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AC CIRCUITS- 2 (THREE PHASE CIRCUITS)
I - BALANCED 3cp CIRCUITS :
The magnitude of the line currents are the same and they have 120 0 phase difference.
(1 ) WYE (STAR) Connected System : One-line equivalent circu it :
T ¥ T 7 ^ T ~ 1 1 -------------
Relationships : V% "'if- u = h
v > o a N it * * m
(2) D E LTA {MESH} Connected System :
(0h€ phase of bolflncfcd
W g| 0 - Connected ^ • s je w
v
\
s
V
Relationships
NOTE : To analyzed De/ra-connected system using one-line equivalent circuits, It is first transforme Into an equivalent vvye-connected load making used o f :
A *
CO
A
W)
se A*
6c
Note : If WMRj+ WMR2 is negative, you can carry the negative sign. Consider the Following :
- If p.f = 1.0 , both Wattmeters reads positive W , = W 2
- If p.f. > 0.50 , both readings are positive PT - W, + W
- If p.f. * 0.50 ,only one wattmeter gives reading, other will not read PT “ W,
- If p.f < 0.50, only one wattmeter will read positive and other will give reversed reading (i.e. negative)
II - UNBALANCED 3<p CIRCUITS :
The line currents will not be equal nor will they have 120 0 phase difference as was the case of with balanced loads.
« - in
A
irr
( b.) For unbalanced 3cp system: Consider the first Wattmeter’s current coil in line a ,and potential coil between fines a and c. While the second Wattmeter’s current coil in line b and potentia coil between lines b and c. The Wattmeter readings will be :
W M R ,= V J , c o s S , }"■ (^) WMR2 = V J , c o s tf, ] J
3****. Three wattmeter Method —used to measure power drawn by a 3(p,, 4 wire system. In the case o f balanced and unbalanced loads : Diagram of Connection as Shown
WMR, = V J a c o s 0 , £ WMR2 =V hllIhco s0 2 f WMR, = V J r c o s 9, f
BLONDEL’S THEOREM :
States that the minimum number of wattmeters required to measure power in a polyphase System is “ONE - LESS ” than the number of wires in the system.
ita i rms a. a B 2
elta
mec ictar line A. B.
120 nne
cii
A.
B
bal< sist /i E
thr X)nn onr
Thr< bate seo the
A t kV
is I
A
1C
ta
0. T
tl
F
- J
S Y M M ETR IC A L C O M PO N EN TS : ( FO R T E SC U E ’S T H E O R E M )
Any unbalanced 3<p system o f vectors can be resolved into three (3) balanced system o f vectors N am ely: (L ) A balanced set o f 3<p vectors having the same phase sequence as the original unbalanced system of vectors. This balanced system o f 3cp vectors is known as “Positive Sequence System ”... (2,) A balanced set o f 3cp vectors having phase sequence which opposite to that o f the original unbalanced system o f vectors. This balanced system o f 3cp vectors is known as “Negative Sequence System”, (3.) A set o f three (3) lq> vectors which are equal in magnitude and which have exactly the same 3cp position with respect to any given reference axis. This system o f lcp vectors is known as “Zero Sequence or Uniphase System *'*
For Positive Sequence components :
Va
v „
I20 I-------- > Vb|= a2 V.,
Vc, =a V„
VH For Zero Sequence components :
v a()= v a()
v b 0 = v a„
Vc0 = Va
For Negative Sequence components :
V b i '
20 Vfc
' b
VC 2 =a2Va 2
Where : a = 1 / 120” = - 0.5 + j 0.
a2 - 1 / 240° - - 0.5 - j 0.
Iden tity :
N O TE : Symmetrical components do not have separate existence , they are only mathematical components which does not exist at a ll
HARMONICS in different 3cp systems :
In 3q> systems, harmonics may be produced in the same way as in l <p system. Hence.for all calculations each harmonic is treated separately. Usually, EVEN harmonics are absent in 3<p system. But care must be exercised when dealing with ODD harmonics , especially with 3rd. and all triple-n harmonics.
The following points are important in harmonic analysis for 3 <psystems :
(i)
(ii) (iii) (iv)
(v)
(vi)
All triple-n harmonics i.e. 3rd.,9* ,15th. ...etc. are in phase. The 7 .,13th.,&19th. Harmonics have positve phase rotation of A-B-C. The 5th,,11th., &17th. Harmonics have negative phase rotation o fA-C-B For Star-connected system ,the line voltages will be the difference between the successive phase voltages and hcnce will contain No 3r<l. harmonic terms, bccausc being identical in each phase, thus it will cancel out. For Delta-connected system ,the resultant emf acting around the closed loop would be the sum of the phase emfs. It is zero in the case of fundamental,5th.,7th., 1i J}, 17,l,ete. but for 3rd.,9th.,15th.,etc,the resultant emf is 3 times the phase value acting in the loop. For 3(p, 4 wire system, there will be No 3rd. harmonic component in the line voltage. If the load is balanced, the resulting 3rd. harmonic line currcnt will all be in phase so that the Neutral wire will have to carry 3 times the 3rd. harmonic line current. •
9. A circuit consisting of three resistors rated: 10 ohms, 15 ohms and 20 ohms are
connected in wye. What would be the resistances of the equivalent Delta
connected load?
A. 33, 23 &15 ohm C. 70, 44.4 & 36.7 ohms
B. 30, 40 & 50 ohms D. 65, 43.3 & 32.5 ohms
10. Three capacitors, each having a capacitance of 75 uF are connected in star to a
400V, 3ph, 50Hz supply. Calculate the delta equivalent capacitance so that when
they are connected in the same supply, the line current will remain the same.
A. 150 |iF C. 225 uF
B. 25 jaF D. 75 |.tF
11. A balanced delta connected load of 4 + y'3 ohms per phase is connected to a 3-
phase 230 V supply. What is the line current?
A^SOA C. 23 A
B. 46 A D. 33 A
12. The metering of a 3-phase inductive load indicated the following values: 4000 kW,
34.5 kV, 80% power factor. What is the kVA? _
A. 3000 C. 5000
B. 4000 D. 7000
13. A balanced wye connected load of impedance 8+j6 ohms is connected to a three
phase balanced source of 400 volts 60 Hz. Find the power per phase.
A. 4267W C. 5333 W
B. 12800W D. 3198 W
14. The phase b voltage and the phase b current of a balance 3-phase system are: V
= 220 sin(oot+210°) and 1=10 sin (ot-1800). What is the power of the system?
A )5 7 16 watts C. 1905 watts
B. 3810 watts D. 3300 watts
15. A balanced delta connected load draws 10 Amp line current and 3 kW at 220V.
The reactance of each phase of the load in ohms is .......
A. 30 C. 38.
B. 23.5 D. 20
16. A three phase source of 230-V supplies the following three phase loads.
20 kW at 90% power factor leading
5p kW at 80% power factor lagging
10 kW at unity power factor
What is the total kVA and current?
A. 85.7 & 225 A C. 84.7 & 212.6 A
B. 94.7& 182.8 A D. 84.7 & 92.88 A
17. Three identical impedances of 15 cis 60° ohms are connected in star to a 2
phase, 3 wire, 240 V system. The lines between the supply and the load hav
impedance of 2 + y'1 ohms. Find magnitude of the line voltage at the load.
A. 220 V C. 240 V
B. 185V D. 213V
18. A balanced delta connected load of impedance 13.5+j9 ohms is connected to a
three phase balanced source of 4160 volts 60 Hz through a line with 0.5 ohm
resistance and 2 ohm reactance. With Van=2400Z0, find the line B current usinc
ABC sequence.
A. 87.8+j327.8 A C. -327.8-j87.8 A
B. 69.3-j52 A D. 240-j240 A
19. A balanced 3-phase, 3 wire, 240 V, 60 Hz source supplies power to a star
connected load ZA = 10 + JO Q, ZB = 10 + y'10 f2 and Zc= 10 + ;0 Q. What is
the line current lB?
A. 11.53 Amp C. 15.93 Amp
B. 10.02 Amp D. 16.97 Amp
20. The power input to a 100 hp, 480 Volt, 85% efficiency and 75% power factor 3-
phase motor at full load is measured by two wattmeter connected at lines A & C.
What are the readings in kW?
A. 66.23 & 21.55 C. 87.76 & 11.
B. 69.5 & 18.26 D. 67.89 & 30.
21 .A three phase induction motor takes 3000 W at 80 percent power factor. Two
wattmeters are used to measure the power, Wa and Wc. Wa reading is _____.
A. 2000 /C:)
B. 3000 D. 2500
22. In a balanced three phase, 208 volts circuit, the line current is 100 amperes. The
power is measured by two wattmeter method, one reads 18 kW and the other is 9
kW, What is the power factor of the load?
A. 0.8 C. 0.
B. 0.5 D. 0.
23. An unbalanced wye connected loads on a 3 phase 4-wire system consist of Zi =
10 + j5, Z 2 = 8 + j4 and Z3 = 20 + jO. The wye connected generator load has
balanced voltage to neutral of 120 volts. What is the wattmeter reading at Z 1?
A. 720 watts C. 1440 watts
B. 3312 watts Q. 1152 watts
24. It is required to increase the power factor of a 750 kW three phase balance load
from 70% lagging to 90 %lagging. The line voltage is 6,900 volts, 60 Hz. Specify
the capacitor required to increase the power factor of this wye connected load in
micro-farad per phase.
A. 20.45 C. 22.
B. 18.58 D. 17.
Supplementary Problems
- In a balanced12-phase; voltage;system, the phase voltages are 100 volts, what is the line to line voltage? A. 120 volts C. 141 volts B. 170volts D. 60 volts
- In a balanced 3-phase voltagje generator, the different phase voltages are____ degrees apart. A. 120 C. 60 B. 240 D. 30
- In a 3-phase 4-wire system, the line voltage is 4160 volts. What is the line to neutral voltage? A. 1390 volts C. 2400 volts B 2080 volts D. 4160 volts
- In ABC sequence, What is the equivalent of Van?
A. VbcZ9Q° C. ~ V b c Z - 9 0 ° a/ 3 V
B. ~ V b c Z \2 0 ° ( D. j = VbcZ - 120°
V3. ■ V _ A / 3
5 In a balance three phase voltage, Van=120/30°, What is Vcn using ABC sequence? A.-103.9+j60 C.-60-j109. B. —60—j103.9 D. 0—j
- The secondary of a 3-phase transformer bank is connected delta, the line to line voltage is 480V, the midpoint of the winding between terminal A & B is grounded. What is the voltage across terminal C & the ground? A. 240 volts C. 720 volts B. 277 volts D. 416 volts
- A three phase, balanced delta connected resistive load draws 30 ampere phase current from balance three phase supply. An open circuit fauit occurs in one of the lines. Determine the line current. A. 45A C. 90A B. 60A D. 100 A
- • Three 30 ohms resistance are connected in delta across 220 V, 3-phase supply. What is its equivalent wye connected resistance? A. 90 Q C. 45 Q B. 30 Q D. 10 Q
- Three identical capacitance, each of 150 ^F are connected in delta. The value of capacitance in each phase of the equivalent star connected load would b e ....... A. 50nF C. 150fiF B. 450fiF D. 100nF
- A balance 3-phase load takes 20 amperes at 230 volts line to line. The phase A line current lags the line voltage Vba by 50 degrees. What is the power factor of the load in percent? A. 86.6 C. 94 B. 64.3 D. 76.
- Three identical resistors are connected in delta across balance, 3-phase source. If one of the resistor is open, A. Powerwill becomedouble C. Power will become three times B. Power is reduce by 33% D Power will increase by 67%
- The secondary of a transformer bank is wye-grounded, it has a balance voltage and the balanced load. The voltage to ground is 132V and its line current is 40A. What is kVA of the load? A. 9.1 C. 27. B. 19.2 D. 15.
- A 170 kV, 3-phase electric source delivers 200 MVA to a balanced load which has a power factor of 90 % lagging. What is the line current? A. 257 A C. 402 A B. 502 A ^ D. 679 A
14. The phase b line voltage and the phase a line current of a balanced 3-phase system are : V =
220 Z 210° Volts and I = 10 Z 30° Amp respectively. What is the power of the system? A. 1905 Watts C. 5716 Watts B. 3300 Watts D. 3810 Watts
15. The phase b voltage and the phase a current of a balanced three-phase system are : V = 220 sin
(a)t + 210°) Volts and I = 10 sin (co t + 60°) Amp respectively. What is the power of the system? A. p C. 1905 B-/3300 D. 6600
- The phase A line voltage and the phase C line current of a balanced three-phase system are : V = 220 sin (a t - 30°) Volts and I = 10 sin (co t + 90°) Amp respectively. What is the impedance of the system? A. 19.1 —J11 C. 19.1 +j B. 33 - j19.1^r D. .33+j19.
- A balanced delta connected load of 4 - j3 ohms/phase is fed by a 380 V, 3-phase system. Find the reactive power. A. 52 kVARS C. 69 kVARS B 29 kVARS D. 87 kVARS /
- A balanced three-phase wye-connected load of 150 kW takes a leading current of 100 A, when the line is 2400 V, 60 Hz. What is the capacitance per phase? A. 21 mF C. 205 mF B. 21 pF D. 205 pF
- A balanced delta connected load consisting of pure resistance of 18 ohms per phase in parallel with a purely resistive balanced wye connected load of 12 ohm per phase. The combination is connected to a three phase balanced supply Of 346.42 volts line to line via a three phase line of inductive reactance j3 ohm per phase. Determine the line to line terminal voltage of the combined load. A. 332.6 V ' C. 277.1 V B. 320.8 V D. 294.3 V
- A symmetrica! three phase 240 volts supplies a balanced delta connected load of 8-j6 ohms per phase. Using ABC sequence with Vab as the reference voltage, What is the line A current? A.-25+J32A C. 16.3-p8.2A B. 24.9+j33/2A D.' 41.27+j4.97A
- A three phase line has an impedance of 2+j4 ohms. The line feeds two balanced three phase loads that are connected in parallel. The first load is wye-connected and has an impedance of 30+j40 and the second is delta connected load and has an impedance of 60-j45 ohms. The line is energized at the sending end from a balanced supply of line voltage 207.85 volts. Taking phase Va as reference determine the line current la. A. 5 - j 8.66 A D. 5+j 8. B. 5 A D. 0 - j
- A delta connected load of 9+j12 ohms per phase is supplied by a 3-phase balance source of 230 volt rms through a line with an impedance of 0.06+j 0.12 Ohm. Find phase voltage at the load. A. 288 volts C. 132 volts B. 224 volts D. 228 volts
- A delta connected load has a per phase impedance of 10 +j 10 ohms. It is supplied by a wye connected balance source through a 3-phase line with 1.0 ohm resistance and 1.0 ohms inductive reactance. The voltage sequence is given by Va=120/-30° Vb=120/-150° V c -120/90°. What is the line A current? A. 13.86+j13.86 A C. 5.07-j18.93 A B. 13.86-j13.86 A D. -18.93+j5.07 A
- A 120 volts per phase, 3-phase wye connected source delivers power to the following delta connected load. Phase 1:40+j0 ohms, Phase 2: 20 cis (-60°)ohms, phase 3: 15 cis45° ohms. Find the current at line 1. (Use1-3-2 sequence) A. 18.5+j 3.57 C.-15.2+jO B .-2.9-j3.57 D. -2.6+j 0.
- A balanced 3-phase, 3 wire, 240 V, 60 Hz, ABC source supplies power to a star connected resistance 10 Q, 15 £2 and 20 Q. What is the current through 20 ohms resistance? A. 11.23 Amp C. 15.93 Amp B. 9.77 Amp /'D ^ 8.05 Amp
- A three phase ABC source with Van= 240cis 30degrees volts supplies unbalanced wye- connected loads: 15 ohms resistance connected across lines AN , a 12 ohms resistance connected across lines BN and 20 ohms resistance across CN. Determine the current at line B. A. 17.32-j10 A C. 0-j20A B.-10.39+j6A D. 13.85+j8 A
- Three unequal single-phase induction motor are connected between lines and the neutral of balanced 3-phase , 202-V line to neutral. The first load takes 20 kW at 0.82 power factor, the second takes 28 kW at 0.75 power factor and the third takes 36 kW at 0.8 power factor. What is the current in the neutral conductor? A. 105.5 A C. 85.6 A B. 125.4 A D. 133.4 A
- A balanced 3-phase, 3 wire system supplies two balance wye-connected loads. The first draws 6 •kW at 0.8 pf lagging while the other requires 12 kW at 0.833 pf leading. If the current in each line is 8 A. Find the supply voltage in kV. A. 1.544 C. 2. B. 1.682 D. 1.
- A balanced 3-phase source supplies three loads. Load 1 is 15 kVAat 75% pf lagging, Load 2 is 10 kW at 90% pf lagging andLoad 3 is unknown.If the generator delivers 35.52kVA at 80% p.f. lagging, what is the unknown load? A. 8.37kVA at 76.4% pf lagging C. 8.37kVA at 76.4% pf lagging B. 9.3kVA at 89.4% pf lagging p>9.7kVA at 73.8% pf lagging
- Three star-connected impedances Zi = 20 + /37.7 ohms per phase are connected in parallel with three delta-connected impedances Z2 = 30 - y'159.3 ohms per phase. The line voltage is 398 V. Find the line current. A. 3.36 A C. 5.8 A B. 7.26 A D. 4.24 A
- A three phase induction motor takes 3000 watts at 80% power factor. Two wattmeter Wa & Wc are used to measure the power. What is the reading of Wc? A. 800 watts C. 950 watts B. 850 watts D. 900 watts
- The power input to 440 Volt 3-phase motor is measured by two wattmeter method, the readings are 460 W and 780 W. What is the input current? A. 1.78 A C 1.63 A B. 0.94 A D. 2.82 A
- A balance three phase resistive load is supplied by 230 volts 60 Hz and draws a current of 90A. If two wattmeter is used, what will be the reading of each wattmeter. A. 20. 7 kW C. 25.7 kW B. 35.9 kW D. 17.93 kW
- The ratio of the wattmeter readings in the 3-phase inductive system is 5:3 when power is measured by these two wattmeter. What is the power factor of the load? AJ 0.92 C. 0. B. 0.50 D. 0.
- A three phase balanced load id connected across 220 volts three phase source. A wattmeter reads 600 watts when its current coil is connected in line a and potential coil across line a and b. Next, the voltage coii is connected between lines b and c, but the current coil remains at line a. The wattmeter agam reads 600 watts. Calculate the load power factor. A. 0.8 C. 0. B. 0.866 D. 0.
- A 4,160 V, 3-phase, 3 wire- ABC source supplies power to an industrial load that draws the following line currents : la = 110 cis (-36.87°) Amp, lb = 166.5 cis (-168.2°) Amp. Using two wattmeter method, what is the total power drawn by the load? A. 882.5 kW C. 1,281 kW 1|) 828.5 kW D. 97.5 kW
47 One wattmeter method of measuring power of a balanced system, The indication of wattmeter by T-method is _____. A. Pt/3 LC> Pt/ B. Pt/CTR D. Vlcos(30+#)
- An unbalanced wye connected loads on a 3 phase four wire system consist of Z1 = 10 + j5, Z2 = 8 + j4 and Z3 = 20 + jO. The wye connected generator supplying the unbalanced load is balancec voltage to neutral of 120 volts. What is the wattmeter^ reading at Z2 , watts? ' A. 720 o < ^ > 4 0 B. 3312 D. 1152
- The 3-phase, 4-wire source voltage of Van =230 sin(cot+30deg.) volts supply loads .with the following line currents: la = 4.7 sin(cot +30 deg.),lb = 6.8 sin(ot-60deg) and Ic = 3. sin(o)t+120deg.) Determine the total power. A.3113 watts C. 2786 watts B. 2768 watts D. 2733 watts
- A three phase system delivers 50 MVA of a balanced delta load of power factor 78%. Determin the size of capacitor bank necessary to raise the power factor to 90%. A. 17.3 MVAR Q, 11.4 MVAR B. 13.5 MVAR D> 12.4Vy MVAR
- A 132 kV line, three phase system delivers 70.7 MVA of a balanced delta load of power factc 70.7%. Determinethe reactance necessary to attain unity power factor. A. 1092 ohms Cs 1142 ohms B. 965 ohms D. ')1045 ohms
