Collection of Problems.
BALANCED POLYPHASE CIRCUITS
(a)
(b)
Fig. 4 coils having induced emf’s shown in part (b)
(a)
(b)
Fig. 5. Resultant emf shown in (b) for connection of coils shown in (a)
Problem 1. In Fig. 4.a connect terminal b to terminal c and compare the resultant voltage
Ead with voltge Ecb of Fig. 5b.
Ans : Ead = Ecb
Problem 2. (a) Connect terminal d to terminal b in Fig. 4a and find the voltage Eca if E = 20
Volts. Eab and Ecd have the same vector relation as shown in Fig. 4b.
Ans : Eca = 120 /-60o Volts.
(b) With terminal d connected to terminal b as above, find Eac.
Ans : Eac = 120 /120o Volts.
1 Fig 6. See Problem 3
Problem 3. Find the magnitude and vector position of voltage Eca in Fig. 6a if Eab and Ecd
are displaced from each other by 30o in time phase as shown in Fig. 65.
Ans : Eca = 51.76 /105o
Oscillogram 1. Illustrating the 30o angular displacement between the phase
voltages and the systematically labeled line-to-line voltages in a balanced,
three-phase, wye-connected load. Effective value of each line-to-line
voltage is 100 volts
Problem 4. (a) Draw a polar (or single-origin) vector diagram which will represent the
same phase voltages and the same line voltages as shown in Oscillogram 1 using Vbn as
reference. Specify the effective magnitude of the phase voltages, the sequence of the phase
voltages, and the seqthe sequence of the line voltages.
2 Ans : V/phase = 57.7 Volts.
Phase voltage sequence : an-bc-ca.
Line voltage sequence : ab-bc-ca.
(b) Draw a polar (or single-origin) vector diagram which will represent the same phase
voltages as shown in Oscillogram 1, namely Van, Vbn, and Vcn , together with the line
voltages Vba, Vcb, and Vac using Vcn as reference. Specify the sequence of these line
voltages.
Ans : Line voltage sequence : ba-cb-ac.
Oscillogram 2. Oscillographic study of a balanced, delta-connected, unitypower-factor load.
Problem 5. Refer to Oscillogram 2. Draw a compete vector diagram of Vab, Vbc, Vca, Iab, Ibc,
Ica, Ia’a, Ib’b, Ic’c employing Vbc as reference. From the scaled ordinates given on
Oscilliogram 2, determine the effective values of line (or phase) voltage, phase current, and
line current.
Ans : V = 100 Volts; Ip =3.5 Ampere; Il = 6 Amperes.
Problem 6. Find the magnitude of the line currents issuing from a balanced six-phase, mesh
connected generator if the phase currents are known to be 100 amperes in magnitude.
Illustrate solution by means of a vector diagram.
Ans : IL = Ip = 100 amperes.
3 Problem 7. Find the voltage between adjacent lines of a balanced twelve-phase, starconnected system if the phase voltages are 50 volts in magnitude. Illustrate solution by
means of a vector diagram.
Ans : 25.88 volts.
Problem 8. Find the voltage between alternate lines of a balanced six-phase, star-connected
system if the phase voltages are 132.8 volts in magnitude.
Ans : 230 volts.
Problem 9. Three-phase line voltages of 2300 volts magnitude are impressed on a balancd
wye-connected load which consists of 100 ohms resistance per phase in series with 173.2
ohms inductive reactance per phase. Find the line current and the total power taken by the
three-phase load. Calculate Pt as 3Ip2Rp as 3 VpIp cos θp1 a…d as
cos θp.
Ans : IL = IP = 6.64 amperes, Pt = 13.22 kw.
Problem 10. Repeat Problem 9, assuming that the three impedances are connected in delta
(rather than in wye) across the same line voltages.
Ans : IL = 19. 92 ampere, Pt = 39.66 kw.
Problem 11. Three-phase line voltages of 410 volts are impressed on a balanced deltaconnected load which consists of 8 ohms resistance in series with 6 ohms inductive
reactance per phase.
(a) Find the volt-amperes per phase, the reactive volt-amperes per phase, and the reactive
factor of each phase.
Ans : vap = 19.360, rvap = 11.616, r.f. = 0.6.
(b)Find the total volt-amperes of the system, the total reactive volt-amperes of the system,
and the reactive factor of each phase.
Ans : vat = 58,080, rvat = 34,848, r.f. = 0,6
4 Oscillogram 3. Ascillographic representation of all voltages and currents
involved in the two-wattmeter method of measuring balanced three-phase
power at unity power factor.
5 Oscillogram 4. Oscillographic representation of all voltages and currents
involved in the two-wattmeter method of measuring balanced three-phase
power at 0.5 p.f. lag, the condition under which one wattmeter reads zero.
Problem 12. Refer to Oscillogram 3. (a) If the line-to-line voltages have instantaneous
maximum values of 155,5 volts and the dlta-line currents have instantaneous maximum
values of 14,14 amperes, find the average power readings of the wattmeters Wab-a’a and
Wcb-c’c.
(b) Draw a vector diagram indicated all currents and voltages show on Oscillogram 3. Use
Vab as reference, and include the delta-phase currents Iab, Ibc, and Ica which are not shown
on the oscillogram but which combine to form the delta-line currents Ia’a and Ic’c.
Ans : (a) Wab-a’a = Wcb-c’c = 952.6 watts.
(b) ab-bc-ca sequence of line-to-line voltages; Iab in time phase with Vab; Ia’a lags Vab
by 30o; Ic’c leads Vcb by 30o.
Problem 13. Refer to Fig 49. Find the ratio of the copper required for two-phase, three-wire
transmission to that required for three-phase, three-wire transmission under the following
conditions, all imposed simultaneously.
6 (a) A fixed amount of power transmitted.
(b) The same distance.
(c) With the same total line losss.
Hint :
From condition (a) : P2t = 2Vp2I2cos θ = P2t = 3Vp3I3cos θ
From condition (d) : I2 =
I3
From condition (c) : 2I22R2 + (
l2)2R2’ = 3I32R3
From condition (e) : Area of R2 wire =
x area of R2 wire
From condition (b) : R2’ =
Ans : 1,94
UNANSWERED PROBLEMS
14. What is the phase voltage and also the voltage berween adjacent line of two-phase star
connection if the greatest voltage between any pair of lines is 156 volts.
15. The voltage between adjacent lines of a twelve-phase star is 400 volts. Find voltage to
neuiral, the voltage between alternate lines, and the greatest voltage between any pair
of lines.
16. Find the phase current in a six-phase mesh if the line current is 10 amperes; for a
twelve-phase mesh for the same line current.
17. Given six coils each having an induced voltage of 63.5 volts. Adjacent coil voltages
are 60o apart. In how many ways may you connect these coils to form a need threephase wye system of voltages if all coils must be used for each system and if the
magnitude of the line voltages of each system must be different? What is the line
voltages for each wye system?
18. A generator has six coils, adjacent coils being displaced 30 electrical degrees, each coil
voltage is 114 volts, show how to connect them and alculat the line terminal voltage
for three-phase star. Repeat for three-phase mesh. Repeat two-phase, where line
voltage is taken as the phase voltage.
19. A generator has six coils, adjacent coils being displaced 30 electrical degrees. All coils
are used to form a three-phase mesh, what must be the emf of each coil yield balanced
7 three-phase voltages of 230 volts each? If all coils are connected three-phase star, what
must be the emf of each coil to give an emf between lines 230 volts?
20. Draw vector diagrams which represent the currents and voltages shown in Oscillogram
3 and 4, pages 355 and 356, and label them in accordance with the beling on the
oscillogram.
21. Three-phase line voltages of 230 volts are impressed on a balanced wye load having
1G ohms resistance and 12 ohms reactance in series in each phase. Find be line current
and total power. If the three impedances are reconnected in delta and placed across the
same line voltages, what are the line and phase currents and be total power?
22. A current of 10 amperes flows in the lines to a twelve-phase mesh-connected having 5
ohms resistance and 8 ohms capacitive reactance in series in each phase. What is the
voltage between alternate lines on the load? Draw the vector diagram of the voltages
and phase currents of two adjacent phases, and also show the line current from the
junction of the two phase.
23. A balanced wye load consists of 3 ohms resistance and 4 ohms capacitive reactance in
series per phase. Balanced three-phase voltages of 100 volts each phase impressed
across the lines at the load. If the load is connectd to a generator through three lines of
wqual impedance, each line containing a resistance of 1 ohm and an inductive
reactance of 4 ohms, find the voltage at the generator terminals.
24. A balanced wye load having 8 ohms resistance and 6 ohms inductive reactance a
series in each phase is supplied through lines each having 1 ohm resistance and 2 ohms
inductive reactance. If the sending-end voltage between lines is 250 volts, what will be
the voltage between lines at the load?
25. A balanced delta load contains a resistance of 12 ohms and a capacitive reactance of 16
ohms in series in each phase. If the balanced impressed line voltages the load are 115
volts each, calculate the line and phase currents.
26. A balanced delta load having 18 ohms resistance and 24 ohms capacitive reactance in
series in each phase is supplied though lines each having 1 ohm resistance and 2 ohms
inductive reactance. If the line-to-line voltage at the sending end is 250 volts, find the
line-to-line voltage at the load terminals. Also find the total power consumed by the
load.
27. A balanced wye inductive load takes 5.4 kw at 0.6 power factor at a line voltage of 200
volts. It is in parallel with a pure resistive balanced wye load taking 5 kw. Find the
resultant line current supplied the combination.
8 28. The total power supplied two balanced three-phase load in parallel is 12 kw at 0.8
power factor lagging. One of the load takes 10 kva at 0.8 power factor load. The
second load is a delta-connected balance load. Find the resistance and the reactance per
phase of the delta load if the line voltage is 230 volts. If the unknown load were wyeconnected, what would be the resistance and reactance per phase.
29. Each phase of a delta load has 6 ohms resistance and 9 ohms capacitive reactance in
series. Each phase of a wye load has 8 ohms resistance and 6 ohms inductive reactance
in series. The two loads are connected in parallel across three-phase line voltages of
100 volts. Calculate the resultant line current, the total power consumed, and the power
factor of the combination.
30. A three-phase, 5 hp, 220 volt induction motor (balanced load has an efficiency of 86
per cent and operate at 86.6 per cent lagging power factor. It is paralleled with a threephase resistance furnace consisting of three 36 ohm resistance connected in delta. Find
the kilo volt-amperes demanded by the combination, the power factor , and the line
current.
31. A three-phase generator supplies balanced voltages of 230 volts. each at its terminals
when it carries a load which requires 10 amperes. If the power factor at the generator
terminals is 0.8 leading, calculate the voltage at the load if the load is connected
though lines each having 1 ohm resistance and 5 ohms inductive resistance.
32. A balanced three-phase load requires 10 kva at 0.5 lagging power factor. Find the kva
size of the condenser bank which may be paralleled with the load to bring the power
factor of the combination to 0.866 lag, and also to 0.866 lead.
33. If the line voltage for problem 32 is 230 volts and the frequency 60 cycles, find the
capacitance in microfarads of capacitors required in each phase of the capacitor bank if
they are delta-connected. What capacitance is required if they are wye-connected.
34. Three 15 /600 ohm load impedances are connected in delta and supplied by lines, each
line containing 1 ohm resistance and 1 ohm inductive reactance. If the line voltages on
the supply side of the line impedances are balanced three-phase of 115 volts each, find
the voltage across the load impedances. Also calculate the power loss in the supply
lines and the power dissipated by load it self.
35. If the current through each of the load impedances in problem 34 is 20 amperes, find
the required voltage on the supply side of the line impedances.
36. A three-phase line has three capacitors, each having a reactance of 300 ohms connected
in delta across the lines at the source. Three similar capacitor are so connected between
9 the line at the load. Between these two sets of capacitors each line a series inductive
reactance of 10 ohm. If a balanced three-phase load of 100 kva at 0.6 power-factor lag
requires 2300 volts between lines. What voltage between lines will be required at the
source? What will be the power input to the lines and the power factor at the source ?
Fig 54. See problem 37 and 38
37. The motor M in Fig.54 has 2300 volts balanced three-phase voltage impressed at each
terminal and takes 120 kva at 0.6 leading power factor. Calculate the line volts, power
input, and the power factor at a, b, c.
38. If the motor in Fig.54 is removed from the circuit and balanced three-phase line
voltages of 2300 volts each are impressed at a, b, c how many volt will appear between
lines at the motor end of the line?
39. A three-phase resonant shunt is connected to three-phase, 2300 volts lines to furnish a
low impedance for a certain frequency so as to reduce the inductive interference with a
telephone line. The shunt consist of three 10 kva, 60 cycle, 2300 volts capacitors
connected in delta. In series with each line terminal from the delta is an inductance of
2.5 millihenrys. At what frequency does this three-phase combination resonate. That is
offer minimum impedance? Assume that resistances of capacitors and inductances are
negligible.
40. (a) Three coils each having 36 ohms resistance and 100 millihenrys inductance are
connected in delta. Find the microfarad capacitance of each capacitor which may be
placed in each of the three lines from the delta to pruduce resonance (unity p.f. ) of the
system as a whole for a frequency of 800 cycles. This is a type of resonant shunt
sometimes connected to power lines to reduce inductive interference with telephone
circuits.
10 (b) Assume the capacitors calculated for each line in (a) are removed and connected in
delta. Find how many henrys of inductance would be required in each line from this
delta to bring the power factor of the combinations to unity at 800 cycles.
D:\Tugas Pak Imam\gbr 55.jpg
Fig 55. See problems 41, 43, and 44
41. Find the readings of Wa and Wb in Fig. 55 for the sequence Va, Vnc, Vnb. Find the power
dissipated in each phase.
42. A balanced three-phase load takes 5 kw and 20 reactive kva. Find the readings of two
wattmeters properly connected to measure the total power.
43. In Fig. 55 find the reading of WR. Also calculate the total reactive volt-amperes taken
by the load. What is the ratio of the total reactive volt-ampere taken to the reading of
WR ?
44. Prove that the ratio of the reading of WR of Fig. 55 to the total reactive volt-amperes
obtained in Problem 43 will obtain for all balanced loads when the impressed voltages
are sinusoidal balance three-phase.
45. (a) Calculate analytically the power-factor angle for a balanced three-phase circuit in
which two wattmeters properly connected to measure three-phase power read + 1000
and + 800 watts, respectively.
(b) Also calculate the angle if the meters read +1000 and -800 watts respectively.
46. Two wattmeters measuring power to a balanced three-phase load read 1200 and -100
watts respectively. How many volt-amperes does the load take? At what power factor?
47. The power to a balanced three-phase leading-power-factor load is measure by two
wattmeters. The wattmeter having its current coil in line A and each potential coil from
line A to line C indicates +1000 watts the other wattmeter with it’s current coil in line
11 B and it’s potential coil from line B to line C indicates +400 watts. What is the voltage
sequence? What is the power factor of the load?
48. Each phase of balanced twelve-phase star-connected load consists of 3 ohms resistance
and 4 ohms inductive reactance in series. Balanced twelve-phase line voltages of 51.76
volts between adjacent lines are applied to the load. Calculate the line current, power
factor, and total power consumed by the load.
49. The voltage induced in phase na of a three-phase wye-connected generator is
If the sequence is
voltage
,
,
find the equation with respect to time of the line
. Note : Phase voltages of polyphase generators differ only in phase angle.
50. If the phases of generator in Problem 49 are reconnected in delta, what will be the
equation with respect to time of the line voltage across phase nα ?
51. A wye-connected generator has a generated voltage per phase which contains only the
fundamental, third, fifth, and seventh harmonies. The line voltage to as measured by a
voltmeter is 230 volts; the voltage to neutral is 160 volts. Calculate the magnitude of
the third harmonies in the generated voltage.
Fig 56. See problems 52 and 53
52. The induced emf of a delta generator with one corner of the delta open as shown in
Fig. 56 contains only odd harmonies up to the seventh. A voltmeter across αc reads
2500 volts, and, across bb’ when negligible current flows, 1800 volts. Find the reading
of a voltmeter connected from α to b’.
53. The induced phase voltage of a delta generator with one corner open as shown in Fig.
56 contains only odd harmonies up to the seventh. A voltmeter connected from α to b’
reads 2500 volts, and from α to α it reads 2200 volts when negligible current flows.
What should it read from b to b’ ?
12 Fig 57. See problem 54
54. Figure 57 shows a generator connected to a balanced pure resistance load. An ammeter
in the reads 15 amperes, and the wattmeter shown reads 600 watts. A voltmeter shows
a balanced line voltage of 230 volts. Find the line currents to the load and the voltage
from the line to neutral at the load. assuming that the generated voltage contains only
fundamental and third-harmonic components.
13 UNBALANCED POLYPHASE CIRCUITS
Problem 1. Determine the values of Van, Vbn, and Vcn in example 2.
Ans : Van = 36.6 /15o ;Vbn = 205.6 /-80.1o ; Vcn = 239.6 /-23.8o
Problem 2. Determine the power dissipated in each of the three phases (an, bn, and cn) of
example 2.
Ans : Pan = 134; Pbn = 2120; Pcn = 0 watts.
Fig 2. Conversion from a wye-connected load to an equivalent deltaconnected load
Problem 3. Find the magnitudes of Ia’a, Ib’b, and Ic’c in Fig. 2 if Vab = 212 /90o, Vbc = 212 /30o, and Vca = 212 /150o volts. As in example 2, Zan = (10+j0), Zbn = (10+j10), and Zcn = (0
- j20) ohms.
Ans : Ia’a = 13.65; Ib’b = 6.20; Ic’c =7.54 amperes.
Problem 4. Solve equation (S) explicitly for In’n and state in words how to find Ia’a, Ib’b, and
Ic’c after In’n has been evaluated.
Ans : In’n =
Problem 5. Show by means of a qualitative vector diagram that the voltmeter (Vm) of Fig.
11a reads below line voltage for voltagesequence ab-ca-bc.
14 (a)
(b)
Fig 11. A voltmeter method of checking phase sequence in three-phase
systems. See example 6 and problems 5 and 6.
Problem 6. What is the magnitude of the voltmeter reading in Fig. 11a if Xc = 100 ohms, R
= 100 ohms, and Vab = Vbc = Vca = 141.4 volts if the voltage sequence is ab-ca-bc?
Ans : 51.8 volts.
Problem 7. Calculate the readings of Wba-wb’b and Wca-wc’c in the above example and
compare the sum if the wattmeter readings this found with the total commected load.
Ans : Wba-wb’b = 5685; Wca-wc’c = 4315 watts.
Fig 22. A particular unbalanced three-phase load
Problem 8. If the reactive volt-ampere meters shown in Fig. 22 are placed so that the
current coils carry Ia’a and Ic’c what will be the individual meter readings in vars? It is
assumed that the potential circuits of the meters are connected in such as manner that the
algebraie sum of the readings will be equal to
?
Ans : Meter a reads -1464 vars; meter a reads zero.
15 Problem 9. What is the power factor of the unbalanced load shown in Fig.22 as determined
from
and
?
Ans : 0.939
Problem 10. Find the magnitudes of Ia’a, Ib’b, and Ic’c in Fig. 26 utilizing the calculationsof
example 13 in so far as they are helpful.
Ans : Ia’a = 55.6; Ib’b = 55.6; and Ic’c = 111.2 amperes.
UNANSWERED PROBLEMS
11. An unbalanced delta system labeled abc at the corners consists of Z ab =
10/-60º,
Z bc = 5 /0 º, Z ca = 10/-60ºohms. If Vab =100/0 º and the voltage sequence is cb-ba-ac,
find the vector expressions for the currents entering the terminals a, b, and c. The
three-phase supply voltages are balanced. Also solve for the opposite sequence.
12. An unbalanced load labeled abc at the corners contains of Z ab = 5/19 º , Z bc = 10 /-30 º,
Z ca = 8 /45º ohms. Three phase balanced line voltages 115 volts each are applied. If the
sequence is cb-ac-ba, calculated the complex expressions for the line currents leaving
terminals a, b, and c for Vab =115/0 º volts.
Fig 27. See problem 13
13. Refer the Fig. 27. VAB and VCB represent a balanced two-phase system of voltage
drops, the magnitude of each being 115 volts. The voltage phase sequence is AB-CB.
VAB is to be used as reference. Find I AB , I CB , I BB ' and draw a vector diagram of the
voltages and currents.
16 14. A wye-connected set of impedances contains of Z an =5 /0 º, Z bn =5 /60 º, Z cn =5 /-60 º
ohms. Find the equivalent delta-connected impedances Z ab , Z bc , Z ca which can be
used to replace the wye-connected set of impedances.
15. Refer to Fig. 23 The terminals a’b’c’ represent a balanced three-phase system voltages
the sequence of wihich is b’c’-a’b’-c’a’. The magnitude of each line-to-line voltage is
230 volts. Find the readings of ammeters placed in the a’a, b’b, and c’c lines.
Fig 7. A three-wire three-phase network
16. In Fig 7. page 380, it will be assumed that the generated voltages are Z n 'a ' =100/0 º,
Z n 'b ' =100/-120 º, Z n 'c ' =100/-240 º volts and that
Z n 'a 'an = (2 − j1)ohms
Z n 'b 'bn = (1 − j 3)ohms
Z n 'c 'cn = (3 ÷ j 4)ohms
Find the lines currents I a 'a , I b 'b , and I c 'c . Draw a vector diagram of line-to-line
voltages and the line currents.
Fig 8. A four-wire three-phase system
17 17. Refer to Fig. 8, page 381. let it be assumed that the following quantities are known:
En’a’ = 1000 + j0 = 1000 /0o volts
En’b’ = -500 – j866 = 1000 /-120o volts
En’c’ = -500 + j866 = 1000 /120o volts
Za’b’ = 20 – j20 = 28.28 /-45o ohms
Zb’c’ = 50 + j0 = 50.0 /0o ohms
Zc’a’ = 30 + j52 = 60.00 /60o ohms
Zg = 2 + j8 = 8.25 /76o ohms
Z1 = 1 + j1 = 1.41/45o ohms
Zn = 2.5 + j1 = 2.70/21.8o ohms
Write the expressions for Iaa’, Ibb’, Icc’, employing determinants and the numerical
values of the E’s and Z’s specified above. Use loop currents I1 = Ia’a, I1 = Ib’b, dan I3 =
Ic’c all returning through line nn’. (Results may left in the for of the ratio of two
matrices.)
18. A delta-connected set of impedance consists of Z an =5 /0 º, Z bc =5 /60 º , dan Z an =5 /60 º ohms. Find the equivalent wye-connected impedances
Z ab =5/0 º, Z bc =5/60
º, and Z ca =5/-60 º which can be employed to replace the above delta-connected
impedances.
Fig 9. A wye-delta circuit arrangement
19. Refer to Fig. 9, page 382. Assume that the generator is capable of maintaining a
balances system three-phase of voltages Eb’c’, Ea’c’, Ec’b’, the sequence of which is b’a’ –
a’c’ –c’b’. The magnitude of each line voltage is 100 volts. Zc’c = Zb’b = Zc’c = 0.5 +
f0,5 ohm. Z ab =5 /0 º, Z bc =5 /60 º , dan Z an =5 /-60 º ohms. Find Ia’a, Ib’b, Iab, dan Ibc,
dan Icb with respect to Va’b’ as a reference.
18 20. Explain, by means of qualitative vector diagrams, the operation of a three-phassequence indicator that employs an inductance coil in place of the condenser shown in
Fig 11a, page 386. Does the voltmeter read above or below line voltage for sequene
ab-ca-bc?
21. Devise some cheme for checking the phase sequence of two-phase-voltages.
22. Find the reading of wattmeter which has its current coil current coil in the A’A line and
its potential coil across the voltage VAC in Problem 13 and Fig. 27.
Fig 13. The three-wattmeter method of measuring individual phase
powers in a delta-connected load
23. Refer to Fig 13, page 386. Vab = 200, Vbc = 141.4, and Vca = 141.4 volts. Sequence abbc-ca. Zab = Zbc = Zca = (S – j6) ohms. Find the reading of each of the watermetters.
Find the reading of a wattmeter with its current coil in line a and potential coil from a
to b; also one with current coil in line c and potential coil from c to b.
Fig 1. Unbalanced delta load
19 24. (a) if a wattmeter Wa has its current coil in line a and its potential coil form line a to b
of Fig. 1, page 373, what will it read for a sequence Vab-Vcz-Vbc? If another wattmeter
Wb has its current coil in line b and its potential coil connected form line b to c, what
will it read?
(b) If Wa and Wb were wattmeters what would they read?
25. (a) Find readings of wattmeter Wa and Wb with its current coils in lines a and b,
respectively, supplying load of Problem 11 if the potential coils are properly connected
so that the sum of the readings will give the total power consumed by the load.
(b) Find the readings if Wa and Wb are varmeters.
Fig 29. See problem 26
26. Refer to Fig. 29. Va’b’ –Vc’z’-Vb’c’ represent a balanced three-phase system of voltage
drops, the magnitude of each being 200 volts. The voltage sequence is a’b’-b’c’-c’a’.
Two balanced three-phase loads indicated by the circles are connected to the terminals
abc as shown in Fig. 29. In addition to the two balanced load, a single-phase, 4-kw,
unity-power-factor load is placed across the bc terminals as indicated.
(a) Find the reading of Wa’a=ab and Wc’c-cb,.
(b) If reactive volt-ampere meters replaced Wa’a=ab and Wc’c-cb,, find their respective
readings.
(c) Find the combined vector power factor of the composite load.
20 Fig 21. The two reactive volt-ampere meter method of measuring
in a three-wire, three-phase system
27. In Fig. 21 page 386, it will be assumed that Va’b’, Vb’c’, and Vc’a’, represent a balanced
three-phas sytem of voltages the sequence of which is a’b-c’a’-b;c;. Z cn = 10 /0 º ,
Z bn = 10 /-60 º , and Z cn = 10 /90 º ohms. Assume that line-to-line voltage of the volts.
(a) Find the readings of the reactive volt-ampere meters shown in Fig. 21.
(b) Find the readings of wattmeters placed at similar posistions in the circuit, namely,
at the a’a-ab and c’c-cb positions.
(c) Find the vector power factor of the unbalanced load as recognized by t’e A.I.E.E.
28. In Fig. 30 Va’b’ , Vb’c’, and Vc’a, are balanced three-phase voltages each having a
magnitude of 200 volts and a phase sequence of ab-bc-ca. Determine the readings of
the two wattmeters shown in the figure.
Fig 30. See Problems 28
29. In Fig. 31, En’a’, En’b’, Ea’c’ are balanced three-phase voltages with magnitudes of 115,4
volts and a phase sequence of n’a’-n’b’-n’c’. Find the following quantities and express
all complex quantities with reference to Vab.
(a) Vab, Vbc, Vca.
(b) Iab, Ibc, Ica
21 (c) Ia’a, Ib’b, Ic’c
(d) The sum of the readings of wattmeters Wa, Wb, Wc when they are connected as
shown.
(e) The individual readings of wattmeters Wa, , Wb’, Wc if the common point O is
connected to line b’b.
Fig. 31. See Problem 29.
30. The line-to-line voltages of a three phase system are Vab = 200, Vab = 150, dan Vca =
120 volts. Write the polar expressions for Vab, Vbc, dan Vca with respect to Vab as
reference for both phase sequences.
31. Refer to Fig 2. In a particular case measurements yield Vcb = 140, Vbc = 120, dan Vca =
150, Van = 200, Vba = 200, Vbn = 80, and Van = 104,2 volts. Draw the qualitative phasor
diagram of the voltages for sequence abc, and determine analytically the complex
expressions for each of the currents with respect to Vab = as a reference.
33. Calculate the line currents in Prooblem 16 by the loop-current method.
34. Refer to example 13, page 402-403, including Fig, 26. Solve for I1, I2, and I3 by the
loop-current method, neglecting the resistive components of all branch impedances for
a voltage sequence Ena-Enc-Eab (Resutls may be left in the form of the ratio of two
matrices.)
35. In Fig. 32, I’ab = Iab = Icb = 0,01 henry and the coefficient of coupling is 0.5. Assume no
resistance or inductances except as indicated driving voltages is n’a’-n’b’-n’c’, and
En’a’ = 57.7/90 º volts. For ω = 1000 radians per seconds calculate the line and phase
currents for the load. Use Maxwell’s cyclic-current method.
22 Fig. 32. See Problems 35 and 36
36. Set up the determinant form of the solution for Iaa’ in Problem 35 if 3 ohms pure
resistance is inserted in each line to the load and the same sequence and reference as
specified in Problem 35 are employed. For uniformity in checking results, use loop
currents as follows:
Loop current I1 = Ia’ncc’
Loop current I2 = Ic’cbb’
Loop current I3 = In’n’b’ba
37. Solve for Ia’a = Ib’b dan Ic’c, in Fig. 33 if Ea’a’ = 1350 + j0 volts, Ea’b’ = -675 – j1170
volts, and En’c’ = -675 + j1170 volts.
Fig 33. See problem 37
23