Sinusoidal
Waveform
The voltage waveform in Figures 1.1, 1.2, 1.3, 1.7 is called a sine wave, sinusoidal wave or a sinusoid because the amount of induced voltage
is proportional to the sine of the angle of rotation in the circular motion
producing the voltage. As angular values vary from 00 through 3600,
the sine of these values will vary from zero for 00, 1800 and
3600 to a maximum value of 1 for 900 and a minimum value
of -1 for 2700. This nature of variation of values of induced
voltage, when plotted gives the shape of a sine curve; hence the name of the
voltage wave is a sine wave.
Frequency
The number of cycles per second is the frequency,
with the symbol f. , if
the loop rotates through 60 complete revolutions, or cycles, during 1 s, the
frequency of the generated voltage is 60 cps, or 60 Hz. You see only one cycle
of the sine waveform, instead of 60 cycles, because the time interval shown
here is '/6o s. Note that the factor of time is involved.
More cycles per second means a higher frequency
and less time for one cycle, as illustrated . Then the changes in values are faster for higher
frequencies.
A complete cycle is measured between two
successive points that have the same value and direction. the cycle is between
successive points where the waveform is zero and ready to increase in the positive
direction. Or the cycle can be measured between successive peaks.
On the time scale of 1 s, waveform a goes
through one cycle; waveform b has much faster variations, with four complete
cycles during 1 s. Both waveforms are sine waves, even though each has a
different frequency.
In comparing sine waves, the amplitude has
no relation to frequency. Two waveforms can, have the same frequency with
different amplitudes, the same amplitude but different frequencies different
amplitudes and frequencies. The amplitude indicates how much the voltage or
current is, while the frequency indicates the time rate of change of the amplitude
variations, in cycles per second.
In a conventional generator, the frequency
is dependent upon the speed of rotor rotation within its stator and the number
of poles. Two poles of a rotor must pass a given point on the stator every
cycle; therefore:
For aircraft constant frequency systems 400
Hz has been adopted as the standard.
Peak
or Maximum Value
The value of the highest point of the
waveform, usually indicated by the symbols Vm (or Vmax )
for voltage, Im (or Imax) for current, etc. For source voltage/EMF,
this value is normally Em or Emax.
Average
or Mean Value
This is the average value of the waveform
taken over one half cycle. If an average value was taken over a full cycle the
positive and negative half cycles would cancel each other out. We usually
represent these with the symbols Vav for voltage, Iav for
current, etc.
Instantaneous
Value
At any given instant of time the actual
value of an alternating quantity may be anything from zero to a maximum in
either a positive or negative direction; such a value is called an Instantaneous Value. The Amplitude or Peak Value is the maximum instantaneous value of an alternating
quantity in the positive and negative directions.
The wave form of an alternating e.m.f.
induced in a single-turn coil, rotated at a constant velocity in a uniform
magnetic field, is such that at any given point in the cycle the instantaneous
value of e.m.f. bears a definite mathematical relationship to the amplitude
value. Thus, when one side of the coil turns through 00 from the
zero e.m.f. position and in the positive direction, the instantaneous value of
e.m.f.(e) is the product of the amplitude (Emax) and the sine of θ,
in symbols:
e = Emax Sin
θ
Similarly, instantaneous value of current
(i) is expressed as follows:
i = Imax
Sin θ
e = Emax
Sin θ
Root
Mean Square Value
This is the value of a.c. current that
produces the same amount of heat or does the same amount of work in the same
time as that of an equivalent d.c.
current. The root mean square (rms) value is sometimes referred to as the
effective or virtual value and is indicated by the symbols V, I, etc. The
calculation of power, energy etc. in an a.c. circuit is not so perfectly
straightforward as it is in a d.c.
circuit because the values of current and voltage are changing throughout the
cycle. For this reason, therefore, an arbitrary "effective" value is
essential. This value is generally termed the Root Mean Square (r.m.s.) value
. It is obtained by taking a number of instantaneous
values of voltage or current, whichever is required, during a half cycle,
squaring the values and taking their mean value and then taking the square root.
Thus, if six values of current "I" are taken, the mean square value
is:
The r.m.s. value of an alternating current is related
to the amplitude or peak value according to the wave form of the current. For a
sine wave the relationship
