Fundamental Engineering Concepts - Mathematics and Statistics for the Environment - page 292

11.7.21.8 Effective or RMS Value
The effective value of an AC voltage or current of sine waveform is defined in terms of an equivalent
heating effect of a direct current. Heating effect is independent of the direction of current flow.

Key Point: Because all instantaneous values of induced voltage are somewhere between 0 and E m
(maximum or peak voltage), the effective value of a sine wave voltage or current must be greater
than 0 and less than E m .
The AC of a sine waveform having a maximum value of 14.14 amps produces the same amount
of heat in a circuit having a resistance of 1 ohm as a direct current of 10 amps. For this reason, we
can work out a constant value for converting any peak value to a corresponding effective value. In
the simple equation below, x represents this constant (solve for x to three decimal places):

14.14 x = 10

x = 0.707

The effective value is also called the root-mean-square (RMS) value because it is the square root of
the average of the squared values between zero and maximum. The effective value of an AC current
is stated in terms of an equivalent DC current. The phenomenon used for standard comparisons is
the heating effect of the current.

Note: Anytime an AC voltage or current is stated without any qualifications, it is assumed to be an
effective value.

In many instances, it is necessary to convert from effective to peak or vice versa using a standard
equation. Figure 11.57 shows that the peak value of a sine wave is 1.414 times the effective value;
therefore, the equation we use is

E m = E × 1.414 (11.51)

where
E m = Maximum or peak voltage.
E = Effective or RMS voltage.

and

I m = I × 1.414 (11.52)

where
I m = Maximum or peak current.
I = Effective or RMS current.

Occasionally, it is necessary to convert a peak value of current or voltage to an effective value.
This is accomplished by using the following equations:

E = E m × 0.707 (11.53)

where
E = Effective voltage.
E m = Maximum or peak voltage.

I = I m × 0.707 (11.54)

where
I = Effective current.
I m = Maximum or peak current.