Chapter Outline CHAPER 14 hree-phase Circuits and Power 14.1 What Is hree-phase? Why Is hree-phase Used? 14.2 hree-phase Circuits: Configurations, Conversions, Analysis 14.2.1 Delta Configuration Analysis 14.2.2 Wye Configuration Analysis 14.2.3 Complex Power in hree-phase Circuits 14.2.4 hree-phase Circuit Analysis 14.1 What Is hree-phase? Why Is hree-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? Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 1
Why is three-phase used? Consider instantaneous power p(t). Is instantaneous power a function of time? he instantaneous power due to all three sources in Figure 14.1 is p(t) = v 1 (t)i 1 (t) + v 2 (t)i 2 (t) + v 3 (t)i 3 (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) = V P sin(t)i P sin(t) + V P sin(t 120)I P sin(t 120) + V P sin(t + 120)I P sin(t + 120) (14.2) How far apart is the phase angle of each source from the other sources? p(t) = V P I P sin 2 (t) + V P I P sin 2 (t 120) + V P I P sin 2 (t + 120) (14.3) 2 1 1 sin ( x) cos(2 x) 2 2 (14.4) 1 1 1 1 1 1 p( t) VPI P cos(2 t) cos(2t 240 ) cos(2t 240 ) 2 2 2 2 2 2 (14.5) 3V PIP VP IP p( t) [cos(2 t) cos(2t 240 ) cos(2t 240 )] (14.6) 2 2 Does +240 = 120 and 240 = +120? Justify if so. 3V PIP VP IP p( t) [cos(2 t) cos(2t 120 ) cos(2t 120 )] (14.7) 2 2 Explain the physical significance of the power represented by the first term (review Chapter 4 if necessary): 3VI P Pave 2 P (14.8) Explain the physical significance of the power represented by any of the terms after the first term in Eq. (14.7). Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 2
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) 3V PIP VP IP p( t) [cos(2 t) + cos(2 t) cos(120 ) sin(2 t) sin(120 ) 2 2 cos(2 t) cos(120 ) sin(2 t) sin(120 )] (14.11) 3V PIP VP IP p( t) [cos(2 t) + cos(2 t) cos(120 ) + cos(2 t) cos(120 )] (14.12) 2 2 What does the cos(120 ) equal? 3V PIP VP IP 1 1 p( t) cos(2 t) cos(2 t) cos(2 t) 2 2 2 2 (14.13) 3VI P P p( t) 3V RMSI RMS Pave (14.14) 2 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? Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 3
14.2 hree-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? hus, is a pure series or parallel connection of sources sufficient to form a three-phase circuit? wo 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: he triangle in Figure 14.4a is called delta because it resembles the uppercase Greek letter. he Y in Figure 14.4b is called wye, which is the noun that describes something shaped like the letter Y. he 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). he wye configuration has an internal node at the junction of the wye called the neutral point. o abeled N for sources (uppercase letter). o abeled n for loads (lowercase letter). Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 4
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. hree-phase circuit analysis: Do the same circuit analysis principles as for single-phase AC circuits apply? hus, is superposition a valid strategy? Good news: he analysis of three-phase circuits is simplified because the RMS voltages and currents of each source (or load) are identical. erminology and conventions are defined to assist three-phase circuit analysis and discussion: he term phase refers to each branch, whether a source or a load, in either the delta or the wye configuration. o o Phase voltages are across these branches and phase currents are through these branches. In the power utility industry, phase always refers to the wye configuration, not the delta configuration. he 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 o he lines are labeled to as Aa, Bb, and Cc, he 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). Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 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 he first letter will be either, 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. he next two letters are the nodes between which the voltage exists: ab, bc, or ca o he 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: V V V 0 ab ab V V V 120 bc bc V V V 120 ca ca (14.15) (14.16) (14.17) Note: he 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) Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 6
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 he first letter will be either, 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. he 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 he 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 he 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 he first letter indicates the source terminal end of the line (where the current enters the line). o he 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: I I I 0 (refer to Fig. 14.6) (14.19) Aa ab ca I I I Aa ab ca (14.20) I Aa V Z ab V Z ca (14.21) V 0 V 120 V (1 0 1 120 ) I Aa Z Z Z (14.22) I Aa V (1.73205 ) 30 Z (14.23) Note: he complex number subtraction in parentheses in Eq. (14.22) produces a magnitude exactly equal to 3. I Aa 3V 30 Z (14.24) Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 7
I 3 (delta) I (14.25) I I Aa Aa 3V ( 30 ) Z 3 ( 30 ) I (14.26) (14.27) What is? I 3( I ) 30 3I 30 (14.28) Aa ab hus 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? hen 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: I 3 30 (14.29) where the plus-or-minus depends on the phase relationship between the phase voltages. I Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 8
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: he first letter will be either or for or currents, respectively. If no other letters are present in this subscript, then he next letter(s) refers to Explain an, bn, or cn: he 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? Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 9
Describe the following equations in words for the load in the wye configuration shown in Figure 14.8: I I an Aa (14.30) I I bn Bb (14.31) I I cn Cc (14.32) I I (14.33) Explain the notation that is needed to distinguish quantities in the wye configuration with respect to voltages: he first letter will be either or for or voltages, respectively. If no other letters are present in this subscript, then he 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: V V V an bn cn V 0 V 120 V 120 (14.34) (14.35) (14.36) Note: he 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: V V V 0 ab an bn (14.37) V V V ab an bn (14.38) V V 0 V 120 V (10 1120 ) (14.39) ab Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 10
V 3 30 ab V (14.40) hus the line voltage across lines ab (leads of lags) the phase voltage in phase ab by 30. V 3 (wye) 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: Why is the plus or minus sign needed? V 3 30 (wye) (14.42) V 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. Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 11
14.2.3 Complex Power in hree-phase Circuits Explain each equation that follows: S S V I (14.43) * 3 3 (delta or wye) S 3V I 3 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, I * I S 3V V I 3V V I 3 3 (14.45) S 3 V I( V I) (14.46) V I (14.47) S 3V I (14.48) For the wye configuration, S V I V I (14.49) * 3 3 V ( I )* V S 3 V ( II )* 3 VV ( I I) (14.50) 3 S 3 V I( V I) (14.51) S 3V 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? Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 12
Can the total complex power be determined strictly from the phasor line voltage and current? Explain the following equations and conclude: V V V 3V 30 V 30 (14.53) 3 30 3 S 3V I * (14.54) S V 3 30I 3 * (14.55) S * 3 V I(1 30 ) (14.56) where the plus or minus depends on the phase relationship between the phase voltages. hus, 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 hree-phase Circuit Analysis Approach to the circuit analysis of three-phase balanced circuits: analyze using only one phase and/or line. he results for one phase/line are applicable to the other phases/lines. he 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 V 3 (wye). V o convert the load configuration, use the relationship between impedances in balanced delta and wye configurations from Section 11.4b: Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 13
Z 3Z (14.57) Y hen, 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 oad impedance in each phase: Z 3.5 j14.8 : a. V b. I c. I d. S Strategy: State each step of the strategy in words. a. V V b. V I Z c. I 3 I d. S 3V I * Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 14
Solution: (Verify the result of each calculation.) Are voltages and currents RMS or peak? a. V 480 V V V b. I 31.562 A 31.6 A Z c. I 3 54.667 54.7 A I d. S 3V I 45.449 76.695 kva How is θ determined? S 10.5 kw j44.2 kvar Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 15
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. V b. I c. 1) S by source Y, Z 2) S by load Y, Z Strategy: State each step of the strategy in words. a. V = V b. source Y I V I Z c.1) S = 3V I Z S S Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 16
2) load Y I I 3 S = 3V I Z S S Solution: (Execute the strategy on separate paper. Intermediate and final results follow.) a. V = V = 480 V b. (Figure 14.13), I = I = 18.223 A c.1) (Figure 14.13), S 15.150 76.695 kva 3.4865 kw j14.743 kvar 15.2 76.7 kva 3.49 kw j14.7 kvar Figure 14.13 Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 17
, I 10.521 A c.2) (Figure 14.14), Z 45.625 76.695, S same answer as c.1). Figure 14.14 earning 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. Contemporary Electric Circuits, 2 nd ed., Prentice-Hall, 2008 Class Notes Ch. 14 Page 18