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Topic 4: AC waveforms
> ACTA 4 -1
Sine Wave (Degrees and Radians)
Circumference of a circle:
= 2
For a circle of radius = 1 unit
!"# = 2
If an object moves 1 complete round outside the
circle, it would have travelled ______ units or
rotated _____
_____ _______ units
If an object moves half a round outside the
circle, it would have travelled ______ units or
rotated _____
_____ _______ units
If an object rotates 1 outside the circle, it would
have travelled ______ units
_____ _______ units
> 1 unit
Topic 4: AC waveforms
ACTA 4 -2
Sine Wave (Degrees and Radians)
Angle in Degree Angle in Radians
360 = _______ radians
180 = _______ radians
1 = _______ radians
_______ radians = _______
_______ radians = _______
_______ radians = _______
Sine waveforms are periodic waveforms that repeats itself every cycle
In 1 cycle, we can say a sine wave rotates ______ degrees or ______ radians
= $ sin
V
> 90 270 0 360 180
Vm
-Vm
V
Vm
-Vm
The unit radian is used to define the _______ covered when walking around the
circumference
Using radian to define the angle of a sine waveform
When an object rotates ______ degrees or ______ radians, we complete 1
cycle Topic 4: AC waveforms
Example 4.1 : Convert the following angles from degrees to radians:
(a) 30
(b) 45
(c) 420
(d) 77
For (a) -(c), leave your answer in terms of
[!
> "
, !
> #
, $!
> %
, 1.34 rad]
ACTA 4 -3
Sine Wave (Degrees and Radians)
Example 4.2 : Convert the following angles from radians to degrees:
(a) 3
(b) $!
> "
(c) 1 radian
(d) 0.32 radian
[540 , 210 , 57.3 , 18.3 ]Topic 4: AC waveforms
> ACTA 4 -4
Sine Wave (Angular Velocity)
In 1 period, sine wave completes __ cycle or _____ or _____ radians
Frequency = 10 Hz
Period = ____ s
___ second => 2 radians
Total angle completed in 1 second
= ________ radians
1 second => 2 __ radians
>
> V
>
> V
> 2
> 0
> 2
> 3
> 2
= $ sin
= $ sin
Angular velocity ( ) =
Sine waveform for Voltage vs Angle (Radian) :
Sine waveform for Voltage vs Time :
In seconds => 2 ___ ___ radians
Using radian to define the angle of a sine waveform
Angular velocity
= Total angle completed by the sine wave in 1 s
= ________
For a sine wave of frequency 10 Hz, the sine wave completes __ cycle or
_____ radians in ____ s Topic 4: AC waveforms
ACTA 4 -5
Sine Wave (Angular Velocit y)
Example 4.3 : Given a sinusoidal waveform of f = 50 Hz, determine the
(a) the time it takes to complete 1 cycle
(b) angular velocity
(c) the time it will take the sinusoidal waveform to pass through an angle of 60 degrees
[20 ms , 314 rad/s, 3.33 ms ]
Example 4.4 : A sinusoidal voltage waveform is given as: v = 325 sin 628 t.
(a) Find the period
(b) Sketch the waveform with the abscissa:
(i) angle in degrees
(ii) angle in radians
(iii) time in seconds
(c) How long does it take to complete 72 ?
(d) How long does it take to complete 1 radian?
(e) How many degrees has passed after 0.4 seconds?
[10ms, -, 2ms, 1.59ms, 14400 ]
V
> 90 270 0 360 180
V
( )
VTopic 4: AC waveforms
ACTA 4 -6
Sine Wave (General form)
Phase Relations
V
> 90 270 0 360 180
V
> 90 270 0 360 180
V
> 90 270 0 360 180
= $ sin +
= / sin = 0
= / sin + = 30
= / sin = 45
_____ is positive
Waveform is shifted _____
_____ is negative
Waveform is shifted _____
The general form of the sine waveform for voltage: Topic 4: AC waveforms
ACTA 4 -7
Sine Wave (General form)
Example 4. 5: What is the phase relationship between the sinusoidal waveforms of each of the
following sets?
> AC TECHNOLOGY & APPLICATIONS
> ACTA Page 98
Example 13.5
What is the phase relationship between the
sinusoidal waveforms of each of the following sets?
a. V VLQ W
i VLQ W
Figure 13.15
b. i VLQ W
V VLQ W -
Figure 13.16
> AC TECHNOLOGY & APPLICATIONS
> ACTA Page 98
Example 13.5
What is the phase relationship between the
sinusoidal waveforms of each of the following sets?
a. V VLQ W
i VLQ W
Figure 13.15
b. i VLQ W
V VLQ W -
Figure 13.16 Topic 4: AC waveforms
> ACTA 4 -8
Sine Wave (General form)
> =125 sin (377 +25 )
= $ sin ( + )
> =125 sin (+25 )
Example 4. 6:
Step 1: Substitute t = 5 ms into v(t)
> Find v(t) when t = 5 ms
Step 2: Convert to degrees Topic 4: AC waveforms
> ACTA 4 -9
Sine Wave (General form)
Example 4.7 :
Step 1: Substitute and find angle
Find the time when v(t) = 75 V
Step 2: Find t Topic 4: AC waveforms
> ACTA 4 -10
Sine Wave (General form)
Equation t Instantaneous
Value
13 ms
15 V
8 ms
6 V
25 ms
8 V
= 30sin (377 + 25 )
= 12sin (314 18 )
= 25 sin (377 +
6)Topic 4: AC waveforms
> ACTA 4 -11
Sine Wave (Average / Effective (R.M.S.) Values)
> Example 4.8 :Determine the average value of the following waveforms:
> ACTA Page
## Period
## Over 1 cycle, the average value is the equivalent
## dc value and it is the va lue indicated on the dc
## meter.
## Example 13.6
## Determine the average value of the following
## waveforms:
## Figure 13.17
Average Value = Algebraic ______ of areas in 1 __________
Period
Average value of a waveform Topic 4: AC waveforms
> ACTA 4 -12
Sine Wave (Average / Effective (R.M.S.) Values)
Effective value of a waveform
An AC supply and a DC supply are both connected to the same load
What voltages for both supplies would give the same amount of power delivered ?
same
power
same
power
= ! For the same amount of power delivered:
R
"#
"#
R
$#
$#
"#
> !
=
"# =
"# =
Effective ly, for an AC supply of Vmax = 1 V, the
equivalent DC supply is Vdc =
"# =Topic 4: AC waveforms
ACTA 4 -13
Sine Wave (Average / Effective (R.M.S.) Values)
Example 4.9 :
(a) Given that an AC supply has a peak voltage of 325 V, find the effective (or r.m.s. ) value of the
voltage. State the voltage of the DC supply that will give the same amount of power to the
load.
(b) A DC supply has a voltage of 7.07 V and is connected to a load. If an AC supply is to supply
the same amount of power to the load, state the effective (or r.m.s. ) value of the voltage.
What is the peak voltage of the AC supply?
Example 4.10 : The 230 V dc source delivers 100 W to a bulb. Determine the rms (effective)
values and peak values of the applied voltage and current of the ac source if it is to deliver the
same power to the bulb to produce the same brightness.
> AC TECHNOLOGY & APPLICATIONS
Example 13.8
The 230 V dc source delivers 100 W to a bulb.
Determine the rms (effective) values and peak
values of the applied voltage and current of the ac
source if it is to deliver the same power to the
bulb to produce the same brightness.
Figure 13.21
Solution :Topic 4: AC waveforms
ACTA 4 -14
Example 4.12 : Determine the average value and effective ( r.m.s ) value of the following voltage
waveform.
0
6 V
6
- 4 V
1
t (ms)
V(v)
6
One
cycle
-4
3
Sine Wave (Average / Effective (R.M.S.) Values)
Example 4.11 : Determine the r.m.s value of the following waveforms:
> ACTA Page
## Example 13.6
## Determine the average value of the following
## waveforms:
## Figure 13.17