What is RMS Voltage : Methods and Its Equations

In the domain of electronics, we frequently hear the terms alternating and direct current. So, an alternating waveform is the one which is related to AC current. This means that it is a periodic kind of waveform that switches in between negative and positive values. And the most general type of waveform used to represent this is a sinusoidal waveform. When coming to direct current waveform, the current and voltage values are basically in stable condition. It is so simplified to represent stable values and their magnitude values too. But as per the above discussion, the magnitude values of AC waveforms are not so simple as because it varies continuously corresponding to time. To know this, there are many methods and the most popular method is “RMS Voltage”. This article clearly explains the entire RMS voltage theory, its equations, applicable methods, and others.

What is RMS Voltage?

Definition: Firstly, it is expanded as Root-Mean-Squared Value. The general definition given by many for this is the amount of calculated AC power which delivers the same amount of heating power corresponding to the DC power, but RMS voltage has additional functionality. It is termed as  √ of the average value of the double function of instant generated values.

The value is represented as V_RMS and RMS current value is I_RMS.

RMS values are calculated only for the time fluctuating sinusoidal voltage or current values where the magnitude of the wave alters in correspondence to time, but not employed for the calculation DC waveform values as the magnitude stays to be constant. By comparing the RMS value of the AC sine wave which delivers a similar amount of electrical power with the provided load as a similar DC circuit, then the value is known as effective value.

Here, the effective current value is represented as I_eff and the effective voltage value is V_eff. Or else, the effective value is also stated as how many amperes or volts for a DC wave are similar to corresponding to the capability to generate a similar amount of power.

Equation

It is more important to know the RMS Voltage equation where it is employed to calculate many values and the basic equation is

V_RMS = V_peak-voltage * (1/ (√2)) = V_peak-voltage * 0.7071

The RMS voltage value is based on the AC wave magnitude value and it is not dependent on either the phase angle or frequency of the alternating current waveforms.

For instance: when the peak voltage of the AC waveform was provided as 30 volts then the RMS voltage is calculated as follows :

V_RMS = V_peak-voltage * (1/ (√2)) = 30 * 0.7071 = 21.213

The resultant value is almost identical in both the graphical and analytical methods. This happens only in the case of sinusoidal waves. Whereas in non-sinusoidal waves, the graphical method is the only option. Instead of using the peak voltage, we can calculate using voltage is exists between two peak values which is V_P-P.

The Sinusoidal RMS values are calculated as follows :

V_RMS = V_peak-voltage * (1/ (√2)) = V_peak-voltage * 0.7071

V_RMS = V_peak-voltage * (1/ 2(√2)) = V_peak-peak * 0.3536

V_RMS = V_average * (∏/ (√2)) = V_average* 1.11

RMS Voltage Equivalent

There exist mainly two general approaches for the calculation of the RMS voltage value of a sine wave or even another complicated waveform. The approaches are

• RMS Voltage Graphical Method – This is used to calculate the RMS voltage of a non-sine wave that varies according to time. This can be done by pointing mid-ordinates in the wave.
• RMS Voltage Analytical Method – This is used to calculate the voltage of the wave through mathematical calculations.

Graphical Approach

This approach shows the same procedure for the calculation of RMS value for the positive and negative half of the wave. So, this article explains the procedure of a positive cycle. The value can be calculated by considering a specific amount of accuracy for a similarly spaced instant all across the waveform.

The positive half cycle is segregated into ‘n’ equal parts which are also called middle ordinates. When there are more middle ordinates, the result will be more accurate. So, the width of every middle ordinate will be n degrees and the height is the instant value of the wave across the x-axis of the wave.

Here, every middle ordinate value in the wave is doubled and then added to the next value. This approach provides the squared value of the RMS voltage. Then the obtained value is divided by the total number of middle ordinates where this gives the Mean value of RMS voltage. So, the RMS voltage equation is given by

Vrms = [total sum of the middle ordinates × (voltage)2]/ number of the middle ordinates

In the below example, there are 12 middle ordinates and the RMS voltage is shown as

V_RMS = √(V₁²+ V₂²+ V₃²+ V₄²+ V₅²+ V₆²+……+ V₁₂²)/12

Let us consider that alternating voltage has a peak voltage value of 20 volts and with the consideration of 10 middle ordinate values, it is given as

V_RMS = √(6.2²+ 11.8²+ 16.2²+ 19²+ 20²+ 16.2²+ 11.8²+ 6.2²+ 0²)/10 = √(2000)/12

V_RMS  = 14.14 Volts

The graphical approach shows excellent results in knowing the RMS values of an AC wave which is either sinusoidal of symmetrical. This means that the graphical method is even applicable to complicated waveforms.

Analytical Approach

Here, this method deals with only sine waves which are easy to find the RMS voltage values through the mathematical approach. A periodic kind of sine wave is constant and given as

V_(t) = V_max*cos(ωt).

In this, the RMS value of the sine voltage V_(t) is

V_RMS = √(1/T ʃ^T₀V_max²*cos²(ωt))

When the integral limits are considered between 0⁰ and 360⁰, then

V_RMS = √(1/T ʃ^T₀V_max²*cos²(ωt))

On the whole, corresponding to AC voltages, RMS voltage is the best way of representation where it represents the signal magnitude, current, and voltage values. The RMS value is not similar to the median of the whole instant values. The proportion to the RMS voltage and to peak voltage value is equivalent to the RMS current and to peak current value.

Many of the multimeter devices either ammeter or voltmeter calculate RMS value in the consideration of accurate sine waves. For measuring the RMS value of the non-sine wave, an “Accurate Multimeter” is necessary. The value which is found by the RMS approach for a sine wave provides a similar heating effect which is for the DC wave.

For example, I²R = I_RMS²R. In the case of AC voltages and currents, they have to be considered as RMS values if not considered as others. So, an AC of 10 amps will provide a similar heating effect as a DC of 10 amps and a peak value of approximately 14.12 amps.

Thus, this is all bout the concept of RMS voltage, its equation, sinusoidal waveforms, methods used for calculation of these voltage values, and the detailed RMS voltage theory of it. Also, know about how peak voltage, average voltage, and peak-to-peak voltages are converted into RMS voltage through an RMS calculator?