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Lesson 15: Polyphase Alternating Current

Recommended Post : 【Circuit Theory】 Circuit Theory Table of Contents


1. Overview

2. Voltage and Current of Balanced Three-Phase Power Circuits

3. Y-Δ Connection of Loads

4. Power in Three-Phase Circuits

5. Symmetrical n-Phase Circuits

6. Three-Phase V-Connection

7. Symmetrical Coordinates



1. Overview

⑴ Polyphase AC System

① Several mechanical powers with the same frequency but different phases coexist in the same circuit

Advantage 1: High efficiency of generation

Advantage 2: Generation and transmission of high voltage

Advantage 3: Generated power remains constant regardless of time

Advantage 4: Low risk of failure

⑵ Symmetrical Polyphase System

① When n polyphase powers have the same magnitude and phase differences between them are equal

② Antonym : Asymmetrical Polyphase System

⑶ Balanced Polyphase System

① When the sum of instantaneous powers of each phase is constant

② Unbalanced Polyphase System

⑷ 3-Phase Generator : A generator with three windings spaced 120° apart on the rotor

Figure. 1. Configuration of a 3-Phase Generator



2. Voltage and Current of Balanced Three-Phase Power Circuits

⑴ Voltage and Current in Y Power Circuits

⑵ Voltage and Current in Δ Power Circuits



##3. Y-Δ Connection of Loads

⑴ Let’s denote the resistances close to terminals a, b, c in a Y circuit as Ra, Rb, Rc, respectively.

⑵ Let’s denote the resistances between terminals a-b, b-c, c-a in a Δ circuit as R1, R2, R3, respectively.

⑶ Total composite resistance between terminals a-b, b-c, c-a

⑷ Y → Δ Equivalence Transformation : Rarely used. Holds for impedance as well.

⑸ Δ → Y Equivalence Transformation : Frequently used. Holds for impedance as well.



4. Power in Three-Phase Circuits



5. Symmetrical n-Phase Circuits

⑴ Formation of Phasors

① Line voltage

② Line current = Phase current

③ Phase relationship : Line voltage leads phase current, and its magnitude is as follows.

⑵ Imaginary Configuration

① Line voltage = Imaginary voltage

② Line current

③ Phase relationship : Line current lags phase current, and its magnitude is as follows.

⑶ Rotating System

① Switching the phase sequence of any two coils in 3-phase AC changes the direction of the rotating system

② Rotating system’s angular velocity is a constant angular velocity ω

③ Symmetrical n-phase AC forms a circular rotating system, asymmetrical n-phase AC forms an elliptical rotating system

④ Power supplied to a symmetrical balanced load in a balanced polyphase AC circuit remains constant regardless of time

⑷ n-Phase Power

① Formulation

② 3-voltage method

Figure. 2. 3-voltage method

③ 3-current method

Figure. 3. 3-current method

④ 2-power method : Method to measure the power of a 3-phase load using two power meters (* Note: Denotes complex conjugate)

Figure. 4. 2-power method

By drawing vector diagrams, the following facts can be observed



6. Three-Phase V-Connection

⑴ Voltage and Current in V-Connection

① In the case of a fault in one power source in a Δ-connected 3-phase power system, switching to a V-connection can supply 3-phase voltage to the load without any issue.

⑵ V-Connection, Y-Connection, Δ-Connection

① Y-Connection ○ Line voltage

○ Line current

○ Output

② Δ-Connection

○ Line voltage

○ Line current

○ Output

③ V-Connection

○ Line voltage

○ Line current

○ Output

⑶ Output Ratio

① Output ratio = V-Connection output ÷ 3-phase output = 1 ÷ √3

⑷ Utilization Factor

① Utilization factor = 3-phase output ÷ Equipment capacity = √3 / 2

② When changing from Δ-Connection to V-Connection, the output decrease is 1/ √3.



7. Symmetrical Coordinate System

⑴ Unbalanced Voltage

① Unbalanced voltage = Zero-sequence voltage + Positive-sequence voltage + Negative-sequence voltage

○ Unbalance factor = Negative-sequence ÷ Positive-sequence × 100 (%)

② Zero-sequence : Components with the same magnitude and in-phase as a neutral component, exist on the central axis of the grounding line

③ Positive-sequence : Voltage with a phase difference of 120˚ between phases a, b, c

④ Negative-sequence : Voltage with a phase difference of 120˚ between phases a, c, b

⑤ Formulation : Using the principle of superposition, taking voltage as an example.

○ Conversely,

⑵ Basic Equation for AC Generators

① Zero-sequence

② Positive-sequence

③ Negative-sequence

⑶ Interpretation of Intertwined Short Circuit : Using Symmetrical Coordinates

① Using Positive Impedance and Negative Impedance



Input : 2016.01.20 18:55

Revised : 2018.01.06 12:49

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