sine waves & phase
DESCRIPTION
Sine Waves & Phase. Sine Waves. A sine wave is the simplest periodic wave there is Sine waves produce a pure tone at a single frequency. Simple Harmonic Motion. Any motion at a single constant frequency can be represented as a sine wave Such motion is known as simple harmonic motion - PowerPoint PPT PresentationTRANSCRIPT
Sine Waves & Phase
Sine Waves
• A sine wave is the simplest periodic wave there is
• Sine waves produce a pure tone at a single frequency
Simple Harmonic Motion
• Any motion at a single constant frequency can be represented as a sine wave
• Such motion is known as simple harmonic motion
• Here the amplitude may vary but the frequency does not
S.H.M.
• A pendulum swings in SHM:
• once it is started off it will take the same time to swing back and forth no matter how high it gets to at the top of its swing
• in other words: its frequency will stay the same no matter how high its amplitude
Electronic Oscillators
• Today electronic oscillators are the principle source of pure tones
• It is easy to specify and vary the frequency of an electronic oscillator precisely
Describing a Sine Wave
• Consider a wheel of radius 1 metre
• There is a line drawn on the wheel from the centre to the edge
• The height of the point where the line touches the edge is plotted as the wheel spins (at say ¼ of a turn per second)
radius = 1
height
Describing a Sine Wave
Describing a Sine Wave
• To create a sine wave the height of the point where the line touches the edge is plotted as the wheel spins clockwise at constant speed
0 45
90 180
270 360
0 seconds
4 seconds
2 seconds1 second
½ a second
3 seconds
Phase Difference
• The phase of periodic wave describes where the wave is in its cycle
• Phase difference is used to describe the phase position of one wave relative to another
½
pressure
time
Phase Difference 180
Phase Difference 90
¼
pressure
time
Phase Difference 45
1/8
pressure
time
Wave A Wave B
Phase Difference
• Is Wave A in front of Wave B or behind it?
• It can be seen either way:
• Wave A leads Wave B by 45; or
• Wave B leads Wave A by 315
The Sine Function
• Sine is a mathematical function
• y = sin(x)
sin(0) = 0 sin(45) = 0.707
sin(90) = 1 sin(180) = 0
sin(270) = -1 sin(360) = 0
0 45
90 180
270 360
x = 0, y = sin(x) = 0 x = 45, y = sin(x) = 0.707
x = 90, y = sin(x) = 1 x = 180, y = sin(x) = 0
x = 270, y = sin(x) = -1 x = 360, y = sin(x) = 0
Radians
One radian is the angle subtended at the centre of a circle by an arc that has a circumference that is equal to the length of the radius of a circle
Radians
arc length
radius
1 radian radius (r) = arc length (s)
angles can be measured in radians:
θ = s / r
Calculating Angles in Radians
angle in radians = arc length / radius
θ = s / r
How Many Radians in a Circle?
• Circumference of a circle = 2 r
• For one complete revolution the arc length is the entire circumference:
θ = s / r = 2 r / r = 2
Radians
2/2 3/2
1
- 1
0phase
Graph showing a sine wave with the y axis giving phase in radians.
Radians & Degrees
2 radians = 360, so /2 radians = 90
1 radian = 90 / * 2 57.5
Common Angles
Cycles 0 1/12 1/8 1/6 1/4 1/2 3/4 1
Degrees 0 30 45 60 90 180 270 360
Radians 0 / 6 / 4 / 3 / 2 3 / 2 2
Time Difference Calculations
Calculating the time difference between waves of identical period:
time difference = * phase difference in cycles
For Example:
If two waves of period 0.05 secs have a phase difference of 45 what is the time difference between them?
0.05 * (1/8) = 0.00625 secs = 6.25ms
45 in terms of cycles
Question 1
If two waves of period 20ms are phase shifted 90 what is the time difference between them?
0.02 * 1/4 = 0.005 secs = 5ms
Question 2
If wave A is leading wave B by 270 degrees and both have a frequency of 200Hz, what is the time difference between the waves?
Question 2 - Solution
0.005 * (3/4) = 0.015 / 4 = 0.00375s (3.75ms)
So: = 1 / f = 1 / 200 = 0.005
Recall: frequency = 1 / period
f = 1 /
Wave A leads Wave B by 270 (3.75ms); or
Question 2 - Discussion
270
Wave A Wave B 90
Wave B leads Wave A by 90 (1.25ms)
Phase Difference Calculations
Calculating the phase difference between waves of identical period:
phase difference = (2 / ) * time difference
For Example:
If two waves of period 0.05 are produced 0.00625 seconds apart what is their phase difference?
(2 / 0.05) * 0.00625 = 0.7853 radians
Question 1
If two waves of frequency 100 Hz are produced 0.005 seconds apart what is their phase difference?
Question 1 - Solution
(2 / 0.01) * 0.005 = radians
So: = 1 / f = 1 / 100 = 0.01
frequency = 1 / period
f = 1 /
phase difference = (2 / ) * time difference
which is 180 degrees
Question 2
If two waves of period 0.009 secs are produced 0.0005 seconds apart what is their phase difference?
(2 / 0.009) * 0.0005 = 0.34906585 radians
phase difference = (2 / ) * time difference
20 degrees (radians * 57.5)
Question 3
If two waves of period 0.03s are produced 0.0025 seconds apart what is their phase difference?
(2 / 0.03) * 0.0025 = 0.523598775 radians
phase difference = (2 / ) * time difference
30 degrees (radians * 57.5)
Question 4
If two waves of period 0.024 s are produced 0.005 seconds apart what is their phase difference?
(2 / 0.024) * 0.005 = 1.308996939 radians
phase difference = (2 / ) * time difference
which is roughly 75 degrees