RMS value of Alternating values

Question

What is a RMS value and what is the point of RMS value? Why do we calculate it? Please be concise and precise and kindly express it in layman terms.

Answer 1

The answer linked explains quite well what RMS is and what it signifies, I will try to explain why it is so important and useful. As Electrical Engineers we often discuss signals in terms of Power and Phase, as opposed to Voltage and Current, and RMS values of Current and Voltage play a critical role in expressing AC Power.

If we have some $V(t)=V_0\sin(\omega t)$ and some $I(t)=I_0\sin(\omega t -\phi)$, then we will obtain for the active power:

$$P_{\text{ave}}=\frac{I_0 V_0}{T}\int_0^T \sin(\omega t) \sin(\omega t -\phi)$$

You can use trigonometric identities to show that :$\frac{1}{T} \int_0^T \sin(\omega t) \sin(\omega t -\phi) = \frac{1}{2} \cos(\phi)$. Notice this would hold if we changed the overall/global phase of $V(t)$ and $I(t)$, so our initial reference does not matter since we are integrating over the full period.

We can now say:

$$P_{\text{ave}}= \frac{1}{2} I_0 V_0 \cos(\phi)= \frac{V_0}{\sqrt2} \frac{I_0}{\sqrt2} \cos(\phi)=V_{\text{RMS}} I_{\text{RMS}} \cos(\phi)$$

A similar procedure can be done for reactive power $Q_{\text{ave}}=V_{\text{RMS}} I_{\text{RMS}} \sin(\phi)$

This expression for average power looks almost exactly like the expression in the DC case, and we can account for the phase factors by expressing the Power in the phasor domain, which will allow us to understand the reactive and active power components of the power vector much better.

$$\tilde{S}=\tilde{V}_{\text{RMS}} \tilde{I}_{\text{RMS}}^* \text{ , where: } P_{\text{ave}}= \text{Re}\{ \tilde{S} \} \text{ , and: } Q_{\text{ave}}= \text{Im}\{ \tilde{S} \}$$

I hope you can now see that the RMS values of current and voltage are not necessary for understanding signals, but they allow us to express AC power in a similar way do DC power, and of course with the use of Fourier Series and superposition, we can then expand this idea to the power of any periodic signal which is incredibly useful in Communication Systems, Control Systems and obviously Power Systems.