AC Generators. Basic Generator

Transcription

1 AC Generators Basic Generator

2 One complete revolution = 1 complete Sinewave

3 2 methods for controlling frequency of generated waveform Frequency is directly proportional to the rate of rotation of the wire loop in an ac generator. Increasing the number of poles increases the frequency in this case doubling the number of poles doubles the frequency of the generated waveform

4 Sinewave Parameters Definition of AC (alternating current) Any waveform which alternates between positive and negative polarity is referred to as an alternating waveform. Unlike DC where the current through a circuit always moves in a single direction, in an AC circuit the current continually changes direction. Period (T) = time required (in seconds) for a waveform to complete one full cycle. Frequency (f) = the number of cycles that a waveform completes in one second. Frequency is measured in hertz (Hz), and one hertz is equivalent to one cycle per second. Hydro provides electricity with a frequency of 60 Hz or 60 cycles per second. Frequency and period are related by the following formula: f 1 T Instantaneous value ( V inst ) = value of the waveform at any given instant in time. Note that instantaneous values are represented by lowercase letters, to distinguish them from constant values. Peak Value ( V p )= maximum positive or negative value of a waveform (measured from a zero reference). For the waveform shown above, the positive and negative peaks have the same absolute value. Peak-to-peak value ( V pp)= the magnitude of voltage (or current) represented between the peaks of a waveform. In the figure above, the peak-to-peak value is simply twice the value of one of the peaks: Vpp 2V p or I pp 2I pp

5 Example: determine the frequency and period of the following waveforms Average value The average value of a waveform is defined as the algebraic sum of all the areas under the curve of the waveform over one complete cycle divided by the period of the waveform. The average value of any waveform is the value indicated on a DC meter. The average value of one complete cycle of a sinewave is zero, since the sum of the positive values equal the sum of the negative values In some cased, the average value of one half-cycle of a sinewave is used, and is given by the formulas: V 2 avg V p and I avg 2 I p

6 RMS or Effective value ( V rms ) = the value of a sinusoidal voltage that produces the same amount of heat (dissipates as much power) in a given resistance as an equivalent DC source would: The peak value ( V p ) of a waveform and its equivalent RMS value are related by a factor of follows: Vp VRMS and Vp 2 VRMS 2 2 as Remember, the RMS value of a sinewave is always less than its peak value (which is why it is divided by the 2 factor). Example: determine the parameters listed below for the following waveform V p = V pp = V rms = F = T = V inst at t 12 s

7 Radians (review) The ancient Babylonians arbitrarily chose to divide a circle into 360 equal parts, a standard which has become widely used to express angular measurements. In the degree measurement system, there are 360 in a complete circle. The radian is a more scientific method of measuring angles, and is based on the basic properties of a circle which state that the length of the circumference of a circle can be found by multiplying the length of its radius by the constant pi: C 2 r Like degrees, radians are angular measurements (they measure the distance traveled around a circle in terms of angles). One radian is defined as the angle formed when a distance equal to the radius of a circle is traveled around the circumference of the circle: Since the distance around the circumference is given by a complete circle. Thus 2 rad 360 or rad r it follows that there must be 2π radians in This means that you can convert from degrees to radians and vice versa by using a conversion factor of 180 or 180 Remember, since there are only 2π radians in a complete circle as opposed to 360 degrees, the 180 appropriate factor would be. Similarly, since there are 360 degrees in a complete circle compared to only 2π radians, the appropriate conversion factor would be. 180

8 Degrees vs. Radians

9 Angular Measurement of a sinewave Using time as a reference for representing sinewaves is intuitive but it does have one drawback since the cycle of a sinewave is frequency dependent, everything you measure on such a graph is relative to the frequency used, which is subject to change. It is therefore often convenient to represent sinewaves in terms of angles, rather than time. The angular measurement can be either degrees or radians. Recall that a sinusoidal voltage can be produced by an AC generator The choice of degrees to represent angular measure in the above diagram is arbitrary, and could easily be measured in radians.

10 Phase The phase of a sinewave is an angular measurement that specifies the position of that sinewave in relation to some reference (usually another sinewave). In part (a) of the above figure, since waveform A occurs 90 before waveform B, waveform A is said to lead waveform B by 90. Similarly, you could say that waveform B lags waveform A by 90. Notice that you could also say that waveform B leads waveform A by 270 or waveform A lags waveform B by 270. In part (b) of the above figure, waveform B leads (occurs earlier than) waveform A by 90, or waveform A lags (occurs later than) waveform B by 90. It would also be correct to say that waveform A leads waveform B by 270 or that waveform B lags waveform A by 270. It is common practice to refer to phase relations between waveforms using angles smaller than 180, although it is not a strict requirement. Thus it would be more common in part (a) to say that waveform A leads waveform B by 90 rather than waveform B leads waveform A by 270. Example: specify the phase angles between the two sine waves shown in the following figure:

11 Review Questions: 1. When the positive-going zero crossing of a sine wave occurs at 0, at what angle do each of the following points occur give values in both degrees and radians? (i) positive peak (ii) negative-going zero crossing (iii) negative peak (iv) end of first complete cycle 2. A half-cycle of a sine wave is completed in degrees, or radians. 3. A full-cycle of a sine wave is completed in degrees, or radians. 4. A quarter-cycle of a sine wave is completed in degrees, or radians. 5. Determine the phase angle (in degrees and radians) between the following two sine waves: A B by degrees OR B A by degrees A B by rads OR B A by rads 6. If the preceding waveforms had an RMS value of 90 V, find: (i) their peak voltage: V p = (ii) their peak-to-peak voltage: V pp =

12 The Sine Wave Expression A sine wave may be expressed mathematically using the following general expression: y Asin( ) where A = the amplitude of the sine wave theta (θ) = the angle (in degrees or radians) of the sine wave at any given instant phi (φ) = the phase angle (if applicable) relative to some reference In the case where the phase angle is zero, the expression becomes simply y Asin( ) Thus the above expression states that the instantaneous value of a sinewave at any point may be found by knowing two things: the angle θ at that point, and the maximum (peak) value of the sine wave. For example, the instantaneous value of the following voltage waveform may be found at any angle (60 in this case) by multiplying the sine of the associated angle with its peak value (10V): v Vp sin( 60 ) 8. 66V Shifted Sine Waves Note the polarities associated with the shifting a positive phase reflects a shift to the left (a leading angle) and a negative phase reflects a shift to the right (a lagging angle).

13 Example: determine the instantaneous values at the 90 reference point on the horizontal axis for each of the following sine waves: v A = v B = v C =

14 Ohm s Law and Kirchoff s Laws in AC Circuits When time-varying ac voltages such as a sinusoidal voltage are applied to a circuit, the circuit laws that were studied earlier still apply. Ohm s law and Kirchhoff s laws apply to ac circuits in the same way that they apply to dc circuits. If a sinusoidal voltage is applied to a resistive circuit, the voltage and current waveforms are both sinusoidal, and both waveforms reach their respective maximum and minimum values at the same time: Voltage and Current waveforms in a resistive circuit are in phase Solving ac circuits When solving ac circuits, it is important to use the same form for both voltage and current. In other words both quantities must be expressed either in peak or rms values. Example: determine the rms voltage across each resistor and the total current in the following circuit: Example: Find the peak voltage drop across R3: