What are Periodic Waves?
A periodic wave is the name given to any repeating pattern with constant wavelength and frequency. We can describe a periodic wave by characteristics such as amplitude, frequency and time period. Examples include sound waves (longitudinal) light waves (transverse) water waves and AC generators.
A sine wave, where y = sin(x) is an example of a periodic wave due to its repeating shape.
The amplitude of a wave, shown as A in the diagram below, is directly related to the energy of a wave, it also refers to the highest and lowest point of a wave.
For instance, if we consider an audio signal, the amplitude is interpreted by the listener as 'loudness' of the noise. The greater the amplitude, the more energy there is to vibrate the ear drum of the listener, which is interpreted as a high energy (loud) noise.
Types of Periodic Waveform
Although most of the waveforms we use are sine waves we need to be aware that others can occur, often associated with sound or digital pulses.
If we consider sound waves we can relate each shape to the following examples
Sine waves are the most recognised waveforms, they have an ideal, pure signal. In reality the closest producer of pure waves is the tuning fork.
Complex waves consist of two or more sine waves combined together, the effect of this when applied to sound is a lowering of quality, although this is often not detected by the human ear
Triangular waves create a much sharper sound and are generated by most string instruments due to friction changes during the movement of the string, either by plucking or using a bow
Square waves represent a digital signal of either 'on' of 'off'.
The AC Waveform
In engineering we mainly focus on AC (alternating current) Sinusoidal Waveforms which are created by rotating a coil within a magnetic field ( as shown below) or by alternating voltages and currents. They vary in both magnitude and direction with respect to time. This can be described as a “Bi-directional” waveform.
An AC waveform can be plotted using revolutions on the x-axis and voltage on the y-axis
The shape of an AC waveform generally follows that of a mathematical sinusoid being defined as:
A(t) = Amax x sin(2πƒt).
The amplitude of the waveform follows a repeated pattern
Angle of rotation | Voltage |
0 | 0V |
1/4 turn | maximum |
1/2 turn | 0V |
3/4 turn | minimum |
complete turn | 0V |
Key points that apply to AC waveforms:
polarity changes every half a cycle
we always draw a central zero line so we can show the change in polarity
It is a time dependant signal and referred to as a periodic waveform
Characteristics of AC Waveform
A waveform can be described using the following characteristics:
Cycle: is a complete wave pattern, from zero, through the positive and negative halves and then back to zero.
Time Period (T): the time in seconds it takes for a whole cycle to be completed, this can be applied to all types of periodic waves.
Frequency (ƒ): is the number of complete waveforms in one second. It can be calculated using the formula ƒ = 1/T with the unit being the Hertz, (Hz).
Amplitude (A): is the magnitude or intensity of the signal waveform measured in volts or amps. Always measured from the central line.
Relationship Between Frequency and Periodic Time
The equation below shows how the frequency and time period are related
Where Frequency (f) is measured in Hertz and periodic Time (T) is measured in seconds
AC Waveform Example
What will be the periodic time of a 50Hz waveform
What is the frequency of an AC waveform that has a periodic time of 10ms.
Solutions 1.
2.
Amplitude of an AC Waveform
Amplitude is also known as Maximum or Peak value, represented by the terms, Vmax for voltage or Imax for current.
For pure sinusoidal waveforms this peak value will always be the same for both half cycles i.e.
( +Vmax = -Vmax )
For non-sinusoidal or complex waveforms the maximum peak value can be very different for each half cycle.
Sometimes, alternating waveforms are given a peak-to-peak, Vp-p value and this is simply the distance or the sum in voltage between the maximum peak value, +Vmax and the minimum peak value,
-Vmax during one complete cycle.
The Average Value of an AC Waveform
For DC voltage, the mean value will be equal to its maximum peak value as a DC voltage is constant, but in a pure sine wave if the average value is calculated over the full cycle, the average value would be equal to zero as the positive and negative halves will cancel each other out.
For an AC supply we use an effective value, this is called the rms or root mean squared.
Average Value of a Non-sinusoidal Waveform
To find the average value of the waveform we need to calculate the area underneath the waveform using the mid-ordinate rule, trapezium rule or the Simpson’s rule. Where we split the waveform into small sections and use the instantaneous voltages, the more sections we use, the more accurate the average value becomes
Average Value of an AC Waveform
Where: n equals the actual number of mid-ordinates used, and the voltages used are shown in the diagram
For a pure sinusoidal waveform this value will always be the same due to the symmetry and can be calculated by:
0.637 x Vmax
This relationship holds true for RMS values of current.
The RMS Value of an AC Waveform
Since the current is constantly changing we can no longer use Power = current squared x resistance (P = I^2 R) unless we specify which value of current to use so we introduce the effective value.
The effective value of a sine wave produces the same heating effect in a load as we would expect to see if the same load was fed by a constant DC supply.
The effective value of a sine wave is more commonly known as the Root Mean Squared or RMS value as it is calculated using the square root of the mean (average) of the square of the voltage or current.
That is Vrms or Irms is given as the square root of the average of the sum of all the squared mid-ordinate values of the sine wave.
The RMS value for any AC waveform can be found from the following modified average value formula as shown.
RMS Value of an AC Waveform
Where: n equals the number of mid-ordinates and V is the corresponding voltages taken form the graph.
For a pure sinusoidal waveform this effective or R.M.S. value will always be equal to:
1/√2 x Vmax or 0.707 x Vmax
This relationship is the same for RMS values of current.
In this example If the peak voltage is 20V then the Vrms = 0.707 x 20 = 14.14V
so a 20V AC supply provided the same effective voltage as a 14.14V DC supply
Phase Difference and Phase Shift
In AC supplies we often experience a phase shift. This is caused by the inductive reactance changing as the current changes.
(electromagnetic induction).
The phase difference or phase shift of a Sinusoidal Waveform is the angle Φ (Greek letter Phi), in degrees or radian that the waveform has shifted from a certain reference point along the horizontal zero axis.
Phase Difference Φ is used to describe the difference in degree or radian when two or more alternating quantities reach their maximum or zero values, this value can range from zero to complete time period T
The wavelength, frequency and time period will remain the same, but the graph will be shifted to either the right or left
A Sinusoidal Waveform will have a positive maximum value at time π/2, a negative maximum value at time 3π/2, with zero values occurring along the baseline at 0, π and 2π.
Phase difference can also be expressed as a time shift of τ in seconds representing a fraction of the time period, T for example, +10ms or – 50µs.
Phase Difference Equation
Where:
● Amax – is the maximum amplitude of the waveform measured in Volts or Amps
● ωt – is the angular frequency of the waveform in radian/sec.
● Φ (phi) – is the phase angle in degrees or radians
If the positive slope of the sinusoidal waveform passes through the horizontal axis “before” t = 0 then we have a positive phase shift. Else if, t = 0 then we have a negative phase shift.
Note: frequency, maximum current and maximum voltage remain the same, only the angle Φ changes.
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