Environmental Engineering Reference
In-Depth Information
For a description of alternating quantities, the definition of an average
value is interesting. Since the positive and negative parts of a sinusoidal
oscillation cancel each other out when calculating the arithmetic average value,
the
root mean square value
(
rms
) is used in electrical engineering. The
definition of the root mean square value of a function
v
(
t
) with the periodic
time
T
=1/
f
is as follows:
(5.51)
For sinusoidal currents and voltages the rms values are:
(5.52)
(5.53)
The vector diagram of the rms values is similar to that of the amplitudes shown
in Figure 5.16, except that the lengths of the vectors are shorter. The vector
diagram is also used in mathematics for complex numbers. Therefore, the
vectors of current and voltage can be interpreted as complex quantities.
However, the real axis (Re) of the vector diagram for the currents and voltages
is drawn vertically, whereas in mathematics it is usually drawn horizontally.
Complex quantities
are underlined in the following expressions.
With the complex voltage
V
=
V
•
e
jϕ
v
=
V
•
e
j0
=
V
of the example above,
the complex value of the rms value of the current with the phase angle
ϕ
i
and
the imaginary unit j (j
2
= -1) becomes:
(5.54)
Electronic components such as capacitors or inductors cause a phase
displacement between current and voltage. For the description in the complex
number system, an imaginary resistance, the so-called reactance
X
, is
introduced.
Figure 5.17 shows the series connection of a resistance and inductance and
the associated vectors of the currents and voltages. The voltage
V
1
is chosen
as a reference value and is drawn onto the real axis (
ϕ
v
= 0). In this example
the current
I
is turned by the zero phase angle
ϕ
i
=3
π
/4. Hence, the phase
angle between current and voltage is
ϕ
=-3
π
/4. This value in the example is
chosen arbitrarily.
With the current
I
=
I
•
e
jϕ
i
, the voltage across the resistance
R
becomes:
(5.55)
