Formula Sheets: Symmetrical Components | Power Systems - Electrical Engineering (EE) PDF Download

 Page 1


Power S ystems F ormula Sheet:
S ymmetrical Component s
1. S ymmetrical Components Basics
• Definition : An y unbalanced three-phase system (voltages or currents) can be
resolved into three balanced sets: positive, negative, and zero sequences.
• Phase Oper ator (a ):
a =e
j120
?
=-
1
2
+j
v
3
2
, a
2
=e
j240
?
=-
1
2
-j
v
3
2
, a
3
= 1
• S ymmetrical Components Tr ansformation :
?
?
V
a
V
b
V
c
?
?
=
?
?
1 1 1
1 a
2
a
1 a a
2
?
?
?
?
V
0
V
1
V
2
?
?
where V
0
, V
1
, V
2
are zero, positive, and negative sequence voltages (similarly
for currents).
• Inverse Tr ansformation :
?
?
V
0
V
1
V
2
?
?
=
1
3
?
?
1 1 1
1 a a
2
1 a
2
a
?
?
?
?
V
a
V
b
V
c
?
?
2. Sequence Components
• Zero Sequence : Equal magnitude, same phase:
V
a0
=V
b0
=V
c0
=V
0
• Positive Sequence : Equal magnitude, 120° phase shift (a-b-c rotation):
V
a1
=V
1
, V
b1
=a
2
V
1
, V
c1
=aV
1
• Negative Sequence : Equal magnitude, 120° phase shift (a-c-b rotation):
V
a2
=V
2
, V
b2
=aV
2
, V
c2
=a
2
V
2
1
Page 2


Power S ystems F ormula Sheet:
S ymmetrical Component s
1. S ymmetrical Components Basics
• Definition : An y unbalanced three-phase system (voltages or currents) can be
resolved into three balanced sets: positive, negative, and zero sequences.
• Phase Oper ator (a ):
a =e
j120
?
=-
1
2
+j
v
3
2
, a
2
=e
j240
?
=-
1
2
-j
v
3
2
, a
3
= 1
• S ymmetrical Components Tr ansformation :
?
?
V
a
V
b
V
c
?
?
=
?
?
1 1 1
1 a
2
a
1 a a
2
?
?
?
?
V
0
V
1
V
2
?
?
where V
0
, V
1
, V
2
are zero, positive, and negative sequence voltages (similarly
for currents).
• Inverse Tr ansformation :
?
?
V
0
V
1
V
2
?
?
=
1
3
?
?
1 1 1
1 a a
2
1 a
2
a
?
?
?
?
V
a
V
b
V
c
?
?
2. Sequence Components
• Zero Sequence : Equal magnitude, same phase:
V
a0
=V
b0
=V
c0
=V
0
• Positive Sequence : Equal magnitude, 120° phase shift (a-b-c rotation):
V
a1
=V
1
, V
b1
=a
2
V
1
, V
c1
=aV
1
• Negative Sequence : Equal magnitude, 120° phase shift (a-c-b rotation):
V
a2
=V
2
, V
b2
=aV
2
, V
c2
=a
2
V
2
1
3. Sequence Impedances
• Tr ansmission Line :
Z
0
=R+jX
0
, Z
1
=Z
2
=R+jX
1
where Z
0
is zero sequence impedance, Z
1
, Z
2
are positive and negative se-
quence impedances (Z
0
>Z
1
=Z
2
).
• Tr ansformer :
– Positive and negative sequence: Z
1
=Z
2
=Z
leakage
.
– Zero sequence: Depends on winding connection (e.g., grounded wye al-
lows zero sequence current).
• S ynchronous Machine :
Z
1
˜X
d
, Z
2
˜X
''
d
, Z
0
(depends on grounding)
whereX
d
is direct-axis synchronous reactance,X
''
d
is subtr ansient reactance.
4. Power in T erms of S ymmetrical Components
• T otal Complex Power :
S =V
a
I
*
a
+V
b
I
*
b
+V
c
I
*
c
S = 3(V
0
I
*
0
+V
1
I
*
1
+V
2
I
*
2
)
whereI
0
,I
1
,I
2
are sequence currents.
• Real Power :
P = Re[S] = 3(|V
0
||I
0
| cos?
0
+|V
1
||I
1
| cos?
1
+|V
2
||I
2
| cos?
2
)
• Reactive Power :
Q = Im[S] = 3(|V
0
||I
0
| sin?
0
+|V
1
||I
1
| sin?
1
+|V
2
||I
2
| sin?
2
)
5. Sequence Networks
• Zero Sequence Network : Includes zero sequence impedances, connected based
on grounding.
• Positive Sequence Network : Includes positive sequence impedances, resem-
bles normal system oper ation.
• Negative Sequence Network : Includes negative sequence impedances, no
voltage source for balanced systems.
• Network Connection for F aults : Depends on fault type (e.g., single line-to-
ground, line-to-line, etc.).
2
Page 3


Power S ystems F ormula Sheet:
S ymmetrical Component s
1. S ymmetrical Components Basics
• Definition : An y unbalanced three-phase system (voltages or currents) can be
resolved into three balanced sets: positive, negative, and zero sequences.
• Phase Oper ator (a ):
a =e
j120
?
=-
1
2
+j
v
3
2
, a
2
=e
j240
?
=-
1
2
-j
v
3
2
, a
3
= 1
• S ymmetrical Components Tr ansformation :
?
?
V
a
V
b
V
c
?
?
=
?
?
1 1 1
1 a
2
a
1 a a
2
?
?
?
?
V
0
V
1
V
2
?
?
where V
0
, V
1
, V
2
are zero, positive, and negative sequence voltages (similarly
for currents).
• Inverse Tr ansformation :
?
?
V
0
V
1
V
2
?
?
=
1
3
?
?
1 1 1
1 a a
2
1 a
2
a
?
?
?
?
V
a
V
b
V
c
?
?
2. Sequence Components
• Zero Sequence : Equal magnitude, same phase:
V
a0
=V
b0
=V
c0
=V
0
• Positive Sequence : Equal magnitude, 120° phase shift (a-b-c rotation):
V
a1
=V
1
, V
b1
=a
2
V
1
, V
c1
=aV
1
• Negative Sequence : Equal magnitude, 120° phase shift (a-c-b rotation):
V
a2
=V
2
, V
b2
=aV
2
, V
c2
=a
2
V
2
1
3. Sequence Impedances
• Tr ansmission Line :
Z
0
=R+jX
0
, Z
1
=Z
2
=R+jX
1
where Z
0
is zero sequence impedance, Z
1
, Z
2
are positive and negative se-
quence impedances (Z
0
>Z
1
=Z
2
).
• Tr ansformer :
– Positive and negative sequence: Z
1
=Z
2
=Z
leakage
.
– Zero sequence: Depends on winding connection (e.g., grounded wye al-
lows zero sequence current).
• S ynchronous Machine :
Z
1
˜X
d
, Z
2
˜X
''
d
, Z
0
(depends on grounding)
whereX
d
is direct-axis synchronous reactance,X
''
d
is subtr ansient reactance.
4. Power in T erms of S ymmetrical Components
• T otal Complex Power :
S =V
a
I
*
a
+V
b
I
*
b
+V
c
I
*
c
S = 3(V
0
I
*
0
+V
1
I
*
1
+V
2
I
*
2
)
whereI
0
,I
1
,I
2
are sequence currents.
• Real Power :
P = Re[S] = 3(|V
0
||I
0
| cos?
0
+|V
1
||I
1
| cos?
1
+|V
2
||I
2
| cos?
2
)
• Reactive Power :
Q = Im[S] = 3(|V
0
||I
0
| sin?
0
+|V
1
||I
1
| sin?
1
+|V
2
||I
2
| sin?
2
)
5. Sequence Networks
• Zero Sequence Network : Includes zero sequence impedances, connected based
on grounding.
• Positive Sequence Network : Includes positive sequence impedances, resem-
bles normal system oper ation.
• Negative Sequence Network : Includes negative sequence impedances, no
voltage source for balanced systems.
• Network Connection for F aults : Depends on fault type (e.g., single line-to-
ground, line-to-line, etc.).
2
6. F ault Analysis Using S ymmetrical Components
• Single Line-to-Ground F ault (Phasea to ground) :
I
a0
=I
a1
=I
a2
=
V
f
Z
1
+Z
2
+Z
0
whereV
f
is pre-fault voltage,Z
0
,Z
1
,Z
2
are sequence impedances.
• Line-to-Line F ault (Phaseb toc ):
I
a1
=-I
a2
=
V
f
Z
1
+Z
2
, I
a0
= 0
• Double Line-to-Ground F ault (Phaseb andc to ground):
I
a1
=
V
f
(Z
2
+Z
0
)
Z
1
Z
2
+Z
2
Z
0
+Z
0
Z
1
, I
a2
=-
Z
0
Z
2
+Z
0
I
a1
, I
a0
=-
Z
2
Z
2
+Z
0
I
a1
3