)`

`\I^2=((i_1^2+i_2^2+⋯+i_n^2)/n)`

`\∴I=\sqrt{(i_1^2+i_2^2+⋯+i_n^2)/n}…(3)`

Hence, the RMS value of AC is numerically equal to the square root of the mean of the square of all instantaneous values. Thus, the name Root Mean Square (RMS) value.

(2). Analytical Method of RMS Value:

The analytical method of determining the RMS value of alternating current is applicable when the equation of the current is known.

In order to determine the RMS value of AC by analytical method, we first have to find the square of the current, then the mean is taken, and finally, the square root is taken.

To understand the concept, let us consider a sine wave expressed by the following equation,

`\i=I_m  sin⁡θ`

Therefore, by taking the square of it, we get,

`\i^2=I_m^2  sin^2⁡ θ`

The waveforms of i and i² are shown in figure-2.

Let us consider an elementary strip of very small thickness dθ on the waveform of i² as shown in figure-2. If i² is the instantaneous magnitude of this strip, then the area of the strip is given by,

`\dA=i^2.dθ`

Therefore, the area of the half cycle (because i² repeats itself after a half cycle of i, hence the time period of i² is equal to half of the time period of i) is given by,

`\A=∫_0^π i^2.dθ`

`\⇒A=∫_0^π I_m^2  sin^2⁡θ.dθ`

`\⇒A=I_m^2 ∫_0^π sin^2⁡θ.dθ`

`\∵sin^2⁡θ=(1-cos⁡2θ)/2`

`\∴A=(I_m^2)/2 ∫_0^π (1-cos⁡2θ ).dθ`

`\⇒A=(I_m^2)/2 [θ-sin⁡((2θ)/2)]_0^π`

`\⇒A=(I_m^2)/2 [π-0]=(πI_m^2)/2`

Therefore, the mean (average) of the wave over half-cycle is,

`\"Mean"=("Area over the half cycle")/"Base"`

`\∴"Mean"=((πI_m^2⁄2))/((π-0) )=(I_m^2)/2`

Thus, the Root of the mean can be expressed as,

`\I=\sqrt{I_m^2/2}`

Therefore, the RMS value of the alternating current is

`\I=I_m/\sqrt{2}=0.707I_m`

In general, the RMS value of sinusoidal AC is,

`\"RMS Value"=("Max Value")/(\sqrt{2})`

Note – For sinusoidal alternating voltage, the RMS value of alternating voltage is given by,

`\V_(RMS)=V_m/\sqrt{2}=0.707V_m`

Note: It is important to note that for DC (Direct Current), average value, RMS value, maximum value, and instantaneous value all are the same.

Now, let us discuss some numerical examples of the RMS value of AC.

Numerical Example (1) – An alternating current wave is represented by

`\i=150 sin⁡314t`

Find the RMS value of this alternating current.

Solution – By inspection, the maximum value of current is 150 A (I_m). Thus, the RMS value of AC is given by,

`\I=I_m/\sqrt{2}=150/\sqrt{2}`

`\∴I=106.08" A"`

Numerical Example (2) – The voltage in an AC circuit is given as,

`\v=200 sin⁡314t`

Determine the RMS value of voltage in the circuit.

Solution – By inspection, the maximum voltage value is 200 V (V_m). Hence, the RMS value of voltage is,

`\V=V_m/\sqrt{2}=200/\sqrt{2}`

`\∴V=141.44 V`

Hence, this is all about the RMS value of alternating current (AC) in electrical circuits.