Advanced Electrical Theory - Three Phase Voltages and Systems

Exam Information

• Exam Date: June 11 (Wednesday)
• Time: 9:00 AM to 3:00 PM
• Duration: One day, one attempt within the specified hours.

Three Phase Voltages

• Definition: Multiple (more than one) sinusoidal voltages separated by a phase angle.
• Common System: The most common polyphase system is the three-phase system (3-Φ).
• Applications: Commonly used in industrial applications.
• Labeling: Voltages can be labeled as:
  • R (Red), B (Blue), W (White)
  • A, B, C
• Phase Angles:
  • V_A: Starts at 0 degrees (reference).
  • VB: Lags VA by 120 degrees.
  • VC: Lags VB by 120 degrees (lags V_A by 240 degrees).

Three-Phase Generator

• Simultaneously produces three sinusoidal voltages separated by 120 degrees.
• Two-pole configuration (North and South).
• Output: Three outputs (A, B, C), with 'A' connected to a common ground.
• Induction Motor Example: Three-phase induction motors produce rotor current through induction, not through slip rings and brushes.

Advantages of Three-Phase Systems

• Reduced Copper Wire Size: Requires less copper to carry current from generator to load, resulting in lighter, smaller wires.
• Constant Power: Produces constant power in the load, resulting in uniform energy conversion.
• Constant Rotating Magnetic Field: Maintains constant shaft speed on motors.

Types of Three-Phase Systems

• Y-Connected (Star Connection)
• Delta-Connected

Y-Connected (Star Connection) System

• Configuration: Generator coils connected in a 'Y' shape with a neutral point.
• Coil Connections:
  • Each coil (A, B, C) has a start and end point (A', B', C').
  • All end points are connected together at the star point.
  • The star point is usually connected to neutral and grounded.
• Components:
  • Three-phase load.
  • Generator (often the secondary side of a three-phase transformer).
• Phase Voltage (V_{phase}): Voltage across the coil itself (from start to end).
• Phase Current (I_{phase}): Current flowing through the phase.
• Line Current (I_{line}): Current flowing out of the star point through the phase and into the line.
• Relationship: In a star-connected system, I{phase} = I{line}.
• Line Voltage (V_{line}): Voltage between any two lines.
• Relationship: V{line} = \sqrt{3} \times V{phase}.

Key Points of Y-Connected Systems

• Magnitude of each line current equals the corresponding phase current.
• Three line voltages exist, one across each phase voltage pair.
• Magnitude of each line voltage equals \sqrt{3} times the magnitude of the phase voltage.
• There is a 30-degree phase difference between each line voltage and the nearest phase voltage.

Voltage Standards

• Australia: Phase voltage is 240V, Line voltage is 415V (240 \times \sqrt{3} = 415).
• United States: Phase voltage is 120V, Line voltage is 208V (120 \times \sqrt{3} \approx 208).

Balanced vs Unbalanced Loads

• Balanced Load: Discussed when loads are perfectly balanced in Y-connected systems.

Advantages of Three-Phase Systems (Revisited)

• Thinner conductors can be used (less copper) to transmit the same kVA (kilovolt-amperes) at the same voltage. Copper usage is reduced by approximately 25%, lowering construction and maintenance costs.
• Lighter lines are easier to install, requiring less massive supporting structures.
• Three-phase equipment and motors have better running and starting characteristics due to a more even flow of power.
• Larger motors are typically three-phase because they are self-starting and don't require additional starting circuits.

Delta-Connected System

• Configuration: The end of one coil is connected to the start of the next coil, forming a closed loop (delta shape).
• Neutral: Delta connections do not have a neutral point.
• Line Voltage (V_{line}): Voltage between any two lines.
• Phase Voltage (V_{phase}): Voltage across the coil itself.
• Relationship: In a delta-connected system, V{line} = V{phase}.
• Line Current (I{line}): I{line} = \sqrt{3} \times I_{phase}.

Phase Diagrams

• Difference of 30 degrees between line and phase voltages is illustrated using phase diagrams.
• Phase sequence: Determined by the order in which the phase voltages pass through a fixed point.

Balanced and Unbalanced Systems

Balanced Systems

• Conditions:
  • Currents in each line are the same.
  • Load connected to each phase has the same power factor.
• Example: A three-phase system where each phase has a purely resistive load, and 150 amperes flow in each phase. The power factor is one (resistive load).
• Advantage: Balanced loads do not require a neutral wire.

Unbalanced Systems

• Single-phase loads:
  • Star-connected systems can power both single-phase and three-phase loads.
  • Delta-connected systems can only power three-phase loads.
• A three-phase four-wire (star-connected) system can supply 240V to single-phase loads by connecting them between any active line and the neutral.
• The neutral is connected between the star point and ground.

Calculations for Unbalanced Systems

• When systems are unbalanced, currents and power factors differ across phases. Vector math is required to analyze such systems.
• Example: Simulating the circuit and calculating values for voltages, current and resistance.

Neutral Current in Unbalanced Systems

• Assumption: Phase currents flowing out of the star point are positive; neutral current flowing into the star point is negative.
• Calculation Method: The neutral current equals the negative of the phasor sum of the line currents.

Power in Three-Phase Systems

• Single-Phase Power (Review):

  • Apparent Power (S) = V × I
  • True Power (P) = V × I × Power Factor
  • Reactive Power (Q)
  • Power Factor = P/S

• Three-Phase Balanced System Apparent Power Calculation:

  • S = \sqrt{3} \times V{line} \times I{line} Where
    • V_{line} is the line voltage.
    • I_{line} is the line current.

• Three-Phase Balanced System True Power Calculation:

  • P = \sqrt{3} \times V{line} \times I{line} \times Power Factor Where
    • V_{line} is the line voltage.
    • I_{line} is the line current.

• Power calculations in both star and delta connections are similar, but formulas vary depending on whether phase or line values are used. In balanced systems, power calculations can be simplified; in unbalanced systems, individual phase powers must be calculated and summed.