2. THREE PHASE CIRCUITS, Electrical Engg 3rd semester notes EE 3002,

 

2.1 Phasor and Complex Representation of Three-Phase Supply

• Three-phase supply: A three-phase system consists of three sinusoidal AC voltages, each 120° apart in phase. This is the most common type of power supply used in industrial and commercial systems because it provides a constant power flow.

Phasor Representation:

• Each phase voltage can be represented as a phasor, which is a rotating vector in the complex plane.
• The three-phase voltages are:
  • $V_A = V_m \angle 0^\circ$
  • $V_B = V_m \angle -120^\circ$
  • $V_C = V_m \angle 120^\circ$

Where $V_m$ is the maximum voltage of each phase.

Complex Representation:

• A three-phase voltage can also be represented as complex numbers, where the real and imaginary parts correspond to the components of the voltage in the rotating vector.

Diagram Suggestion:

• Three phasors (rotating vectors) representing the three-phase voltages, each 120° apart.

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2.2 Phase Sequence and Polarity

• Phase Sequence: This refers to the order in which the voltages in a three-phase system reach their peak. There are two possible phase sequences:

  • Positive Phase Sequence: $A, B, C$ (voltage A leads B by 120°, B leads C by 120°).
  • Negative Phase Sequence: $A, C, B$ (voltage A leads C by 120°, C leads B by 120°).

• Polarity: The polarity of each phase in the three-phase system tells you which direction the current is flowing at any given time. This is critical for ensuring proper operation of motors and equipment connected to the supply.

Diagram Suggestion:

• A simple line diagram showing the voltages for both positive and negative phase sequences, with arrows showing the direction of rotation.

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2.3 Types of Three-Phase Connections

There are two common ways to connect the three phases: Star (Wye) connection and Delta connection.

1. Star (Wye) Connection:

• In this connection, one end of each of the three windings is connected to a common point (neutral point).
• The other end of each winding is connected to a line.
• The line voltage is $\sqrt{3}$ times the phase voltage.
• The current in each phase is the same as the line current.

2. Delta Connection:

• In this connection, the three windings are connected end-to-end in a triangle, forming a closed loop.
• The line voltage is equal to the phase voltage.
• The current in each line is $\sqrt{3}$ times the phase current.

Diagram Suggestion:

• Diagrams of both star and delta connections with labeled line and phase points.

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2.4 Phase and Line Quantities in Three-Phase Star and Delta System

Star Connection:

• Line Voltage (V_L): The voltage between any two lines in the system.
  $V_L = \sqrt{3} \times V_{\text{phase}}$
• Line Current (I_L): The current flowing through the lines.
  $I_L = I_{\text{phase}}$

Delta Connection:

• Line Voltage (V_L): The voltage across any two lines, which is equal to the phase voltage.
  $V_L = V_{\text{phase}}$
• Line Current (I_L): The current flowing through the lines, which is $\sqrt{3}$ times the phase current.
  $I_L = \sqrt{3} \times I_{\text{phase}}$

Diagram Suggestion:

• For Star Connection: Show the relationship between phase voltage, line voltage, and phase current, line current.
• For Delta Connection: Show similar relationships, emphasizing that $V_L = V_{\text{phase}}$ and $I_L = \sqrt{3} \times I_{\text{phase}}$.

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2.5 Three-Phase Power, Active, Reactive, and Apparent Power in Star and Delta System

In three-phase systems, the total power is calculated by considering the voltages and currents in both the line and phase quantities.

Active Power (Real Power):

• Active power (P) is the power that does useful work, measured in Watts (W). It is the product of the voltage, current, and the power factor (cosine of the phase angle between voltage and current).

  $P = \sqrt{3} \times V_L \times I_L \times \cos(\phi)$
  In a star connection, $V_L = \sqrt{3} \times V_{\text{phase}}$, so the active power can also be expressed in terms of phase quantities.

Reactive Power:

• Reactive power (Q) is the power that alternates between the source and the load but doesn't perform any useful work. It is measured in Volt-Amperes Reactive (VAR).

  $Q = \sqrt{3} \times V_L \times I_L \times \sin(\phi)$

Apparent Power:

• Apparent power (S) is the total power supplied to the circuit, measured in Volt-Amperes (VA). It is the combination of active and reactive power.

  $S = \sqrt{3} \times V_L \times I_L$

• The Power Factor $pf$ is the ratio of active power to apparent power: $pf = \frac{P}{S}$

In Star Connection:

• Active power, reactive power, and apparent power can be calculated in terms of phase quantities, but for line quantities, the formulas above apply.

In Delta Connection:

• The total active, reactive, and apparent power are calculated the same way, but using the relationships for line and phase quantities.

Diagram Suggestion:

• A power triangle showing active, reactive, and apparent power. For both star and delta systems, depict the relationships between these powers and their phase differences.

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Summary Diagrams:

1. Phasor Representation: Show three sinusoidal waveforms 120° apart, or three rotating vectors.
2. Phase Sequence: Show the voltages for positive and negative phase sequences.
3. Star and Delta Connection: Illustrate the wiring diagrams for both star and delta configurations.
4. Line and Phase Quantities: Diagram the relationships between line and phase voltages and currents in both connections.
5. Power Triangle: Show the relationship between active, reactive, and apparent power for star and delta systems.