1. Lecture 10 - Transformers/Three-Phase Circuits

2. Lecture 10

3. Outline Mutual inductance (review) Transformers
Balanced Three-Phase System
Balanced sources
Balanced loads
Circuit analysis
Phase voltage/current
Line voltage/current
Lecture 10

4. Review: Self Inductance
Self inductance: reaction of the inductor to the change in current through itself.
Flux 𝜙=𝒫𝑁𝑖
𝒫: permeance
𝑣=𝑁 𝑑𝜙 𝑑𝑡 =𝐿 𝑑𝑖 𝑑𝑡 ,𝐿=𝑁 𝑑𝜙 𝑑𝑖
Permeance: 磁导率, 磁导
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5. Review: Mutual Inductance
Mutual inductance: reaction of the inductor to change in current through another inductor.
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6. Mutual Inductance: General Case
Two circuits lined by a magnetic field
𝐿 1 , 𝐿 2 : self-inductances
𝑀: mutual inductance
Dots: indicating polarity of mutually induced voltages.
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7. Exercise
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8. Energy in a Coupled Circuit
The energy stored in an inductor is
For coupled inductors, the total energy stored is
The positive sign is selected when the currents both enter or leave the dotted terminals.
[Text, Ch. 6.5]
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9. Coupling Coefficient 𝒌
Define a parameter describes how closely M approaches upper limit.
Coupling coefficient, 0≤𝑘≤1.
determined by the physical configuration of the coils.
The system cannot have negative energy
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10. Linear Transformers
A transformer is a magnetic device that takes advantage of mutual inductance.
Called linear if the coils are wound on a magnetically linear material, i.e. permeability 𝜇 is constant.
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11. Transformer Impedance
An important parameter to know for a transformer is how the input impedance Zin is seen from the source.
Zin is important because it governs the behavior of the primary circuit.
Reflected impedance from secondary to primary
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12. Ideal Transformers The ideal transformer has:
Coils with very large reactance
(L1, L2, M →)
Coupling coefficient k=1.
Primary and secondary coils are lossless, 𝑅 1 = 𝑅 2 =0.
𝐕 𝟐 𝐕 𝟏 = 𝑁 2 𝑁 1 =𝑛
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13. Ideal Transformers II The current is related as: Reflected impedance
𝐕 𝟐 𝐕 𝟏 = 𝑁 2 𝑁 1 =𝑛
The current is related as:
Reflected impedance
𝐈 𝟐 𝐈 𝟏 = 𝑁 1 𝑁 2 = 1 𝑛
1 𝑛 2 𝑍 𝐿
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14. Ideal Autotransformer
Autotransformer uses one winding for primary & secondary
It does not offer isolation!
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15. Single Phase vs. Polyphase
Circuits that operate at the same frequency but with multiple sources at different phases are called polyphase.
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16. Balanced Three-Phase Sources
Three phase voltages are typically produced by a three-phase AC generator.
The output voltages look like below.
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17. Balanced Sources A source is said to be balanced when
Two sequences for the phases:
and 120°out of phase
Positive
Negative
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18. Connecting the Sources
Three phase voltage sources can be connected to the loads by either three or four wire configurations.
Three-wire configuration accomplished by Delta connected source.
Four-wire system accomplished using a Y connected source.
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19. Line-to-Line Voltage Use the positive sequence:
The line to line (or line in short) voltages:
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20. Balanced Loads
Similar to the source, a balanced load is one that has the same impedance presented to all three voltage sources.
They may also be connected in either Delta or wye
For a balanced wye connected load:
For a balanced delta connected load:
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21. Source-Load configurations
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22. Source-Load Configurations
Y-Y
Y-Delta

23. Source-Load Configurations
Delta-Y
Delta-Delta

24. Balanced Y-Y connection
Any three-phase system can be reduced to an equivalent Y-Y system.
The load impedances Zy is assumed to be balanced.
This can be the source 𝑍 𝑠 , line 𝑍 𝑙 and load 𝑍 𝐿 together.
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25. Example
Calculate the line currents.
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26. Y and ∆, Which One Better?
a balanced delta-connected load is more common than a balanced wye-connected load. This is due to the ease with which loads may be added or removed from each phase of a delta-connected load. This is very difficult with a wye-connected load because the neutral may not be accessible.
On the other hand, delta-connected sources are not common in practice because of the circulating current that will result in the delta-mesh if the three-phase voltages are slightly unbalanced.
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27. Content for the Discussion Session
Analysis of other 3 cases:
Y-∆
∆−∆
∆−Y
Equivalent circuits
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