App-28
IM 253710-01E
AC power
Because a phase difference depending on the load exists between the voltage and
current, AC power is not as easily determined as DC power.
When the instantaneous voltage is u = U
m
sin
ω
t and the instantaneous current is
i = I
m
sin(
ω
t –
φ
), the instantaneous power, p, is
p = u
×
i = U
m
sin
ω
t
×
I
m
sin(
ω
t –
φ
) = UIcos
φ
– UIcos(2
ω
t –
φ
)
where U and I represent the rms values of the voltage and current, respectively.
p becomes the sum of the time-independent term, “UIcos
φ
” and the AC component of
twice the frequency of the voltage or current, “–UIcos(2
ω
t –
φ
).”
The mean value of power over one period is called the AC power. Taking the mean
over one period, the AC power, P, becomes
P = UIcos
φ
[W]
The power varies depending on the phase difference,
φ
, even when the voltage and
current are the same. As shown in the figure below, the section above the horizontal
axis represents positive power (power supplied to the load) and the section below the
horizontal axis represents negative power (power delivered from the load). The
difference between the two is the power dissipated by the load. As the phase
difference between the voltage and current increases, negative power increases.
At
φ
=
π
/2, the positive and negative powers become equal and no power is dissipated
in the load.
p
0
π
2
π
i
u
ω
t
Mean power P = UI
When the phase difference between the voltage and current is 0
Positive power
p
0
π
i
u
ω
t
Mean power P = UIcos
φ
φ
Positive power
When the phase difference between the voltage and current is
φ
2
π
Negative power
p
0
2
π
i
u
ω
t
Positive and negative powers are equal
Mean power P = UIcos
2
π
= 0
2
π
When the phase difference between the voltage and current is
2
π
π
Appendix 7 Power Basics (Power/Harmonics/Three Constants Related to the AC Circuit)
