AC Generators
Basic Generator
One complete revolution = 1 complete Sinewave
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
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 ( Vinst ) = 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:
V pp  2V p
or
I pp  2I pp
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:
2
Vavg   V p
 
and
2
I avg    I p
 
RMS or Effective value ( Vrms ) = 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
Vp  2  VRMS
and
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 = _____________
Vrms = ____________
F = ____________
T = _____________
Vinst at t  12s _____________
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 2r it follows that there must be 2π radians in
a complete circle. Thus
2 rad  360
 rad  180
or
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
.
Degrees vs. Radians
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.
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:
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 = ____________
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).
Example: determine the instantaneous values at the 90° reference point on the horizontal axis for each
of the following sine waves:
v A = ___________
vB = ___________
vC = ___________
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: