6.3 Balanced and unbalanced three-phase power calculations

Three-phase power calculations are crucial for understanding electrical systems. Balanced systems have equal loads across phases, simplifying analysis. Unbalanced systems require more complex techniques, but both types involve calculating active, reactive, and apparent power.

Power factor plays a key role in system efficiency. We'll explore methods for analyzing balanced and unbalanced loads, including symmetrical components and per-phase analysis. These concepts are essential for designing and troubleshooting three-phase circuits.

Power Quantities

Understanding Active, Reactive, and Apparent Power

• Active power (P) represents real power consumed by the load measured in watts (W)
• Reactive power (Q) denotes power oscillating between source and load measured in volt-amperes reactive (VAR)
• Apparent power (S) encompasses total power supplied by the source measured in volt-amperes (VA)
• Power triangle visually represents relationship between P, Q, and S
• Calculate apparent power using formula $S = \sqrt{P^2 + Q^2}$

Power Factor and Its Significance

• Power factor (PF) indicates efficiency of power utilization in electrical systems
• Defined as ratio of active power to apparent power: $PF = \frac{P}{S} = \cos\theta$
• Ranges from 0 to 1, with 1 being ideal (purely resistive load)
• Low power factor results in increased current flow and higher power losses
• Improve power factor through capacitor banks or synchronous condensers

Total Three-Phase Power Calculations

• Calculate total active power for balanced three-phase system: $P_{total} = 3V_{L}I_{L}\cos\theta$
• Determine total reactive power: $Q_{total} = 3V_{L}I_{L}\sin\theta$
• Compute total apparent power: $S_{total} = 3V_{L}I_{L}$
• $V_{L}$ represents line voltage, $I_{L}$ denotes line current
• For unbalanced systems, sum individual phase powers: $P_{total} = P_a + P_b + P_c$

Load Types

Balanced Load Characteristics and Analysis

• Balanced load exhibits equal impedances in all three phases
• Voltage and current magnitudes remain constant across phases
• Phase angles maintain 120° displacement between phases
• Simplifies calculations allows use of single-phase equivalent circuit
• Calculate line and phase quantities using relationships: $V_{L} = \sqrt{3}V_{ph}$ and $I_{L} = I_{ph}$ (for Y-connected systems)

Unbalanced Load Challenges and Solutions

• Unbalanced load results from unequal impedances across phases
• Causes voltage and current imbalances leading to increased losses and equipment stress
• Requires individual analysis of each phase
• Utilize symmetrical components method for simplified analysis
• Implement load balancing techniques (redistributing loads, using balancing transformers)

Analysis Techniques

Symmetrical Components Method

• Decomposes unbalanced three-phase system into three balanced systems
• Consists of positive sequence, negative sequence, and zero sequence components
• Positive sequence represents balanced three-phase system
• Negative sequence indicates degree of unbalance
• Zero sequence appears in systems with neutral or ground path
• Apply transformation matrix to convert phase quantities to sequence components

Per-Phase Analysis Approach

• Analyze each phase of three-phase system independently
• Suitable for both balanced and unbalanced systems
• Convert delta-connected loads to equivalent wye configuration if needed
• Calculate individual phase voltages, currents, and powers
• Sum results to obtain total system power and other parameters
• Verify calculations using power conservation principles