AC WAVEFORMS
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Alternating Current (AC) is current that periodically
reverses direction
If the switch changes position every second:
10V
1A
10V
10Ω
10V
10V
1A
10Ω
i
+1A
0
1
2
3
4
T(s)
-1A
6. Alternating Current
Waveforms
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AC WAVEFORMS
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The plot, or graph, of a current (or voltage) versus time
is called a waveform
The magnitude is the size of current or voltage (y-axis)
Waveforms where the current changes magnitude, but
not direction (all the values remain positive or negative)
are referred to as pulsating DC
Such waveforms can also be regarded as the
superposition (addition) of an AC waveform and a DC
level
An AC voltage is one that periodically reverses polarity
The voltage across the 10Ω resistor reverses polarity
as 1s intervals
An AC voltage source produces an EMF whose polarity
reverses at periodic intervals
The AC waveform used the most in circuit theory is the
sinusoidal waveform or sine wave
6. Alternating Current
Waveforms
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AC WAVEFORMS
i(t)
AC increasing and decreasing
linearly with time, triangular waveform
t
Pulsating DC. Current does not
reverse direction, doesn’t go negative
t
Sawtooth waveform
t
Pulsating DC.
Not AC.
AC superimposed
on DC level
t
Max current in positive direction
Sinusoidal AC
t
Min current in
negative direction
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Waveforms
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REVIEW OF TRIG FUNCTIONS
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Sine is a function of angle
y
Positive angle
x
Negative angle
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Positive angles are measured in the anticlockwise
direction from the positive x-axis. Negative angles are
measured in a clockwise direction.
In AC circuit theory, angles greater than 180°are
expressed as negative equivalent angles.
Eg. 225°= -135°(225 - 360)
Angles more negative than -180°are expressed in
equivalent positive angles
Top two quadrants expressed in positive angles
Bottom two quadrants expressed in negative angles
Angle on negative x-axis is ±180°
200°=200°-360°= -160°; -250°= -250°+360°= 110°
400°=400°-360°= 40°; 800°=800°-2(360°) = 80°
Sin(-θ) = -sin(θ)
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PROJECTION OF ROTATING
RADIUS
90°
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90°
As the radius of the circle rotates anticlockwise, the
angle it generates between itself and the positive x-axis
varies from 0 to 360°
At any instant, the radius is the hypotenuse of a rightangled triangle, containing the angle θ
Sin θ = 0 when θ = 0 and θ = 180°
Sin (θ+90°) = cos θ
Sin θ = cos (θ - 90°)
Cos(- θ) = cos θ
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Waveforms
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USEFUL TRIG RELATIONSHIPS
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Sin(θ±180°) = -sin θ
Cos(θ±180°) = -cos θ
Sin(θ±ϕ±180°) = -sin(θ±ϕ)
Cos(θ±ϕ±180°) = -cos(θ±ϕ)
Tan90°= +∞
Tan(-90°) = -∞
θ
Tan θ = b/a
Sin θ = b/r; b = r sin θ
Cos θ = a/r; a = r cos θ
Sin-1(b/r) = θ or arcsin(b/r) = θ
Cos-1(a/r) = θ or arcos(a/r) = θ
Tan-1(b/a) = θ or arctan(b/a) = θ
6. Alternating Current
Waveforms
r
b
a
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WAVEFORM PARAMETERS:
PERIOD AND FREQUENCY
i
T = 0.4s
0.3
0.4
0.6
t
0
0.1
0.2
0.5
T = 0.4s
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An AC waveform can be considered to exist for all time
Yet we need a reference time, t =0, where plot begins
Helps us to express waveform mathematically
Periodic AC waveform repeats at regular intervals
The time required to complete a cycle is called the
period T
Period can be measured between any two
corresponding points on successive cycles
Frequency is 1/T (Hertz – Hz)
1 Hz = 1 cycle/second
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Waveforms
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WAVEFORM PARAMETERS:
RADIANS & ANGULAR FREQUENCY
y
+1
y = sinθ = sin ωt
π
0
π/2
3π/2
2π
θ = ωt rad
-1
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Radian is the SI unit of angle: π radians = 180°
45°= 45(π/180) = π/4 rad = 0.7854 rad
1 rad = 1(180/π) = 57.296°
Above is a sine function plot versus angle in radians
1 cycle repeats at every 2nπ radians intervals
Angular velocity (ω) is the amount of angle the plot
sweeps through in a given amount of time
ω = θ/t rad/s (rad s-1) angular frequency = 2πf
θ = ωt rad
Sine wave can be expressed as a function of time
Sin θ = sin ωt = sin (2πf)t
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Waveforms
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WAVEFORM PARAMETERS:
PEAK & INSTANTANEOUS VALUES
v(t)
+3V
Peak-to-peak
value = 6V
Peak
value
=3V
t
0
-3V
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Max value reached by AC waveform - peak value (pk)
Peak-to-peak (p-p) value is the difference between
positive peak and negative peak values (3 - -3 = 6)
The peak value is also called amplitude
Any sin function can be expressed as
v(t ) = V p sin ωt
i (t ) = I p sin ωt
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Vp and Ip are the peak values
Lowercase i and v used for AC quantities
Uppercase I and V used for DC quantities
Instantaneous value of AC waveform is the value at
specific instant of time: i(t) = 3sin100t at t = 2ms?
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PHASE RELATIONS
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Adding angle ϕ to angle θ in the sine function: sin(θ± ϕ)
causes to sine waveform to shift left (+ ϕ) or right (- ϕ)
v(t ) = V p sin (ωt + φ )
i(t ) = I p sin (ωt + φ )
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ωt is in radians, but ϕ is expressed in degrees
v(t) = 5 sin (100t + 30°) V means v(t) is shifted left by
30°
Example: Find the instantaneous value at t = 0.25µs of
i(t) = 0.5 sin (8×105t + 50°) A
Answer: 0.439 A
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Waveforms
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LAG AND LEAD
• When two waveforms have different phase
angles, the one shifted farthest to the left is said
to lead the other
• v1(t) = 6 sin(ωt + 50°) leads v2(t) = 0.1 sin(ωt +
20°) because v1 is shifted left by 50°, while v2 is
shifted left by 20°
• v1 has a phase shift 30°greater than that of v2, i.e.
v1 leads v2 by 30°, or v2 lags v1 by 30°
• Lead-lag terminology derived from observation of
relative positions of the waveforms when plotted
versus time
• The waveform with greater positive phase
reaches its peak first (earliest in time), i.e. it leads
the other
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Waveforms
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LAG AND LEAD EXAMPLE
v(t ) = 10 sin (ωt + 30°)V
80°
i (t ) = 10 sin (ωt − 50°)A
ωt
30°
50°
80°
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v(t) is shifted left by 30°, i(t) is shifted right by 50°
v(t) lies to the left by 80°, thus v(t) leads i(t) by 80°
Equivalently i(t) lags v(t) by 80°
Notice how v(t) reaches its peak value 80°before i(t)
Phase comparisons of this type can only be made if the
two waveforms have the same frequency ω
i1 = 75sin(ωt - 18°)A; i2 = 3sin(ωt - 31°)A
i1 shifted right by 18°, i2 shifted right by 31°. i1 lies 3118=13°to left of i2. i1 leads i2 by 13°
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Waveforms
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ADVANTAGES OF AC VOLTAGES
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AC voltages in the form of sine waves can be
transmitted over long distances with minimal power
losses
Electrical power transmitted along power lines from
power stations and overhead railway cables are AC
type
AC voltages can also be “stepped up” or “stepped
down” using transformers
AC voltages are more efficient to tramsmit
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Waveforms
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