Calculating Root Mean Square (RMS) Voltage

In the preceding AC Waveform analysis we introduced the concept of RMS (Root-Mean- Square) voltage applicable to sinusoidal waveforms.

We learned that, the RMS voltage of a sine wave corresponds to the DC voltage level that would produce identical heating in a resistive element.

This tutorial expands upon this foundation, by investigating into a more comprehensive examination of RMS voltages and currents.

It’s important to acknowledge that while many texts define RMS value solely in terms of its heating equivalence to DC, its significance extends beyond this specific application.

The RMS (Root-Mean-Square) value signifies the square root of the average of the squared instantaneous values of a waveform.

It is typically denoted by V_RMS or I_RMS. Its crucial to recognize that, the RMS concept is exclusively applicable to time-varying waveforms.

These can be sinusoidal voltages, currents, or even more complex periodic signals where the magnitude fluctuates over time. Conversely RMS has no relevance in DC (Direct Current) circuit analysis, where the magnitude remains constant.

In the context of equating the heating effect of an AC (Alternating Current) sinusoidal waveform to a DC circuit supplying the same electrical power to a load, the RMS value becomes the “effective value.”

This effective value is commonly symbolized by V_eff or I_eff. Actually, the effective value is a DC equivalent that reflects the ability of a time-varying sinusoidal waveform to deliver the same average power as a steady DC voltage.

It expresses the AC voltage in terms of its heating effect, which is comparable to a specific DC voltage. For instance the UKs domestic mains supply is rated at 240 Vac.

This value represents the effective value or “240 Volts RMS.” Therefore the sinusoidal RMS voltage from UK household wall sockets can deliver the same average heating power as a constant DC voltage of 240 volts, as illustrated below.

Equivalent of RMS Voltage

Finding the Root Mean Square (RMS) voltage of a sine wave can be done in two main ways:

• Graphical Method: This method is applicable to any irregular waveform and not just sine waves. It involves dividing the waveform into sections and then estimating the average value over a period.
• Analytical Method: This mathematical approach uses calculus to determine the RMS voltage of any periodic waveform including sine waves.

In simpler terms, RMS voltage represents the equivalent value of a DC voltage, that would produce the same heating effect as the given AC sine wave.

Finding RMS voltage with the Graphical Method:

This method works for any waveform, but we’ll use a sine wave’s positive half for simplicity.

The key idea is, to approximate the average effect of the voltage by taking readings at specific points.

We can divide the positive half cycle into equal sections called mid-ordinates.

The more mid-ordinates we use, the more accurate our answer will be.

Each mid-ordinate will have a width representing a certain time interval (not degrees), and a height representing the voltage at that specific time.

Let’s understand the process in a step-wise manner:

1. Squaring the Mid-Ordinates: Each voltage value at the midpoint of a time interval (mid-ordinate) is squared. This emphasizes the magnitude of the voltage regardless of positive or negative sign.
2. Summing the Squares: The squared values from all mid-ordinates are added together. This captures the total squared voltage over the waveform.
3. Finding the Mean of the Squares: The sum of the squared values is divided by the total number of mid-ordinates used. This provides the average squared voltage.
4. Taking the Root: The square root of the mean squared voltage is calculated. This process cancels out the squaring done earlier and provides the RMS voltage.

Basically here, the RMS voltage represents the equivalent value of a DC voltage that would produce the same heating effect in a resistor as the AC voltage waveform.

The mid-ordinate method offers a graphical approach so that we can calculate RMS voltage without needing complex mathematical analysis of the entire waveform.

V_RMS = √[sum of mid-ordinate (voltage)²] / number of mid-ordinates.

For the above straightforward example, the RMS voltage could be computed as follows:

V_RMS = √(V²₁ + V²₂ + V²₃……..+ V²₁₁ + V²₁₂ / 12)

For illustration purposes, let’s consider an AC voltage with a peak value (V_pk) of 20 volts.

We’ll use, the mid-ordinate method to estimate its RMS voltage. Here’s a sample data set:

Then, we can calculate the RMS voltage as shown below:

• √(6.2²+11.8²+16.2²+19²+20²+19²+16.2²+11.8²+6.2²+0²) / 10
• VRMS = √2000 / 10 = √200 = 14.14 Volts

The analysis reveals an RMS voltage of 14.14 volts, based on the mid-ordinate graphical approach.

Using Analytical Method to Determine RMS Voltage

The previously discussed graphical method using mid-ordinates is a powerful tool for determining the effective or Root Mean Square (RMS) value of alternating waveforms, that deviate from a pure sine wave. These waveforms are often referred to as complex waveforms, due to their irregular shapes.

But with sinusoidal waveforms, which are characterized by a smooth, repetitive pattern, a more efficient method exists – the analytical method. This method leverages mathematical calculations to obtain the RMS value.

A periodic sinusoidal voltage can be represented by the equation V_(t) = V_max * cos(ωt), where:

• V_(t) represents the instantaneous voltage at a specific time (t).
• V_max signifies the peak voltage of the waveform.
• ω denotes the angular frequency.

The analytical method allows us to calculate the RMS value (V_RMS) of this sinusoidal voltage using the following formula:

V_RMS = √[(1/T)∫^T₀ V²_mcos²(ωt)dt]

After performing the integration (which involves calculating the area under a curve), we take the limits of integration from 0 to 360 degrees (or T, representing the period of the waveform). This integration process provides the following result:

V_RMS = √(V²_m / 2T)[t + (1/2ωt)(sin(2ωt)]^T₀

• Vm: This represents the peak or maximum value of the waveform, also known as the peak voltage.
• ω: This symbol denotes the angular frequency. We can rewrite it using the relationship ω = 2π/T, where T represents the period of the waveform (time for one complete cycle).

Formula for RMS Voltage

• V_RMS = V_pk(1/√2)
• = V_pk * 0.7071

The RMS (Root Mean Square) voltage of a sinusoidal waveform, can be efficiently determined using a simple formula. This formula multiplies the peak voltage (V_pk) of the wave by 0.7071 which is mathematically equivalent to one divided by the square root of two (1/√2).

It’s important to note, that RMS voltage, also referred to as the effective value, represents the magnitude of the AC voltage and its heating effect in a circuit. It’s independent of both the waveform’s frequency and its phase angle.

Let’s for example, consider the graphical example mentioned earlier where the peak voltage (Vpk) was 20 volts.We can calculate the RMS voltage (V_RMS) using the formula:

V_RMS = V_pk * 0.7071 = 20 x 0.7071 ≈ 14.14 volts

The voltage of 14.14 volts corresponds to the previous graphical approach. Therefore we have the option to employ either the graphical mid-ordinate method, or the analytical approach for determining the RMS voltage or current of a sinusoidal waveform.

It’s important to note that, the multiplication of the peak or maximum value by the constant 0.7071 is applicable solely to sinusoidal waveforms. Non-sinusoidal waveforms necessitate the use of the graphical method.

Also besides utilizing the peak or maximum value of the sinusoid, we can also employ the peak-to-peak (V_P-P) value or the average (V_AVG) value to ascertain that the equivalent root mean squared value of the sinusoid, as demonstrated below:

Sine wave RMS Calculations

• V_RMS = (1 / √2) * V_pk = 0.7071 * V_pk
• V_RMS = (1 / 2√2) * V_p-p = 0.3536 * V_p-p
• V_RMS = (π / 2√2) * V_avg = 1.11 * V_avg

Conclusions

When addressing alternating voltages (or currents) we encounter the challenge of representing their magnitudes. One approach is to utilize peak values for the waveform while another common method involves using the effective value, also referred to as the Root Mean Square (RMS) value.

The RMS value of a sinusoid differs from the average of its instantaneous values. The ratio of the RMS voltage to its maximum value is identical to that of the RMS current to its maximum value.

Most multi-meters whether voltmeters or ammeters, gauge RMS values presuming a pure sinusoidal waveform. However for assessing the RMS value of non-sinusoidal waveforms, a “True RMS Multimeter” is necessary.

The RMS value of a sinusoidal waveform generates the same heating effect as a DC current of equivalent magnitude. Take for example if a direct current (I) passes through a resistance (R) in ohms, the resulting DC power consumed by the resistor as heat equals I²R watts.

Same way if an alternating current (i = I_max * sinθ) flows through the same resistance, the AC power converted into heat amounts to: I²rms * R watts.

When dealing with alternating voltages and currents, it’s advisable to treat them as RMS values, unless specified otherwise. Thus an alternating current of 10 amperes yields the same heating effect as a direct current of 10 amperes and a maximum value of 14.14 amperes.

Having established the RMS value of an alternating voltage (or current) waveform our next tutorial investigates calculating the Average value (V_AVG) of an alternating voltage and ultimately compare the two.

References: Root mean square

Definition and facts