Electric Circuit Theory - KOCWcontents.kocw.net/KOCW/document/2015/korea_sejong/... · 2016. 9....

Electric Circuit Theory

Nam Ki Min

010-9419-2320 [email protected]

Balanced Three-Phase Circuits

Chapter 12

Nam Ki Min

010-9419-2320 [email protected]

CIEN 202 Electric Circuits II Nam Ki Min 010-9419-2320 [email protected]

Chapter 12 Balanced Three-Phase Circuits 3 Contents and Objectives

Chapter Contents

12.1 Balanced Three-Phase Voltages

12.2 Three-Phase Voltage Sources

12.3 Analysis of the Wye-Wye Circuit

12.4 Analysis of the Wye-Delta Circuit

12.5 Power Calculations in Balanced Three-Phase Circuits

12.6 Measuring Average Power in Three-Phase circuits

Chapter Objectives

1. Know how to analyze a balanced, three-phase wye-wye connected circuit

2. Know how to analyze a balanced, three-phase wye-delta connected circuit

3. Be able to calculate power (average, reactive, and complex) in any three-phase circuit


Chapter 12 Balanced Three-Phase Circuits 4 Introduction

Single-phase System

A single-phase ac power system consists of a generator connected through a pair of wires (a transmission line) to a load.

• Single-phase two-wire system(Fig. a)

• Single-phase three-wire system(Fig. b):

- contains two identical sources (equal magnitude and the same phase) which are connected to two loads by two outer wires and the neutral.

- allows the connection of both 120-V and 240-V appliances.

Fig.1 Single-phase systems

Polyphase System

Polyphase : Circuits or systems in which the ac sources operate at the same frequency but different phases.

- Two-phase

- Three-phase

It is more advantageous and economical to generate and transmit electric power in the polyphase mode than with single-phase systems.



Fig.3 Three-phase systems

Two-phase three-wire system

Three-phase system

• A two-phase system is produced by a generator consisting of two coils placed perpendicular to each other so that the voltage generated by one lags the other by 90◦.

• A three-phase system is produced by a generator consisting of three sources having the same amplitude and frequency but out of phase with each other by 120◦.

• Three-phase systems are important for at least three reasons.

- Nearly all electric power is generated and distributed in three-phase.

- The instantaneous power in a three-phase system can be constant (not pulsating). This results in uniform power transmission and less vibration of three-phase machines.

- For the same amount of power, the three-phase system is more economical than the single-phase. The amount of wire required for a three-phase system is less than that required for an equivalent single-phase system.

Fig.2 Two-phase systems



• Since the three-phase system is by far the most prevalent and most economical polyphase system, discussion in this chapter is mainly on three-phase systems.

• From a such a system, power can be supplied as single phase (load connected between a line and neutral) or three phase (load connected between all three lines).


Chapter 12 Balanced Three-Phase Circuits 7 11.1 Balanced Three-Phase Voltages

Balanced Three-Phase Voltages

If the three sinusoidal voltages have the same magnitude and frequency and each voltage is 120° out of phase with the other two, the voltages are said to be balanced.

If the loads are such that the currents produced by the voltages are also balanced, the entire circuit is referred to as a balanced three-phase circuit.

A balanced set of three-phase voltages can be represented in the time domain.

𝑣𝑎𝑛 𝑣𝑏𝑛 𝑣𝑐𝑛

𝑣𝑎𝑛 𝑡 = 𝑉𝑚 cos 𝜔𝑡

𝑣𝑏𝑛 𝑡 = 𝑉𝑚 cos(𝜔𝑡 − 120°)

𝑣𝑐𝑛 𝑡 = 𝑉𝑚 cos(𝜔𝑡 − 240°) = 𝑉𝑚 cos(𝜔𝑡 + 120°)


Chapter 12 Balanced Three-Phase Circuits 8 Balanced Three-Phase Voltages


Balanced three-phase voltages can be represented in the frequency domain

(a)abc or positive sequence (b)acb or negative sequence

Since the three-phase voltages are 120◦ out of phase with each other, there are two possible combinations.

𝐕an = 𝑉𝑚∠0°

𝐕bn = 𝑉𝑚∠ − 120°

𝐕c𝑛 = 𝑉𝑚∠ + 120°

(11.1)


𝐕bn = 𝑉𝑚∠ − 120°

𝐕c𝑛 = 𝑉𝑚∠ + 120°

(11.1)


𝐕bn = 𝑉𝑚∠ + 120°

𝐕c𝑛 = 𝑉𝑚∠ − 120° (11.2)


Chapter 12 Balanced Three-Phase Circuits 9 Balanced Three-Phase Voltages


An important property of the balanced voltage set is

𝑣𝑎𝑛 + 𝑣𝑏𝑛 + 𝑣𝑐𝑛 = 0

𝐕an + 𝐕bn + 𝐕cn = 0 (11.3)

(11.4)

𝐕an + 𝐕bn + 𝐕cn = 𝑉𝑚∠0° + 𝑉𝑚∠ − 120° + 𝑉𝑚∠ + 120°

= 𝑉𝑚 1.0 − 0.5 − j0.866 − 0.5 + j0.866

= 0


Chapter 12 Balanced Three-Phase Circuits 10

Three-Phase Generator

Three-Phase Voltage Sources

• The rotor is an electromagnet driven at synchronous speed by a prime mover, such as a steam or gas turbine.

• Each winding comprises one phase of the generator. Because the coils are placed 120° apart, the induced voltages in the coils are equal in magnitude but out of phase by 120°◦.

• Since each coil can be regarded as a single-phase generator by itself, the three-phase generator can supply power to both single-phase and three-phase loads.

𝑎 𝑏 𝑐

𝑛 𝑛 𝑛

𝑛

𝑐 𝑏

𝑎

𝑣𝑎𝑛

𝑣𝑏𝑛

𝑣𝑐𝑛



Three-Phase Generator


Three phase output

𝑛 𝑛 𝑛

𝑐 𝑏

𝑎

𝑣𝑎𝑛

𝑣𝑏𝑛

𝑣𝑐𝑛

𝑣𝑎𝑛

𝑣𝑏𝑛

𝑣𝑐𝑛



Two Basic Connections of an Ideal Three-phase Source


Wye(Y) Connection :

There are two ways of interconnecting the separate phase windings to form a three-phase sources:

Neutral terminal (neutral line)

𝐕𝑎𝑛

𝐕𝑏𝑛 𝐕𝑐𝑛

Y-connected source

• Fig. (a) : consists only of ideal voltage sources.

• Fig. (b) : model with winding impedance in series with an ideal sinusoidal voltage source.

Winding impedance

• The winding impedance of a three-phase generator is inductive.

(a) (b)



Two Basic Connections of an Ideal Three-phase Source


Δ-connected source

• Fig. (a) : consists only of ideal voltage sources.

• Fig. (b) : model with winding impedance in series with an ideal sinusoidal voltage source.

Winding impedance

Delta(Δ) Connection :

(a) (b)



Two Possible Three-Phase Load Configurations


• Like the generator connections, a three-phase load can be either wye-connected or delta-connected, depending on the end application.

Y-connected load Δ-connected load

𝐙𝑐𝑎

𝐙𝑎

𝐙𝑐

𝐙𝑏 𝐙𝑎𝑏

𝐙𝑏𝑐

Possible Three-Phase Source- Load Connections

• Y - Y Connection

• Y – Δ Connection

• Δ – Y Connection

• Δ – Δ Connection



General Y – Y Circuit

Analysis of the Y-Y Circuit

• Using the source neutral(n) as the reference node and letting 𝐕N denote the node voltage between the node N and n,

𝐈0 =𝐕N

Z0

𝐈aA Zga + Z1a + ZA = 𝐕a′n − 𝐕N →

𝐈bB Zgb + Z1b + ZB = 𝐕b′n − 𝐕N →

𝐈cC Zgc + Z1c + ZC = 𝐕c′n − 𝐕N →

𝐈aA =𝐕a′n − 𝐕N

ZA + Z1a + Zga

𝐈bB =𝐕b′n − 𝐕N

ZB + Z1b + Zgb

𝐈cC =𝐕c′n − 𝐕N

ZC + Z1c + Zgc

(2)



General Y – Y Circuit


𝐕N

Z0+

𝐕N − 𝐕a′n

ZA + Z1a + Zga+

𝐕N − 𝐕b′n

ZB + Z1b + Zgb+

𝐕N − 𝐕c′n

ZC + Z1c + Zgc= 0

• From Eqs.(1) and (2), the node voltage equation is

𝐈0 − 𝐈aA − 𝐈bB − 𝐈𝐜𝐂 = 0

(11.5)



Balanced Y – Y Circuit


• A balanced Y-Y system is a three-phase system with a balanced Y-connected source and a balanced Y-connected load.

• 𝐙𝜙: the total load impedance per phase= the sum of the source impedance 𝐙𝑠, line impedance, 𝐙𝑙 and load impedance 𝐙𝐿 for each phase.

𝐙ϕ = 𝐙s + 𝐙l + 𝐙L (3)





• For the balanced Y-Y circuit,

𝐕N

1

Z0+

3

Z∅=

𝐕a′n + 𝐕b′n + 𝐕c′n

Z∅

𝐕N

Z0+

𝐕N − 𝐕a′n

ZA + Z1a + Zga+

𝐕N − 𝐕b′n

ZB + Z1b + Zgb+

𝐕N − 𝐕c′n

ZC + Z1c + Zgc= 0

(11.6)

𝐙ϕ = 𝐙s + 𝐙l + 𝐙L ≠ 0

(11.5)

𝐕N

Z0+

𝐕N − 𝐕a′n

Zϕ+

𝐕N − 𝐕b′n

Zϕ+

𝐕N − 𝐕c′n

Zϕ= 0 →

𝐕N

Z0+

𝟑𝐕N

Zϕ=

𝐕a′n + 𝐕b′n +𝐕c′n

Zϕ →

𝐕a′n + 𝐕b′n + 𝐕c′n = 0

𝐕N

1

Z0+

3

Z∅= 0 → 𝐕N = 0 (11.7)

Eq.(11.7) is extremely important. For the balanced Y-Y circuit, the voltage across the neutral wire(n-N) is zero. The neutral line can thus be removed without affecting the system.

In fact, in long distance power transmission, conductors in multiples of three are used with the earth itself acting as the neutral conductor. Power systems designed in this way are well grounded at all critical points to ensure safety.



• The line currents are


ZA + Z1a + Zga


ZB + Z1b + Zgb


ZC + Z1c + Zgc

(2)


ZA + Z1a + Zga=

𝐕a′n

Z∅


ZB + Z1b + Zgb=

𝐕b′n

𝐙∅


ZC + Z1c + Zgc=

𝐕c′n

Z∅

(11.8)

(11.9)

(11.10)

𝐙ϕ = 𝐙s + 𝐙l + 𝐙L

𝐙ga = 𝐙ga = 𝐙ga = 𝐙s

𝐙la = 𝐙lb = 𝐙lc = 𝐙l

𝐙A = 𝐙B = 𝐙C = 𝐙𝐿

𝐕N = 0

General Y-Y connection




Chapter 12 Balanced Three-Phase Circuits 20 Analysis of the Y-Y Circuit


• An alternative way of analyzing a balanced Y-Y system is to do so on a “per phase” basis.

𝐙ϕ = 𝐙ga + 𝐙la + 𝐙𝐀 = 𝐙s + 𝐙l + 𝐙L


ZA + Z1a + Zga=

𝐕a′n

Z∅

• As long as the system is balanced, we need only analyze one phase. We may do this even if the neutral line is absent, as in the three-wire system.

• Line Current

𝐈0 = 𝐈aA + 𝐈bB + 𝐈cC

• The current in the neutral conductor of the balance Y-Y circuit is

(11.11)




• Line Voltages

(11.13)

𝐕AB = 𝐕AN − 𝐕BN

𝐕BC = 𝐕BN − 𝐕CN

𝐕CA = 𝐕CN − 𝐕AN

(11.12)

(11.14)

𝐕AN, 𝐕BN , 𝐕CN : Phase voltags(line − to − neutral voltages)

𝐕AN = V∅∠0°

𝐕BN = V∅∠ − 120°

𝐕CN = V∅∠ + 120°

(11.15)

(11.16)

(11.17)

𝐕AB = V∅∠0° − V∅∠ − 120° = 3V∅∠30°

𝐕BC = V∅∠ − 120° − V∅∠120° = 3V∅∠ − 90°

𝐕CA = V∅∠120° − V∅∠0° = 3V∅∠150°

(11.18)

(11.19)

(11.20)

Homework




• For the balance Y-Y circuit

1. The magnitude of line-to-line voltage(line voltage) is 3 times the magnitude of the line-to-neutral voltage (phase voltage).

2. The line-to-line voltages(line voltages) form a balanced three-phase set of voltages(phase voltages).

3. The set of line-to-line voltages leads the set of line-to-neutral voltages by 30°.



Balanced Y – Δ Circuit

Analysis of the Y-Δ Circuit

• A balanced Y-Δ system consists of a balanced Y-connected source feeding a balanced-connected load.

• This is perhaps the most practical three-phase system, as the three-phase sources are usually Y-connected while the three-phase loads are usually Δ-connected.





• The Δ load can be transformed into a wye by using the delta-to-wye transformation.

• When the load is balance, the impedance of each leg of the wye is

Z1 =𝑍𝑏𝑍𝑐

𝑍𝑎 + 𝑍𝑏 + 𝑍𝑐

, (9.51)

Z2 =𝑍𝑐𝑍𝑎

𝑍𝑎 + 𝑍𝑏 + 𝑍𝑐 , (9.52)

Z3 =𝑍𝑎𝑍𝑏

𝑍𝑎 + 𝑍𝑏 + 𝑍𝑐 , (9.53)

𝐙1 =𝐙∆𝐙∆

𝐙∆ + 𝐙∆ + 𝐙∆=

𝐙∆

3

𝐙3 =𝐙∆𝐙∆

𝐙∆ + 𝐙∆ + 𝐙∆=

𝐙∆

3

𝐙2 =𝐙∆𝐙∆

𝐙∆ + 𝐙∆ + 𝐙∆=

𝐙∆

3 𝐙Y = 𝐙1 = 𝐙1 = 𝐙1 =

𝐙∆

3

(11.21)





• The relationship between the phase currents and the line currents

𝐈AB = 𝐼∅∠0°

𝐈BC = 𝐼∅∠ − 120°

𝐈CA = 𝐼∅∠ + 120°

(11.23)

(11.22)

(11.24)

Using KCL

𝐈aA = 𝐈AB − 𝐈CA = 𝐼∅∠0° − 𝐼∅∠120° = 3𝐼∅∠ − 30°

𝐈bB = 𝐈BC − 𝐈AB = 𝐼∅∠ − 120° − 𝐼∅∠0° = 3𝐼∅∠ − 150°

𝐈cC = 𝐈CA − 𝐈BC = 𝐼∅∠120° − 𝐼∅∠ − 120° = 3𝐼∅∠90°

(11.25)

(11.26)

(11.27)

• The magnitude of line current is 3 times the magnitude of the phase current and the set of line currents lags the set of phase currents by 30°.





• The magnitude of line current is 3 times the magnitude of the phase current and the set of line currents lags the set of phase currents by 30°.



Average Power in a Balanced Y Load

Power Calculations in Balanced Three-Phase Circuits

• The average power associated with each phase

𝑃𝐴 = 𝐕AN 𝐈aA cos (𝜃vA − 𝜃𝑖A)

𝑃𝐵 = 𝐕BN 𝐈bB cos (𝜃vB − 𝜃𝑖B)

𝑃𝐶 = 𝐕CN 𝐈cC cos (𝜃v𝐶 − 𝜃𝑖C)

(11.28)

(11.29)

(11.30)

𝜃v

𝜃𝑖

: phase angle of phase voltage

: phase angle of phase current(line current)

𝐕AN , 𝐈aA : rms values

• In a balanced three-phase system,

𝑉∅ = 𝐕AN = 𝐕BN = |𝐕CN|

𝐼∅ = 𝐈AN = 𝐈BN = |𝐈CN|

(11.31)

(11.32)

𝜃∅ = 𝜃𝑣A − 𝜃𝑖A = 𝜃𝑣B − 𝜃𝑖B = 𝜃𝑣C − 𝜃𝑖C (11.33)

The power delivered to each phase of the load is the same, so

𝑃𝐴 = 𝑃𝐵 = 𝑃𝐶 = 𝑃∅ = 𝑉∅𝐼∅ cos 𝜃∅ (11.34)



Average Power in a Balanced Y Load


The total average power delivered to the balanced Y-connected load is

𝑃𝑇 = 3𝑃∅ = 3𝑉∅𝐼∅𝑐𝑜𝑠𝜃∅ (11.35)

= 3𝑉𝐿

3𝐼𝐿 cos 𝜃∅

= 3𝑉𝐿𝐼𝐿 cos 𝜃∅

The total power in terms of the rms value of the line voltage and current is

(11.36)

𝑉𝐿 = 3𝑉∅

𝐼𝐿 = 𝐼∅

𝑃𝑇 = 3𝑉∅𝐼∅𝑐𝑜𝑠𝜃∅



Complex Power in a Balanced Y Load

• For a balanced load, the reactive power is for each phase

(11.37) 𝑄∅ = 𝑉∅𝐼∅ sin 𝜃∅

𝑄𝑇 = 3𝑄∅ = 3𝑉𝐿𝐼𝐿 sin 𝜃∅ (11.38)

The total reactive power is

• The complex power is for each phase

𝑆∅ = 𝐕∅ 𝐈∅∗

= 𝑃∅ + 𝑗𝑄∅

The total complex is

𝑆𝑇 = 3𝑆∅ = 3𝑉𝐿𝐼𝐿∠𝜃∅° (11.41)

(11.40)




Power Calculations in a Balanced Δ Load

(11.42)

(11.43)

𝑃𝐴 = 𝐕AB 𝐈AB cos 𝜃vAB − 𝜃𝑖AB

𝑃𝐵 = 𝐕BC 𝐈BC cos 𝜃vBC − 𝜃𝑖BC

𝑃𝐶 = 𝐕CA 𝐈CA cos 𝜃vCA − 𝜃𝑖CA (11.44)

• For a balanced load,

• For each phase,

𝐕AB = 𝐕BC = 𝐕CA = 𝑉∅

𝐈AB = 𝐈BC = 𝐈CA = 𝐼∅

𝜃𝑣AB − 𝜃𝑖AB = 𝜃𝑣B𝐶 − 𝜃𝑖BC = 𝜃𝑣CA − 𝜃𝑖CA = 𝜃∅

𝑃𝐴 = 𝑃𝐵 = 𝑃𝐶 = 𝑃∅ = 𝑉∅𝐼∅ cos 𝜃∅

(11.45)

(11.46)

(11.47)

(11.48)



Chapter 12 Balanced Three-Phase Circuits 31 Power Calculations in Balanced Three-Phase Circuits

Power Calculations in a Balanced Δ Load

• The total power delivered to a balanced Δ-connected load is

(11.49)

𝑃𝑇 = 3𝑃∅ = 3𝑉∅𝐼∅ cos 𝜃∅

= 3𝑉𝐿

𝐼𝐿

3cos 𝜃∅

= 3𝑉𝐿𝐼𝐿 cos 𝜃∅

𝑉𝐿 = 𝑉∅

𝐼𝐿 = 3𝐼∅

• Reactive power

𝑄∅ = 𝑉∅𝐼∅ sin 𝜃∅;

𝑄𝑇 = 3𝑄∅ = 3𝑉∅𝐼∅ sin 𝜃∅;

𝑆∅ = 𝑃∅ + 𝑗𝑄∅ = 𝐕∅𝐈∅∗

𝑆𝑇 = 3𝑆∅ = 3𝑉𝐿𝐼𝐿∠𝜃∅

• Complex power

(11.50)

(11.51)

(11.52)

(11.53)


Chapter 12 Balanced Three-Phase Circuits 32 Power Calculations in Balanced Three-Phase Circuits

Instantaneous Power in Three-Phase Circuits

• In a balanced three-phase circuit, this power has an interesting property:

It is invariant with time! Thus the torque developed at the shaft of a three-phase motor is constant, which in turn means less vibration in machinery powered by three-phase motors.

• Let the instantaneous line-to-neutral voltage 𝑣𝐴𝑁 be the reference, and, as before, 𝜃∅ is the phase angle Then, for a positive phase sequence, the instantaneous power in each phase is

𝑉𝑚 , 𝐼𝑚=the maximum amplitude of the phase voltage and line current, respectively.

• The total instantaneous power is the sum of the instantaneous phase powers;

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