ࡱ> WYVa JjbjbA]A] +?+?r7KLLL4h,*`r^0 0 0 0 "^#H*******,+R!.~@*]L#0 0 ##@*(%jL0 0 *(%(%(%#L0 L0 *(%LL#*(%(%'LL'T ?$|''*0*',/(%/'(%LD(cover sheet: Article) Three-Phase and Six-Phase AC at the Lab Bench George M. Caplan (Authors bio) George M. Caplan is in his eighth year as Instructor in Physics Laboratory at Wellesley College. Before joining the Wellesley faculty, he worked for more than twenty years as a software engineer at Nova Biomedical. In 1993 and 1994, he taught evening physics courses at Massachusetts Bay Community College. Address: Department of Physics Wellesley College 106 Central St. Wellesley, MA 024818203 email: gcaplan@wellesley.edu Three-Phase and Six-Phase AC at the Lab Bench George M. Caplan Utility companies generate threephase electric power, which consists of three sinusoidal voltages with phase angles of0,120, and240 degrees., The AC generators described in most introductory textbooks are singlephase generators, so physics students are not likely to learn much about threephase power. I have developed a simple way to display the waveforms of the three phase power supplied to my lab and to demonstrate some interesting features of threephase power. The waveform displays require three small transformers, a Vernier LabPro( datacollection device, and a computer; but the demonstrations require only the transformers and some miniature light bulbs. I have also developed a way to demonstrate how six-phase AC can be derived from three-phase AC. Compared to singlephase power, three-phase power has many advantages. Three-phase transmission lines require fewer wires than an equivalent set of three singlephase lines. For single-phase power, the instantaneous power delivered to a load pulsates, but for threephase power, it is constant, which decreases generator vibration and increases generator life. Also, three-phase motors are more efficient and more reliable than singlephase motors. Although single-family houses are usually connected to only one of the three phases, most schools and other large buildings are connected to all three. In my lab, eachduplex outlet at a lab bench is wired to one of two phases, and some of the wall outlets are wired to the third phase (see Fig.  SEQ Figure Lab_Bench_Photo_1 \* MERGEFORMAT 1), so I have access to all three phases. I use three small transformers plugged into outlets in my lab, one per phase. The transformers are center tapped and lower the 120VAC from the outlet toabout7.5VAC. In each transformer, the primary is electrically isolated from the secondary. This isolation is important for safety. I cannot use "Variacs" or other variable autotransformers in place of the transformers because an autotransformer does not provide the needed isolation. To show students the waveforms of all three phases, I use a Vernier LabPro datacollection device along with Vernier Logger Pro software and a personal computer. Figure SEQ Figure LabPro_Connection_Dwg \* MERGEFORMAT 2 shows how I connect the three phases to the LabPro. To assure that the peak voltage will not exceed the10volt limit of the Vernier voltage probe, I use half of each secondary. (For a transformer output of 7.5VACRMS, the peak voltage is(2(7.5volts(10.6 volts.) Figure SEQ Figure Vernier_Three_Phase_graph \* MERGEFORMAT 3 shows a Logger Pro graph of the three voltages. To create Fig. SEQ Figure Vernier_Three_Phase_graph \* MERGEFORMAT 3, I set the LabPro's data collection time to20milliseconds and the sample rate to3.333msecs/sample. To get a graph like Fig. SEQ Figure Vernier_Three_Phase_graph \* MERGEFORMAT 3, I sometimes have to move one of the red leads in Fig. SEQ Figure LabPro_Connection_Dwg \* MERGEFORMAT 2 to the opposite end of a transformer secondary.  In Fig. SEQ Figure Vernier_Three_Phase_graph \* MERGEFORMAT 3, the period of each waveform is16.7 msec, which corresponds to the60Hz frequency of the alternating current. Once I have the data shown in Fig. SEQ Figure Vernier_Three_Phase_graph \* MERGEFORMAT 3, I can calculate the difference between any two of the voltages. I can do the calculation within Logger Pro, or I can first export the data to another program, such as Excel. Figure SEQ Figure Vernier_Voltage_Difference_graph \* MERGEFORMAT 4 is a Logger Pro graph of two of the phase voltages and the calculated difference between them. As I can show using phasors or the trigonometric method described in the Appendix, the amplitude of this voltage difference ought to be(3 times the amplitude of either of the voltages, and the phase ought to be/6 radians. Using the data shown in Fig. SEQ Figure Vernier_Three_Phase_graph \* MERGEFORMAT 3, I can also calculate the sum of the three voltages. Figure SEQ Figure Vernier_Voltage_Sum_graph \* MERGEFORMAT 5 is a Logger Pro graph of the three voltages and the calculated sum. I can again use phasors or the trigonometry found in the Appendix to show that this sum ought to be zero. The sum in Fig.  SEQ Figure Vernier_Voltage_Sum_graph \* MERGEFORMAT 5 is not zero because the amplitudes of the three sine waves are not exactly equal.  In commercial wiring, three transformers are often connected in a bank, like the one shown in Fig. 6, with one primary per phase and with the secondaries connected in a"Y". Figure SEQ Figure Y_connected_Fil_Xfrmrs_Dwg \* MERGEFORMAT 7 shows how I connect the secondaries of my three transformers in aY. The common point of theY connects all of the secondary leads where no voltage probe lead was connected when I used the circuit in Fig. SEQ Figure LabPro_Connection_Dwg \* MERGEFORMAT 2 to record the waveforms in Fig. SEQ Figure Vernier_Three_Phase_graph \* MERGEFORMAT 3; this point is the neutral. With the secondaries connected this way, the voltagesVAN, VBN, and VCN shown in Fig. 7 are called phase voltages or linetoneutral voltages, and the voltagesVAB, VBC, and VCA are called line voltages or linetoline voltages. In my Yconnected circuit, I use the entire secondary of each transformer, so all the linetoneutral voltages are7.5VAC, and all the linetoline voltages are(3(7.5VAC. In commercial buildings (in the U.S.), thelinetoneutral voltage is 120 VAC, and the linetoline voltage is 208VAC ((3(120=208). In those buildings, linetoneutral voltage is available at standard duplex outlets (like the ones in Fig. SEQ Figure Lab_Bench_Photo_1 \* MERGEFORMAT 1), and the linetoline voltage is often available at special threephase outlets like the one shown in Fig. SEQ Figure Hubbell_Three_Phase_Outlet_Photo \* MERGEFORMAT 8. To demonstrate what happens when three-phase power is applied to a load, I connectthree #47 pilot bulbs as shown in Fig. SEQ Figure Pilot_Bulbs_Schematic \* MERGEFORMAT 9. With three bulbs connected this way, equal currents flow through each one, and because of the phase relationship between the currents, no current flows in the neutral wirea fact wellknown to electrical engineers and electricians. (Because the bulbs are of equal resistance, the current waveforms look like the voltage waveforms shown in Fig. SEQ Figure Vernier_Three_Phase_graph \* MERGEFORMAT 3, and their sum, which flows in the neutral wire, is zero.) In aY arrangement like this one, the neutral wire can sometimes be eliminated when the load will always be balanced. In the circuit shown in Fig. SEQ Figure Pilot_Bulbs_Schematic \* MERGEFORMAT 9, the load is balanced, and if I remove the neutral wire, nothing changes! When I place an additional bulb in parallel with one of the bulbs and then remove the neutral wire, the paralleled bulbs dim and the others brighten, showing that a current was flowing in the neutral wire and that wire is now needed. TableI lists results for three bulb combinations. The transformers are center tapped, and for a center tapped transformer, the potential difference between the center tap and one end of the secondary is180degrees out of phase with the potential difference between the center tap and the other end. With the center taps connected as shown in Fig. SEQ Figure Six_Phase_Schematic \* MERGEFORMAT 10, I can use this180degree difference to convert threephase AC to six-phase AC., (The generation of six phases is explained mathematically in the Appendix.) By connecting together all the center taps, I get the original three phases (A, B, and C) and three new phases (D, E, and F). The six linetoneutral voltages are all 3.75 VAC, but the voltage between any two lines depends on the phase difference between them and can be 3.75 VAC (for 60 degrees), (3(3.75 VAC (for 120 degrees), or 7.5VAC (for 180 degrees). Sixphase AC is not just a laboratory curiosity. Sixphase, and even twelvephase, AC is used when AC is converted to DC for transmission as high voltage DC (HVDC). Sixphase and twelvephase power transmission lines also have advantages of their own, and experimental transmission lines like the ones shown in Fig.  SEQ Figure DOESix_Phase_Tower_Photo \* MERGEFORMAT 11 have been built and tested. A LabPro plot of the six linetoneutral voltages created using the circuit in Fig. SEQ Figure Six_Phase_Schematic \* MERGEFORMAT 10 can be found at www.wellesley.edu/Physics/gcaplan along with a description of how I created the plot and a discussion of other methods for viewing the six waveforms. I thank George Dikmak and Elaine Igo of the Wellesley College Science Center for constructing much of the equipment; Garry Sherman for creating many of the figures; and the Wellesley College physics department for help with expenses. APPENDIX The three voltages, VA, VB, and VC, all have the same amplitude, A, but they differ in phase by EMBED Equation.3  radians (120degrees), so  EMBED Equation.3  (A1)  EMBED Equation.3  (A2)  EMBED Equation.3  (A3) The potential differenceVAVB is  EMBED Equation.3 . (A4) Using the trig identity  EMBED Equation.3  (A5) gives  EMBED Equation.3  (A6)  EMBED Equation.3  (A7)  EMBED Equation.3  (A8)  EMBED Equation.3  (A9)  EMBED Equation.3  (A10) This shows that the amplitude ofVA-VB is (3A and thatVA-VB reaches its peak30degrees (/6 radians) before the peak ofVA. The sum of the three voltages is  EMBED Equation.3 . (A11) Using the trig identity  EMBED Equation.3  (A12) gives  EMBED Equation.3 , (A13) which simplifies to  EMBED Equation.3 , (A14) and then  EMBED Equation.3 . (A15) This shows that the sum of the three voltages is zero. In addition to VA, VB, and VC, the center tapped transformer can provide VD, VE, and VF where VD = - VA (A16) VE = - VB (A17) VF = - VC (A18) Combining equations REF V_A_defn \h A1,  REF V_B_defn \h A2,  REF V_C_defn \h A3,  REF V_D_defn A16,  REF V_E_defn A17, and  REF V_F_defn A18 with the trig identities  EMBED Equation.3  (A19) and  EMBED Equation.3  (A20) gives  EMBED Equation.3  (A21)  EMBED Equation.3  (A22)  EMBED Equation.3  (A23) showing that VA, VF, VB, VD, VC, and VE are six discrete phases, each one separated from the next by60degrees (/3 radians). Phase APhase BPhase CNeutral Wire InstalledNeutral Wire Removed11 Bulb1 Bulb1 BulbAll bulbs equal brightnessSame as with neutral installed21 Bulb2 Bulbs in parallel1 BulbAll bulbs equal brightnessParallel bulbs dimmer, others brighter.3No Bulb2 Bulbs in parallel1 BulbAll bulbs equal brightnessParallel bulbs dimmer, other brighter. Table I Bulb Brightness for Three Configurations of the Circuit in Fig.  SEQ Figure Pilot_Bulbs_Schematic \* MERGEFORMAT 9 Figure Captions Fig.  SEQ Figure \* MERGEFORMAT 1 Lab outlets that are wired to the three phases of the power line. Fig.  SEQ Figure \* MERGEFORMAT 2. Schematic diagram of LabPro connection for observing three-phases. Fig.  SEQ Figure \* MERGEFORMAT 3. Vernier three-phase graph. Fig.  SEQ Figure \* MERGEFORMAT 4. Vernier voltage difference graph, showing two of the three phases and the calculated difference between those two. Fig.  SEQ Figure \* MERGEFORMAT 5. Vernier voltage sum graph, showing all three phases and their calculated sum. Fig.  SEQ Figure \* MERGEFORMAT 6. Bank of three transformers for lowering the voltage of three-phase power before it enters a building. (Photo courtesy of NSTAR.) Fig.  SEQ Figure \* MERGEFORMAT 7. Y-connected transformer secondaries. (The common point of theY connects all of the secondary leads where no voltage probe lead was connected in Fig. SEQ Figure LabPro_Connection_Dwg \* MERGEFORMAT 2.) Fig.  SEQ Figure \* MERGEFORMAT 8. Three-phase, 208V outlet. (Photo courtesy of Hubbell Wiring Device-Kellems.) Fig.  SEQ Figure \* MERGEFORMAT 9. Schematic diagram of pilot bulbs used as load for threephase power. (Transformer primaries are not shown.) Fig.  SEQ Figure \* MERGEFORMAT 10. Schematic diagram of transformer secondaries connected to convert threephase power to sixphase power. Fig.  SEQ Figure \* MERGEFORMAT 11. Six-phase (left) and twelve-phase (right) electric transmission towers. These were constructed in the 1980s at a test site in Malta, NY. (Photo courtesy of the U.S. Department of Energy.)     PRELIMINARY George M. Caplan, Wellesley College  SAVEDATE \@ "MMMM d, yyyy" January 6, 2008  SAVEDATE \@ "HH:mm" 18:22 PRELIMINARY Page  PAGE 4 of  NUMPAGES 10 George M. Caplan  FILENAME Three_Phase_Paper_1n.doc  E. Hecht, Physics: Calculus, 2nd ed., (Brooks/Cole, Pacific Grove, CA. 2000), p. 843.  E. Hecht, Physics Algebra/Trig, 3rd ed., (Brooks/Cole, Pacific Grove, CA. 2003), p. 745.  D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, 8th ed., (John Wiley and Sons, Hoboken, NJ, 2008), p. 836. See also, R. Wolfson and J. M. Pasachoff, Physics for Scientists and Engineers, 3rd ed., (AddisonWesley, Reading, MA, 1999), p. 813.  Stancor part number P-6466.  The secondary voltage of the transformers is nominally 6.3 VAC, but with no load, the voltage is about 7.5 VAC.  I used Vernier VP-BTA voltage probes which have an input range of +10 to -10 volts. The Vernier VPDIN voltage probe has an input range of 0 to 5 volts and cannot be used.  Instead of moving the red lead at the secondary of the transformer, I could have reversed the wiring to the transformer primary. If a non-polarized, two-wire line cord is used, this can be done by simply reversing the plug at the120VAC socket.  See, for example, R. L. Boylestad, Introductory Circuit Analysis, 11th ed. (Pearson Education, Upper Saddle River, NJ, 2007), p.1032.  See, for example, R. L. Boylestad, op. cit., p.1031.  The amplitude inequality is probably caused by slight differences in the transformers. I have measured the input voltage to the transformers and  See, for example, L. S. Bobrow, Elementary Linear Circuit Analysis, 2nd ed., (Oxford University Press, NY, 1987), p. 414.  See R. L. Boylestad, op. cit., pp.1030 - 1033. See also L. S. Bobrow, op. cit., pp. 414  415.  A.R. Bergen, Power Systems Analysis, (Prentice-Hall, Englewood Cliffs, NJ, 1986), p.34.  See, for example, R. L. Boylestad, op. cit., p.1035.  See J. Keljik, Electricity 3 - Power Generation and Delivery, 8th ed., (Thomson Delmar Learning, Clifton Park, NY, 2006), p. 42.  See also P. R. Clement and W. C. Johnson, Electrical Engineering Science, (McGraw-Hill, NY, 1960), p.568.  See, for example, N. G. Hingorani, "Highvoltage DC transmission: A power electronics workhorse," IEEE Spectrum 33(4), 63-72 (April 1996). 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