The EUREC M.Sc. In Renewable Energy
Power take-off electrical components - Problems
António Dente, Célia de Jesus, Gil Marques
1
PROBLEM 1 - RL series circuit in steady state
Determine the steady state time evolution
of voltages and currents in each of the
elements of the circuit below, when an
alternate sinusoidal voltage with the
following RMS value Eef  230V , f  50 Hz
is applied
e(t )  2 Eef cos(t )  E  Eef e j 0
Z  Z R  Z L  R  j L  Ze j
I 
Z  R 2  ( L ) 2  96   arctan(
L
R
)  78º
º
E
 I ef e  j  2.4e  j 78 A
Z
2
º
E
I   I ef e j  2.4e j 78 A
Z
U R  RI  48e
 j 78
º
V
U L  j LI  225e V
j12º
i (t )  2 2.4 cos(t  78º

180
uR (t )  2 48cos(t  78º
)

180
uL (t )  2 225cos(t  12º
)

180
)
3
4
PROBLEM 2 – - RLC series circuit in steady state
Sketch the vector diagrams and the steady
state time evolution of voltages and
currents in each of the elements of the
series RLC circuit below, when it fed by an
alternate sinusoidal voltage with the
following RMS value and for the following
values of the RLC circuit parameters
Eef  230V , f  50 Hz, R  20, C  300  F , L  300mH
Z  Z R  ZC  Z L  R  j
1
 j L  R 
C
I  2 2.7e  j 77 A U R  2 53e  j 77
º
º
1 

j L 
C 

Z  86e
j 77 º
U L  2 252e j13 V U C  2 28e  j167 V
º
º
5
RLC series circuit in steady state
1
2
• 𝑆 = 𝑈 𝐼 ∗ = 𝑆𝑐𝑜𝑠 𝜑 + 𝑆𝑠𝑖𝑛 𝜑 = P+jQ
𝑆 = 621𝑉𝐴 𝑃 = 140𝑊
𝑄 = 605𝑉𝐴𝑟
6
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PROBLEM 3 – Power Factor Compensation
Consider the following circuit, fed by an AC network and with the following parameters:
Eef  230V f  50 Hz R1  R2  0,5 L1  5mH L2  4mH LM  50mH Rext  10
Determine the capacitance value of a capacitor, to put at the circuit’s entry, in order to
assure a unitary power factor
 I1  I M  I 2  0

( R1  j L1 ) I1  j LM I M  E
( R  R  j L I  j L I  0
2
2 2
M M
 ext
I1  2 23.6e  j 44
I 2  2 18.6e
º
 j12.2º
I M  2 12.5e  j 95.4
º
8
1
S  E I1*  P  jQ  3.91 kW  j 3.77 kVAr
2
1
SC  E I C *  jQC
2
P  jQ  jQC  P
jCE 2  j 3.77 kVAr
C  227  F
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PROBLEM 4 – Three-phase circuits
Consider the circuit shown below:
The voltage sources form a three
phase symmetrical and balanced
system of voltages, of RMS value
230/400 V and frequency f = 50
Hz. The inductive and balanced
load has the value of
R  6 L  25,5mH
a) Compute the phase and neutral currents
b) Sketch the vector diagram for the complex voltages and currents.
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Z  R  j L  6  j8.01
E1
 j 53.2º
I1 
 2 23e
A
Z
 j120º
Ee
I2  1
Z
 2 23e
 j (120  53.2)º
A
I N  I1  I 2  I 3  0
E1e  j 240
 j (240  53.2) º
I3 
 2 23e
A
Z
º
U3
I3
I2
U1
U2
I1
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PROBLEM 5 – Synchronous Machine
Consider an alternator with the following nominal values:100 MVA / 6kV
The synchronous reactance of the machine is XS = 0,31 
The alternator is connected to a network with constant voltage and frequency.
a) Assume that the network is inductive and that the alternator is working as a
synchronous compensator. Represent qualitatively in a vector diagram the
characteristic quantities that represent this kind of machine behavior.
b) In the previous question conditions determine the value of electromotive force
when an apparent power of 40 MVA is delivered to the network.
c) Consider that the alternator should change its operating regime and operate
now as a generator delivering to the network an active power of 80 MW.
Assuming that the value of reactive power is kept constant determine the change
in the excitation current needed to assure this new operating regime.
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I
U
E  jXI  U
E
UE
UE
U2
S 3
sin( )  j 3(
cos( ) 
)
X
X
X
40106
Q  3UI sin(90º ) I 
 3.85kA
3
3 6 10 / 3
S  P  jQ  j 40MVAr
P  0    0 E U 
P/3
tg ( ) 
Q / 3 U 2 / X
XQ / 3
U
  27.1
º
E  4.66kV
E
XP / 3
sin( )U
E  5.23e j 27.1 kV
º
E
E  If
I
U
I fb  I fa
Eb
Ea
I fb  1.12% I fa
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PROBLEM 6 – Asynchronous Machine
Consider a 2.0 kW three-phase induction machine with the following rated values
U N  400V
f N  50Hz
N N  1420r. p.m.
a) Suppose that this machine is connected to the low voltage network and that
it drives a constant torque load with the value T  8Nm
Determine the machine’s speed, electromagnetic torque and mechanic
power in these conditions
b) Consider now that, with the load specified in the previous question the
machine is fed by an electronic system that imposes the relation V f  8
Determine the machine’s speed, electromagnetic torque and mechanic
power for a frequency of 25 Hz. Suggestion: use the model of the ideal
asynchronous machine to answer this question.
c) Repeat questions a) and b) but for a torque T  8Nm
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PN  2000   N TN 
T
2 1420
TN
60
 TN  13.45 Nm
TN
( N  N S ) N S  1500 rpm (50 Hz )
NN  NS
T  0.1681( N  N S )
TL  8  0.1681( N  N S )  N L  1452rpm PL  1.22kW
N L  1548rpm PL  1.3kW (Generator )
N S  750 rpm (25 Hz )
N L  702rpm PL  0.59kW
N L  798rpm PL  0.67kW (Generator )
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