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Unformatted text preview: +1; Substituting these values in the equations determined in Prob. 1.25, compute
the voltages at the nodes of Fig. 1.23. Numerically determine the corresponding Zbus matrix. Solution:
Using the Ybus solution of Problem 1.25, substitute the given admittance values:
—314.5 38.0 34.0 32.5 v1 0
38.0 —jl7.0 340 35.0 V2 _ 0
34.0 34.0 38.8 30 V3 1.0 3—90°
32.5 35.0 30 —3‘8.3 V4 0.68/~135"
Compute voltages:
YbusV = I
Ybuslybusv = Y'bus”:lI
30.7187 30.6688 30.6307 30.6194
1 __ 30.6688 30.7045 30.6242 30.6258
Where .Ybus ‘ zbus ‘ 30.6307 30.7045 30.6840 30.5660
30.6194 30.6258 30.5660 30.6840
v = Ybus'lI
V1 30.7187 30.6688 30.6307 30.6194 0
V2 _ 30.6688 30.7045 30.6242 30.6258 0
V3 — 30.6307 30.7045 30.6840 30.5660 101—90"
V4 30.6194 30.6258 30.5660 30.6840 0.68 —135°
V1 0.9285 — 30.2978 0.9750 (—17.780
V2 _ 0.9251 —jO.3009 _ 0.9728 g —18.02°
V3 — 0.9562 — 30.2721  0.9941 5—15.89"
V4 0.8949 — 30.3289 0.9534 (—20.18c’ Chapter 2 Problem Solutions 2.1 A singlephase transformer rated 7.2 kVA, 1.2kV/120 V has a primary wind
ing of 800 turns. Determine (a) the turns ratio and the number of turns in the secondary winding, (b) the currents carried by the two windings when the
(2.7). transformer delivers its rated kVA at rated voltages. Hence, verify Eq. Solution:
(0)
N1 __ V1 __
N3 _ V2 —
N1 II T herefore, N2 11 A V MAMHW.—~mmwmw 12 Srnwd = lvllratedlIllrated = ll/eratedllerated
7.2x103 = 1.2x103111,a,ed = 120121md
7.2 x 103
Ull'md = 1.2x103 = 6 A
7.2 x 103
IIQIrated = 120 = A
11 6
 ' . .7: — = — = 0.]
Left hand Slde of Eq (2 ) 12 60
N2 1 Righthand side of Eq. (2.7):
Lefthand side of Eq. (2.7) N—1 —— 10 = 0.1
Righthand side of Eq. (2.7) 2.2 The transformer of Prob. 2.1 is delivering 6 kVA at its rated Voltages and
0.8 power factor lagging. (a) Determine the impedance 22 connected across its
secondary terminals. (b) What is the value of this impedance referred to the
primary side (i.e. Z5)? (c) Using the value of Z; obtained in part ((2), determine
the magnitude of the primary current and the kVA supplied by the source. Solution: (0) $2 = 152mg = 6x103436.9° VA
52 ‘
12 r (72)
22 = n = V2 = MK
12 Sgt/V; 35
_ (120)2 Q
’ 6x1034—36.9° = 2.44369" :2 = (1.92+j1.44) Q , N1 2 V1 2
22 = — Z1 = —. Zl = 100x2.4g36.9° 0
N2 V2
= 2404369" Q = .192+j144 n
(e)
W 1.2 x 103 I = — = ———— = '
I ll  él 240 A o A
151) = mm = 1.2x 103x5 VA = 6 kVA 13 2.3 With reference to Fig. 2.2, consider that the ﬂux density inside the center—leg of
the transformer core, as a function of time t, is B (t) = Bm sin (27rft) where Bm
is the peak value of the sinusoidal flux density and f is the operating frequency
in Hz. If the flux density is uniformly distributed over the crosssectional area
A m2 of the centerleg, determine (a) the instantaneous ﬂux <25 (t) in terms of Bm, f , A and t, ( b) the instantaneous induced—voltage 61 (t), according to Eq. (2.1). (c) Hence show that the rms magnitude of the induced voltage of the primary
_ is given by lEll = ﬁﬁleBmA. (d) If A = 100 ch, f = 60 Hz, Bm = 1.5 T and N1 = 1000 turns, compute [E1[.
Solution:
(0)
¢o(t) = B(t)A = BmAsin(27rft)
(5)
e1 (t) = NI d2?) = NleAg {sin(27rft)} = 27rleBmAcos(27rft)
(C)
E1 = ﬁgklﬁﬂmaz = Eff—JE—T—M = ﬁﬂvaleA (c) With given values, E1 ﬁrrx60x1000x15x100x104 v = 4.0 kV 2.4 For the pair of mutually coupled coils shown in Fig. 2.4, consider that L11 =
1.9 1'1,ng 2 L21 =1 H, L22 2 H and T1 = T2 = 0 The system is
operated at 60 Hz. ' ((2) Write the impedance form [Eq. (2.24)] of the system equations (1)) Write the admittance form [Eq. (2.26)] of the system equations (c) Determine the primary voltage V1 and the primary current 11 when the
secondary is (2') open circuited and has the induced voltage V2 2 100 ﬂ V
(ii) short circuited and carries the current 12 = 21 90° A 14 Solution:
(a) From Eq. (2.22) and (2.23),
V __ ‘ L11 L12 [1 _  1.9 0.
ll’él * le L21 L22ll12l _ 31207Tl09 0. 3102 [ $83 i233? l l l; l ONO
—_J
rﬁ
3‘5
I—g (6) From Eq. (2.25), [5;] = alewe Harm] = aleMas; Ham] (CW)
3 [100%] = mossy] I hence 11 = 0.295i0o A
1 V1 = 211.114): V
l (ii)
[2&] = 10‘2x1ﬂliﬁéélvl
hence V1 = 117.30L0° V 11 = 1.111—90" A 2.5 For the pair of mutually coupled coils shown in Fig. 2.4, develop an equivalent—T
network in the form of Fig. 2.5. Use the parameter values given in Prob. 2.4
and assume that the turns ratio a equals 2. What are the values of the leakage
reactances of the windings and the magnetizing susceptance of the coupled coils? Solution:
L1, = LH—aL21 =1.9—2x0.9 H = 0.1 H
L2; = L22—L12/a = 0.5—2x0.9/2 H = 0.05 H
a2L21 = 4x005 H = 0.2 H
Ln = aL21= 2x0.9 H = 1.8 H
w = 1207 rad/sec j 31m j 75.4 n Leakage reactances: x1 = 37.? Q 15 $2 = 75.4 9
TA
$2 = .3—9=18.850
4
1
Magnetizing susceptance: Brrl = 'wL'm = 12O7rlx 1.8 s
= 1.474x10"3 s 2.6 A single—phase transformer rated 1.2 kV / 120 V, 7.2 kVA has the following wind
ing parameters: r1 = 0.8 0, x1 = 1.2 9, 7‘2 : 0.01 Qand x2 = 0.01 9. Deter
mine (a) the combined winding resistance and leakage reactance referred to the pri
mary side, as shown in Fig. 2.8,
(b) the values of the combined parameters referred to the secondary side (c) the voltage regulation of the transformer when it is delivering 7.5 kVA to
a load at 120 V and 0.8 power factor lagging. Solution:
(a) With turns ratio a = 1.2 x 103/120 = 10, R1 = n+8” = 0.8+100x001 n = 1.8 :2
X1 = x1+a2z2 = 1.2+100x0.01 n = 2.2 n
(b)
32 2 Rl/a2 = 1.8/100 n = 0.018 :2
X2 2 Xl/a2 = 22/100 0 = 0.022 n
21
11 R. 5x1 ‘3;
h—p—WYWW
Iv! 13V:
(c) o———_______¢
21 = (1.8+j2.2) n 7200 o 4
IQ,FL = Isg/VQ; 5—9 = ﬁg—369 A = 60536.? A z
I \
Irn = 2'0” = 6.05—369" A J
OVQ’FL = V
V1,FL = 0V2,FL+Ii,i*LZ1 16 ll 1200+6.0/—36.9‘(1.8+j2.2) V = 1216.57/O.19° v
120v l/le/a = 121.66 v (l21.66—120)/120 = 1.38% lV2.FLl IVQ, NLl
% Regulation 2.7 A single—phase transformer is rated 440/220 V. 5.0 kVA. When the low—voltage
side is short circuited and 35 V is applied to the highvoltage side, rated current
flows in the windings and the power input is 100 W. Find the resistance and
reactance of the high and lowvoltage windings if the power loss and ratio of
reactance to resistance is the same in both windings. :1 Solution: i '000 Rated] = ——0220 = 22.73 A (low voltage) ~ = % = 11.36 A (high voltage)
100 = = 0.774 9
R 11.362
2 = Egg = 3.08 n (R, z, X highvoltage) X 2.98 = . 2 . 2 = . — = _ = , X v308 0774 298 Q R 0774 385 For equal loss in high and lowvoltage windings, ."7
High voltage: 7 = 213—4 = 0.387 52
:c = 3.85 x0.387 = 1.49 9
Low volta e' r — 0387x 220 2 — 0097 Q
g ‘ _ ' 440 7 '
220 2
a: 14 (440) 0373 $2 2.8 A singlephase transformer rated 1.2 kV/ 120 V, 7.2 kVA yields the follOwing
test results: OpenCircuit Test (Primary Open)
Voltage V2 = 120 V; Current 12 z 1.2 A; Power W2 2 40 W ShortCircuit Test (Secondary Shorted) Voltage V1 = 20 V; Current I] = 6.0 A; Power W1 2 36 W 17 Determine (a) the parameters R1 = r; + azrgT X1 = 3:; + a222, GC and B7" referred to the
primary side, Fig. 2.7 (b) the values of the above parameters referred to the secondary side (c) the efﬁciency of the transformer when it delivers 6 kVA at 120 V and
0.9 power factor. Solution: (a) From open—circuit test, G; = Wg/Vg = 40/12023 = 2.78x10’3 5
11;] = IQ/VQ = 12/1205 = 0.01 s
B; = Inf—cg? = 9.606x10—3 s a = 1.2x103/120 = 10 Therefore, cc = (ye/cl2 = 2.78x105 3
Bm Ema? = 9.606x10'5 s From the shortcircuit test, R = Wl/If = 36/002 :2 = 1.0 9 Z = V1/11 = 20/609 = 3.33 o
X = \/Z2—R2 = 3.18 n
(b)
R’ = R/a2 = 0.01 9 X’ = X/a2 = 0.0318 (2
G; = 2.78x10'3 8 3;" = 9.606x103 s (c) When SQ = 6.0 kVA and V2 = 120 V, 6 x 103 ,
I2 — 120 ‘A — 00 A
Core loss ath = 120V = 40W Winding loass at 12 50 A = 1212R' = 502 x0.01W = 25 W 1+"
Power output at 52 = 6.0 kVA at 0.9 pf. = 6 x 103 x 0.9 W = 5400 W 5
5400 ’7 ' 5400+40+25 ” 98'81 % 2.9 A singlephase transformer rated 1.2 kV/ 120 V, 7.2 kVA has primary—referred
parameters R1 = T1+Q2T2 = 1.0 Q and X1 = x1+a2$2 = 4.0 $2. At rated voltage
its core loss may be assumed to be 40 W for all values of the load current. 18 Solution: (a) (2') c059 # 0.8, lagging 6 = —36.9° V2 = 120i° V
I — 7200 36 9°  60 36 9° A
2 ’ mg~  g’
Total loses = 40 + 602 x W = 76 W
Output power = 7.2 x 103 x 0.8 W = 5760 W
5760
77 _— m  98.698%
I .
Vi,FL = aV2,FL+ 2:1 (R1+JX1)
VLFL = 120 x 10L0°+m4—36.9° (1.0+j4.0) V = 1219.3¢O.73°
lV2,FLl = 120 V lV2,FLl = lV1,FLl/a = 121.93 V
%Reg‘ulation = 121.323120 = 1.61 % (ii) 0050 = 0.8, leading 0 = 36.9°
1; = 9869896 because it does not depend on whether 9 is leading or lagging. I .
V1,FL — 0V2,FL+ 25L (31+JX1)
Vin = 120x10ﬂ+f—3436.9° (1.0+j4.0) V = 1190.611.1°
% Regulation = = —0.78%
(b) Load current at which 77 is maximum is given by
, R
lI2l2 a7 = Pcore
Therefore,
. 40
[12] = 10/100 A = 63.245 A 40W Winding loss at [12‘] = 1;}? 19
Output = 120 x 63.245 x 0.8 W = 6071.57 W
6071.57
"m r — 9870093
Corresponding kVA level = 120 x 63.245 VA = 7.589 kVA 2.10 A singlephase system similar to that shown in Fig. 2.10 has two transformers
A—B and BC' connected by a line B feeding a load at the receiving end C. The
ratings and parameter values of the components are Transformer AB: 500 V/ 1.5 kV, 9.6 kVA, leakage reactance = 5%
Transformer 3—0: 1.2 kV/ 120 V, 7.2 kVA, leakage reactance = 4°70
Line B: series impedance = (0.5 + 33.0) 9 Load C: 120 V, 6 kVA at 0.8 power factor lagging ((1) Determine the value of the load impedance in ohms and the actual ohmic
impedances of the two transformers referred to both their primary and
secondary sides. (b) Choosing 1.2 kV as the voltage base for circuit B and 10 kVA as the
systemwide kVA base, express all system impedances in per unit. (c) What value of sendingend voltage corresponds to the given loading condi
tions? Solution: (a) Ohmic impedances Transformer AB Primary: $1165 x j0.05 = 3' 1.302 9
1.52 x 106 , .
Secondary: W x_70.05 = 311.719 9
2 6
Transformer BC Primary: x j0.04 = 3'80 9
2 _
Secondary: $21—iomg x 30.04 = 30.08 0
V 2 02
Load: li?ll/_9 = 61:1034cos”10.8 = 2.4436.9° Q a
~46
Q)
(b) Impedance bases . _ 1.22 x 106 “J”
Circu1t B: W (2 = 144 Q
2
Circuit C: ﬂ— 9 = 1.44 0 10 x 103 20 2.11 V 2.12 Per unit impedances on new bases: . 11.719
Transformer AB. j 144 . 8
J 144
(1.5+j3.0) 144 2.4
: — 369°
Load 1.44 [—— = 30.08138 per unit Transformer B—C: 30.0556 per unit Line B: 0.0104 + 30.0208 per unit 1.66M 369° per unit (c) Sendingend voltage calculations 5.5 0.0104 + j 0.0208 R.E. j 0.08138 1.667 4369" p.u.
(0.0104 +j 0.15778) p.u. VR = 120 V = 1.0 per unit
. 7 6. ° 0.0104 0.15778
VS = x W = per unit
1.667 g 36.9°
The sendingend voltage base is
500 3
vsybne = mxl.2x10 = V
Therefore, the required sendingend voltage is
V5 = 400x 1.0642 = 425.69 V A balanced Aconnected resistive load of 8000 kW is connected to the low
voltage, Aconnected side of a YA transformer rated 10,000 kVA, 138/ 13.8 kV.
Find the load resistance in ohms in each phase as measured from line to neutral
on the high—voltage side of the transformer. Neglect transformer impedance and
assume rated voltage is applied to the transformer primary. Solution:
8 000
I“, = ’ = 33.47 A
I 1 9' «3 x 138
138,000N5
R _ 3347 _ 2380 n Solve Prob. 2.11 if the same resistances are reconnected in Y. Solution: If the Aconnected resistors are reconnected in
times as great and Y, then the resistance to neutral will be three R = 3x2380 = 7140 Q 21 2.13 Three transformers, each rated 5 kVA, 220 V on the secondary side, are conected
AA and have been supplying a balanced 15 kW purely resistive load at 220 V.
A change is made which reduces the load to 10 kW, still purely resistive and
balanced. Someone suggests that, with twothirds of the load, one transformer
can be removed and the'system can be operated openA. Balanced three—phase
voltages will still be supplied to the load since two of the line voltages (and thus also the third) will be unchanged. ﬁt
5
Z
2
3.
‘1“
a’.
'i To investigate further the suggestion ((1) Find each of the line currents (magnitude and angle) with the 10 kW load
and the transformer between a and c removed. (Assume Val, = 220 ﬂ V, sequence a b c.)
(27) Find the kilovoltamperes supplied by each of the remaining transformers. (c) What restriction must be placed on the load for openA operation with
these transformers? ((1) Think about why the individual transformer kilovoltampere values include
a Q component when the load is purely resistive. Solution: —>
1b (a) Vab and V4,c remain the same after removing the third transformer, so Vm is also the
same and we have a threephase supply, and these voltages are: Vab = 220 L_° V, Vbc =
22012400 V and V“ = 2201 120° V. Then, V,m = 1275—30° V, V5,, = 127/ 210° V and
Va, = 127430: V. The line currents are 10 000
I = ———’—— —30° = 26.24 —30° A
“ \/§ x 2201—— L—~
1,, = 26241210" A
Ic = 26.24490° A (b) kVAsupplied = 220 x 26.24 x 103 = 5.772 kVA
(c) The load must be reduced to (5.0/ 5.772) x 100 = 86.6% or 4.33 kW for each transformer. (d) The current and voltage in each of the remaining two transformers are not in phase.
Output of each transformer before the reduction in load is, 51 = VabI; = 220ﬂix2624430° = 5000 + 32886 VA
52 = VcbI; = 220mm x26.244270° = 5000—32886 VA 22 Note that Q is equal in magnitude but opposite in sign. There is no Q output from the
open delta. After the load reduction, 51 = 4333+j2500 VA
52 = 4333—32500 VA 2.14 A transformer rated 200 MVA, 345Y/20.5A kV connects a balanced load rated
180 MVA, 22.5 kV, 0.8 power factor lag to a transmission line. Determine (a) the rating of each of three singlephase transformers which when properly
connected will be equivalent to the above threephase transformer and (b) the complex impedance of the load in per unit in the impedance diagram
if the base in the transmission line is 100 MVA, 345 kV. Solution: (a) Each single—phase transformer is rated 200/3 = 66.7 MVA. Voltage rating is (345/ \/§) / 20.5
or 199.2/20.5 kV. (b) 2
LoadZ = (2128:) (cos—10.8 = 2.81436.87° Q (lowvoltage side)
At the load,
BaseV = 20.5 kV
_ (20.5)? _
BaseZ — 100 — 4.20 O .8
LoadZ = 373436.87o = 0.669436.87° per unit 2.15 A threephase transformer rated 5 MVA, 115/132 kV has perphase series
impedance of (0.007 + 3' 0.075) per unit. The transformer is connected to a short
distribution line which can be represented by a series impedance per phase of
(0.02 + 30.10) per unit on a base of 10 MVA, 13.2 kV. The line supplies a bal
anced three—phase load rated 4 MVA, 13.2 kV, with lagging power factor 0.85. ((1) Draw an equivalent circuit of the system indicating all impedances in per
unit. Choose 10 MVA, 13.2 kVA as the base at the load. (b) With the voltage at the primary side of the transformer held constant at 115 kV, the load at the receiving end of the line is disconnected. Find the
voltage regulation at the load. 23 Solution: (a) Base voltages are shown on the single—line diagram. 115 kV 13.2 RV
Load Transformer Z = 2 (0.007+j0.075) = 0.014 +j0.150 per unit
0
V5 = 1.0 per unit
Line Z = 0.02+j0.10 per unit
(13.2)2 x 1000 _
= ———— = 43. 6 Q
L°ad '2' 3400/0235 °
.2 2
BaseZat load = (13 ) = 17.42 9
10
43.56 _1 , o
LoadZ = 1742[cos 0.85 = 250(318 ll 2.125 +j1.317 per unit 0.014 + p.150 0.02 + 10.10 2.125 11.317 (values are in per unit) (b) Voltage regulation calculations 1.0 _ 1.0 0.014 + 0.02 + 2.125 +j(0.150 + 0.10 + 1.317) 2.6684 35.97°
= 0.375 {—35.97° per unit VRYFL = 0.3755—35.97° x 2.5131.8° = 09371—417" per unit
VR'NL = V5 = 1.0
1 — 0.937 I = 2.16 Three identical singlephase transformers, each rated 1.2 kV/ 120 V. 7.2 kVA
and having a leakage reactance of 0.05 per unit, are connected together to
form a threephase bank. A balanced Yconnected load of 5 9 per phase is
connected across the secondary of the bank. Determine the Yequivalent per—
phase impedance (in ohms and in per unit) seen from the primary side when the transformer bank is connected (0) YY, (b) YA, (c) AY and (d) A—A.
Use Table 2.1. < W12“? (LE 1 24
Solution: (a) Y—Y connection: 1.213'kw12013'v
IVLLI = 1.2x 103x\/§ v jx‘ 1mm Wu] = 120\/§V '——"“"——§
1, 2
3 mom/5 r r
R, = — x __ = Q , ,
g L O (um/3) ZL Rt R‘
' 2
. 06
i Zb = M = 200 g _
7.2 x 103 x 3
X, = 0.05 per unit = 200x0.05§2 = 10 Q
E Z'L = (500+j10) Q
5 (b) Y—A connection: 1.213'kV/120v ix, 7.2x3kVA
i VLL = 1200xx/5 v ___nrvwx__§
I qul = 120 V ' I“ , I"
E ' 2
1200¢§ _ 2‘. a,
E ’ = = 00 9
RL 5 X ( ) 10
X1 = 10 erom part (a) _
Z’L = (1500+j10) o
(c) A—Y connection: 12 kV/izorz'v
jx1 7.2x3kVA
[VLLI = 1200 V ___nmm___§
114,] = 120%)? v —~ r
1200 2 500
R’ = 5x = — = 166.67 n 25 ac R
L (120%) 3
12002
= ————— = , 7. _
Z" 7.2x3x103 666 Q
X; = 0.05 per unit 2 66.67x0.05§2 = 3.33 0
Z}, = (166.67+j3.33) Q
(d) A~A connection: jx, 1va:41:2va [VLLI = 12oov ‘—‘ “H—g qul = 120 V r r
’ 1200 2 ' 21” RL. 12L
RL — 5 X —— 000 X; = 3.33 9 from part (c) _
Zi = (500 +3333) {2 2.17 Figure 2.17a shows a threephase generator supplying a load through a three
phase transformer rated 12 kVA/600 V Y, 600 kVA. The transformer has per
phase leakage reactance of 10%. The linetoline voltage and the line current at 25 the generator terminals are 11.9 kV and 20 A, respectively. The power factor
seen by the generator is 0.8 lagging and the phase sequence of supply is ABC. ((1) Determine the line current and the line—to—line voltage at the load, and the
per—phase (equivalentY) impedance of the load. (1)) Using the lineto—neutral voltage VA at the transformer primary as refer
ence, draw complete perphase phasor diagrams of all voltages and currents.
Show the correct phase relations between primary and secondary quanti
ties. (6) Compute the real and reactive power supplied by the generator and con
sumed by the load. Solution: (0) 12 103
Voltage ratio = a = —6):)T—130° = 20430"
Current ratio = (—11: = 0.054 30°
2
(12 x 103)
X‘ 600x103 x01 2
LetV  11'9/0° kV  687 kV
s «5* — ~
Then, 15 = I’L = 201—36.9° A IL _ I’La‘ = 20x20/—36.9°—30° A = 4005—66.? A VL' = VS—leIs = ssng— (31%) kV l
= 6.5935—3.34° kV VL = me; = M kV = 329655—3334“ v 20 4 30° Line voltage at the load @1le = 571 v Line current at the load = [Id = 400 A
 32. _ , o
L°ad1mpedance ZL = VL/IL = .365— L—z’ﬂ 9
4005—6690 r V 54:41. = 0.8241336" Q 26 (b) ref
.34= v5 = 6.87 kV
30n
VL’ = 6.593 kV VL: 329.65 V (c) Pg + ng from the generator is 3VsIg, where 31/51; = 3x 6.87./_Q‘:x 20436.9c kVA = 412.24 369° kVA
= 329.8 kW+ 3247.3 kvar PL +jQL by the load is 3VLIZ, where 3 x 329.65 (—33.340 x 4004 66.9" 1000
329.7 kW + j218.7 kvar ll 3VLIZ kVA = 395643356" kVA 2.18 Solve Prob. 2.17 with phase sequence ACB. Solution: (a) Final answers remain the same except for the following intermediate results: D a = 201—30 l/a‘ = 0.051—30°
I], = Iia' = 4005—36.9°+30° A = 400469" A VL = Vé/a = 329655—334o + 30° V = 32965126.?" V
(b) vL= 329.65 V
30° ref “' ref
.340 V = 6.87 kV
W = :593 kV 30" XL A
IL' =15 = 20A (c) Same results as in Problem 2.17. 2.19 A single—phase transformer rated 30 kVA, 1200/ 120 V is connected as an auto
transformer to supply 1320 V from a 1200 V bus. (a) Draw a diagram of the transformer connections showing the polarity marks
on the windings and directions chosen as positive for current in each wind
ing so that the currents will be in phase. "avWM uAWr 1" ,1 K. . “Washe ‘»’:"r 27 (6) Mark on the diagram the values of rated current in the windings and at
the input and output. ( 0) Determine the rated kilovoltamperes of the unit as an autotransformer. (d) If the efﬁciency of the transformer connected for 1200/ 120 V operation
at rated load unity power factor is 97%, determine its efﬁciency as an
autotransformer with rated current in the windings and operating at rated
voltage to supply a load at unity power factor. Solution: 30: 000 rated IHV = 1200 = 25 A
30, 000 ,
rated ILV — 120 — 250 A Connected for 1200/ 120V operation (regular transformer), Pam = 30,000 W En =' 30, 928 W
Loss = 928 W ' Loss remains the same in the autotransformer because current in the windings and voltage
across the windings are unchanged. For the autotransformer, Pm,t = 250x 1320 = 330,000 W H = 330,928 W
330 000
= a = . V =
7; 330,928 x100 99 7% Ratedlr A 330,000» Note that, once we consider loss, we no longer have an ideal transformer; and both winding
resistance and reactance as well as magnetizing current and core loss must be considered.
The applied voltage and input current will be greater than the values shown to achieve rated
output, in which case the equivalent circuit corresponding to Fig. 2.7 would be used. 28 2.20 2.21 Solve Prob. 2.19 if the transformer is to supply 1080 V from a 1200 V bus. Solution: 250 A As in Prob. 2.19, Loss = 928 W. As an autotransformer, Pm.t = 250x 1080 = 270,000 W
1% =: 270,928 \v
270.000 x 100 = 99.7% '7 270,928 Rated kVA = 270, 000, but see the note which accompanies the solution of Problem 2.19. Two buses @ and @ are connected to each other through impedances X1 = 0.1
and X2 = 0.2 per unit in parallel. Bus 0 is a load bus supplying a current
I 2 105—30" per unit. The perunit bus voltage Vb isl.0ﬂ. Find P and Q
into bus 0 through each of the parallel branches (0) in the circuit described, (b)
if a regulating transformer is connected at bus b in the line of higher reactance
to give a boost of 3% in voltage magnitude toward the load (a = 1.03), and
(c) if the regulating transformer advances the phase 2° (a = ejﬂgo). Use the
circulatingcurrent method for parts (6) and (c), and assume that V; is adjusted
for each part of the problem so that Vb remains constant. Figure 2.26 is the
singleline diagram showing buses a and b of the system with the regulating
transformer in place. Neglect the impedance of the transformer. Solution: ® X1 =j0.l @ 11.04—30° (a) Thru X1 the current is 11 = g— x 1.0 5—30" = 0.577 —j0.333 and thru X2 the current is
12 = % x 1.05—30" = 0.289 j0.167. Into bus @ thru X1, P+jQ = Vb]; = 0.577+j0.333 per unit 29 and into bus @ thru X2,
p+jQ = Vb]; = 0.289+j0.167 per unit 0.03
=.  . =———=—'0.1
(b) AV 003,1m JM 2
I1 = 0.577—j0.333—(—j0.1) = 0577—20233
12 = 0.289—j0.167+(—j0.1) = 0289—30267 Into bus @ thru X1, P +jQ = 0.577 +j0.233 per unit
and into bus @ thru X2,
P +jQ = 0.289 +j0.267 per unit
(6)
AV = 1.042 — 1.0 = 0.9994 +j0.0349 — 1.0 = —0.0006 +j0.0349
Im = 33w = 0.116 + 30.002
30.3
I1 — 0.577 j0.333 — (0.116 +j0.002) = 0.461 —j0.335
I2 = 0.289 —j0.167 +0.116+j0.002 = 0.405 —j0.165 Into bus @ thru X1,
P + .70
and into bus ([2) thru X2,
P+jQ = VbI‘ = 0.405+j0.165 per unit
Note: Compare P and Q found in parts (b) and (c) with part (a). 14,? 0.461 +jO.335 per unit 2.22 Two reactances X1 = 0.08 and X2 = 0.12 per unit are in parallel between two
buses @ and @ in a power system. If V, = 1.05 410: and Vb = 1.0Llﬁ per unit,
what should be the turns ratio of the regulating transformer to be inserted in
series with X2 at bus @ so that no vars ﬂow into bus @ from the branch whose
reactance is X1? Use the circulatingcurrent method, and neglect the reactance
of the regulating transformer. P and Q of the load and Vb remain constant. Solution: In reactance X], 1.05( 10° — 1.0 1.034 +j0.1823 1.0 I = ————————— = ——————— = , 7 — ' ,
ab 1.008 1.008 2 2 9 30 425
To eliminate vars to bus (9 thru X1, we need in the X2 branch
Iab,circ = AV .
j0.8+j0.12 ' "30425 a—l = AV = —j0.425(j0.08+ 20.12) = 0.0850 a 1.085 turns ratio 'VAW are = (ZO'I/QS) 0: p901 aanpau 91 oz
I 'I = —— 'I = ——
9 0 6I‘SI 030 1703
VAW 6I'QI = QEXO‘IXVEVO = I‘sl
VAN v'oz = 98x0'tx989'0 = I‘SI
17970 = Isst'oF—EGS‘OI = °“~‘°I'‘I
8890 = luv'ofLov'ol = 1°"°I+‘Il
(gmo + 99910)! '
‘ [— = ———————— = 3'45
mo. 920.0 I
ﬁsooq apnnuﬁvw %9's Imm
QBZ'OFSBS'O = (9'of—s‘o)5§ = ‘I
an
98
QOS‘OF—L V = ; ' __ = I
o o (90 so)“I I fpeol VAW'QS mm memo 'VAW 9'08 = 98 X (Z‘LI/QI) 03 p201 aonpaJ aJogaxaqa ptm ‘pap'eopaAo s; z# nun
x 98910 + QLQI'O v Z'LI = 9 = 3
AW 9 9mm IS'
98910 + 92.91'0
v 811 = 9 x———— = I
M 8 mm '5‘
nmnzad gggm = (91/92)><2.0‘0 = zX
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 Spring '09
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