Sinusoidal Waveform

 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 V_m (or V_max ) for voltage, I_m (or I_max) for current, etc. For source voltage/EMF, this value is normally E_m or E_max. 

 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 V_av for voltage, I_av 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 (E_max) and the sine of θ, in symbols:

e = E_max Sin θ

Similarly, instantaneous value of current (i) is expressed as follows:

i = I_max Sin θ

e = E_max 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