POLYPHASE CIRCUITS Three Phase Circuits Advantages of polyphase circuits Steady power Less wear and tear on the lines Fewer conductors and so more economical

THREE PHASE CIRCUITS Theorem For a balanced three phase circuit the instantaneous power is constant

Proof of Theorem For a balanced three phase circuit the instantaneous power is constant

THREE-PHASE CONNECTIONS Positive sequence a-b-c Y-connected loads Delta connected loads

Advantages of wye Access to two voltages, the line-to-line and line-to-neutral Provides a convenient place to connect to ground  Advantages of delta Stays in balance better when serving unbalanced loads Capable of trapping the third harmonic

SOURCE/LOAD CONNECTIONS BALANCED Y-Y CONNECTION Line voltages Positive sequence phase voltages For this balanced circuit it is enough to analyze one phase

LEARNING EXAMPLE For an abc sequence, balanced Y - Y three phase circuit Balanced Y - Y Determine the phase voltages Positive sequence a-b-c The phasor diagram could be rotated by any angle Positive sequence phase voltages Relationship between phase and line voltages

LEARNING EXAMPLE For an abc sequence, balanced Y - Y three phase circuit Because circuit is balanced data on any one phase is sufficient Determine line currents and load voltages Chosen as reference Abc sequence

LEARNING EXTENSION For an abc sequence, balanced Y - Y three phase circuit Relationship between phase and line voltages

LEARNING EXTENSION For an abc sequence, balanced Y - Y three phase circuit Determine source phase voltages Currents are not required. Use inverse voltage divider Positive sequence a-b-c

DELTA CONNECTED SOURCES Convert to an equivalent Y connection Relationship between phase and line voltages Example

Three-Phase Balanced AC Power Circuits Step 1. Convert the source/load connection to a wye–wye connection if either the source, load, or both are connected in delta since the wye–wye connection can be easily used to obtain the unknown phasors. Step 2. Only the unknown phasors for the a -phase of the circuit need be determined since the three-phase system is balanced. Step 3. Finally, convert the now known phasors to the corresponding phasors in the original system.

LEARNING EXAMPLE Determine line currents and line voltages at the loads Source is Delta connected. Convert to equivalent Y Analyze one phase Determine the other phases using the balance

LEARNING EXTENSION Compute the magnitude of the line voltage at the load Source is Delta connected. Convert to equivalent Y Analyze one phase Only interested in magnitudes!