RMS Voltage

Root Mean Square is referred to as RMS. The square root of the average of the squares of the values or the square of the function defining the continuous waveform is the Root Mean Square (RMS) of a collection of values or a continuous-time waveform. In this article, we will be going through what is RMS Voltage, How to calculate RMS Voltages using Graphical and Analytical Methods, Then we will see Different RMS Voltage Formula, At last we will conclude our Article with Applications, Advantages, Disadvantages and Some Applications.

What is RMS Voltage?

The RMS voltage is determined by taking the square root of the mean square of the instantaneous values of the voltage signal. It can also be defined as a continuously varying voltage in terms of an integral of the squares of the instantaneous values during a cycle. RMS voltage is the effective value of alternating voltage or alternating current. It is also known as the effective voltage. The RMS voltage value is represented as V_RMS.
RMS voltage is also known as the effective voltage or the equivalent DC voltage. This is because the RMS value indicates the amount of alternating current drawn by a resistor, similar to the amount drawn by a direct current source. AC signals are sinusoidal signals under most conditions. For sinusoidal signals, the instantaneous value changes, so the instantaneous value cannot be used for power calculations.

RMS Voltage

How to Calculate RMS Voltage

The RMS value is only calculated for time-varying waveforms in which the magnitude of a quantity fluctuates with time.

There are two approaches or methods to calculate RMS value.

• Graphical Method
• Analytical Method

Graphical Method

It can be used to find the RMS value of any non-sinusoidal time-varying waveform by drawing a number of mid-ordinates onto the waveform. The effectiveness of this approach is contingent upon the quantity of data points extracted from the waveform.

In the graphical method, several mid-ordinates are selected, and their instantaneous values are Recorded. Each mid-ordinate’s value in a waveform is squared and added consecutively. The sum of the squared values is divided by the total number of mid-ordinates, and taking the square root of this result provides the RMS value.

Let us consider an example, V1 ,V2 ,….,V12 are the instantaneous values of voltages.

Graphical Method

VRMS = √(V12+V22+V32+.....+V122)/12

Analytical Method

The analytical method, on the other hand, is a mathematical procedure that utilizes calculus to find the effective or RMS value of any periodic voltage or current. This method is more accurate for pure sinusoidal waveforms. It involves integrating the squared waveform function with respect to time and then simplifying the equation to derive the RMS value.

where,

Vm is the maximum value or peak value of the waveform.

ω is the angular frequency which is equal to 2Л/T.

Now, we can calculate the rms value of a sinusoidal voltage as:

VRMS = √1/T∫0T Vm2 cos2 (ωt)dt

Integrating through with limits taken from 0 to 360^o or T, the period gives:

VRMS  = √Vm2 /2T[t + sin(2ωt)/2ω]0T

The above equation is reduces down to:

VRMS = √V m2 /2
VRMS = V m /√2
VRMS = V m x 0.7071

RMS Voltage Formula

To calculate the RMS voltage from a sinusoidal waveform, the peak value or maximum value of the waveform is divided by the square root of 2. Similarly, the RMS voltage can also be calculated from the peak-to-peak voltage or the average voltage by applying appropriate formulas.

From peak voltage

VRMS = V Peak/√2 = 0.7071VPeak

From peak-to-peak voltage

VRMS = VP-P /2√2 = 0.3536VP-P

From average voltage

VRMS = π/2√2 x V Average = 1.11V Average 

Problem

Lets consider a voltage waveform given by V(t) = 10cos(2π25t) volts. Then find the RMS voltage.

Given, Voltage waveform
V(t) = 10cos(2π25t)
Here, 10 is the amplitude of the cosine wave and 25 Hz is the frequency.
By using the formula,
VRMS = VP /√2
In this case, the peak voltage or maximum voltage(VP) is the amplitude of the cosine wave which is 10 volts.
VRMS = 10/√2  ≈ 7.071 volts
Therefore, the RMS voltage for the given waveform is approximately 7.071 volts.

Significance of RMS Voltage

Here are some key points regarding the significance of RMS voltage:

• One of the most important parameter that is used to describe the strength of an AC(alternating current).
• In resistive elements like heating elements in appliances, the heat generated is proportional to the square of the RMS current or voltage. RMS values accurately reflect the heating effects in such devices.
• RMS voltage provides a convenient way to compare DC and AC values. It simplifies the analysis and design of AC circuits by allowing engineers to use familiar DC concepts.
• In AC circuits, when we calculating power using the formula P = V_RMS . I_RMS ,where V_RMS and I_RMS are the RMS voltage and RMS current. Utilizing RMS values ensures that the calculated power is equal to what would be obtained in a DC circuit, simplifying power analysis.
• Many electrical devices such as lights, bulbs and household appliances are rated in terms of RMS voltage. Allowing users to understand the equivalent DC voltage for the performance.

Derivation of RMS Voltage for Sinusoidal Waveform

Consider a sinusoidal voltage waveform V(t) = V _m sin(ωt).

where, V _m is the maximum amplitude and ω is the angular frequency(ω=2Лf)

V2 (t) = Vm2 sin2 (ωt)
V2 = 1/T ∫0T Vm2 sin2(ωt)dt
V2 = Vm2 /T ∫0T (1-cos2ωt)/2 dt
V2 = Vm2 /2T ∫0T (1-cos2ωt)dt
V2 = Vm2 /2T [T-0-0+0]
V2 = Vm2 /2
VRMS = V m /√2

Applications of RMS Voltage

• Electronic engineers use RMS values to design and analyze electronic circuits. Ensuring that devices like oscillators and amplifiers etc.,
• RMS voltage is used in finding the heating effect of an AC current in resistive elements such as in stovetops or electric heaters.
• In electric motors, the RMS voltage is plays a major role for finding the RMS voltage that contributes to the motors mechanical power output.
• RMS voltage is used for finding the luminosity or brightness of the light produced.
• RMS values are also used for audio engineers for designing and analyzing components such as speakers, microphones and amplifiers to provide optimal sound quality.
• In Telecommunication, RMS values are used to design and study the systems like telephone networks and data networks.

Advantages and Disadvantages of RMS Voltage

Some of the Advantages and Disadvantages are Given Below :

Advantages

• RMS voltage reflects the effective value of an AC signal over time, which only captures the highest amplitude of a sinusoidal wave. RMS voltage plays a major role for calculations involving power dissipation and heat generation.
• In AC(alternating current), power is directly proportional to the square of the RMS voltage and current. It simplifies power calculations and ensures accurate sizing of electrical equipment like motors and heaters.
• RMS voltage is the universally accepted by the magnitude of AC voltage for both single-phase and three-phase systems. This facilitates consistent communication and understanding across various electrical applications.

Disadvantages

• In waveforms, calculating the RMS value requires squaring and averaging. This will increase complexity by additional steps than simply compared to simple peak voltage measurement.
• In some applications, the use of RMS voltage will not be necessary such as Dc circuits or low-power circuits.
• Understanding the concepts of RMS and its relationship to others can be difficult for individuals without grasp of electrical concepts.

Conclusion

RMS voltage with its standardized representation and suitability for power calculations is a vital role in AC systems. RMS voltage is a crucial parameter for characterizing and analyzing AC circuits, accurately represent heating effects and serve as a standard for electrical equipment and systems. By understanding the concepts of RMS voltage and current, engineers can effectively design a wide range of electrical applications.

FAQs on RMS Voltage

Why do we use RMS?

RMS values ​​are used to determine average current or instantaneous voltage. This is necessary because AC directly averages to zero.

Why is RMS voltage used instead of peak voltage?

RMS voltage provides a consistent measurement of voltage that corresponds to the heating effects of DC voltage, so Useful for comparison.

What is the RMS value of a sinusoidal waveform?

The RMS value of the waveform of sinusoidal voltage is 0.7071 of its peak value or maximum value.