Formula Sheets: Three Phase Circuits (Star-Delta Transformations) | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

 Page 1


Net w ork Theory: Three Phase Circuits (Star-Delta
T r ansformations) F orm ula Sheet for Electrical GA TE
Three-Phase Circuit Basics
• Phase V oltage (V
p
) : V oltage across a single phase.
• ( Line V oltage (V
L
): V oltage b et w een t w o lines.
• Phase Curren t (I
p
) : Curren t through a single phase.
• Line Curren t (I
L
) : Curren t through a line conductor.
• Three-Phase P o w er : T otal p o w er in a balanced three-phase system.
P =
v
3V
L
I
L
cos? (W)
Q =
v
3V
L
I
L
sin? (-reactive)
S =
v
3V
L
I
L
()
where ? is the p o w er factor angle.
Star (Y) Connection
• V oltage Relationship :
V
L
=
v
3V
p
, V
p
=
V
L
v
3
, V
L
?30
?
leads V
p
• Curren t Relationship :
I
L
=I
p
• Imp edance p er Phase : Z
Y
(imp edance of eac h branc h in star).
• P o w er in Star :
P = 3V
p
I
p
cos? =
v
3V
L
I
L
cos?
Delta (?) Connection
• V oltage Relationship :
V
L
=V
p
• Curren t Relationship :
I
L
=
v
3I
p
, I
p
=
I
L
v
3
, I
L
?-30
?
leads I
p
• Imp edance p er Phase : Z
?
(imp edance of eac h branc h in delt a).
• P o w er in Delta :
P = 3V
p
I
p
cos? =
v
3V
L
I
L
cos?
1
Page 2


Net w ork Theory: Three Phase Circuits (Star-Delta
T r ansformations) F orm ula Sheet for Electrical GA TE
Three-Phase Circuit Basics
• Phase V oltage (V
p
) : V oltage across a single phase.
• ( Line V oltage (V
L
): V oltage b et w een t w o lines.
• Phase Curren t (I
p
) : Curren t through a single phase.
• Line Curren t (I
L
) : Curren t through a line conductor.
• Three-Phase P o w er : T otal p o w er in a balanced three-phase system.
P =
v
3V
L
I
L
cos? (W)
Q =
v
3V
L
I
L
sin? (-reactive)
S =
v
3V
L
I
L
()
where ? is the p o w er factor angle.
Star (Y) Connection
• V oltage Relationship :
V
L
=
v
3V
p
, V
p
=
V
L
v
3
, V
L
?30
?
leads V
p
• Curren t Relationship :
I
L
=I
p
• Imp edance p er Phase : Z
Y
(imp edance of eac h branc h in star).
• P o w er in Star :
P = 3V
p
I
p
cos? =
v
3V
L
I
L
cos?
Delta (?) Connection
• V oltage Relationship :
V
L
=V
p
• Curren t Relationship :
I
L
=
v
3I
p
, I
p
=
I
L
v
3
, I
L
?-30
?
leads I
p
• Imp edance p er Phase : Z
?
(imp edance of eac h branc h in delt a).
• P o w er in Delta :
P = 3V
p
I
p
cos? =
v
3V
L
I
L
cos?
1
Star-Delta T ransformations
• Star to Delta T ransformation :
Z
?
= 3Z
Y
whereZ
Y
is the imp edance of eac h branc h in star, Z
?
is the imp edance of eac h branc h
in delta.
• Delta to Star T ransformation :
Z
Y
=
Z
?
3
• General T ransformation (Delta to Star) :
Z
Y
=
Z
ab
Z
bc
+Z
bc
Z
ca
+Z
ca
Z
ab
Z
i
where Z
ab
, Z
bc
, Z
ca
are delta imp edances, Z
i
is the opp osite delta imp edance for the
corresp onding star branc h.
• General T ransformation (Star to Delta) :
Z
?
=
Z
Y1
Z
Y2
+Z
Y2
Z
Y3
+Z
Y3
Z
Y1
Z
Yi
whereZ
Y1
,Z
Y2
,Z
Y3
are star imp edances,Z
Yi
is the star imp edance opp osite the delta
branc h.
Balanced vs Un balanced Systems
• Balanced System : Equal imp edances and 120° phase difference b et w een phases.
• Neutral Curren t (Star) : Zero in balanced systems; non-zero in un balanced systems.
• Un balanced System Analysis : Use p er-phase analysis or symmetrical comp onen ts
(not co v ered here).
Key Notes
• Use SI units: V oltage (V ), Curren t (A ), Imp edance (? ), P o w er (W , ,-reactive ).
• F or balanced systems, use
v
3 factor in v oltage/curren t relations.
• Star-Delta transformations simplify un balanced or complex three-phase net w orks.
• P o w er form ulas apply to b oth star and delta for balanced systems.
• F or GA TE, fo cus on balanced systems and simple star-delta con v ersions.
2