chapter 14 three-phase circuits and power chapter...

Contemporary Electric Circuits, 2nd ed., ©Prentice-Hall, 2008 Class Notes Ch. 14 Strangeway, Petersen, Gassert, and Lokken

CHAPTER 14 Three-Phase Circuits and Power

Chapter Outline

14.1 What Is Three-Phase? Why Is Three-Phase Used?

14.2 Three-Phase Circuits: Configurations, Conversions, Analysis

14.2.1 Delta Configuration Analysis

14.2.2 Wye Configuration Analysis

14.2.3 Complex Power in Three-Phase Circuits

14.2.4 Three-Phase Circuit Analysis

14.1 What Is Three-Phase? Why Is Three-Phase Used? Consider Figure 14.1. What is the horizontal axis grid increment? _____________________________________

Are there special relationships between the three signals in Figure 14.1? _______ If so, identify them.

Figure 14.1 (three sinusoidal signals generated by three synchronized separate sources)

What does “balanced” mean in the context of three-phase circuits?


Why is three-phase used? Consider instantaneous power p(t). Is instantaneous power a function of time? ________

The instantaneous power due to all three sources in Figure 14.1 is p(t) = v1(t)i1(t) + v2(t)i2(t) + v3(t)i3(t) (14.1)

Assume each source is connected to a purely resistive load. What is the phase relationship between the voltage and

the current for each source? Why?

Explain how each equation is obtained in the following development.

p(t) = VP sin(t)IP sin(t) + VP sin(t – 120)IP sin(t – 120) + VP sin(t + 120)IP sin(t + 120) (14.2)

How far apart is the phase angle of each source from the other sources? ______________

p(t) = VPIP sin2(t) + VPIP sin

2(t – 120) + VPIP sin

2(t + 120) (14.3)

2 1 1sin ( ) cos(2 )

2 2x x ________________________________________________________ (14.4)

1 1 1 1 1 1( ) cos(2 ) cos(2 240 ) cos(2 240 )

2 2 2 2 2 2P Pp t V I t t t

(14.5)

3( ) [cos(2 ) cos(2 240 ) cos(2 240 )]

2 2

P P P PV I V Ip t t t t (14.6)

Does +240 = –120 and –240 = +120? ________ Justify if so.

3( ) [cos(2 ) cos(2 120 ) cos(2 120 )]

2 2


Explain the physical significance of the power represented by the first term (review Chapter 4 if necessary):

ave

3

2

P PV IP (14.8)

Explain the physical significance of the power represented by any of the terms after the first term in Eq. (14.7).


Continue explaining each step in the development:

cos(a + b) = cos(a) cos(b) – sin(a) sin(b) (14.9)

cos(a – b) = cos(a) cos(b) + sin(a) sin(b) (14.10)

3( ) [cos(2 ) + cos(2 ) cos(120 ) sin(2 ) sin(120 )

2 2

cos(2 ) cos(120 ) sin(2 ) sin(120 )]

P P P PV I V Ip t t t t

t t

(14.11)

3( ) [cos(2 ) + cos(2 ) cos(120 ) + cos(2 ) cos(120 )]

2 2


What does the cos(120 ) equal? __________________

3 1 1( ) cos(2 ) cos(2 ) cos(2 )

2 2 2 2

P P P PV I V Ip t t t t

(14.13)

RMS RMS ave

3( ) 3

2

P PV Ip t V I P (14.14)

What is the significance of this result with respect to the instantaneous power?

What is the relationship between the average power and the instantaneous power in this case?

Consider Figure 14.2. What is the remarkable feature of the total instantaneous power of three AC sinusoidal

steady-state signals that are 120 apart?

Figure 14.2

What can pulsation of power cause in large motors and industrial machinery? _____________________________

Does three-phase have an advantage over single-phase? _______ If so, identify it. __________________________

How much power does three-phase have relative to a single-source, everything else being equal? ______________


14.2 Three-Phase Circuits: Configurations, Conversions, Analysis

Note: Effective (RMS) voltage and current values will be assumed in this section.

Consider how to connect three sources to form a three-phase circuit.

If three voltage sources are connected in series, what is the phasor sum? _________

Why? Make a phasor sketch.

If three current sources are connected in parallel, what is the phasor sum? ________

Thus, is a pure series or parallel connection of sources sufficient to form a three-phase circuit? ____________

Two configurations are primarily used in three-phase to connect three sources (or loads) together (Figure 14.4).

Figure 14.4 (sources shown)

Notation used in this chapter:

The “triangle” in Figure 14.4a is called delta because it resembles the uppercase Greek letter .

The “Y” in Figure 14.4b is called wye, which is the noun that describes something shaped like the letter Y.

The terminals of either the delta or wye configurations are called nodes.

o External terminal nodes are labeled A, B, and C for sources (uppercase letters).

o External terminal nodes are labeled a, b, and c for loads (lowercase letters).

The wye configuration has an internal node at the junction of the wye called the neutral point.

o Labeled N for sources (uppercase letter).

o Labeled n for loads (lowercase letter).


Identify an amazing consequence from the three-phase circuit shown in Figure 14.5? (Hint: lines)

Figure 14.5

Hence, the transmission of electrical energy over a three-phase system requires only ____________ lines.

Three-phase circuit analysis: Do the same circuit analysis principles as for single-phase AC circuits apply? _______

Thus, is superposition a valid strategy? ___________

Good news: The analysis of three-phase circuits is simplified because the RMS voltages and currents of each

source (or load) are identical.

Terminology and conventions are defined to assist three-phase circuit analysis and discussion:

The term phase refers to each branch, whether a source or a load, in either the delta or the wye configuration.

o Phase voltages are across these branches and phase currents are through these branches.

o In the power utility industry, phase always refers to the wye configuration, not the delta configuration.

The term line refers to the voltages that exist between the lines connecting the sources to the loads and the

currents through those lines in a three-phase circuit.

o The lines are labeled to as Aa, Bb, and Cc,

o The first letter is on the source connection to the line and the second letter is on the load connection to the

line (see Figure 14.5).


14.2.1 Delta Configuration Analysis

Consider the delta configuration in Figure 14.6. Is this a source or a load? _____________________ Why?

Figure 14.6

Notation to distinguish quantities in three-phase circuits: a two-part subscript for phasor voltages

The first letter will be either L, for line voltages, or for phase voltages.

o If no other letters are present in this subscript, then the line or phase voltage is being referred to generically.

The next two letters are the nodes between which the voltage exists: ab, bc, or ca

o The first of these letters indicates the node for the positive side of the voltage and the second letter indicates

the negative side of the voltage for a load.

o For a source, the notation would be AB, BC, and CA.

Describe the following equations in words for the load in the delta configuration shown in Figure 14.6:

0ab LabV V V (14.15)

120bc LbcV V V (14.16)

120ca LcaV V V (14.17)

Note: The ab phase voltage is arbitrarily assumed to be the reference (any phase could have been the reference).

What is the relationship between the phase voltages and the line voltages? ___________________ Why? (14.18)


Are the phase currents and the line currents identical? _______________ Why or why not?

Notation to distinguish the currents in three-phase circuits: a two-part subscript for phasor currents

The first letter will be either L, for line currents, or , for phase currents.

o If no other letters are present in this subscript, then the line or phase current is being referred to generically.

The second part of the subscript (next two letters) is labeled as follows:

o For phase currents in a load, they are the nodes through which the current flows: ab, bc, or ca

The first of these letters indicates the node that the current enters the phase (branch) and the second

letter indicates the node that the current leaves the phase.

o For phase currents in a source, they are the nodes through which the current flows: AB, BC, or CA

The first of these letters indicates the node that the current leaves the phase and the second letter

indicates the node that the current enters the phase

o Phase current directions in the source are opposite those in the load.

For line currents,

o The first letter indicates the source terminal end of the line (where the current enters the line).

o The second letter indicates the load end of the line (where the current leaves the line).

What is the relationship between the line and phase currents in a delta configuration? Explain each step that follows:

0LAa ab caI I I (refer to Fig. 14.6) (14.19)

LAa ab caI I I (14.20)

ab ca

LAa

V VI

Z Z

(14.21)

0 120 (1 0 1 120 )

LAa

V V VI

Z Z Z

(14.22)

(1.73205 ) 30

LAa

VI

Z

(14.23)

Note: The complex number subtraction in parentheses in Eq. (14.22) produces a magnitude exactly equal to 3.

330LAa

VI

Z

(14.24)


3 (delta)LI I (14.25)

3( 30 )LAa

VI

Z

(14.26)

3 ( 30 )LAaI I (14.27)

What is –? __________________________________________________________________________________

3( ) 30 3 30LAa abI I I (14.28)

Thus the line current in line Aa ____________________ (leads or lags) the phase current in phase ab by 30.

State in words the relationships in Figure 14.7 (for the case of a resistive load) between:

phase voltages and phase currents:

line currents and phase currents:

What if the phase shift of the bc

phase voltage were +120 and the

phase shift of the ca phase voltage

were –120?

Then the line current in line Aa

would lead the phase current in

phase ab by 30

Do not memorize! Figure 14.7

If phases are needed, draw the phasor diagram and note the phase relationships.

General relationship between the line and phase currents in the delta configuration: 3 30LI I (14.29)

where the plus-or-minus depends on the phase relationship between the phase voltages.


14.2.2 Wye Configuration Analysis

Consider the wye configuration in Figure 14.8. Is this a source or a load? _____________________ Why?

Figure 14.8

Explain the notation that is needed to distinguish quantities in the wye configuration with respect to currents:

The first letter will be either _______ or _______ for ______________ or ______________ currents, respectively.

If no other letters are present in this subscript, then ___________________________________________________

The next letter(s) refers to

Explain an, bn, or cn:

The junction of the wye, labeled n in Figure 14.8, is called the _____________________________________

Explain AN, BN, or CN:

Explain the notation for line currents and line voltages:

Does this notation differ from that for the line connections in the delta configuration? ________________________


Describe the following equations in words for the load in the wye configuration shown in Figure 14.8:

an LAaI I (14.30)

bn LBbI I (14.31)

cn LCcI I (14.32)

LI I (14.33)

Explain the notation that is needed to distinguish quantities in the wye configuration with respect to voltages:

The first letter will be either _______ or _______ for ______________ or ______________ voltages, respectively.

If no other letters are present in this subscript, then ___________________________________________________

The next letter(s) refers to:

an, bn, or cn:

AN, BN, and CN:

Collectively identify the following three equations as they relate to Figure 14.8:

0anV V (14.34)

120bnV V (14.35)

120cnV V (14.36)

Note: The an phase voltage is arbitrarily assumed to be the reference (any phase could have been the reference).

What is the relationship between the line and phase voltages in a wye configuration? Explain each step that follows:

0Lab an bnV V V (14.37)

Lab an bnV V V (14.38)

0 120 (1 0 1 120 )LabV V V V (14.39)


3 30LabV V (14.40)

Thus the line voltage across lines ab ____________________ (leads of lags) the phase voltage in phase ab by 30.

3 (wye)LV V (14.41)

State in words the relationships in Figure 14.9 (for the case of a resistive load) between:

phase voltages and phase currents:

line voltages and phase voltages:

If the phase shift of the bn phase

voltage were +120 and the phase

shift of the cn phase voltage were

-120, then the line voltage across

lines ab would lag the phase

voltage in phase an by 30.

Do not memorize!

Figure 14.9

If phases are needed, draw the phasor diagram and note the phase relationships.

General relationship between the line and phase voltages in the wye configuration:

3 30 (wye)LV V (14.42)

Why is the plus or minus sign needed?

Perspective: Most of our calculations will concentrate on voltage and current magnitudes, but knowledge of the

phases is required for complex power, the next topic.


14.2.3 Complex Power in Three-Phase Circuits

Explain each equation that follows:

*3 3 (delta or wye)TS S V I (14.43)

* *3 3 ( )T V IS V I V I (14.44)

where the phase subscript on the angles is used to distinguish them from line phase angles.

For the delta configuration,

*3 3

3 3

L L

T L V I L V I

I IS V V

(14.45)

3 ( )T L L V IS V I (14.46)

V I (14.47)

3T L LS V I (14.48)

For the wye configuration,

*3 3 ( )*T V IS V I V I (14.49)

3 ( )* 3 ( )3

L

T V L I L V L I

VS I V I (14.50)

3 ( )T L L V IS V I (14.51)

3T L LS V I (14.52)

What does the equality of Eq. (14.48) and Eq. (14.52) mean?

Are the complex power calculation techniques in Chapter 10 applicable? _______________________________


Can the total complex power be determined strictly from the phasor line voltage and current? Explain the following

equations and conclude:

3 30 303 30 3

L L

L

V VV V V

(14.53)

*3TS V I (14.54)

*3 303

L

T L

VS I

(14.55)

*3 (1 30 )T L LS V I (14.56)

where the plus or minus depends on the phase relationship between the phase voltages.

Thus, can total complex power can be determined using line voltages and currents? _________________________

Does this result have any practical measurement implications? If so, explain.

What is the reason why the additional angle of ±30 must be incorporated into the complex power calculation?

14.2.4 Three-Phase Circuit Analysis

Approach to the circuit analysis of three-phase balanced circuits: analyze using only one phase and/or line.

The results for one phase/line are applicable to the other phases/lines.

The phase angle may need to be adjusted for any given quantity (120 apart), but the magnitudes are identical.

How are three-phase circuits analyzed for voltage, current, and power values? Strategy:

If the source and the load are both wye (or both delta), analyze the three-phase circuit as is.

If the source is in a delta configuration and the load is in a wye configuration, or vice versa, convert the load or

the source to match the configuration of the other, and then analyze the circuit.

Source voltages are converted using the line-phase voltage relationship 3 (wye)LV V .

To convert the load configuration, use the relationship between impedances in balanced delta and wye

configurations from Section 11.4b:


Y3Z Z

(14.57)

Then, convert resultant voltages and currents as needed to express them in the original delta or wye

configuration for the source or load that was converted.

Example 14.2.1 (Explain each step.) [Note: Example changed from text to emphasize magnitudes.]

Determine (a) the line voltage magnitude, (b) the line current magnitude, (c) the phase current magnitude in the load,

and (d) the total load complex power for the circuit shown in Figure 14.10.

Figure 14.10

___________: Source phase voltages per Figure 14.10

Load impedance in each phase: 3.5 14.8 Z j

___________: a. VL b. IL c. I d. TS

Strategy: State each step of the strategy in words.

a.

LV V

b. V

IZ

c. 3LI I

d. *3TS V I


Solution: (Verify the result of each calculation.)

Are voltages and currents RMS or peak? _____________________________

a. 480 VLV V

b. 31.562 A 31.6 AV

IZ

c. 3 54.667 54.7 ALI I

d. 3 45.449 76.695 kVAT L LS V I

How is θ determined?

10.5 kW 44.2 kVARTS j


Example 14.2.2 Determine (a) the line voltage magnitudes, (b) the line current magnitudes, and (c) the total load complex power for

the circuit shown in Figure 14.12 by (1) converting the source to a wye configuration.

(2) Repeat part (c) by converting the load to a delta configuration. Compare the answers from both approaches.

Figure 14.12

Given: _________________________________________________________________________________

Desired: a. VL b. IL c. 1) ST by source Y, Z 2) ST by load Y , Z

Strategy: State each step of the strategy in words.

a. VL = V __________________________________________________________________

b. source Y ____________________________________________________________

L

VI I

Z

_____________________________________________________________

c.1) ST = 3VI ________________________________________________________________

Z __________________________________________________________________

T TS S _______________________________________________________________


2) load Y

3

LII __________________________________________________________________

ST = 3VI ________________________________________________________________

Z __________________________________________________________________

T TS S _______________________________________________________________

Solution: (Execute the strategy on separate paper. Intermediate and final results follow.)

a. VL = V = 480 V

b. (Figure 14.13), IL = I = 18.223 A

c.1) (Figure 14.13),

15.150 76.695 kVA 3.4865 kW 14.743 kVAR 15.2 76.7 kVA 3.49 kW 14.7 kVARTS j j

Figure 14.13


c.2) (Figure 14.14), 45.625 76.695 Z , 10.521 AI , TS same answer as c.1).

Figure 14.14

Learning Objectives Discussion: Can you perform each learning objective for this chapter? (Examine each one.)

As a result of successfully completing this chapter, you should be able to:

1. Describe what three-phase circuits are and why they are used.

2. Draw three-phase delta and wye configurations.

3. State the phase relationships among the voltages and currents in balanced three-phase delta and wye

configurations.

4. Convert between balanced three-phase delta and wye configurations.

5. Determine line and phase voltages and currents in balanced three-phase circuits.

6. Determine the complex power in both balanced three-phase delta and wye configurations.

7. Analyze delta source–delta load, wye source–wye load, delta source–wye load, and wye source–delta load

balanced three-phase circuits.

chapter 14 three-phase circuits and power chapter...

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