ECE 3364 Circuits II Power Formulas
Instantaneous power )(*)()( titvtp =
In AC-circuits )cos()( 0vm tvtv
θ
ω
+= and )cos()( 0im titi
θ
ω
+= Inductors: 2/
π
=
θ
−
θ
iv . Capacitors: 2/π−=θ−θ iv
[]
avt
iv
t
iv
tp iv
mm
iv
mm ⋅ωθ−θ−ω+θ−θ= )2sin()sin(
2
)2cos(1)cos(
2
)( 00 where )(ti is the reference°0
[]
avt
iv
tp iviv
mm ⋅θ+θ+ω+θ−θ= )2cos(1)cos(
2
)( 0 where iv
θ
θ
& can be referenced with respect to any reference°0
Average power watts
iv
Piv
mm )cos(
2
θθ
−= Reactive power s
iv
Qiv
mm var)sin(
2
θθ
−=
Power factor angle iv
pfa
θ
θ
−= Power factor )cos( iv
pf
θ
θ
−
=
Reactive factor )sin( iv
rf
θ
θ
−=
Lagging (current-phase lags voltage-phase) iv
θ
θ
> leading (current-phase leads voltage-phase) iv
θ
θ
<
RMS values 2
m
rms v
v= and 2
m
rms i
i= so that wattsivP ivrmsrms )cos(
θ
θ
−
=
Note: "rms" and "effective" are synonymous, so you often see eff
v instead of rms
v, etc.
Complex power avjQPS ⋅+= Apparent power avQPS ⋅+= 22
av
iv
e
iv
Siv
mmmm iv ⋅−∠== −)(
22
)(
θθ
θθ
wattsSP iv )cos( θ−θ= sSQ iv var)sin( θ−θ=
Given S , a lagging pf , wattspfSP ||= and spfSQ var))(cossin( 1−
+=
Given S , a leading pf , wattspfSP ||= and spfSQ var))(cossin( 1−
−=
In phasor notation (AC-circuits)
vm
vV
θ
∠=
~
and im
iI
θ
∠=
~
or in RMS units vrmsv
m
rms v
v
V
θθ
∠=∠= 2
~ and irmsi
m
rms i
i
I
θθ
∠=∠= 2
~
()
Load
rms
Load
rms
rmsrms
iv
mm ZI
Z
V
avIV
IV
iv
S2
*
2
*
*|
~
|
)(
|
~
|
~~
2
~
~
)(
2==⋅==θ−θ∠=
Balanced 3-phase circuits (AC)
Pos phase seq:
()
ANBN VV
~
1200.1
~
°−∠= ;
(
)
ANCN VV
~
1200.1
~
°+∠= and
[
]
ABBC VV
~
1200.1
~
°−∠= ;
[
]
ABCA VV
~
1200.1
~
°+∠=
Neg phase seq:
()
ANBN VV
~
1200.1
~
°+∠= ;
(
)
ANCN VV
~
1200.1
~
°−∠= and
[
]
ABBC VV
~
1200.1
~
°+∠= ;
[
]
ABCA VV
~
1200.1
~
°−∠=
Pos phase seq:
(
)
ANAB VV
~
303
~
°+∠= aAAB II ~
30
3
1
~⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛°+∠= note: aAAN II
~
~
=
Neg phase seq:
(
)
ANAB VV
~
303
~
°−∠= aAAB II ~
30
3
1
~⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛°−∠=
wattsPPIVP loadY
C
loadY
BIV
rms
aA
rms
AN
loadY
AaAAN
−−− ==−= )cos(
~
~
θθ
ATTL PP 3
=
sQQIVQ loadY
C
loadY
BIV
rms
aA
rms
AN
loadY
AaAAN var)sin(
~
~
−−− ==−=
θθ
ATTL QQ 3
=
()
CBIV
rms
aA
rms
AN
rms
aA
rms
AN
ASSIVIVS aAAN ==−∠== )(
~~~~ *
θθ
Voltage source transformation: Δ→Y and Y→Δ
Δ
→Y:
(
)
naab vv '' 303 o
+∠= Y
ZZ 3
=
Δ
Y→
Δ
: '' 30
3
1ab
o
na vv ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−∠= Δ
=ZZY3
1
For
Δ
→Y load transformation, just let voltages =0
For Y→
Δ
load transformation, just let voltages=0
+
-
+
-
+
-
ZΔ
a’
a
b
c
vca’
vbc’
vab’
b’
c’
+
-
+
-
+
-
+
-
+
-
+
-
ZΔ
ZΔ
a’
a
b
c
vca’
vca’
vbc’
vbc’
vab’
vab’
b’
c’
+
-
+
-
+
-
ZYa’
a
b
c
va’n
nvb’n
vc’n
+
-
+
-
+
-
+
-
+
-
+
-
ZY
ZYa’
a
b
c
va’n
va’n
nvb’n
vb’n
vc’n
vc’n