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Phase Opposition When two sine waves begin exactly 1 2 cycle, or 180 , apart, they are said to be in phase opposition. This is illustrated by the drawing of Fig. 12-8. In this situation, engineers sometimes say that the waves are out of phase, although this expression is a little nebulous because it could be taken to mean some phase difference other than 180 . If two sine waves have the same amplitudes and are in phase opposition, they cancel each other out. This is because the instantaneous amplitudes of the two waves are equal and opposite at every moment in time. If two sine waves are in phase opposition, and if neither wave has dc superimposed, then the resultant is a sine wave with positive or negative peak amplitudes equal to the difference between the positive and negative peak amplitudes of the composite waves. The phase of the resultant is the same as the phase of the stronger of the two composite waves. Any sine wave without superimposed dc has the unique property that, if its phase is shifted by 180 , the resultant wave is the same as turning the original wave upside down. Not all waveforms have this property. Perfect square waves do, but some rectangular and sawtooth waves don t, and irregular waveforms almost never do. Intermediate Phase Differences Two sine waves can differ in phase by any amount from 0 (phase coincidence), through 90 ( phase quadrature, meaning a difference a quarter of a cycle), 180 (phase opposition), 270 (phase quadrature again), to 360 (phase coincidence again).

Back in 9, degrees of phase were discussed. If you wondered then why phase is spoken of in terms of angular measure, the reason should be clearer now. A circle has 360

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degrees. A sine wave can be represented as circular motion. Exact moments along the sine curve correspond to specific angles, or positions, around a circle. Figure 12-5 shows the way a rotating vector is used to represent a sine wave. At A, the vector points east, and this is assigned the value of 0 degrees, where the wave amplitude is zero and is increasing positively. At B, the vector points north ; this is the 90-degree instant, where the wave has attained its maximum positive amplitude. At C, the vector points west. This is 180 degrees, the instant where the wave has gone back to zero amplitude, and is getting more negative. At D, the wave points south. This is 270 degrees and represents the maximum negative amplitude. When a full circle has been completed, or 360 degrees, the vector once again points east. Thus, 360 degrees is the same as 0 degrees. In fact, a value of x degrees represents the same condition as x plus or minus any multiple of 360 degrees. The four points for the model of Fig. 12-5 are shown on a sine wave graph in Fig. 12-6. You can think of the vector as going around and around, at a rate that corresponds to one revolution per cycle of the wave. If the wave has a frequency of 1 Hz, the vector goes around at a rate of a revolution per second (1 rps). If the wave has a frequency of 100 Hz, the speed of the vector is 100 rps, or a revolution every 0.01 second. If the wave is 1 MHz, then the speed of the vector is 1,000,000 or 106 rps, and it goes once around every 10 6, or 0.000001, second.

by 90 of phase (1 4 of a cycle).

the mobile appliance receives a transmission from the store, wakes up, and notifies you to stop and buy the items on the list. Good application Maybe. Bluetooth, named for a tenth-century Danish king (no, I don t know whether he had a blue tooth), is experiencing growing pains and significant competition for many good reasons from 802.11. Whether Bluetooth succeeds or not is a matter still open for discussion. It s far too early to tell.

At current loops, the loss in the line conductors is exaggerated. At voltage loops, the loss in the line dielectric is increased. At minima, the losses are decreased. But overall, in a mismatched line, the losses are greater than they are in a perfectly matched line.

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