0195136047.pdf

1.1 ELECTRICAL QUANTITIES 15

Theeffective,orroot-mean square(rms), value is the square root of the average off^2 (t),

Frms=

√√

√

√

√

1

T

∫T

0

f^2 (t) dt (1.1.23)

Determining the square of the functionf(t), then finding the mean (average) value, and finally
taking the square root yields the rms value, known as effective value. This concept will be seen
to be useful in comparing the effectiveness of different sources in delivering power to a resistor.
The effective value of a periodic current, for example, is a constant, or dc value, which delivers
the same average power to a resistor, as will be seen later.
For the special case of a dc waveform, the following holds:
f(t)=F; Fav=Frms=F (1.1.24)

For the sinusoid or cosinusoid, it can be seen that

f(t)=Asin(ωt+φ); Fav= 0 ; Frms=A/

√

2 ∼= 0 .707 A (1.1.25)

The student is encouraged to show the preceding results using graphical and analytical means.
Other common types of waveforms areexponentialin nature,

f(t)=Ae−t/τ (1.1.26a)

f(t)=A( 1 −e−t/τ) (1.1.26b)

whereτis known as thetime constant. After a time of one time constant has elapsed, looking at
Equation (1.1.26a), the value of the waveform will be reduced to 37% of its initial value; Equation
(1.1.26b) shows that the value will rise to 63% of its final value. The student is encouraged to
study the functions graphically and deduce the results.

EXAMPLE 1.1.5

A periodic current waveform in a rectifier is shown in Figure E1.1.5. The wave is sinusoidal for
π/ 3 ≤ωt≤π, and is zero for the rest of the cycle. Calculate the rms and average values of the
current.

i

10

π

3

π 2 π ωt

Figure E1.1.5

Solution

Irms=

√√

√

√

√

√

1

2 π







∫π/^3

0

i^2 d(ωt)+

∫π

π/ 3

i^2 d(ωt)+

∫^2 π

π

i^2 d(ωt)





