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## BASIC AC THEORY## AC Generation
The elementary AC generator (Figure 1) consists of a conductor, or loop of wire in a magnetic field that is produced by an electromagnet. The two ends of the loop are connected to slip rings, and they are in contact with two brushes. When the loop rotates it cuts magnetic lines of force, first in one direction and then the other. Figure 1 Simple AC GeneratorMenu
## Development of a Sine-Wave OutputAt the instant the loop is in the vertical position (Figure 2, 0°), the coil sides are moving parallel to the field and do not cut magnetic lines of force. In this instant, there is no voltage induced in the loop. As the coil rotates in a counter-clockwise direction, the coil sides will cut the magnetic lines of force in opposite directions. The direction of the induced voltages depends on the direction of movement of the coil. The induced voltages add in series, making slip ring X (Figure 1) positive (+) and slip ring Y (Figure 1) negative (-). The potential across resistor R will cause a current to flow from Y to X through the resistor. This current will increase until it reaches a maximum value when the coil is horizontal to the magnetic lines of force (Figure 2, 90°). The horizontal coil is moving perpendicular to the field and is cutting the greatest number of magnetic lines of force. As the coil continues to turn, the voltage and current induced decrease until they reach zero, where the coil is again in the vertical position (Figure 2, 180°). In the other half revolution, an equal voltage is produced except that the polarity is reversed (Figure 2, 270° 360°). The current flow through R is now from X to Y (Figure 1). Figure 2 Developing a Sine-Wave VoltageThe periodic reversal of polarity results in the generation of a voltage, as shown in Figure 2. The rotation of the coil through 360° results in an AC sine wave output. ## AC Generation Analysis
## Effective ValuesThe output voltage of an AC generator can be expressed in two ways. One is graphically by use of a sine wave (Figure 3). The second way is algebraically by the equation e = E Figure 3 Voltage Sine WaveWhen a voltage is produced by an AC generator, the resulting current varies in step with the voltage. As the generator coil rotates 360°, the output voltage goes through one complete cycle. In one cycle, the voltage increases from zero to E One way to refer to AC voltage or current is by peak voltage (E Another value, the peak-to-peak value (E The values of I are plotted on the upper curve, and the corresponding values of I There are six basic equations that are used to convert a value of AC voltage or current to another value, as listed below. The values of current (I) and voltage (E) that are normally encountered are assumed to be RMS values; therefore, no subscript is used. Figure 4 Effective Value of CurrentAnother useful value is the average value of the amplitude during the positive half of the cycle. Equation (7-7) is the mathematical relationship between I Equation (7-8) is the mathematical relationship between E Example 1: The peak value of voltage in an AC circuit is 200 V. What is the RMS value of the voltage? Example 2: The peak current in an AC circuit is 10 amps. What is the average value of current in the circuit? ## Phase AnglePhase angle is the fraction of a cycle, in degrees, that has gone by since a voltage or current has passed through a given value. The given value is normally zero. Referring back to Figure 3, take point 1 as the starting point or zero phase. The phase at Point 2 is 30°, Point 3 is 60°, Point 4 is 90°, and so on, until Point 13 where the phase is 360°, or zero. A term more commonly used is phase difference. The phase difference can be used to describe two different voltages that have the same frequency, which pass through zero values in the same direction at different times. In Figure 5, the angles along the axis indicate the phases of voltages e 180°). The voltage e Figure 5 Phase RelationshipPhase difference is also used to compare two different currents or a current and a voltage. If the phase difference between two currents, two voltages, or a voltage and a current is zero degrees, they are said to be "in-phase." If the phase difference is an amount other than zero, they are said to be "out-of-phase." ## Voltage CalculationsEquation (7-9) is a mathematical representation of the voltage associated with any particular orientation of a coil (inductor). Example 1: What is the induced EMF in a coil producing a maximum EMF of 120 V when the angle from reference is 45°? The maximum induced voltage can also be called peak voltage E Using substitution laws, a relationship between the voltage induced, the maximum induced voltage, and the angular velocity can be expressed. Equation (7-11) is the mathematical representation of the relationship between the voltage induced, the maximum voltage, and the angular velocity, and is equal to the output of an AC Generator. ## Current CalculationsMaximum induced current is calculated in a similar fashion. Equation (7-12) is a mathematical representation of the relationship between the maximum induced current and the angular velocity. ## Frequency CalculationsThe frequency of an alternating voltage or current can be related directly to the angular velocity of a rotating coil. The units of angular velocity are Example 1: The frequency of a 120 V AC circuit is 60 Hz. Find the following: - Angular velocity
- Angle from reference at 1 msec
- Induced EMF at that point
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