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Light Sensing
How can the level of light be measured
and how can it be used to control a
device?
Contents
Initial Problem Statement 2
Narrative 3-11
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Notes 12-15
Appendix 16
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Initial Problem Statement
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Light Sensing
There are many occasions when the amount of
light available is important. These range from
activities such as taking a photograph, where
the light is metered in order to produce the right
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exposure, to sporting activities such as cricket,
where “bad light” can stop play, through to
control applications such as turning on a night
or security light.
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How can the level of light be measured
and how can it be used to control a
device?
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Student
Narrative
Introduction
The unit for measuring the amount of light falling on a unit area of surface is the lux. The following
table shows typical lux values for various forms of illumination. It shows that our eyes are very
sensitive to a broad range of lux levels.
Illumination Source
Typical lux value
Starlight
0.00005
Moonlight
0.1
60 W bulb at 1 m
50
Bright sunlight
500
on
Fluorescent Lighting, e.g. classroom
30,000
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The value of the light level can be measured using a device called a light dependent resistor or LDR.
An example of this kind of sensor is shown below.
Figure 1.
An LDR has a resistance that is high when it is dark and low when it is placed in bright light.
Multimedia
The video Light Sensing Video is available to demonstrate the behaviour of an LDR.
Activity 1
The following data have been collected from an LDR at various light levels for which it
might be used. Plot the data on the given axes.
Discussion
What do you notice about the numbers being plotted? Is the plot useful?
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An LDR has a resistance value of 1.5 kΩ. Use your graph to estimate the light level in
lux.
Student
Activity 2
Discussion
How could you make the data clearer?
Measured resistance (kΩ)
0.1
248.87
0.5
92.15
1
60.07
on
Light level (lux)
10
14.50
100
3.50
1,000
0.84
10,000
0.20
30,000
0.10
280
260
240
200
180
160
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Measured
resistance (k Ω)
120
100
80
60
40
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0
0
5000
10000
15000
20000
25000
30000
35000
Light level (lux)
Figure 2.
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You might be tempted to zoom in on part of the graph, particularly if you have
plotted the data with a spreadsheet. Look at the following zoom-in of the plot to
answer the question,
Student
Discussion
“An LDR has a resistance value of 1.5 kΩ. Use your graph to determine the light
level in lux.”
Do you trust the plot to be accurate?
Measured
resistance (k Ω)
15
14
13
12
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11
10
9
8
7
6
3
2
0
0
100
200
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1
300
400
500
600
700
800
900
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5
1000
Light level (lux)
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Figure 3.
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There are many problems in engineering that use a large range of values. To make the range more
manageable a mathematical technique called changing the variables is often used. To see how this
might work consider the following table
x
log10 x
0.1
-1
1
0
10
1
100
2
1000
3
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Discussion
Student
2. Changing the variables
Activity 3
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The range of both measured resistance and the light level in the LDR data is very
large. One way to reduce the range is to take the log of both sets of data.
Mathematically, call the light level l and the resistance r. Take the log base 10 (log10 )
of these values and fill in the table. (Report values to 2 d.p.)
0.1
0.5
1
10
log10 r
248.87
92.15
60.07
14.50
3.50
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100
log10 l
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l
0.84
10,000
0.20
30,000
0.10
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1,000
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What do you notice about how log10 x changes when x changes? Discuss the
range of x and compare it with the range of log10 x.
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Plot log10 r vs log10 l. What do you notice about the range of the variables? What do
you notice about the shape of the curve?
Student
Activity 4
log10 r
3.00
2.50
2.00
1.50
0.00
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
-1.00
-1.50
Activity 5
4.00
4.50
5.00
log10 l
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Figure 4.
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An LDR has a resistance value of 1.5 kΩ. Use your graph to determine the light
level in lux to the nearest integer value. Compare your answer with 520 lux found by
zooming in on the graph.
Discussion
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Under what circumstances should you use a logarithmic graph
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Light Sensing
0.50
on
1.00
The last activity showed that by transforming the variables by taking their logarithms, the relationship
between light level and measured resistance for an LDR became linear. This section will perform
some mathematical analysis of the data.
Student
3. Mathematical analysis
Look again at the plot using the logarithm of the variable values:
3.00
2.50
2.00
0.50
0.175
0.00
-0.50
0.00
0.50
1.00
1.50
-0.50
2.50
3.00
3.50
4.00
4.50
5.00
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-1.00
-1.50
2.00
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Figure 5.
The first thing to do is assign variable name for the transformed variables to make them easier to
deal with.
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Discussion
Introduce two new variables x and y and write:
x = log10 l
y = log10 r
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What are the axes labels on the above graph using these new variables?
Activity 6
Call the gradient of the line m and the y-intercept c.
Write an equation relating x and y.
Determine the values of m and c and write out the full equation relating x and y.
Discussion
Do you expect the gradient to be positive or negative?
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Use the transformed variable and your equation to find the light level corresponding
to a resistance of 1.5 kΩ. Compare this with the previously found value of 398 lux.
Student
Activity 7
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You have previously determined that for the LDR being considered, the light level, l, and resistance,
r, are related through the expression:
Student
4. Use in a device
y = mx + c
where
and
x = log10 l
y = log10 r
m = −0.62
c = 1.78
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find transformed
unknown variable from
y = mx + c
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transform value by
using the inverse log
function
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To make the calculation take fewer steps (and be easier to implement in your device) you will
investigate the possibility of finding a direct relationship between r and l.
Activity 8
Substitute x = log10 l and y = log10 r into your equation and use the laws of logs to
find a direct relationship between r and l.
Discussion
Did taking the logarithms of the values help?
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transform known
variable by taking logs
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This solution leads to the following sequence of calculations
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Compare the smooth curve provided by the spreadsheet with the actual curve
produced from your mathematical expression. What does this tell you about
automatically generated curves in spreadsheets?
Student
Discussion
Measured
resistance (k Ω)
15
14
13
12
11
10
9
6
5
4
3
Spreadsheet "smooth" line
2
1
0
0
100
200
300
400
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Actual line
500
600
700
800
900
1000
Light level (lux)
Figure 6.
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Activity 9
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The expression that you have just derived gives the resistance as a function of the light, which is
what would typically be published by a manufacturer as they are showing their calibration data for
standard light levels. However, this expression is not usable in a device as you wish to give light level
as a function of resistance.
m
c
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Rearrange r = l ×10 to make l the subject of the equation. In a previous activity
you have determined that l = 386 when r = 1.5. Verify your expression agrees with
this.
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Student
Notes
Logarithmic graphs
Many engineering problems have number ranges that scale many orders of
magnitude, such as the resistance of a LDR, see table below.
Measured resistance (kΩ)
0.1
248.87
0.5
92.15
1
60.07
10
14.50
100
3.50
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Light level (lux)
1,000
0.84
10,000
0.20
0.10
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One method of plotting these is to transform the variables using the log function. While this gives
a clearer graph, it means that you have to convert from the original variables to the log variables in
order to be able to use the graph. To avoid this the data can be plotted on a graph with logarithmic
scales.
3.00
2.50
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2.00
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In the following graph, the variables have been transformed by taking their log and plotted on a
graph with linear scales, i.e. a scale where each division represents a fixed unit.
1.50
1.00
0.50
0.00
0.00
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-1.50
-1.00
-0.50
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
-0.50
-1.00
-1.50
Figure 7.
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Student
Notes
Logarithmic graphs
If, instead of taking the log of the values, you plot the original data on a graph
with logarithmic axes you obtain the following.
Measured
resistance (k Ω)
1000
0.1
10
10 0
1 00 0
10 0 00
1 0 00 0 0
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Figure 8.
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Notice the similarity in the shape of the two graphs. On this graph each major division represents
a value that is ten times greater than the previous one. This leads to the odd spacing of the subdivision lines. These log graphs can take practice to plot and to read. For example, in the above a
light level of 10,000 lux corresponds to a resistance of 0.2 kΩ. To see this either:
• Count down from the horizontal line r = 1 in units of 0.1 (the size of the subdivision in this region
of the graph).
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or
• Count up from the horizontal line r = 0.1 in units of 0.1 (the size of the subdivision in this region of
the graph).
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Light level (lux)
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Notes
Logarithmic graphs
Activity 2 uses a log-log graph. There are also many occasions where a
semi-log plot is used. Common examples include
The Richter Magnitude scale for earth quakes. This measures the deflection of a seismic trace
away from the centre line. Each 10-fold increase in deflection gives an increment in the magnitude
of the earth quake. For example, if the seismometer trace gave a 0.1 cm deflection for a magnitude
5.0 quake it would give a 1 cm deflection for a magnitude 6.0 quake and a 10 cm deflection for a
magnitude 7.0 quake.
Radioactivity. The activity of a radioactive source follows and exponential decay with time. If
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the activity relative to the initial activity is plotted using a logarithmic scale a straight line graph is
obtained that is easier to read for large values of time.
For example, the following graph shows the decay of radioactive material with a half-life of 10 days
(i.e. the amount of material present halves every 10 days). While it is reasonably easy to see how
much material is present for the first 50 or so days it becomes increasingly more difficult to read the
graph for larger values of time; how much material is present after 90 days?
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Radio active
material present
1.0
0.9
0.8
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0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
10
20
30
40
50
60
70
80
90
100
time
(days)
Figure 9.
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Solutions that have a lower pH are more acidic, meaning a higher concentration of hydrogen
ions. A solution with a pH of 4 is 10 times more acidic than a solution with a pH of 5, i.e. it has a
concentration of hydrogen ions that is 10 times larger. Similarly, a solution with a pH of 11 is 10
times less acidic, commonly stated as 10 times more alkaline, than a solution with a pH of 10, i.e. it
has a concentration of hydrogen ions that is 10 times smaller.
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The pH scale. This measures the acidity and alkalinity of a solution and represents the ratio of
hydrogen ions relative to pure water. Pure water has a pH of 7.
Student
Notes
Logarithmic graphs
Plotting the values with a logarithmic y-axis allows this to be seen more easily.
The graph below shows that the amount remaining after 90 days about 0.002
(units not specified).
Radio active
material present
0
10
20
30
40
50
60
70
80
90
100
0.1
0.0001
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0.01
time
(days)
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Figure 10.
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Appendix
mathematical coverage
PL objectives
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Use algebra to solve engineering problems
• Be able to work with numbers in index form
• Simplify and evaluate expressions involving the use of indices
• Change the subject of a formula
• Be able to solve linear equations
• Be able to solve simultaneous equations
• Know how to check answers by substitution
• Be able to plot data
• Be able to draw graphs by constructing a table of values
• Be able to construct and use conversion graphs
• Be able to extract information from a graph
• Plot a straight line graph from given data and use it to deduce the gradient, intercept and
equation of a line
• Solve problems using the laws of logarithms
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