App-17
IM 760301-01E
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App
App
Index
Appendix
In the case of the mean value, simply taking the average over 1 period of the sine
wave results in zero. Therefore, the average is taken over 1 period of absolute values.
As with the rms value, if we denote the average value of the current of instantaneous
values i = I
m
sin
ω
t as I
mn
, we obtain
I
mn
= i The average of i over one cycle =
1
π
0
2
π
i d
ω
t
I
m
2
=
2
π
These relationships also apply to a sinusoidal voltage.
The maximum value, rms value, and average value of a sinusoidal alternating current
are related as shown below. Crest factor and form factor are used to define the
tendency of the AC waveform.
Crest factor (crest factor) =
Maximum value
Rms value
Form factor (form factor) =
Rms value
Average value
• Vector Display of Alternating Current
In general, the instantaneous value of voltage and current is expressed with the
following equations.
Voltage: u = U
m
sin
ω
t
Current: i = I
m
sin(
ω
t –
φ
)
The time offset between the voltage and current is called a phase difference, and
φ
is
called a phase angle. This time difference is mainly caused by the load to which the
power is supplied. In general, the phase difference is zero when the load consists of
only resistance. If the load consists of inductance (coiled object), the current lags the
voltage. If the load consists of capacitance, the current leads the voltage.
0
π
2
π
i
u
ω
t
φ
When the current lags the voltage
0
π
2
π
i
u
ω
t
φ
When the current leads the voltage
Vector display is used to easily convey the size and phase relationships between the
voltage and current. The upward direction on the vertical axis is the reference, and a
positive phase angle is represented by an angle in the counterclockwise direction.
Normally, a dot is placed above the symbol representing the quantity when explicitly
indicating that it is a vector. The size of the vector represents the rms value.
φ
U
I
When the current lags the voltage
φ
U
I
When the current leads the voltage
Appendix 4 Power Basics (Power, Harmonics, and RLC of AC Circuits)