Formula Sheets: Single-Phase AC Circuits | Basic Electrical Technology - Electrical Engineering (EE) PDF Download

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Basic Electrical T ec hnology: Single-Phase A C
Circuits F orm ula Sheet for Electrical GA TE
Sin usoidal Quan tities
• V oltage/Curren t Expression :
v(t) =V
m
sin(?t+?), i(t) =I
m
sin(?t+?)
where V
m
, I
m
are p e ak v alues (V , A ), ? is angular frequency (rads
-1
), ? , ? are phase
angles.
• RMS V alue :
V
rms
=
V
m
v
2
, I
rms
=
I
m
v
2
• Angular F requency :
? = 2pf
where f is frequency (Hz ).
Imp edance
• Resistor Imp edance :
Z
R
=R
where R is resistance (? ).
• Inductor Imp edance :
Z
L
=j?L
where L is inductance (H ).
• Capacitor Imp edance :
Z
C
=
1
j?C
= -j
1
?C
where C is capacitance (F ).
• T otal Imp edance (Series) :
Z =R+j(X
L
-X
C
)
where X
L
=?L , X
C
=
1
?C
are inductiv e and capacitiv e reactances (? ).
• Imp edance Magnitude and Phase :
|Z| =
v
R
2
+(X
L
-X
C
)
2
, ? = tan
-1
(
X
L
-X
C
R
)
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Basic Electrical T ec hnology: Single-Phase A C
Circuits F orm ula Sheet for Electrical GA TE
Sin usoidal Quan tities
• V oltage/Curren t Expression :
v(t) =V
m
sin(?t+?), i(t) =I
m
sin(?t+?)
where V
m
, I
m
are p e ak v alues (V , A ), ? is angular frequency (rads
-1
), ? , ? are phase
angles.
• RMS V alue :
V
rms
=
V
m
v
2
, I
rms
=
I
m
v
2
• Angular F requency :
? = 2pf
where f is frequency (Hz ).
Imp edance
• Resistor Imp edance :
Z
R
=R
where R is resistance (? ).
• Inductor Imp edance :
Z
L
=j?L
where L is inductance (H ).
• Capacitor Imp edance :
Z
C
=
1
j?C
= -j
1
?C
where C is capacitance (F ).
• T otal Imp edance (Series) :
Z =R+j(X
L
-X
C
)
where X
L
=?L , X
C
=
1
?C
are inductiv e and capacitiv e reactances (? ).
• Imp edance Magnitude and Phase :
|Z| =
v
R
2
+(X
L
-X
C
)
2
, ? = tan
-1
(
X
L
-X
C
R
)
1
P o w er i n A C Circuits
• Instan taneous P o w er :
p(t) =v(t)·i(t)
• A v erage (Real) P o w er :
P =V
rms
I
rms
cos?
where cos? is the p o w er factor.
• Reactiv e P o w er :
Q =V
rms
I
rms
sin? (VA)
• Apparen t P o w er :
S =V
rms
I
rms
=
v
P
2
+Q
2
(VA)
• P o w er F actor :
cos? =
P
S
=
R
|Z|
Series and P arallel Circuits
• Series Circuit Curren t :
I =
V
|Z|
• P arallel Circuit A dmittance :
Y =
1
Z
=G+jB
where G =
R
R
2
+X
2
is conductance, B =
-X
R
2
+X
2
is susceptance (S ).
• T otal A dmittance (P arallel) :
Y
total
=Y
1
+Y
2
+···+Y
n
Resonance in Series RLC Circuit
– Resonan t F requency :
f
0
=
1
2p
v
LC
– Angular Resonan t F requency :
?
0
=
1
v
LC
– Qualit y F actor (Q) :
Q =
?
0
L
R
=
1
R
v
C
L
– Bandwidth :
BW =
f
0
Q
=
R
2pL
2
Page 3


Basic Electrical T ec hnology: Single-Phase A C
Circuits F orm ula Sheet for Electrical GA TE
Sin usoidal Quan tities
• V oltage/Curren t Expression :
v(t) =V
m
sin(?t+?), i(t) =I
m
sin(?t+?)
where V
m
, I
m
are p e ak v alues (V , A ), ? is angular frequency (rads
-1
), ? , ? are phase
angles.
• RMS V alue :
V
rms
=
V
m
v
2
, I
rms
=
I
m
v
2
• Angular F requency :
? = 2pf
where f is frequency (Hz ).
Imp edance
• Resistor Imp edance :
Z
R
=R
where R is resistance (? ).
• Inductor Imp edance :
Z
L
=j?L
where L is inductance (H ).
• Capacitor Imp edance :
Z
C
=
1
j?C
= -j
1
?C
where C is capacitance (F ).
• T otal Imp edance (Series) :
Z =R+j(X
L
-X
C
)
where X
L
=?L , X
C
=
1
?C
are inductiv e and capacitiv e reactances (? ).
• Imp edance Magnitude and Phase :
|Z| =
v
R
2
+(X
L
-X
C
)
2
, ? = tan
-1
(
X
L
-X
C
R
)
1
P o w er i n A C Circuits
• Instan taneous P o w er :
p(t) =v(t)·i(t)
• A v erage (Real) P o w er :
P =V
rms
I
rms
cos?
where cos? is the p o w er factor.
• Reactiv e P o w er :
Q =V
rms
I
rms
sin? (VA)
• Apparen t P o w er :
S =V
rms
I
rms
=
v
P
2
+Q
2
(VA)
• P o w er F actor :
cos? =
P
S
=
R
|Z|
Series and P arallel Circuits
• Series Circuit Curren t :
I =
V
|Z|
• P arallel Circuit A dmittance :
Y =
1
Z
=G+jB
where G =
R
R
2
+X
2
is conductance, B =
-X
R
2
+X
2
is susceptance (S ).
• T otal A dmittance (P arallel) :
Y
total
=Y
1
+Y
2
+···+Y
n
Resonance in Series RLC Circuit
– Resonan t F requency :
f
0
=
1
2p
v
LC
– Angular Resonan t F requency :
?
0
=
1
v
LC
– Qualit y F actor (Q) :
Q =
?
0
L
R
=
1
R
v
C
L
– Bandwidth :
BW =
f
0
Q
=
R
2pL
2
Key Notes
– Phasor Represen tation : V =V
rms
?? , I =I
rms
?? .
– P o w er F actor : Leading (capacitiv e, ?< 0 ), lagging (inductiv e, ?> 0 ).
– Resonance : A t resonance, X
L
=X
C
, Z =R , p o w er factor = 1.
– GA TE F o cus : Solv e for imp edance, p o w er, resonance frequency , and phasor cal-
culations.
– Units : Use SI units for GA TE (e.g., V , A , ? , H , F , Hz ).
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