1. The document discusses electrical engineering concepts related to three-phase systems including voltages, currents, and phasor diagrams. 2. Equations are presented relating voltages and currents in three-phase systems to phase angles and a common voltage or current. 3. Diagrams illustrate the phase relationships between voltages and currents in three-phase systems under different conditions.

2. !quot; # $
!quot; $
% &

3. '
( # )*
# )*
+ ,
+ ) /01*
-.
2-
2
23
2

4. # )
-*
4 ,
(56(∠/0
(6 (∠ 5 /0
(-6(∠7 /0
5
(
#
4#
#

5. 89
u
u( t ) = U MAX sen(ωt + 0°)
N
t
-90° -60° -30° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
ω
S
TENSION
# :
# )*

6. 89
u( t ) = U MAX sen(ωt − 120°)
u, i
N
t
-90° -60° -30° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
ω
S
5 /0: ;
# < )*

7. 89
u( t ) = U MAX sen(ωt + 120°)
u
N
t
-120° -90° -60° -30° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
ω
S
5 /0:
# < )*

8. u
N
t
-120° -90° -60° -30° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
S
4
120° 120°
== =
#

9. T1
)*
R1
0°
ϕ6
12
/0
12
0°
N
S2 T2
R2 R1
T2
S1
S2
S
T1
R2
S1
120°
#
#

10. UTT` ) *
)
*
12
0°
U RR´ = U F ∠0°
120°
URR`
USS´ = U F ∠ − 120°
0°
12
UTT ´ = U F ∠ + 120°
USS`

11. ! .
+ $
+T1 +S1 +R1
!
8 $
-T2 -S2 -R2
# +
#
# +
!quot;
< % $

12. 8 %
+ :
)# *

13. 8 %
+R1
+R1 -T2
-R2
-S2 -R2
+T1
-T2
+S1
+T1 -S2
+S1
8 % & 8 %

14. IR
R
R
+
Punto centro
de la estrella
- URS
O
- UTR
- IS
T S
+
S
+
UST
IT
T
-+ :4 - 4-
+

15. > 4 +
R
R
+
+
URO
-
URS
O
-
- USO
T S
+
S
+ +
T
+
U RS = U RO − USO
+ #
?4 ?

16. > 4 +
U RS
-U SO
U RS = U RO − USO U LINEA
+
60°
30°
U RO
U RO
U FASE
U LINEA = 2cos 30°U FASE
120°
U SO
U LINEA = 3 U FASE

17. -U
-U U RS URS
RO U TO
RO U TO U RS
-USO
+ U TR -USO
U TR
UTR
120°
#
60°
60°
30°
30°
U RO
U RO
120°
0°
120°
12
+
12
0°
USO
USO
5 /0 +
-UTO U
-UTO UST
U
ST
ST

18. -URO U TO U RS
( ?6 >∠/0
( -U SO
U TR
( ?6 >∠ 5 /0
(
60°
( ?6 >∠7 /0 30°
( 5 U RO
120°
+ USO
( 6 ∠-/0
(
( 6 ∠@
( /0
( 6 ∠7 /0 -U TO
( 5A UST

19. 8
IR
R
R
+ ;
quot;
IR
-
IT O
- <
- IS
T S # +
+
S
+ IS
IT
ILINEA = IFASE
T
-+ :4

20. R
R
+
B
)*
B - URS
N
- UTR
-
T S
+ URO
S
+
UST
USO
T
UTO
N
- +
+ : : 4B
4-

21. IR
T2
R1
- + I SR
U U
T1T2 R1R2 U
RS
I TS
+
I RT
- IS
U TR
T1 R2
- +
S2 S1
U
S1S2
U
ST
IT
4 +

22. UR1R2 =U RS
+ 4
120°
(6(∠/0
(6(∠ 5 /0 120°
UR1R2 =U RS
(6(∠7 /0
5
120°
U LINEA = U FASE
US1S2 =U ST

23. 8
IR -I TS IT
R
T2
R1
- + I SR
U U I RT
T1T2 R1R2
I TS
+
I RT
- IS
T1 R2 120° 120°
S ISR
- +
S2 S1
U 30°
S1S2
120°
IT IS -I RT
T I TS
-I SR IR
< # #
+ :# C #, 6

24. IR
R
+
U RS
IS
U TR
S
GENERADOR CARGA
TRIFASICO TRIFASICA
+
U ST
IT
T
+
( # #
+ 5 /0 +4

25. 8 ;
IR
- +
D
U RS IS
#
- CARGA
U TR ,
+ TRIFASICA
(E
U ST IT FG
7
-
+ (E
FG
7
(E
FG
7

26. 8
quot; 4 4
#
=
!
H $8
! $

27. 8
# IR
+
- +
8 #
U RS IS
% - CARGA
U TR
8 + TRIFASICA
8 U ST IT
8 # -
+
8 H
8 =

28. 8
H #
I
I
R R
ZØ ZØ
ZØ
ZØ S
S ZØ
ZØ
T
T
8 = 8 =
8 8 +

29. 8
I LINEA
R
UFASE ZØ
ULINEA
ZØ
I LINEA
S ZØ
I LINEA
T

30. 8
8 + F
> G
IR
R
+
U RN Z
IN
N
Z
IS
+
Z
+ S
IT
T
# < #
8 # #

31. 8
8 +
IR
R
+
U RN Z
IN
N
U F ∠0°
U RN
IR = IR =
Z∠φ
Z
#
UF
(> 4 #
∠0° − φ
IR =
J
Z

32. 8
8 +
U F ∠ − 120°
IS =
N
Z∠φ
IS
+
UF
Z
∠ − 120° − φ
IS =
S
Z
IN
N
U F ∠ + 120°
Z
IT =
Z∠φ
+
IT
UF
∠ + 120° − φ
IT =
Z
T

33. 8
8 +
U TN
U RN = U F ∠0° IT
Ø
IR = IF ∠0° − φ
USN = U F ∠ − 120° U RN
IS = IF ∠ − 120° − φ Ø
IS IR
Ø
UTN = U F ∠ + 120°
IT = IF ∠ + 120° − φ
USN

34. 48
UU TN
TN IT
U TN
Ø
IT
120°
U RN
120° U RN IR
Ø
IS IR IS U RN
120°
Ø
USN
Ix = 0
USN
USN
=
8 ?! quot; + $

35. 8
8 +
IR
R
+
U RN Z
IN
N N´
Z
IS
+
Z
+ S
IT
T
IN + IR + IS + IT = 0
# K
IN = 0
# BL
#

36. 8
8 +
IR
R
+
U RN´ Z
U RS
N´
Z
IS U SN´
+
Z
+ S
IT
T
+ #
4 # : #
J #
+

37. 8
8 +
IR
4
U RN´ Z
+ U RS
N´
#
Z
IS U SN´
< Z
#
IT
#

38. 8
I LINEA
R
ZØ ZØ
ULINEA UFASE
=
I LINEA I FASE
S
ZØ
I LINEA
T

39. 8
8 +
IR
R
+
ITR
IRS
U RS
Z
U TR Z
IS
S
Z
U ST
IT
T
+
+ # #
:

40. 8
8 +
R
+
ITR
IRS
U RS
Z
U TR Z
TR
S
T
+
UTRU RS − ∠ + ∠0°°I U FFU F ∠120° − φ
UF U F φ F 120 U
Uy
IRS = ∠0° =
= = = ∠0° − ° − φ
ITR ∠120 φ
= = =
IRS TR
ZZ ZZ
Zφ
Z∠∠φ
Z Z

41. 8
8 +
IR
R
+
ITR
IRS
U RS
Z
U TR Z
IS
S
Z
U ST
IT
T
+
+
IR = IRS − ITR
K

42. 8
8 +
U TR
I TR
IR = IRS − ITR Ø
I LINEA = 2 cos 30°I FASE U RS
IFASE
Ø
I RS
30°
I LINEA = 3 I FASE ILINEA
-I TR
IR

43. 8
8 +
-I ST
U TR IT
I TR
+
# 30°
Ø
IS
U RS
I R = I RS − ITR 30° Ø
-IRS
I RS
IS = IST − I RS IST Ø 30°
IT = ITR − IST -ITR
UST IR

44. 8
8 +
U TR
IT
+ Ø +30°
#
IS
+
U RS
φ 7 -/0
Ø +30° Ø +30°
M
8 I
C #:
IR
#
# 7N
C
F ;G UST

45. <; &
R R
S S
T T
1 1
ZY Z∆ Z∆
ZY
Z∆
ZY
3 2 3 2
1
ZΥ = Z∆
3

46. #
8
R R
S S
T T
N N
M 3Ø
3Ø
Motor trifásico Horno trifásico Motor trifásico Horno trifásico
15 kW 4,5 kW 15 kW 4,5 kW
F.P.=0.86 conectado conectado en
en estrella triángulo
:
# ##