fundamentals of engineering supplied-reference handbook - Ventech!

AC POWER
Complex Power
Real power P (watts) is defined by
P = (½)VmaxImax cos θ
= VrmsIrms cos θ
where θ is the angle measured from V to I. If I leads (lags)
V, then the power factor (p.f.),
p.f. = cos θ
is said to be a leading (lagging) p.f.
Reactive power Q (vars) is defined by
Q = (½)VmaxImax sin θ
= VrmsIrms sin θ
Complex power S (volt-amperes) is defined by
S = VI* = P + jQ,
where I* is the complex conjugate of the phasor current.
S=VI
θ
P=VIcosθ
Q=VIsin θ
Complex Power Triangle (Inductive Load)
For resistors, θ = 0, so the real power is
P = V I =
2
V
2
/ R = I R
rms rms rms rms
Balanced Three-Phase (3-φ) Systems
The 3-φ line-phase relations are
for a delta
for a wye
VL = Vp
V = 3V = 3V
I = 3I
IL = Ip
L p
L p LN
where subscripts L/P denote line/phase respectively.
A balanced 3-φ delta-connected load impedance can be
converted to an equivalent wye-connect load impedance
using the following relationship
Z ∆ = 3Z Y
The following formulas can be used to determine 3-φ power
for balanced systems.
S = P+ jQ
S = 3VI = 3VI
P P L L
3 P
*
P
3VI L Lcos P jsin
P
( )
S= V I = θ + θ
For balanced 3-φ wye- and delta-connected loads
2
= *
L V
S
Z
2
L = 3 *
V
S
Z
Y
∆
171
ELECTRICAL AND COMPUTER ENGINEERING (continued)
where
S = total 3-φ complex power (VA)
S = total 3-φ apparent power (VA)
P = total 3-φ real power (W)
Q = total 3-φ reactive power (var)
θ P = power factor angle of each phase
V L = rms value of the line-to-line voltage
V LN = rms value of the line-to-neutral voltage
I L = rms value of the line current
I P = rms value of the phase current
For a 3-φ wye-connected source or load with line-to-neutral
voltages
Van
= VP∠°
0
Vbn
= VP∠−
120°
V = V ∠+ 120°
cn P
The corresponding line-to-line voltages are
V
V
V
= 3V∠ 30°
ab P
= 3V∠− 90°
bc P
= 3V∠+ 150°
ca P
Transformers (Ideal)
Turns Ratio
a = N1
N2
V p
a =
V
=
I
I
s
The impedance seen at the input is
ZP = a 2 ZS
s
p