Question

Root Mean Square (RMS) Value

Answer 1

Let's do the exact same process with a sine wave, which is the perfect real-world example. The principle is identical.

The Goal is the Same

Remember our goal: We have an AC sine wave voltage. We want to find the equivalent DC voltage that produces the same amount of heat. This effective value is the RMS value.

And we will use the same recipe: SQUARE -> MEAN -> ROOT.

Visualizing the Sine Wave

Imagine the voltage from your wall outlet. It's a smooth, flowing wave.

• It starts at 0 Volts.
• It smoothly rises to a Peak Voltage (let's call it Vp).
• It smoothly falls back to 0 Volts.
• It then goes negative, smoothly down to a negative peak (-Vp).
• Finally, it comes back to 0 Volts to complete the cycle.

Let's use a clear example. Imagine we have an AC signal with a Peak Voltage (Vp) of 10 Volts. This means it swings between +10V and -10V.

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Let's Follow the RMS Recipe: Square -> Mean -> Root

Step 1: S for SQUARE

We take every single instantaneous value of the sine wave and square it.

• What happens to the positive half of the wave (0V to +10V and back)? It gets squared. The shape changes a bit—it becomes a taller, pointier "hump." The peak of this new squared hump is now 10² = 100.
• What happens to the negative half of the wave (0V down to -10V and back)? When you square a negative number, it becomes positive! So, the negative part flips up and becomes an identical positive hump.

The result: Our original up-and-down sine wave is transformed into a new wave that is always positive. It looks like a series of identical, connected "humps" that bounce off the zero line. The peak value of this new squared wave is (Vp)², which is 100 in our example.

Step 2: M for MEAN (Average)

Now we need to find the average value of this new "bouncy hump" wave.

Look at the squared wave. It goes from 0 up to a peak of 100, and back to 0. It does this over and over. What would its average height be?

This requires a little bit of calculus, but the result is very simple and intuitive: The average value of a squared sine wave is exactly half of its peak value.

So, the peak of our squared wave was (Vp)² = 100.

The Mean (average) is: (Vp)² / 2 = 100 / 2 = 50.

So, the average of our squared sine wave is 50. This represents the average heating power.

Step 3: R for ROOT

Finally, we take the square root of the Mean we just found. This brings the value back into the correct unit (Volts).

The Mean was 50.

The Root is: √50 ≈ 7.07 Volts.

There it is! The RMS value of a 10V peak sine wave is 7.07V.

This means our AC sine wave that swings from +10V to -10V delivers the exact same heating power as a steady 7.07V DC battery.

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The Magic Formula for Sine Waves

From the steps above, we can create a simple formula that works for any sine wave:

1. We started with the Peak Voltage, Vp.
2. We squared it: (Vp)²
3. We took the mean (divided by 2): (Vp)² / 2
4. We took the square root: √( (Vp)² / 2 )

The math simplifies to: Vrms = Vp / √2

Since 1 / √2 is approximately 0.707, the formula is often written as:

Vrms = Vp × 0.707

Bringing It Back to Your Wall Outlet

Now we can finally understand the voltage in our homes.

Let's say you're in the US, and the standard voltage is 120V AC. This 120V is the RMS value. It's the "effective" voltage.

What is the actual peak voltage coming out of the wall? We can use our formula in reverse:

Vp = Vrms × √2
Vp = 120V × 1.414
Vp ≈ 170 Volts

This is amazing! It means the voltage from a standard 120V outlet is actually a sine wave that swings all the way up to +170V and down to -170V, 60 times every second. But its effective heating power is the same as a 120V DC battery.

That's why a multimeter set to "AC Volts" will read 120V—it's built to show you the useful RMS value.