The mean value and the root-mean-square value of a function 14.2
Introduction Currents and voltages often vary with time and engineers may wish to know the average value of such a current or voltage over some particular time interval. The average value of a time-varying function is defined in terms of an integral. An associated quantity is the root-mean-square (r.m.s) value of a current which is used, for example, in the calculation of the power dissipated by a resistor.
① be able to calculate definite integrals Prerequisites ② be familiar with a table of trigonometric Before starting this Section you should ... identities
✓ calculate the mean value of a function Learning Outcomes After completing this Section you should be ✓ calculate the root-mean-square value of a able to ... function 1. Average value of a function Suppose a time-varying function f(t)isdefined on the interval a ≤ t ≤ b. The area, A, under the graph of f(t)isgiven by the integral b A = f(t)dt a This is illustrated in Figure 1.
f(t)
h
a b t Figure 1. the area under the curve from t = a to t = b and the area of the rectangle are equal On Figure 1 we have also drawn a rectangle with base spanning the interval a ≤ t ≤ b and which has the same area as that under the curve. Suppose the height of the rectangle is h. Then area of rectangle = area under curve b h(b − a)= f(t)dt a 1 b h = − f(t)dt b a a The value of h is the average or mean value of the function across the interval a ≤ t ≤ b.
Key Point The average value of a function f(t)inthe interval a ≤ t ≤ b is 1 b − f(t)dt b a a
The average value depends upon the interval chosen. If the values of a or b are changed, then the average value of the function across the interval from a to b will change as well.
HELM (VERSION 1: March 18, 2004): Workbook Level 1 2 14.2: The mean value and the root-mean-square value of a function Example Find the average value of f(t)=t2 over the interval 1 ≤ t ≤ 3.
Solution Here a =1and b =3. 1 b average value = f(t)dt b − a a 3 3 3 1 2 1 t 13 = − t dt = = 3 1 1 2 3 1 3
Find the average value of f(t)=t2 over the interval 2 ≤ t ≤ 5.
Here a =2and b =5. Your solution
average value =
2 2 5
−
t d t
2 1 5
Now evaluate the integral. Your solution
average value = 13
3 HELM (VERSION 1: March 18, 2004): Workbook Level 1 14.2: The mean value and the root-mean-square value of a function Exercises
1. Calculate the average value of the given functions across the specified interval.
(a) f(t)=1+t across [0, 2] (b) f(x)=2x − 1 across [−1, 1] (c) f(t)=t2 across [0, 1] (d) f(t)=t2 across [0, 2] (e) f(z)=z2 + z across [1, 3]
2. Calculate the average value of the given functions over the specified interval.
(a) f(x)=x3 across [1, 3] (b) f(x)= 1 across [1, 2] √x (c) f(t)= t across [0, 2] (d) f(z)=z3 − 1 across [−1, 1] 1 − − (e) f(t)= t2 across [ 3, 2] 3. Calculate the average value of the following: π (a) f(t)=sin t across 0, 2 (b) f(t)=sin t across [0,π] (c) f(t)=sin ωt across [0,π] π (d) f(t)=cos t across 0, 2 (e) f(t)=cos t across [0,π] (f) f(t)=cos ωt across [0,π] (g) f(t)=sin ωt + cos ωt across [0, 1]
4. Calculate the average value of the following functions: √ (a) f(t)= t +1across [0, 3] (b) f(t)=et across [−1, 1]
(c) f(t)=1+et across [−1, 1]
9
.(a) 4. 1752 . 2 (c) 1752 . 1 (b)
14
ω
(g)
ω cos ω sin 1+ −
πω π πω π π
e (f) 0 (e) (d) )] πω cos( [1 (c) (b) .(a) 3.
−
) πω sin( 2 1 2 2
6
(e) 1 (d) 0.9428 (c) 0.6931 (b) 10 (a) 2.
−
1
3 3 3
(e) (d) (c) 1 (b) 2 (a) 1. Answers
−
19 4 1
HELM (VERSION 1: March 18, 2004): Workbook Level 1 4 14.2: The mean value and the root-mean-square value of a function 2. Root-mean-square value of a function.
If f(t)isdefined on the interval a ≤ t ≤ b, the mean-square value is given by the expression: b 1 2 − [f(t)] dt b a a
This is simply the average value of [f(t)]2 over the given interval. The related quantity: the root-mean-square (r.m.s.) value is given by the following formula.
Key Point b 1 2 r.m.s value = − [f(t)] dt b a a
The r.m.s. value depends upon the interval chosen. If the values of a or b are changed, then the r.m.s value of the function across the interval from a to b will change as well. Note that when finding an r.m.s. value the function must be squared before it is integrated.
Example Find the r.m.s. value of f(t)=t2 across the interval from t =1tot =3.
Solution
1 b r.m.s = [f(t)]2dt b − a a 1 3 = [t2]2dt 3 − 1 1 1 3 1 t5 3 = t4dt = =4.92 2 1 2 5 1
5 HELM (VERSION 1: March 18, 2004): Workbook Level 1 14.2: The mean value and the root-mean-square value of a function Example Calculate the r.m.s value of f(t)=sin t across the interval 0 ≤ t ≤ 2π.
Solution Here a =0and b =2π. 1 2π r.m.s = sin2 tdt 2π 0 The integral of sin2 t is performed by using trigonometrical identities to rewrite it in the alter- 1 − native form 2 (1 cos 2t). This technique was described in Chapter 13 section 7. 1 2π (1 − cos 2t) r.m.s. value = dt 2π 2 0 1 2π = (1 − cos 2t)dt 4π 0 1 sin 2t 2π = t − 4π 2 0 1 1 = (2π)= =0.707 4π 2 Thus the r.m.s value is 0.707.
In the previous example the amplitude of the sine wave was 1, and the r.m.s. value was 0.707. In general, if the amplitude of a sine wave is A, its r.m.s value is 0.707A.
Key Point
The r.m.s value of any sinusoidal waveform taken across an interval equal to one period is 0.707 × amplitude of the waveform.
HELM (VERSION 1: March 18, 2004): Workbook Level 1 6 14.2: The mean value and the root-mean-square value of a function Exercises
1. Calculate the r.m.s. values of the functions in question 1 of the previous Exercises.
2. Calculate the r.m.s. values of the functions in question 2 of the previous Exercises.
3. Calculate the r.m.s. values of the functions in question 3 in the previous Exercises.
4. Calculate the r.m.s. values of the functions in question 4 in the previous Exercises.
.()151 b .46()2.2724 (c) 1.3466 (b) 1.5811 (a) 4.
ω
1+ (g)
ω sin 2
πω 2 2 πω 2 2
+ (f) 0.7071 (e) 0.7071 (d) .()077 b .01(c) 0.7071 (b) 0.7071 (a) 3.
−
πω cos πω sin 1 πω cos πω sin 1
.()1.97()077 c d .60()0.1712 (e) 1.0690 (d) 1 (c) 0.7071 (b) 12.4957 (a) 2.
.()201 b .25()047 d .89()6.9666 (e) 1.7889 (d) 0.4472 (c) 1.5275 (b) 2.0817 (a) 1. Answers
7 HELM (VERSION 1: March 18, 2004): Workbook Level 1 14.2: The mean value and the root-mean-square value of a function