1. ELECTRICAL CIRCUIT ET 201
Define and explain characteristics of sinusoidal wave, phase relationships and phase shifting

2. SINUSOIDAL ALTERNATING
WAVEFORMS
(CHAPTER 1.1 ~ 1.4)

3. Understand Alternating Current
DIRECT CURRENT (DC) – IS WHEN THE CURRENT FLOWS IN ONLY ONE DIRECTION. Constant flow of electric charge
EX: BATTERY
ALTERNATING CURRENT AC) – THE CURRENT FLOWS IN ONE DIRECTION THEN THE OTHER.
Electrical current whose magnitude and direction vary cyclically, as opposed to direct current whose direction remains constant.
EX: OUTLETS

4. Sources of alternating current
By rotating a magnetic field within a stationary coil
By rotating a coil in a magnetic field

5. Generation of Alternating Current
A voltage supplied by a battery or other DC source has a certain polarity and remains constant.
Alternating Current (AC) varies in polarity and amplitude.
AC is an important part of electrical and electronic systems.

6. Faraday’s and Lenz’s Law involved in generating a.c current
Faraday’s Laws of electromagnetic Induction.
  Induced electromotive field
Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil.
e.m.f, e = -N d N = Number of turn
dt  = Magnetic Flux
Lenz’s law
An electromagnetic field interacting with a conductor will generate electrical current that induces a counter magnetic field that opposes the magnetic field generating the current.

7. Sine Wave Characteristics
The basis of an AC alternator is a loop of wire rotated in a magnetic field.
Slip rings and brushes make continuous electrical connections to the rotating conductor.
The magnitude and polarity of the generated voltage is shown on the following slide.

8. Sine Wave Characteristics

9. Sine Wave Characteristics
The sine wave at the right consists of two, opposite polarity, alternations.
Each alternation is called a half cycle.
Each half cycle has a maximum value called the peak value.

10. Sine Wave Characteristics
Sine waves may represent voltage, current, or some other parameter.
The period of a sine wave is the time from any given point on the cycle to the same point on the following cycle.
The period is measured in time (t), and in most cases is measured in seconds or fractions thereof.

11. Frequency
The frequency of a sine wave is the number of complete cycles that occur in one second.
Frequency is measured in hertz (Hz). One hertz corresponds to one cycle per second.
Frequency and period have an inverse relationship. t = 1/f, and f = 1/t.
Frequency-to-period and period-to-frequency conversions are common in electronic calculations.

12. Peak Value
The peak value of a sine wave is the maximum voltage (or current) it reaches.
Peak voltages occur at two different points in the cycle.
One peak is positive, the other is negative.
The positive peak occurs at 90º and the negative peak at 270º.
The positive and negative have equal amplitudes.

13. Average Values
The average value of any measured quantity is the sum of all of the intermediate values.
The average value of a full sine wave is zero.
The average value of one-half cycle of a sine wave is:
Vavg = 0.637Vp or Iavg = 0.637Ip
Chapter 6 - Alternating Current

14. rms Value
One of the most important characteristics of a sine wave is its rms or effective value.
The rms value describes the sine wave in terms of an equivalent dc voltage.
The rms value of a sine wave produces the same heating effect in a resistance as an equal value of dc.
The abbreviation rms stands for root-mean-square, and is determined by: Vrms = 0.707Vp or Irms = 0.707Ip
Chapter 6 - Alternating Current

15. Peak-to-Peak Value
Another measurement used to describe sine waves are their peak-to-peak values.
The peak-to-peak value is the difference between the two peak values.

16. Form Factor
Form Factor is defined as the ratio of r.m.s value to the average value.
Form factor = r.m.s value =  peak value
average value  peak valur =

17. Peak Factor
Crest or Peak or Amplitude Factor
Peak factor is defined as the ratio of peak voltage to r.m.s value.

18. 13.1 Introduction Alternating waveforms
Alternating signal is a signal that varies with respect to time.
Alternating signal can be categories into ac voltage and ac current.
This voltage and current have positive and negative value.

19. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Voltage and current value is represent by vertical axis and time represent by horizontal axis.
In the first half, current or voltage will increase into maximum positive value and come back to zero.
Then in second half, current or voltage will increase into negative maximum voltage and come back to zero.
One complete waveform is called one cycle.
volts or amperes
units of time

20. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Defined Polarities and Direction
The voltage polarity and current direction will be for an instant in time in the positive portion of the sinusoidal waveform.
In the figure, a lowercase letter is employed for polarity and current direction to indicate that the quantity is time dependent; that is, its magnitude will change with time. 20

21. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Defined Polarities and Direction
For a period of time, a voltage has one polarity, while for the next equal period it reverses. A positive sign is applied if the voltage is above the axis.
For a current source, the direction in the symbol corresponds with the positive region of the waveform. 21

22. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
There are several specification in sinusoidal waveform:
1. period
2. frequency
3. instantaneous value
4. peak value
5. peak to peak value
6. angular velocity
7. average value
8. effective value

23. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Period (T)
Period is defines as the amount of time is take to go through one cycle.
Period for sinusoidal waveform is equal for each cycle.
Cycle
The portion of a waveform contained in one period of time.
Frequency (f)
Frequency is defines as number of cycles in one seconds.
It can derives as
f = Hz
T = seconds (s)

24. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
The cycles within T1, T2, and T3 may appear different in
the figure above, but they are all bounded by one period of time
and therefore satisfy the definition of a cycle.

25. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Signal with lower frequency
Signal with higher frequency
Frequency = 1 cycle
per second
Frequency = 21/2 cycles
per second
Frequency = 2 cycles
per second
1 hertz (Hz) = 1 cycle per second (cps)

26. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Instantaneous value
Instantaneous value is magnitude value of waveform at one specific time.
Symbol for instantaneous value of voltage is v(t) and current is i(t).

27. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Peak Value
The maximum instantaneous value of a function as measured from zero-volt level.
For one complete cycle, there are two peak value that is positive peak value and negative peak value.
Symbol for peak value of voltage is Em or Vm and current is Im .
Peak value, Vm = 8 V

28. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Peak to peak value
The full voltage between positive and negative peaks of the waveform, that is, the sum of the magnitude of the positive and negative peaks.
Symbol for peak to peak value of voltage is Ep-p or Vp-p and current is Ip-p
Peak to peak value, Vp-p = 16 V

29. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Angular velocity
Angular velocity is the velocity with which the radius vector rotates about the center.
Symbol of angular speed is and units is radians/seconds (rad/s)
Horizontal axis of waveform can be represent by time and angular speed.

30. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Angular velocity
Degree
Radian
90°
(π/180°) x ( 90°) = π/2 rad
60°
(π/180°) x ( 60°) = π/3 rad
30°
(π/180°) x (30°) = π/6 rad
Radian
Degree
π /3
(180° /π) x (π /3) = 60° π
(180° /π) x (π ) = 180°
3π /2
(180°/π) x (3π /2) = 270°

31. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Plotting a sine wave versus (a) degrees and (b) radians.

32. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
The sinusoidal wave form can be derived from the length of the vertical projection of a radius vector rotating in a uniform circular motion about a fixed point.
Waveform picture with respect to angular velocity

33. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Angular velocity
Formula of angular velocity
Since (ω) is typically provided in radians/second, the angle ϴ obtained using ϴ = ωt is usually in radians.

34. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Angular velocity
The time required to complete one cycle is equal to the period (T) of the sinusoidal waveform.
One cycle in radian is equal to 2π (360o).
(rad/s) or

35. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Angular velocity
Demonstrating the effect of  on the frequency f and period T.

36. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Example 13.6
Given  = 200 rad/s, determine how long it will take the sinusoidal waveform to pass through an angle of 90
Solution

37. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Example 13.7
Find the angle through which a sinusoidal waveform of 60 Hz will pass in a period of 5 ms.
Solution

38. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Average value
Average value is average value for all instantaneous value in half or one complete waveform cycle.
It can be calculate in two ways:
Calculate the area under the graph:
Average value = area under the function in a period
period
2. Use integral method
For a symmetry waveform, area upper section equal to area under the section, so just take half of the period only.

39. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Average value
Example: Calculate the average value of the waveform below.
Solution:
For a sinus waveform , average value can be calculate by

40. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Effective value
The most common method of specifying the amount of sine wave of voltage or current by relating it into dc voltage and current that will produce the same heat effect.
Effective value is the equivalent dc value of a sinusoidal current or voltage, which is 1/√2 or of its peak value.
The equivalent dc value is called rms value or effective value.
The formula of effective value for sine wave waveform is;

41. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Example 13.21
The 120 V dc source delivers 3.6 W to the load. Find Em and Im of the ac source, if the same power is to be delivered to the load.

42. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Example – solution
and

43. 13.2 Sinusoidal AC Voltage Characteristics and Definitions
Example – solution

44. Basic sine wave for current or voltage
General Format for the Sinusoidal Voltage or Current
The basic mathematical format for the sinusoidal waveform is:
where:
Am : peak value of the waveform
 : angle from the horizontal axis
volts or amperes
Basic sine wave for current or voltage

45. 13.5 General Format for the Sinusoidal Voltage or Current
The general format of a sine wave can also be as:
General format for electrical quantities such as current and voltage is:
where: and is the peak value of current and voltage while i(t) and v(t) is the instantaneous value of current and voltage.
α= ωt

46. 13.5 General Format for the Sinusoidal Voltage or Current
Example 13.8
Given e(t) = 5 sin, determine e(t) at  = 40 and  = 0.8.
Solution
For  = 40,
For  = 0.8

47. 13.5 General Format for the Sinusoidal Voltage or Current
Example 13.9
(a) Determine the angle at which the magnitude of the sinusoidal function v(t) = 10 sin 377t is 4 V.
Determine the time
at which the magnitude
is attained.

48. 13.5 General Format for the Sinusoidal Voltage or Current
Example solution
Hence,
When v(t) = 4 V,

49. 13.5 General Format for the Sinusoidal Voltage or Current
Example 13.9 – solution (cont’d)
(a) But α is in radian, so α must be calculate in radian:
(b) Given, , so

50. 13.6 Phase Relationship Phase angle
Phase angle is a shifted angle waveform from reference origin.
Phase angle is been represent by symbol θ or Φ
Units is degree ° or radian
Two waveform is called in phase if its have a same phase degree or different phase is zero
Two waveform is called out of phase if its have a different phase.

51. Phase Relationship
The unshifted sinusoidal waveform is represented by the expression:

52. Phase Relationship
Sinusoidal waveform which is shifted to the right or left of 0° is represented by the expression:
where  is the angle (in degrees or radians) that the waveform has been shifted.

53. Phase Relationship
If the wave form passes through the horizontal axis with a positive-going (increasing with the time) slope before 0°:

54. Phase Relationship
If the waveform passes through the horizontal axis with a positive-going slope after 0°:

55. Phase Relationship

56. Phase Relationship
The terms leading and lagging are used to indicate the relationship between two sinusoidal waveforms of the same frequency f （or angular velocity ω) plotted on the same set of axes.
The cosine curve is said to lead the sine curve by 90.
The sine curve is said to lag the cosine curve by 90.
90 is referred to as the phase angle between the two waveforms.

57. 13.6 Phase Relationship Note: sin (- α) = - sin α cos(- α) = cos α
cos (α-90o)
sin (α+90o)
Note:
sin (- α) = - sin α
cos(- α) = cos α
Start at + sin α position;

58. 13.6 Phase Relationship If a sinusoidal expression should appear as
the negative sign is associated with the sine portion of the expression, not the peak value Em , i.e.
And, since;

59. 13.6 Phase Relationship Example 13.2
Determine the phase relationship between the following waveforms

60. 13.6 Phase Relationship i leads v by 40 v lags i by 40
Example 13.2 – solution
i leads v by 40 or
v lags i by 40

61. 13.6 Phase Relationship i leads v by 80 v lags i by 80
Example 13.2 – solution (cont’d)
i leads v by 80 or
v lags i by 80

62. 13.6 Phase Relationship i leads v by 110 v lags i by 110
Example 13.2 – solution (cont’d)
i leads v by 110 or
v lags i by 110

63. 13.6 Phase Relationship Example 13.2 – solution (cont’d) OR
v leads i by 160 Or
i lags v by 160
i leads v by 200 Or
v lags i by 200