Single Phase AC:

AC signal in a pure Resistive circuit:

Resistors dissipate the power.
The phase of the current and voltage remain the same.
The current and the voltage achieve their maximum value simultaneously.
Resistor is a passive component, and it neither produces nor consumes electrical power. So, it converts the available energy into heat.
Resistor resistance value will remain constant regardless of the supply frequency.
Power factor of Resistor is unity always as the phase difference between current and voltage is zero.
The power consumed in the Resistor which is the real power is same as apparent power that is the product of rms value of voltage and current.

AC signal in a pure Inductive circuit:

When an alternating voltage is applied to a purely inductive circuit, the current lags the voltage by 90 degrees.
Pure Inductor has only reactance which is purely imaginary in nature.
The nature of the reactance of Pure Inductor is positive in nature.(+jX)
The reactance of pure inductor is directly proportional to frequency of generated single phase AC.
As current lags the voltage by 90 degree, the phase difference between current & voltage is 90 degree but in negative.
So the power factor is zero as cos(90)=0.
As the power factor is zero, so the power consumed=0.
So pure inductor is always loss less.

AC signal in a pure Capacitive circuit:

When an alternating voltage is applied to a purely inductive circuit, the current leads the voltage by 90 degrees.
Pure capacitor has only reactance which is purely imaginary in nature.
The nature of the reactance of Pure capacitor is negative in nature.(-jX)
The reactance of pure capacitor is inversely proportional to frequency of generated single phase AC.
As current leads the voltage by 90 degree, the phase difference between current & voltage is 90 degree but in positive.
So the power factor is zero as cos(90)=0.
As the power factor is zero, so the power consumed=0.
Hence pure capacitor is always loss less.

RL Series Circuit:

A circuit that contains a pure resistance R ohms connected in series with a coil having a pure inductance of L (Henry) is known as RL Series Circuit.
A choke coil can be considered as an example of RL series circuit.

In this circuit current I lags behind the applied voltage V by an angle less than 90 degree or in other words voltage leads the current I.
Current I is taken as a reference and the Voltage drop across the resistance is IR is drawn in phase with the current I.
The voltage drop across the inductive reactance is +jIXL is drawn ahead of the current I as the current lags voltage by an angle of 90 degrees in the pure Inductive circuit.
The vector sum of the two voltages drops VR and VL is equal to the applied voltage V.
Here total Impedance Z is R+jX, where X is Reactance of Inductor.
The phase angle is same as angle of impedance.

As the current is lagging here so the power factor for this is also lagging.
Power consumed in a RL circuit is the power across the Resistance only as the inductor is lossless and hence consumes a zero power.
So the power factor here is the ratio of power consumed across Resistance to the total Volt Ampere Power Supplied to the circuit.
More is the power factor more is the power consumed, as the power across resistance is proportional to power factor.
Therefore a component with more resistive part consumes more power as compare to the component with less resistive part.
This is the reason filament lamps consume more power than CFL or LED bulbs.

RC Series Circuit:

A circuit that contains pure resistance R ohms connected in series with a pure capacitor of capacitance C farads is known as RC Series Circuit.
In this circuit current I leads the applied voltage V by an angle less than 90 degree or in other words voltage lags behind the current I.

Current I (r.m.s value) is considered as a reference vector.
Voltage drop in resistance VR equal to IR, is taken in phase with the current vector.
Voltage drop in capacitive reactance VC equal to IXC is drawn 90 degrees behind the current vector, as current leads voltage by 90 degrees (in the pure capacitive circuit)
The vector sum of the two voltage drops is equal to the applied voltage V (r.m.s value).

Here total Impedance Z is R-jX, where X is Reactance of Capacitor.
The phase angle is same as angle of impedance.

As the current is leading here, so the power factor for this is also leading.
Power consumed in a RC circuit is the power across the Resistance only as the capacitor is lossless and hence consumes a zero power.
So the power factor here is the ratio of power consumed across Resistance to the total Volt Ampere Power Supplied to the circuit.
More is the power factor more is the power consumed, as the power across resistance is proportional to power factor.

RLC Series Circuit:

When a pure resistance of R ohms, a pure inductance of L Henry and a pure capacitance of C farads are connected together in series combination with each other then RLC Series Circuit is formed.

Here total Impedance Z is R+jXL-jXc, where XL is Reactance of Inductor and Xc is Reactance of Capacitor.
The phase angle is same as angle of impedance.
Here 3 different cases appears: (a) if XL=Xc(Inductive Reactance=Capacitive Reactance) (b) if XL is greater than Xc (c) if Xc is greater then XL
(a) if XL=Xc then overall reactance becomes zero and the circuit becomes pure resistive. So voltage remains in phase with current. Power factor is unity.
(b) if XL is greater then Xc then the circuit becomes inductive. Hence Voltage leads the current by angle which is less than 90 degree. Power factor is 0 to 1 and it is lagging in nature.
(c) if Xc is greater then XL then the circuit becomes capacitive. Hence current leads the voltage by angle which is less than 90 degree. Power factor is 0 to 1 and it is leading in nature.

AC in any combination of impedance circuit:

For any combination of impedance, the nature of circuit can be found from the overall equivalent impedance.
The equivalent impedance of circuit can be found in the same way as finding the equivalent impedance.
Equivalent impedance of series impedances is sum of all impedances.
Equivalent impedance of parallel impedances is found as below:
If the equivalent impedance has positive imaginary part then it is inductive in nature & the power factor is lagging.
Similarly if the equivalent impedance has negative imaginary part then it is capacitive in nature & the power factor is leading.

Three Phase AC

Three phase systems are well suited to electricity supply applications because of their ability to transmit high powers efficiently and to provide powerful motor drives.

Polyphase Voltage:

A single phase alternator has one armature winding only.
But if the number of armature windings is increased, then it becomes polyphase alternator and it produces as many independent voltage waves as the number of windings or phases.
These windings are displaced from one another by equal angles, the values of these angles being determined by the number of phases or windings and it is equal to 360p. (Where p is the number of phases)
The word polyphase means poly (many or numerous) and phases (winding or circuit).
In a two phase alternator, the armature windings are displaced 90 electrical degrees apart.
A 3 phase alternator has three independent armature windings which are 120 electrical degrees apart, so the voltages induced in the three windings are 120 degree apart in phase.

Reasons for switching to PolyPhase AC:

The earliest application of alternating current was for heating the filaments of electric lamps. For this purpose the single phase system is enough. But when a.c. motors were developed, and it was found that for this application the single phase system was not very satisfactory.
For instance, the single phase induction motor, the type most commonly employed is not self starting unless it was fitted with an auxiliary winding.
Then two separate windings with currents differing in phase by a quarter of a cycle were used.
Gradually three windings with currents differing in phase by a third of a cycle was also used
It was found that the induction motor becomes self starting by using the above 2 methods and had better efficiency and power factor than the corresponding single phase machine.
The system utilizing two windings is referred to as a two phase system and that utilizing three windings is referred to as a three phase system.

Importance of 3 phase system:

For generation for same amount of Electric Power, the size of 3 phase alternator is small as compare to single phase alternator.
Overall cost of alternator for 3 phase is less then that of single phase alternator for generation of same amount of power.
Transportation and installation of 3 phase alternator is convenient due to the invention of transformers.
Less space is required to accommodate 3 phase alternator in power houses.
For electric power transmission and distribution of same amount of power, the requirement of conductor material is less in 3 phase system as compare to single phase system.
Hence, the 3 phase transmission and distribution system is economical as compare single phase system.
In the 3 phase system, the instantaneous power is almost constant over the cycle results in smooth and vibration free operation of the machine.
A 3 phase system can be used to feed a single phase load, whereas vice versa is not possible

Generation of 3 phase AC:

Normally for 3 phase AC generation Armature is kept stationary as the weight of windings, the voltages generated and complexities are more.
Magnetic field is kept as rotating as it works in low voltage with less complexity.
It has three armature coils placed with 120 degree displaced apart from one another.
Thus the three coils have three emf induced in them which are similar in all respects except that they are 120 degree out of phase with one another.
Each of the voltage wave is considered as sinusoidal having maximum value of Em.

Phase Sequence:

It is the order in which the three phases attain their peak or maximum values.
(a) part has phase sequence A B C where as (b) part has sequence as ACB
The phase sequence can be reversed by interchanging any pair of lines.
For induction motor, reversal of sequence results in the reversed direction of motor rotation.

Numbering of Phases:

The three phases are named as per three colors.
The colors used commercially are red, yellow (or sometimes white) and blue.
Phase sequence can be considered as RYB or RBY.

Interconnection of Three Phases

If the three armature coils of the 3 phase alternator as shown in the figure are not interconnected but are kept separate, then each phase or circuit would need two conductors.

Total number of conductors, in this case, will become six.
It means that each transmission cable would contain six conductors which will make the whole system complicated and expensive.
Hence, the three phases are generally interconnected for substantial saving of copper.
The general methods of interconnection are (a) Star or Wye connection and (b) Mesh or Delta connection.

Delta or Mesh Connection:

In this form, of interconnection the dissimilar ends of the three phase winding are joined together i.e. the starting end of one phase is joined to the finishing end of the other phase and so on.
In a way the three windings are joined in series to form a closed mesh.
Instantaneous value of total e.m.f is always zero here.
It does not have any neutral point.
It can be seen properly in the below circuit diagram in a form of delta.

Star or Wye (Y) Connection:

In this method of interconnection, the similar ends say, which is R1, Y1 and B1 terminal ends of three coils (it could be finishing ends also) are joined together at point N known as neutral point.
The three conductors meeting at point N are replaced by a single conductor known as neutral conductor.
Such an interconnected system is known as four wire, 3 phase system and is diagrammatically shown here.

If these three loads are exactly alike, the phase currents have the same peak value, and differ in phase by 120 degree.
Hence if the instantaneous value of the current in load L1 is represented as below:

The instantaneous value of the resultant current in neutral conductor is zero.

Three Phase Voltage & Currents:

Voltages and Currents measured at 3 phase supply side are known as Line Parameters.
Voltages and Currents measured at 3 phase load side are known as Phase Parameters.

Voltages and Currents in Star(Y) Connection:

   - Voltage between any 2 supply lines is known as Line Voltage. For example between RY or RB or YB.
   - Current in any supply line is known as Line current. For Example Ir or Iy or Ib.
   - Voltage across any load is known as phase voltage or here it can be considered as voltage between the load and neutral point.
   - The voltage induced in each winding is called the phase voltage and current in each winding is likewise known as phase current. 
   - However, the voltage available between any pair of terminals (or outers) is called line voltage (VL) and the current flowing in each line is called line current (IL ).

   - Line voltage VRY between line 1 and line 2 is the vector difference of ER and EY.
   - Line voltage VYB between line 2 and line 3 is the vector difference of EY and EB.
   - Line voltage VBR between line 3 and line 1 is the vector difference of EB and ER.
   - Hence, VRY is found by compounding ER and EY reversed and its value is given by the diagonal of the parallelogram as shown in the figure. It can be seen as subtraction of ER and EY.

   - Line voltages are 120 degree apart.
   - Line voltages are 30 degree ahead of their respective phase voltages.
   - The angle between the line currents and the corresponding line voltages is 30 degree plus phase angle between phase voltage and phase current with current lagging.
   - each line is in series with its individual phase winding, hence the line current in each line is the same as the current in the phase winding to which the line is connected. (So line current is same as phase current)
   - The total active or true power in the circuit is the sum of the three phase powers.
   - The total active or true power can be expressed as 3 times of product of Phase voltage, current and power factor of each phase load.
   - The total active or true power can be expressed as root three times of product of line voltage, current and power factor of each phase load.

Voltages and Currents in Delta Connection:

It is seen from the figure that there is only one phase winding completely included between any pair of terminals.
– Hence, in Delta connection, the voltage between any pair of lines is equal to the phase voltage of the phase winding connected between the two lines considered. (So the Phase voltage is same as Line voltage)

    - current in each line is the vector difference of the two phase currents flowing through that line.
    - Current in line 1 is vector difference of IR IB, Current in line 2 is vector difference of IY IR, Current in line 3 is vector difference of IB IY.

    - Line currents are 120 degree apart 
    - Line currents are 30 degree behind the respective phase currents 
    - The angle between the line currents and the corresponding line voltages is 30 degree plus phase angle between phase voltage and phase current with the current lagging.
    - The total active or true power in the circuit is the sum of the three phase powers.
    - The total active or true power can be expressed as 3 times of product of Phase voltage, current and power factor of each phase load.
    - The total active or true power can be expressed as root three times of product of line voltage, current and power factor of each phase load.