CHAPTER 2
Engineering Functions
Curve drawing in general
In this chapter you will be shown how to draw the graphs of the functions:
The contents of
is referred to as the argument of the function. Take
as an example, if
you
have the natural or unshifted form of the function with a typical period starting at
degrees or radians and
ending at
degrees or
radians. In this chapter we will use the radian measure for the trigonometric
functions. If you instead replace
, the typical period will instead start at
and end at
but if you replace
, the typical period will start at
and end at
. On
the other hand, adding a constant to the entire function
, will move
upwards by units but
will move
downwards units. In general, adding/subtracting constants to the argument causes
left/right shifts and adding/subtracting constants externally causes up/down shifts in the graph.
2.1 Waves and Vibrations
2.1.1 The phasor diagram and wave expression
a) Phasor diagrams: A phasor diagram is a rotating vector with constant length that represents or translates
into a sine wave if you follow the
coordinate. Follow the coordinate in the animated version of the diagram
below at https://en.wikipedia.org/wiki/Phasor .
Suppose the arow in the above diagram has magnitude (length) . This arrow is moving counter clockwise at an
angular velocity
(rad/s) and its angular displacement will be
after seconds. If, at time zero, the
arrow was already at
radians (called the phase angle), the radian displacement travelled by the arrow after
time is
. If
, the arrow started on the horizontal axes and the sine wave plotted is the unshifted
version starting at
and ending at
. Each point on the circle locus this arrow is tracing
out has rectangular
coordinates which can be calculated by converting the polar coordinates
to rectangular coordinates
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In the top part of the diagram above, only the coordinate is plotted as a function of time defining the wave
. If you instead plot the coordinate as a function of time, you will generate the wave
. At
, the
coordinate of the arrow is calculated as
and represents the frozen phasor
which can be graphically represented by an arrow of length
situated
radians with regards to the positive axes (either clockwise or counter clockwise depending on the
sign of ). The frozen phasor
can be used to determine the resultant wave when more than one sine
wave or more than one cosine wave traveling at the same angular velocity
are superimposed (added). It can
also be used to determine lead/lag between two waves traveling at the same angular velocity
(who is leading
and by how many radians).
Example
i) Graphically represent the frozen phasor of the wave
. Calculate the corresponding
coordinates on the phasor diagram and the
coordinate on the sine graph.
Step 1: You must substitute
from the positive
to determine that the arrow representing the frozen phasor
has magnitude 2 and is situated
radians measured clockwise (as it is negative)
axes (1.2 radians is equivalent to
).
Step 2: Draw an arrow with magnitude 2 and
Step 3: The corresponding
clockwise from the positive
coordinates corresponding to
axes
can be calculated by converting the polar
from
to rectangular form using the Rec function on your calculator
This is the coordinate of the head of the arrow in the above diagram.
Step 4: The corresponding coordinate on the sine graph is
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as indicated below.
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ii) Determine the phase distance between the two out of phase waves
Step 1: The respective frozen phasors are
and
(counter clockwise) and
(clockwise)
Step 2: The total phase distance between these two waves is
b) Graphical representation of a typical cycle/period
To avoid confusion, the phase angle in the sine wave will be indicated as
The sine and cosine functions are periodic and have an oscillation axes.
and for the cosine wave,
.
The sine wave
For the simple case
, the start, midpoint and ending of a typical cycle is at
all corresponding to
. When
, this translates into the three
significant time values corresponding with
(also referred to as the time displacement)
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Note that
• If is negative and you have
, the time displacement is positive and the cycle starts
to the right of the axes. If is positive and you have
, the time displacement is
negative and the cycle starts to the left of the axes. If
and you have
,
, the
cycle starts at the origin and represents the unshifted version of the graph.
• The period is
or
seconds. The frequency is
hertz.
• If you want to calculate where the sine graph reaches its optimum values of
respectively calculate the
values corresponding to
and
, you would
or determine the midpoint of
for
and the midpoint of
for
.
• If the typical period ranges from negative values to positive values, the graph will intercept the
axes at the point corresponding to
which is at
.
• The graph
oscillates about the axes
. When adding a constant to the graph
of
, that is, when drawing
, you are raising (positive ) or lowering
the graph and its oscillation axes is at
.
Examples
i) Plot one cycle/period of the graph of
determine the period and frequency of
. Compare this graph to that of
.
and
Step 1: The amplitude is but the graph has been moved upwards by 1 unit, therefore, the cycle starts at
with corresponding
the next zero is at
with
and the last zero is at
with
Step 2: As
is negative and
.
Step 3: The graph of
been moved to the left by
period is
is positive, the graph will intercept the y axes. The
intercept is at
is the graph of the zero phase angle version
units and moved upwards by 1 unit. The oscillation axes is
which has
. The
and the frequency is
syms t
y(t)=2*sin(3*t+2)+1;
fplot(y,[-0.67,1.43])
grid on
legend('show')
title('classic period for a sine graph')
xlabel('time')
ylabel('y')
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ii) Given the graph of a classic window of a sine graph. Determine the equation of this graph.
Step 1: The amplitude is
Step 2: From the graph,
and the graph is oscillating about the time axes
. From this,
.
but from the formula
,
can be used to calculate the angular velocity
Step 3: From the graph, the period starts to the left of the
. From
axes and the graph will be of type
, the value of the phase angle can be calculated as
Step 4: The equation is
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iii) Given the graph of a classic window of a sine graph. Determine the equation of this graph.
Step 1: The amplitude is
and the graph is oscillating about the time axes
Step 2: From the graph,
. From this,
.
but from the formula
,
which can be used to calculate the angular velocity
Step 3: From the graph, the period starts to the right of the
axes (
and the graph will be of type
, the value of the phase angle can be calculated as
. From
has been moved to the right)
Step 4: The equation is
iv) Calculate the angular displacement and the time displacement for
to those of the unshifted version
. Compare your results
Step 1: The angular displacement is
. When determining the frozen phasor, the polar expression is
. The angle 1.3 rad is the angular displacement of the phasor from the positive axes counter
clockwise.
Step 2: The time displacement is
starts at
which is to the left of the
. On the graph of the sine wave, a typical cycle/period
axes.
Step 3: For the unshifted version,
and
. In comparing this to the shifted graph,
seconds: when the periodic clock started,
was already ahead of
would say that in comparison,
leads
by 1.3 rad or by 0.65 seconds.
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The cosine wave
For the simple case
maximum
. When
, the start, midpoint and ending of a typical cycle is at
respectively corresponding to the maximum
, minimum
and
, this translates into the three significant time values corresponding with
(also referred to as the time displacement)
Note that
• If is negative and you have
, the time displacement is positive and the cycle
starts to the right of the axes. If is positive and you have
, the time displacement
is negative and the cycle starts to the left of the axes. If
and you have
,
, the cycle starts at the origin and represents the unshifted version of the graph.
• The period is
or
. The frequency is
hertz.
• If you want to calculate where the cosine graph intercepts with the
calculate the
values corresponding to
axes, you would respectively
or determine the midpoint of
and the midpoint of
.
• If the typical period ranges from negative values to positive values, the graph will intercept the
axes at the point corresponding to
which is at
.
• The graph
oscillates about the axes
. When adding a constant to the graph
of
, that is, when drawing
, you are raising (positive ) or lowering
the graph and its oscillation axes is at
.
Examples
i) Plot one cycle/period of the graph of
determine the period and frequency of
. Compare this graph to that of
.
and
Step 1: The amplitude is but the graph has been lowered 2 units, therefore, the cycle starts at the maximum
when
the minimum
is at
and
the cycle ends at the maximum
when
.
Step 2: As
is negative and
.
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is positive, the graph will intercept the x axes. The
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intercept is at
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Step 3: The graph of
been moved to the left by
period is
is the graph of the zero phase angle version
units and then downwards 2 units. The oscillation axes is
which has
. The
and the frequency is
syms t
x(t)=2*cos(3*t+2)-2;
fplot(x,[-0.67,1.43])
grid on
legend('show')
title('classic period for a cosine graph')
xlabel('time')
ylabel('x')
annotation('line',[0.3755 0.3794],[0.9236 0.1184])
ii) Given the graph of a classic window of a cosine graph. Determine the equation of this graph.
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Step 1: The amplitude is
Step 2: From the graph,
. From this,
but from the formula
,
which can be used to calculate the angular velocity
Step 3: From the graph, the period starts to the right of the
. From
axes and the graph will be of type
, the value of the phase angle can be calculated as
Step 4: The equation is
iii) If a classic period starts at the maximum (0.4 , 2) , has midpoint at the minimum (0.9, 0) and ends at the
maximum
what is the equation?
Step 1: If the maximum is at
and the minimum is at
Step 2: This describes the cosine graph
amplitude is
and the period
, the oscillation axes is in between at
as it starts to the right of the
.
axes. The
from which
Step 3: The phase angle is calculated from
The equation is
syms t
x(t)=cos(2*pi*t-2.51)+1;
fplot(x,[0.4,1.4])
grid on
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iv) Calculate the time displacement between a typical starting point of a cycle for
Step 1: The respective time displacements are
and
. The total time displacement between
these two waves is
Clearly
.
Putting lead and lag into perspective
In an electric circuit containing an alternating energy source and components consisting of a resistor , inductor
and a capacitor , the relationship between the current flowing through the components and the voltage
measured across the component can be related as follows:
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Paced Exercise (2A)
1. If a classic cycle starts at the zero
, determine the equation.
• This describes the sine graph
• You should find that
with
• The equation is
and ends at the zero
,
with
values ranging between
and
2. What is the angular displacement and the time displacement of
• You angular displacement corresponds with the frozen phasor
with polar representation
which is an arrow of magnitude 2 situated 1.3 radians clockwise with regards to the
positive time axes. The angular displacement is -1.3 rad or
.
• The time displacement is
seconds.
3. Draw one cycle/period of the graph
• You should find that the graph is that of
axes
. Significant times are
• The amplitude is 1 and as the oscillation axes is
• The graph is
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moved one unit downwards with oscillation
, this graph will oscillate between
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4. Draw one cycle of the graph
• You should find that the oscillation axes is at
and that the negative in front of the graph results in
the graph being mirrored about this axes. The significant times are
corresponding to
.
• The amplitude is 1 and as the oscillation axes is
, this means that the graph will oscillate between
.
• The graph is
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2.1.2 Superimposing waves
a) Adding
to
Two sine waves (or two cosine waves) with the same angular velocity can be added by adding their frozen
phasors.
Example
Calculate the resultant wave when adding
and
Step 1: The resultant wave is calculated by determining the resultant frozen phasor by adding the individual
frozen phasors. From Chapter 1 Section 1.4.2, we must first convert polar forms to rectangular forms before
adding (set calculator to radians and use the Rec function)
The polar form for
(set calculator to radians and use the Pol function)
Step 2: The resultant frozen phasor transfers back to the wave form
• If both waves have
, the resultant wave will also have
• The above addition is also referred to as the superimposing of waves. This method can be used when
both waves are of the same type and both waves have the same angular velocity
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b) Adding
to
When you are adding two waves that have the same angular velocity but are not of the same type, you must
use one of the following two formulas depending on the required outcome.
where, using your calculator,
or
where using your calculator
Note that the order of placing
in the Pol command will make a difference to the calculation. The signs in
the shifted formula
are negative.
are fixed and will only change to the opposite sign if the calculated
and
Examples
i) Calculate expressions for the resultant wave when adding
.
Step 1: Set your calculator to radians and use the command Pol(2,-3) to determine the shifted cosine
expression. You should find that
The resultant is therefore
Step 2: If it is preferred to represent the graph as a shifted sine graph, use the command Pol(-3,2) you should
find that
The resultant is therefore
ii) Express the following superimposed wave as a shifted sine wave
We have formulas for adding an unshifted sine and unshifted cosine with the same angular velocity. We must
therefore first convert the above shifted expression to their unshifted versions by applying the given
trigonometric identities below.
Step 1: Use the above identities to simplify the expression for
Step 2: For the sine version use
Step 3: The shifted sine graph is
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c) Adding
When adding waves of the same type with the same amplitude but different angular velocities, it results in a
wave with a modulated amplitude. The formula for calculating the resultant wave in this case is
The sine component has a larger velocity
than the cos component velocity
resulting in a
modulated sine graph where sine's amplitude changes according to the slower cosine's velocity. When listening
to music, the frequency of the sine component is what we hear and the frequency of the cosine component is
what we perceive as the "beat".
Examples
i) Determine the resultant expression for
the frequency of the "beat".
Step 1: Since
Step 2: You will hear
, substitute
. Calculate the frequency you will "hear" and
into the modulating formula
, the beat
syms t
y1(t)=4*sin(6*t)*cos(4*t);
fplot(y1,[-pi,pi],'k')
grid on
legend('show')
title('Modulated wave')
xlabel('time')
ylabel('y')
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Note the changing amplitude (amplitude modulation)
Paced Exercise (2B)
1. Calculate the resultant wave when the waves of
(added).
are superimposed
• The respective frozen phasors are
• You must add the respective rectangular forms to determine the resultant. You must then convert the
result to polar form
• You should find that the resultant phasor is
• The resultant wave is
with polar form
2.What were the two graphs resulting in the superimposed wave
• From
by using
by using the function
back to
• From
• The two graphs were
,
3. Calculate the resultant wave for
, we should be able to reverse
.
and
then represent one cycle of the resultant graphically
• Since you are adding two different types of functions with the same angular velocity, you must use one
of the two given formulas. If you use the sine version, you should find that
• The graph has significant coordinates
with intercept at
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4. Calculate the resultant wave for the following then represent one cycle of the resultant graphically
• Since you are adding the same type of functions with the same angular velocity, you must use their
frozen phasors
• You should find that
• The graph is
• Since you are adding the same type of functions with the same angular velocity, you must use their
frozen phasors
• You should find that
• With graph
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5. Calculate the resultant wave for the following
• You are adding two sine graphs with the same amplitude but different angular velocities. The resultant
will be a modulated graph.
• You should find that
• You are adding two sine graphs with the same amplitude but different angular velocities. The resultant
will be a modulated graph.
• You should find that
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