1. Chapter 12 Polyphase Circuits
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2. Polyphase / Three-phase Sources
Polyphase (in practice, three-phase) voltage sources are important for transmission over the grid, and for large industrial loads.
These three sources are 120° out of phase with each other.
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3. Double-Subscript Voltage
The notation Vab indicates the voltage between point a (+) and point b (−).
Indicating the + and − terminals is redundant if double-subscript voltages are used.
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4. KVL with Double-Subscripts
KVL equations can be written correctly without reference to a circuit diagram:
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5. Example: Voltage Calculations
This is an example three-phase source with a neutral.
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6. Currents and Double-Subscripts
The double subscript notation Iab means the current from a to b by the direct path
When the direct path
is not obvious
(e.g. between c and d)
we need to use a different
notation.
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7. Household Power: not Polyphase
The typical North American household is provided single-phase 3-wire power, where V1≅110 V and Vab=2V1.
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8. Current in Neutral Wire
When the A and B loads are balanced, the neutral wire from n to N carries no current:
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9. Example: Effect of Line Resistance
Determine the power delivered to each of the three loads as well as the power lost in the neutral wire and each of the two lines.
Answer:
P50 = 206 W
P100 =117 W
P20+j10 = 1763 W
PaA = 126 W
PbB = 108 W
PnN =3 W
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10. Three-Phase Y-Y Connection
A balanced three phase source has and
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11. Phase Sequences The abc (positive) and cba (negative) sequences:
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12. Line-to-Line Voltages
The line voltages have amplitude
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13. Balanced Y-Y: Zero Neutral Current
For a balanced source and balanced loads, the neutral wire current INn=0 and so the neutral wire could have any impedance, including ∞.
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14. Example: Y-Y Power Find the total power delivered to the loads.
Answer: P=600 W
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15. Example: Per-phase Analysis
A balanced three-phase system with a line voltage of 300 V is supplying a balanced Y-connected load with 1200 W at a leading PF of 0.8.
Find the line current and the per-phase load impedance.
Answer: IL = 2.89 A, Zp = 60ej36.9◦ Ω
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16. The Delta Connection
A delta-connected load is also commonly used (note the absence of the neutral wire).
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17. Line and Phase Currents for a Delta Load
An example phasor diagram for an inductive phase impedance. Note that
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18. Power Measurements: The Wattmeter
The wattmeter is a four-terminal device that measures power delivered to the network if connected as shown:
the wattmeter
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19. Measuring Three-Phase Power
The sum of the powers measured by the wattmeters is the total power delivered. The common node x is arbitrary.
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20. The Two-Wattmeter Method
One wattmeter can be eliminated if the point x is moved to a line (as shown, B).
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