## Introduction

**RMS (Root Mean Square) value** of any time-varying signal, in general, is related to the amount of heat being produced in the circuit or across the particular element in the circuit. Do you know how to calculate the root mean square or rms voltage of a sine wave given the peak voltage. Here this article helps you calculate the RMS value of AC circuits, and it also provides the process that can help you derive the formula using basic math and calculus.

What is RMS Value? | Easiest Explanation

**Catalog**

## Ⅰ RMS Value of AC Circuits

The RMS value is an important term in electrical engineering. What is the exact meaning of RMS value? And why it is so much important? This RMS value represents the effective value of the given AC signal which will produce the same amount of heat (and hence power) across particular element if it is being replaced by the DC source of equivalent value. The physical meaning of RMS is the effective voltage value of alternating current signal, that is, to do work in physics. The work done in the process of converting electric current into other forms of energy is called electrical work, which is related to current, voltage, and energizing time. The higher the voltage applied to electrical appliances, the greater the energized current, and the longer the energizing time, the more work the current will do. The RMS value is also called effective value, which is considered from the perspective of electrical work. The effective value of alternating current is equal to the direct current/voltage that obtains the same power consumption (heating) on the same resistance. Because it is AC, the correct result must be obtained after time averaging (integration), and the instantaneous value of direct current must not be used to replace the effective value. In statistical analysis of data, the square of all values is summed, then the mean value is calculated, finally the square root is taken to obtain the root mean square value.

Figure 1. Peak to Peak Voltage

In physics, we often use the RMS value to analyze the noise of power supply. When the voltage of the resistor is the AC voltage V(t), the power V2/R changes with time. If the energy consumed per cycle is divided by the number of cycles, this is the average power. Further more, if the power of a DC voltage applied to the same resistor is the same as the average power of the AC voltage, the DC equivalent voltage is the root-mean-square value of the AC voltage.

## Ⅱ Determine RMS Value of AC Signals

Literally RMS means take the SUM, get the MEAN and then take the ROOT; in that order, and you'll get the RMS value. As shown here, the general equation of RMS value has been derived, which can be used to find the RMS value of any time-varying signal. Just obey the rules.

### 2.1 Sinusoidal Waveform

The following formulas only applied to PURE sinewave signal. Nowsaday many electrical appliances, e.g. adjustable speed drives, tend to introduce harmonics into electrical system. Consequently, the signal might not be PURE sinewave anymore.

Calculate the effective value from the definition, that is, alternating current and direct current respectively pass through the same resistor. If the two consumes the same electric energy (or produces the same Joule heat) in the same time, then the direct current value is called the effective value of the alternating current. The accumulation of signal power over time is the work done by the signal. The most primitive is derived for sine waves, but in fact it is applicable to all waveforms.

Use definite integral to calculate the work by the AC signal in the load R at one cycle. It is equal to the work done by a DC quantity (effective value) in one cycle of the load R.

VR:

Instsient power of R:

Average power of R:

The power of the stable DC voltage Vdc is , if this power is the same as the average power of AC, then

Where, Vdc is called the RMS value of the AC signal (Vrms)

With Vrms, calculate the average power of the load resistor R:

The RMS voltage calculation is ultimately used to give a measure of the average continuous power carrying capability of a signal. The instantaneous voltage values are squared (the V^2 term) which is then summed up (the integration) before converting back to voltage by the square root operation. Once the RMS voltage value is known, then you can make accurate estimates on true power delivery over time, independent of the signal's polarity.

To put it simply, for example, a square wave signal with an amplitude of 100V and a duty cycle of 0.5, if calculated based on the average value, its voltage is only 50V, and calculated according to the root mean square value is 70.71V. Why is this? For example, there is a set of 100-volt battery packs, which will stop for 10 minutes after each power supply for 10 minutes, which means that the duty cycle is half. If this batteries set drives a 10Ω resistor, 10A of current and 1000W of power will be generated in 10 minutes, and the current and power will be zero during a power failure.

Then in a period of 20 minutes, the average power is 500W, which is equivalent to the power generated by 70.71V charging a 10Ω resistor directly. The 50V DC voltage can only produce 250W of power when charges a 10Ω resistor. For motors and transformers, as long as the root mean square current does not exceed the rated current, they will not burn out even if they are overloaded within a certain period of time.

### 2.2 RMS Value Equation Steps

To calculate the RMS of y(t), the steps are as follows:

1) Calculate the square of y(t).

2) Take the average value of y2(t) for a period.

3) Take the square root of the average.

**Example:**

(1) a^{2} is a constant, so .

(2) cos(ωt) is a complete cosine curve.

(3)

### 2.3 RMS Mean on a Multimeter

In daily life, ordinary voltmeters are scaled according to the effective value of the sine wave.

The effective value of the sine wave U=maximum Um×0.707

The average value is generally not used, it refers to the average value of each instantaneous value in the positive half cycle or the negative half cycle.

The average value of the sine wave Up=Maximum Um×(2/π)=0.637Um

Note that the measurement of alternating current with a voltmeter and a multimeter is based on the effective value scale. If it is a non-sine wave, such as a pulse wave, the meter reading obtained is meaningless.

### 2.4 The Other Waveform

1) Half Sinusoidal Waveform

2) Square Wave

When we want to average the electrical signal, if the process is completed over the entire period or less, we need to give accuracy. For basic and symmetrical AC signals, regardless of frequency, peak value or period, averaging over a complete period always results in 0V. Therefore, it is more appropriate to average these signals during the half period.

In short, average voltage tells you that your voltage fluctuates around some average value, while RMS voltage shows you how much is that fluctuation. In addition, Square of RMS could be understood as the average power on a resistor of 1 Ohm.

↪️Recommended RMS Value Calculation Tool: *RMS Voltage Calculator – From Average Value, Peak & Peak to Peak Value*

**Frequently Asked Questions About RMS Voltage Calculation**

1. What does RMS mean?

Root mean square

Root mean square or simply RMS watts refers to continuous power handling of a speaker or a subwoofer or how much continuous power an amplifier can output. RMS values are usually lower than peak watts ratings, but they represent what a unit is truly capable of handling.

2. How do you calculate RMS value?

Square each value, add up the squares (which are all positive) and divide by the number of samples to find the average square or mean square. Then take the square root of that. This is the root mean square (rms) average value.

3. Why is RMS used?

Attempts to find an average value of AC would directly provide you the answer zero... Hence, RMS values are used. They help to find the effective value of AC (voltage or current). This RMS is a mathematical quantity (used in many math fields) used to compare both alternating and direct currents (or voltage).

4. What is rms noise?

RMS or root mean square is defined as the average. In terms of noise, it is defined as the process used to determine the average power output (continuous waveform) over a long period of time.

5. Does RMS really matter?

RMS values are typically much lower than peak power ratings, but they more accurately represent what an amplifier or speaker is truly capable of. Think of RMS as a true listening rating. Although not a perfect means of comparison, most RMS ratings are comparable – especially when measuring among name brand products.

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