1. 2. A.C.Fundamentals

2. Course outcome
Determine voltage and current in A.C.circuits

3. Introduction
In the day to day life, we use electrical power for various applications including the domestic & industrial applications.
For most of the domestic applications, we use a single phase ac supply.
For high power industrial applications, the three phase ac supply is used.
For certain domestic applications such as telephones, the dc supply is used.
For certain applications such as electric trains, a high voltage DC system is used.

4. Difference between AC & DC Quantities
Sr. No.
Parameter AC DC 1.
Waveform 2.
Definition
It is a signal which changes its magnitude as well as polarity.
It is a signal which changes its magnitude but does not change its polarity. 3.
Use of transformer
Possible
Not possible 4.
Distribution efficiency
High
Low 5.
Design of machines
Easy
Not easy

5. Continued…….. Sr. No. Parameter AC DC 6. Generation Easy
From the ac waveform using commutator or rectifier 7.
Applications
AC motors, domestic & industrial supply etc.
DC machines, HVDC system

6. Advantages and disadvantages of A.C. over D.C.
The generation of A.C. is cheaper than that of D.C.
A.C. machines are simple , robust and do mot require much attention for their repairs and maintainance during their use.
Wide range of voltages are obtained by the use of transformer.
The magnitude of current can be reduced by using an inductance or a conductor without any appreciable loss of energy
A.C. can easily be converted into D.C. with the help of rectifiers.
When A.C. is supplied at higher voltages in long distance transmission , the line losses are small compared to a D.C. transmission

7. Disadvantegs
Peak value of A.C. is high and it is dangerous to use so better insulation is required.
It attracts person who touches it unlike D.C. which gives a repelling shock.
An A.C. is transmitted from surface of the conductor and hence need several strands of thin wires insulated from each other.

8. Electric potential
When a body is charged, it can attract an oppositely charged body and can repulse a similar charged body. That means, the charged body has ability of doing work. That ability of doing work of a charged body is defined as electrical potential of that body.
Definition:
Electric potential at a point in an electric field is defined as the amount of work to be done to bring a unit positive electric charge from infinity to that point.
The unit of electric potential is volt (V)

9. Potential difference:
Difference in the potential of the two points is called as potential difference.
Electric current:
Electric current is nothing but the rate of flow of electric charge through a conducting medium with respect to time.
Unit of current is Coulombs per second or Ampere(A)

10. waveform of sinusoidal A.C. cycle

11. Faraday Law of Electromagnetic Induction
Faraday's Experiment:

12. RELATIONSHIP BETWEEN INDUCED EMF AND FLUX
Position of magnet
Deflection in galvanometer
Magnet at rest
No deflection in galvanometer
Magnet moves towards the coil
Deflection in galvanometer in one direction
Magnet is held stationary at same position (near the coil)
Magnet moves away from the coil
Deflection in galvanometer but in opposite direction
Magnet is held stationary at same position (away from the coil)

13. Faradays law of electromagnetic induction
Conclusion:
From this experiment, Faraday concluded that whenever there is relative motion between conductor and a magnetic field, the flux linkage with a coil changes and this change in flux induces a voltage across a coil.

14. Generation of single phase A.C. by elementary alternator

15. Single phase generator
Single-phase generator (also known as single-phase alternator) is an alternating current electrical generator that produces a single, continuously alternating voltage.

16. Single phase generator
The design of revolving armature generators is to have the armature part on a rotor and the magnetic field part on stator.
A basic design, called elementary generator,is to have a rectangular loop armature to cut the lines of force between the north and south poles.
By cutting lines of force through rotation, it produces electric current. The current is sent out of the generator unit through two sets of slip rings and brushes, one of which is used for each end of the armature.
In this two-pole design, as the armature rotates one revolution, it generates one cycle of single phase alternating current (AC).
To generate an AC output, the armature is rotated at a constant speed having the number of rotations per second to match the desired frequency (in hertz) of the AC output.

17. Step 1. Armature at 0 degrees.

18. Continued…..
At zero degrees, the rectangular arm of the armature does not cut any lines of force, giving zero voltage output.

19. Step 2. Armature at 90 degrees.

20. As the armature arm rotates at a constant speed toward the 90° position, more lines are cut. The lines of force are cut at most when the armature is at the 90° position, giving out the most current on one direction

21. Step 3. Armature at 180 degrees.

22. As it turns toward the 180° position, lesser number of lines of force are cut, giving out lesser voltage until it becomes zero again at the 180° position.

23. Step 4. Armature at 270 degrees.

24. The voltage starts to increase again as the armature heads to the opposite pole at the 270° position. Toward this position, the current is generated on the opposite direction, giving out the maximum voltage on the opposite side

25. Step 4. Armature at 360 degrees.

26. The voltage decrease again as it completes the full rotation
The voltage decrease again as it completes the full rotation. In one rotation, the AC output is produced with one complete cycle as represented in the sine wave.

27. Definitions related with A.C. signal
Waveform – The waveform is a graph of magnitude of an AC quantity against time.
Cycle - Each repetition of one positive and one identical negative part is called a cycle.

28. 3. Instantaneous value - Instantaneous value of an AC quantity is defined the value of that quantity at that particular instant of time. E.g. i1, i2 etc.
4. Angular velocity (Angular frequency)( )- Angular displacement is rate of change of angle with respect to time. Unit is rad/sec.

29. Relationship between Frequency and Periodic Time
5. Time period or Periodic Time (T)- Time period is defined as the time taken by alternating quantity to complete its one cycle.
6. Frequency (f) – Frequency is the no of cycles completed within one second. Unit is herz (Hz) 
Relationship between Frequency and Periodic Time

30. 7. Peak to Peak Value
The PEAK TO PEAK value is the vertical distance between the positive maximum and negative maximum of the wave. It will be measured in volts on a voltage waveform, and may be labelled VPP or VPK−PK.
8. Amplitude/ Peak Value
The AMPLITUDE of a sine wave is the maximum vertical distance reached, in either direction from the centre line of the wave. In a voltage waveform the peak value may be labelled VPK or VMAX (IPK or IMAX in a current waveform).

31. 9. Average value –
Definition: The average of all the instantaneous values of an alternating voltage or currents over one half cycle is called Average Value.
The average value of a whole sinusoidal waveform over one complete cycle is zero as the two halves cancel each other out, so the average value is taken over half a cycle. The average value of a sine wave of voltage or current is times the peak value.
Average value = 0.637 × maximum value

32. 10. RMS value(Root mean square value) OR Effective value –
Definition: RMS value of an ac current is the dc current required to produce same amount of heat as that of ac current provided resistance and the time for which these current flow are identical.
In other words, the R.M.S value is defined as the square root of means of squares of instantaneous values.
RMS value = 0.707 × maximum value

34. 11. Form Factor - Form Factor is the ratio of the RMS value to the average value and is given as.
For a pure sinusoidal waveform the Form Factor will always be equal to 1.11.
12. Crest Factor - Crest Factor is the ratio the Peak value to the R.M.S. value of the waveform and is given as.
For a pure sinusoidal waveform the Crest Factor will always be equal to 1.414.

35. Phasor representation of sinusoidal A.C. quantity,

36. CONTINUED…….
Each instantaneous values of an ac signal can be represented as a phasor.
In the above fig phasor OA represents instantaneous value at an angle ɵ
1. Phasor - A phasor is a vector that has an arrow head at one end whose length represents the maximum value of the vector quantity ( V or I ) and one end of the vector is fixed that rotates in anticlockwise direction.

37. 2. Phase – The fraction of the time period that elapses in achieving certain instantaneous value is known as phase of that alternating quantity. 3. Phase angle- It is the angle made by the conductor axis with the reference axis. 4.

38. Phase difference - Phase difference is angular displacement between waveforms of same frequency. Phase Difference Equation Where: Am - is the amplitude of the waveform. ωt - is the angular frequency of the waveform in radian/sec. Φ (phi) - is the phase angle in degrees or radians that the waveform has shifted either left or right from the reference point.

39. Phase Relationship of a Sinusoidal Waveform

41. Leading phase difference – A leading alternating quantity is one which attains its zero or maximum value earlier as compared with the other quantity.
Waveform:

42. Equations: Phasor Diagram: Im Vm

43. Lagging phase differance – A lagging alternating quantity is one which attains its zero or maximum value after the other quantity.
Waveform:

44. Equations:
Phasor diagram: Vm Im

45. Types of AC series circuit -
1. Purely resistive circuit –
Circuit diagram –
Let the alternating voltage applied across the circuit be given by the equation

46. The instantaneous value of voltage across the resistance is given by
Then the instantaneous value of current flowing through the resistor shown in the figure below will be

47. Phase Angle and Waveform of Resistive Circuit
From equation (1) and (3), it is clear that there is no phase difference between applied voltage and current flowing through a pure resistive circuit, i.e. phase angle between voltage and current is zero.hence, in an AC circuit containing pure resistance, current is in phase with the voltage as shown in the waveform figure below.
  Waveform Phasor diagram

48. 2. Purely inductive circuit Circuit Diagram -
Equations :

49. Waveform Phasor diagram
The current in the pure inductive AC circuit lag the voltage by 90 degrees

50. 3. Pure capacitive circuit
Circuit Diagram
Equations :

51. Waveform Phasor Diagram
In the pure Capacitor circuit, the current flowing through the capacitor(IC) leads the voltage(VC) by an angle of 90 degrees.

52. Inductive Reactance:
Inductive reactance ( )- Inductive reactance is defined as the opposition offered by the pure inductor to the flow of current.
It is represented as,
Inductive reactance X L is proportional to the sinusoidal signal frequency (f ) and the inductance (L) (measured in HENRY), which depends on the physical shape of the inductor.
Inductive reactance has unit as ohm (Ω)

53. Capacitive Reactance
Capacitive reactance ( Xc ) - Capacitive reactance is defined as the opposition offered by the pure capacitor to the flow of current.
Capacitive Reactance is measured in ohm (Ω)
It is represented as,
Where:
 Xc = Capacitive Reactance in Ohms, (Ω)
  π (pi) = (decimal) or 22÷7 (fraction)
 ƒ = Frequency in Hertz, (Hz)
 C = Capacitance in Farads, (F) 

54. Impedance Z = lZl tan¯¹ (X/R) = IZI Ѳ Where ,
Impedance : Impedance is the combination of a resistance, inductive reactance and capacitive reactance.
It is also defined as the total opposition offered by resistor, inductor and capacitor in an ac circuit. Unit is ohm.
It is also represenred as
Z = lZl tan¯¹ (X/R) = IZI Ѳ
Where ,

55. Impedance Triangle
Impedance triangle- Impedance triangle is the right angled triangle whose base represents resistance, altitude represents resultant reactance and hypogenous represents impedance.

56. Types of power in ac circuits. Active, Reactive, and Apparent Power.
1. Active power (True Power or Real power or Watt-full Power, Useful Power)(P)- True power is the average power consumed by the given circuit. Active - or real or true - power is the power that is used to do work on the load. Active power is measured in watts (W) and is the power drawn by the electrical resistance of a system doing useful work. P = VI Cosθ (Watt)

57. 2. Reactive Power
Reactive Power (Use-less Power, Watt less Power, Imaginary power): (Q)
The powers that continuously bounce back and forth between source and load is known as reactive Power (Q)
Reactive power represent that the energy is first stored and then released in the form of magnetic field or electrostatic field in case of inductor and capacitor respectively.
Reactive power is given by Q = V I Sinθ
which can be positive (+ve) for inductive, negative (-ve) for capacitive load.
The unit of reactive power is Volt-Ampere reactive. I.e. VAR

58. 3.Apparent power
Apparent power (Total power)(S) – It is the total power supplied by the source to a circuit and it is equal to the vector sum of real power(P) and imaginary power(Q).
Apparent power formulas:
S = V I (VA)
Unit is volt-ampere(VA)
Apparent Power = √ (True power2 + Reactive Power2)

59. Power triangle Power Triangle –
Power triangle is the right angled triangle in which base represents the active power, altitude represents the reactive power and hypotenuse represents apparent power.

60. Power factor
1. The Cosine of angle between Current and Voltage is called Power Factor.
2. The ratio between Active Power and Apparent Power is called power factor.
Active power P = VI Cosθ
3.The ratio between resistance and Impedance is Called Power Factor Cosθ = R/Z

61. Importance of power factor
Power factor play an important role in AC circuits and power dissipation depends on this factor. For instant, we know that;
Power in a Single Phase AC Circuits = P = V x I CosФ
And Current in a Three phase AC Circuits = I = P / (V x CosФ)
I ∝ 1/CosФ……… (2)
It is clear from both equations (1) an (2) that Current “I” is inversely proportional to CosФ i.e. Power Factor. In other words, When Power Factor increases, Current Decreases, and when Power Factor decreases, Current Increases.
Now, In case of Low Power Factor, Current will be increased, and this high current will cause to the following disadvantages.

62. Disadvantage of low power factor -
1.) Large Line Losses (Copper Losses):
We know that Line Losses is directly proportional to the squire of Current “I2”
Power Loss = I2xR i.e., the larger the current, the greater the line losses i.e. I>>Line Losses
2.) Large kVA rating and Size of Electrical Equipments:
we know , CosФ = kW / kVA
Therefore, The Lower the Power factor, the larger the kVA rating of Machines also, the larger the kVA rating of Machines, The larger the Size of Machines and The Larger the size of Machines, The Larger the Cost of machines.
3.) Greater Conductor Size and Cost:
In case of low power factor, current will be increased, thus, to transmit this high current, we need the larger size of conductor. Also, the cost of large size of conductor will be increased.
4.) Poor Voltage Regulation and Large Voltage Drop:
Voltage Drop = V = IZ.
Now in case of Low Power factor, Current will be increased. So the Larger the current, the Larger the Voltage Drop.

63. Continued………… 5.) Low Efficiency:
In case of low Power Factor, there would be large voltage drop and large line losses and this will cause the system or equipments efficiency too low.
6.) Penalty from Electric Power Supply Company on Low Power factor Electrical Power supply Company imposes a penalty of power factor below 0.95 lagging in Electric power bill. So you must improve Pf above 0.95. 

64. Following are the causes of low Power factor:
1. Single phase and three phase induction Motors
2.  Varying Load in Power System)
3.  Industrial heating furnaces
4.  Electrical discharge lamps (High intensity discharge lighting) Arc lamps (operate at a very low power factor)
5.  Transformers
6.  Harmonic Currents

65. Advantages of Power factor improvement and Correction:
Increase in efficiency of system and devices
Low Voltage Drop
Reduction in size of a conductor and cable which reduces cost of the Cooper
An Increase in available power
Line Losses (Copper Losses) I2R is reduced
Appropriate Size of Electrical Machines (Transformer, Generators etc)
Eliminate the penalty of low power factor from the Electric Supply Company
Low kWh (Kilo Watt per hour)Saving in the power bill
Better usage of power system, lines and generators etc
Saving in energy as well as rating and the cost of the electrical devices and equipment is reduced.

66. RL Series Circuit
Circuit diagram-

67. Vector Diagrams for the Two Pure Components

68. Vector Diagram of the Resultant Voltage

69. Where,

70. The RL Impedance Triangle

71. Phase Angle From voltage triangle, Power factor :
Power in R L Series Circuit:

72. RC Series Circuit Circuits Diagram Where,
VR – voltage across the resistance R
VC – voltage across the capacitor C
V – total voltage across the RC Series circuit

73. Vector Diagrams for the Two Pure Components

74. Vector Diagram of the Resultant Voltage

76. The RC Impedance Triangle

77. Phase angle From voltage triangle: Power factor
Power in RC Series Circuit

78. RLC Series Circuit Circuit diagram = In the RLC Series Circuit
XL = 2πfL and XC = 1/2πfC
When the AC voltage is applied through the RLC Series Circuit the resulting current I flows through the circuit, and thus the voltage across each element will be
VR = IR that is the voltage across the resistance R and is in phase with the current I.
VL = IXL that is the voltage across the inductance L and it leads the current I by an angle of 90 degrees.
VC = IXC that is the voltage across the capacitor C and it lags the current I by an angle of 90 degrees.

79. Phasor Diagram of RLC Series Circuit
The phasor diagram of the RLC Series Circuit when the circuit is acting as an inductive circuit that means (VL>VC) is shown below and if (VL< VC) the circuit will behave as a capacitive circuit. Current is taken as reference
VR is drawn in phase with it.
VL is drawn leading by 90,
Vc is drawn lagging by 90,

80. Since VL and Vc are in opposition to each other there can be two cases:
1) XL > XC
2) XL < XC
3) XL = XC

81. Case-1: When XL > XC
Phasor diagram

82. Steps to draw the Phasor Diagram of the RLC Series Circuit
Take current I as the reference as shown in the figure above
The voltage across the inductor L that is VL is drawn leads the current I by a 90-degree angle.
The voltage across the capacitor c that is Vc is drawn lagging the current I by a 90 degree angle because in capacitive load the current leads the voltage by an angle of 90 degrees.
The two vector VL and VC are opposite to each other.

83. Phase Angle From the phasor diagram, the value of phase angle will be

84. If the inductive reactance is greater than the capacitive reactance than the circuit reactance is inductive giving a lagging phase angle.
Impedance Triangle

85. Case-2:When XL < XC
Phasor diagram
Impedance Triangle

86. Case-3) : When XL = XC Phasor diagram When XL = XC,
I XL = I XC , VL = VC
The opposing and equal voltages VC and VL now completely cancel each other out. The supply voltage and the circuit current must now be in phase, so the circuit is apparently entirely resistive! L and C have completely "disappeared". 
VS = VR = I R

87. Impedance Triangle As, XL = XC, XL-XC =0 So, Z = R Cosѳ = R/Z = 1
Ѳ = 0
Power factor is unity. Phase angle is zero.

88. Resonance in Series RLC Circuit

89. Variation of Inductive Reactance Vs Frequency

90. Variation of Capacitive Reactance Vs Frequency

91. Inductive Reactance and Capacitive Reactance Vs Frequency

92. Variation of Impedance Vs Frequency

93. Resonant Current

94. Resonance in parallel circuit
A parallel circuit containing a resistance, R, an inductance, L and a capacitance, C will produce a parallel resonance (also called anti-resonance) circuit when the resultant current through the parallel combination is in phase with the supply voltage. At resonance there will be a large circulating current between the inductor and the capacitor due to the energy of the oscillations, then parallel circuits produce current resonance.

95. Parallel circuit

96. Impedance in a Parallel Resonance Circuit

97. Parallel Circuit Current at Resonance