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The basic alternating waveform for all complex waveforms is the sine wave. Therefore, an understanding of sine wave characteristics and how they can be acted upon is essential for you to understand modulation. You may want to review sine waves in chapter 1 of NEETS, Module 2, Introduction to Alternating Current and Transformers at this point. Since numbers represent individual items in a group, arrows can be used to represent quantities that have magnitude and direction. This may be done by using an arrow and a number, as illustrated in figure 1-1, view (A). The number represents the magnitude of force and the arrow represents the direction of the force. Figure 1-1A. - Vectors representing magnitude and direction.
View (B) illustrates a simpler method of representation. In this method, the length of the arrow is proportional to the magnitude of force, and the direction of force is indicated by the direction of the arrow. Thus, if an arrow 1-inch long represents 50 pounds of force, then an arrow 2-inches long would represent 100 pounds of force. This method of showing both magnitude and direction is called a VECTOR. To more clearly show the relationships between the amplitude, phase, and frequency of a sine wave, we will use vectors. Figure 1-1B. - Vectors representing magnitude and direction.
Vector Applied to Sine-Wave Generation As covered, in NEETS, Module 2, Introduction to Alternating Current and Transformers, an alternating current is generated by rotating a coil in the magnetic field between two magnets. As long as the magnetic field is uniform, the output from the coil will be a sine wave, as shown in figure 1-2. This wave shape is called a sine wave because the voltage of the coil depends on its angular position in the magnetic field. Figure 1-2. - Sine-wave generator.
This relationship can be expressed mathematically by the formula:
You should recall that the trigonometric ratio (inset in figure 1-3) for the sine in a right triangle (a triangle in which one angle is 90 degrees) is:
When an alternating waveform is generated, the coil is represented by a vector which has a length that is equal to the maximum output voltage (Emax). The output voltage at any given angle can be found by applying the above trigonometric function. Because the output voltage is in direct relationship with the sine of the angle q, it is commonly called a sine wave.You can see this relationship more clearly in figure 1-3 where the coil positions in relation to time are represented by the numbers 0 through 12. The corresponding angular displacements, shown as q, are shown along the horizontal time axis. The induced voltages (V1 through V12) are plotted along this axis. Connecting the induced voltage points, shown in the figure, forms a sine-wave pattern. This relationship can be proven by taking any coil position and applying the trigonometric function to an equivalent right triangle. When the vector is placed horizontally (position 0), the angle q is 0 degrees. Since e = Emax sine q, and the sine of 0 degrees is 0, the output voltage is 0 volts, as shown below:
Figure 1-3. - Generation of sine-wave voltage.
At position 2, the sine of 60 degrees is 0.866 and an output of 86.6 volts is developed.
This relationship is plotted through 360 degrees of rotation. A continuous line is drawn through the successive points and is known as 1 CYCLE of a sine wave. If the time axis were extended for a second revolution of the vector plotted, you would see 2 cycles of the sine wave. The 0-degree point of the second cycle would be the same point as the 360-degree point of the first cycle. Q.5 What waveform is the basis of all complex waveforms? |
 
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