1 ECE 421/599 FALL 2014 - MIDTERM EXAM Problem 1 (20 points): A load Z L =10+j5is supported by a 100V rms, 60-Hz source through a line with impedance Z T =j5. a. Find the current I, voltage V L , real power, reactive power and power factor of load Z L b. Find the capacitance of the shunt capacitor making overall unity power factor (PF=1) on the load size. What is the new voltage V L on the load side? c. Respectively for the conditions with and without the capacitor, draw two phasor diagrams about V, V L , I, and the voltage of Z T . a. Z=Z T +Z L =10+j10(1) I=V/Z=100/(10+j10)=5-j5A (2) V L =IZ L =75-j25=79.057-18.435 o V (2) S L =P L +jQ L =V L I*=(75-j25)(5+j5)=500+j250 VA P L =500W (1) Q L =250var (1) |S L |=559.0W PF=P L /|S L |=0.894 (1) b. If the load with capacitor has a terminal voltage =V L Q L +Q C =0 Q C =-QL=-250Mvar (1) X C =|V L | 2 /Q C =-25(1) C=-1/(X C )= -1/(377X C )=106.1F (1) Z Lnew =Z L //jX C =12.5 (1) I new =V/(Z T +Z Lnew )=6.90-j2.76=7.43-21.8 o A (1) V Lnew =Z Lnew xI new = 86.20-j34.48 =92.85-21.80 o V (2) c. V ZT =Z T xI=j5(5-j5)=25+j25 V (1) V ZTnew = Z T xI new = j5x(6.90-j2.76)=13.8+j34.5V (1) Phasor diagrams: (2 each)
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ECE 421/599 FALL 2014 - MIDTERM EXAM
Problem 1 (20 points): A load ZL=10+j5 is supported by a 100V rms, 60-Hz source
through a line with impedance ZT=j5.
a. Find the current I, voltage VL, real power, reactive power and power factor of load ZL
b. Find the capacitance of the shunt capacitor making overall unity power factor (PF=1)
on the load size. What is the new voltage VL on the load side?
c. Respectively for the conditions with and without the capacitor, draw two phasor
diagrams about V, VL, I, and the voltage of ZT.
a.
Z=ZT+ZL=10+j10 (1)
I=V/Z=100/(10+j10)=5-j5A (2)
VL=IZL=75-j25=79.057-18.435o V (2)
SL=PL+jQL=VLI*=(75-j25)(5+j5)=500+j250
VA
PL=500W (1)
QL=250var (1)
|SL|=559.0W
PF=PL/|SL|=0.894 (1)
b.
If the load with capacitor has a terminal voltage
=VL
QL+QC=0 QC=-QL=-250Mvar (1)
XC=|VL|2/QC=-25 (1)
C=-1/(XC)= -1/(377XC)=106.1F (1)
ZLnew=ZL//jXC=12.5 (1)
Inew=V/(ZT+ZLnew)=6.90-j2.76=7.43-21.8 o A (1)
VLnew=ZLnewxInew= 86.20-j34.48
=92.85-21.80o V (2)
c.
VZT=ZTxI=j5 (5-j5)=25+j25 V (1)
VZTnew= ZTxInew= j5x(6.90-j2.76)=13.8+j34.5V (1)
Phasor diagrams: (2 each)
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Problem 2 (25 points): Short answers
a. True or false for each statement? Briefly explain why. (5 points)
(1 each; 0.5 for T/F, and 0.5 for reason)
i. The top-3 energy resources of electricity generation for the overall US power grid are coal, nature
gas and hydro. False. Coal, Nature Gas and Nuclear
ii. A “bulk electric system” defined by NERC generally means the portion of system operated at
voltage of 69kV or higher. False. 100kV instead of 69kV
iii. The generation resources utilizing steam turbines are such as combined-cycle power plants,
parabolic dish concentrators, pressurized water reactors, geothermal power plants and tidal power
plants. False. Exclude tidal power.
iv. Hydroelectric power plants usually adopt round rotor generators for high speed operation
False. Salient-pole generators
v. Reactive power sources include capacitors, generators and regulating transformers
False. Transformers are not.
b. What are reliability concerns in integration of wind generation into a power grid? Give two concerns.
(2.5 points for one and 5 points for any two)
1) Congestion issues due to long distance transmission between wind generation and load centers.
2) Mismatch in supply-time and demand-time curves
3) Errors in wind forecast
3
c. Which statements about the per-unit system are correct? (Total 6 points)
i. The circuit laws are still valid in per-unit systems.
Correct (1)
ii. A minimum of four base quantities, i.e. SB, VB, IB and ZB, have to be given for a per-unit system
Incorrect (1). Giving any two is the minimum (1)
iii. A bus with voltage at 0.8pu must mean that its actual voltage is as low as 80% of the nominal value
Incorrect (1). Only when the nominal value is chosen as the base (1)
iv. The per-unit values of impedance, voltage and current of a transformer are the same regardless of
whether they are referred to the primary or the secondary side
Correct. (1)
d. For a 3-phase transmission line with line-to-line voltages Vab , Vbc, and Vca, phase voltages Van , Vbn, and
Vcn, and currents Ia, Ib and Ic, which of the following formulas can calculate its 3-phase complex power?
(Total 4 points)
i. *
3a n a
V I Yes
ii. *
3b n b
V I Yes
iii. *
3 ( 1 3 0 )a b a
V I No
iv. *
3 ( 1 9 0 )a b c
V I Yes
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e. A 3-phase transposed transmission line has 2 bundled identical conductors
per phase placed at a distance of d. Which of the following will necessarily
increase its inductance? Briefly explain why. (total 5; 0.5+0.5 each)
i. Moving the left line to the left while fixing the other two
Yes. GMD L
ii. Moving the middle line to the left while fixing the other two
No. L if D12 becomes too small
iii. Moving the middle line upward while fixing the other two
Yes. GMD L
iv. Increasing d between the two bundled conductors of each phase
No. GMR L
v. Add a 3rd conductor which is identical to the existing two conductors
such that each phase has symmetric 3 bundled conductors.
No. GMR L
23
2 3
1 / 61 / 3 2 / 3 1 / 6
3
1 / 2 1 / 2 1 / 6
2
1
s s
s
s s s
G M R D d G M R D d
G M R D d d d
G M R D d D D
3
1 2 2 3 1 30 .2 ln 0 .2 ln m H /k m
L L
DL
G M R
D
G R
DG
M
M D
5
This question is for ECE599 students only (5 points):
f. A salient-pole synchronous machine has internal emf E, Ra0, and Xd=2Xq. It is operated as a
synchronous motor load. Draw phasor diagrams with E, V, Id, Iq, jXdId and jXqIq to respectively
illustrate two conditions having |E|=|V| and |E|<|V|. Does the machine supply or absorb reactive
power for each condition?
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TAKE-HOME PROBLEMS (Due by today midnight 12:00AM)
Problem 3 (15 points): As shown in the figure, a three phase bundled line has two conductors per phase. D=10m and d=0.1m.
Each conductor has radius r=0.02m. The line spacing is shown in the figure, which is measured from the center of the bundle.
Consider transposition,
a. Calculate the GMR of each conductor, and GMD, GMRL
and GMRC of the line.
b. Determine the inductance and capacitance per phase per