Fundamental Engineering Concepts - Mathematics and Statistics for the Environment - page 294

Voltage wave

Current wave

270° 360°

0° 90° 180°

Time
axis

FIGURE 11.58 Voltage and current waves in phase.

E 2

E 1

90°

Time

0° 90° 180° 270° 360°

Time
one

Time
two

FIGURE 11.59 Voltage waves 90° out of phase.

Note: It is important to remember that Ohm's law for DC circuits is applicable to AC circuits with
resistance only.

Voltage waves are not always in phase. Figure 11.59 shows a voltage wave ( E 1 ) considered to start
at 0° (time 1). As voltage wave E 1 reaches its positive peak, a second voltage wave ( E 2 ) begins to rise
(time 2). Because these waves do not pass through their maximum and minimum points at the same
instant of time, a phase difference exists between the two waves. The two waves are said to be out
of phase . For the two waves in Figure 11.59, this phase difference is 90°.

11.7.11.11 Pha se Relationships
In the preceding section, we discussed the important concepts of in phase and phase difference .
Another important phase concept is phase angle . The phase angle between two waveforms of the
same frequency is the angular difference at a given instant of time. As an example, the phase angle
between waves B and A (see Figure 11.60) is 90°. Take the instant of time at 90°. The horizontal axis
is shown in angular units of time. Wave B begins at maximum value and reduces to 0 value at 90°,
whereas wave A begins at 0 and increases to maximum value at 90°. Wave B reaches its maximum
value 90° ahead of wave A, so wave B leads wave A by 90° (and wave A lags wave B by 90°). This
90° phase angle between waves B and A is maintained throughout the complete cycle and all suc-
cessive cycles. At any instant of time, wave B has the value that wave A will have 90° later. Wave B
is a cosine wave because it is displaced 90° from wave A, which is a sine wave.