9 Week 9

In this section, we consider the mean and root mean square values of a function, which is important in studying sinusoidal type functions.

9.1 Mean and RMS Values

The mean value for a function $y = f(x)$ for the range $a$ to $b$ is the mean value of all the ordinates in that range. Consider a function $y = f(x)$ for the values $x = a$ to $x = b$, divided into a number of strips of equal width $\delta x$. Let the mid-ordinate for each strip be $y_1, y_2, y_3,\ldots,y_n$, see Figure 9.1.

Figure 9.1: $f(x)$

The mean or average value of the mid-ordinates is \[ \overline y=\frac{y_1+\ldots+y_n}{n} \]

Multiplying the numerator and denominator by $\delta x$ gives \[ \overline y=\frac{y_1\delta x+\ldots+y_n\delta x}{n\delta x} \] and since $n\delta x = b - a$ we have \[ \overline y = \frac{\sum y\delta x}{b-a}. \] When the number of strips increases indefinitely such that $\delta x \to 0$ \[ \overline y = \frac{1}{b-a}\int_a^b y\ \mathrm{d}x. \] That is, the mean value of a function between two limits is the area under the curve between those limits divided by the range of $x$ between those limits.

Example 9.1 If the velocity $v$ in $m/s$ of a body is given by $v=4t+3$ where $t$ is in seconds, determine the mean velocity of the body from $t=2\ $s to $t=6\ $s.

Solution. \[\begin{align*} \text{Mean velocity}& = \frac{1}{6-2}\int_2^6(4t+3)\ \mathrm{d}t = \frac 14\left[2t^2+3t\right]_2^6\\ &= \frac 14\left[(2\times36+3\times6) -(2\times4+3\times2)\right] = \frac{76}{4}=19\ m/s. \end{align*}\]

9.1.1 RMS Value

The RMS (or root mean square) value is obtained by taking $f(x)$, squaring it, finding the mean value and then taking the square root.

Its importance in engineering is demonstrated when a sinusoidal function is investigated, for example when the average value of the current or voltage in a circuit is to be found. For all sine or cosine functions, the mean value over a complete cycle is zero because the curve is equally distributed above and below the $x-$axis. However, if the ordinates are squared first, there are then no negative ordinates and an average may be found. Ammeters and voltmeters for a.c. circuits are usually calibrated to find the RMS value.

To summarise, the RMS value is defined to be the number \[ \text{RMS} = \sqrt{\frac{1}{b-a}\int_a^b y^2\ \mathrm{d}x}. \]

Example 9.2 Determine the RMS value of the current in an a.c. circuit given by $i=20+100\sin(100\pi t)$ between $t=0$ and $t=0.02$ s.

Solution. \[\begin{align*} \text{RMS}&= \sqrt{\frac{1}{b-a}\int_a^b i^2\ \mathrm{d}t}\\ i&=20+100\sin(100\pi t)\\ i^2&=400+4000\sin(100\pi t)+10000\sin^2(100\pi t)\\ &=400+40000\sin(100\pi t)+10000\times\frac 12(1-\cos(200\pi t))\\ \text{RMS}^2&=\frac1{0.02-0}\int_0^{0.02}(400+40000\sin(100\pi t)+10000\times\frac 12(1-\cos(200\pi t)))\ \mathrm{d}t\\ &=\frac1{0.02}\left[400t-\frac{40000\cos(100\pi t)}{100\pi}+5000t-\frac{5000\sin(200\pi t)}{200\pi}\right]_0^{0.02}\\ &=\frac1{0.02}\left[(8-12.732+100-0)-(0-12.732+0-0)\right] = \frac1{0.02}\times108\\ \text{RMS}&=\sqrt{\frac{108}{0.02}}=73.5\text{ A} \end{align*}\]