Root Mean Square Values

Root Mean Square Values and Alternating Current

Recall that in the case of alternating current (AC) the flow of electric charge periodically reverses direction. Unlike direct current (DC), where the currents and voltages are constant, AC currents and voltages vary over time. Recall that most residential and commercial power sources use AC. It is often the case that we wish to know the time averaged current, or voltage. Given the current or voltage as a function of time, we can take the root mean square over time to report the average quantities.

Definition

The root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when the function alternates between positive and negative values, e.g., sinusoids.The RMS value of a set of values (or a continuous-time function such as a sinusoid) is the square root of the arithmetic mean of the squares of the original values (or the square of the function). In the case of a set of n values {x₁,x₂,....,x_n} , the RMS value is given by this formula:

$x_{rms}=\sqrt{\frac{1}{n}(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2})}$

The corresponding formula for a continuous function f(t) defined over the interval T₁ ≤ t ≤ T₂ is as follows:

$f_{rms}=\sqrt{\frac{1}{T_{2}-T_{1}}\int_{T_{1}}^{T_{2}}[f(t)]^{2}dt}$

The RMS for a function over all time is below.

$f_{rms}=\lim_{T \to \infty }\sqrt{\frac{1}{T}\int_{0}^{T}[f(t)]^{2}dt}$

The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a series of equally spaced samples.

Application to Voltage and Current

Consider the case of sinusoidally varying voltage :

Sinusoidal Voltage and Current

(a) DC voltage and current are constant in time, once the current is established. (b) A graph of voltage and current versus time for 60-Hz AC power. The voltage and current are sinusoidal and are in phase for a simple resistance circuit. The frequencies and peak voltages of AC sources differ greatly.

$V=V_{0}sin(2\pi ft)$

V is the voltage at time t, V₀ is the peak voltage, and f is the frequency in hertz. For this simple resistance circuit, I=V/R, and so the AC current is as follows:

$I=I_{0}sin(2\pi ft)$

Here, I is the current at time t, and I₀=V₀/R is the peak current. Now using the definition above, let's calculate the rms voltage and rms current. First, we have

$V_{rms}=\sqrt{\frac{1}{T_{2}-T_{1}}\int_{T_{1}}^{T_{2}}[V_{0}sin(\omega t)]^{2}dt}$

Here, we have replaced 2πf with ω. Since V₀ is a constant, we can factor it out of the square root, and use a trig identity to replace the squared sine function.

$V_{rms}=V_{0}\sqrt{\frac{1}{T_{2}-T_{1}}\int_{T_{1}}^{T_{2}}\frac{1-cos(2\omega t)}{2}dt}$

Integrating the above, we have:

$V_{rms}=V_{0}\sqrt{\frac{1}{T_{2}-T_{1}}[\frac{t}{2}-\frac{sin(2\omega t)}{4\omega }]_{T_{1}}^{T_{2}}}$

Since the interval is a whole number of complete cycles (per definition of RMS), the terms will cancel out, leaving:

$V_{rms}=V_{0}\sqrt{\frac{1}{T_{2}-T_{1}}[\frac{t}{2}]_{T_{1}}^{T_{2}}}=V_{0}\sqrt{\frac{1}{T_{2}-T_{1}}\frac{T_{2}-T_{1}}{2}}$

$=\frac{V_{0}}{\sqrt{2}}$

Similarly, you can find that the RMS current can be expressed fairly simply:

$I_{rms}=I_0/\sqrt{2}$

Updated Circuit Equations for AC

Many of the equations we derived for DC current apply equally to AC. If we are concerned with the time averaged result and the relevant variables are expressed as their rms values. For example, Ohm's Law for AC is written as follows:

$I_{rms}=\frac{V_{rms}}{R}$

The various expressions for AC power are below:

$P_{ave}=I_{rms}V_{rms}$

$P_{ave}=\frac{V_{rms}^{2}}{R}$

$P_{ave}=I_{rms}^{2}R$

We can see from the above equations that we can express the average power as a function of the peak voltage and current (in the case of sinusoidally varying current and voltage):

Average Power

AC power as a function of time. Since the voltage and current are in phase here, their product is non-negative and fluctuates between zero and I0V0. Average power is (1/2)I0V0.

$P_{ave}=I_{rms}V_{rms}=\frac{I_{0}}{\sqrt{2}}\frac{V_{0}}{\sqrt{2}}=\frac{1}{2}V_{0}I_{0}$

The RMS values are also useful if the voltage varies by some waveform other than sinusoids, such as with a square, triangular or sawtooth waves .

Waveforms

Sine, square, triangle, and sawtooth waveforms