Digital Signal Processing Reference
In-Depth Information
At this point it is necessary to address an apparent inconsistency. In Illustration 308,
something is used as the “generator voltage” and total current that physically or by
measurement techniques does not exist at all:
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
jt
(
ωϕ
+
)
j
ϕ
j t
ω
j t
ω
jt
(
ωϕ
+
)
j
ϕ
j t
ω
j t
ω
u
t
( )
=
Ue
=
Ue
e
=
U
e
respectively ( )
i t
=
Ie
=
Ie
e
=
Ie
u
u
i
i
Here,
i
are „zero phase angles”, the angles of the vectors in the moment of t = 0.
This phase information together with the amplitude information results in the
complex
amplitude
both for voltage and current.
ϕ
and
ϕ
u
Properly, however, sinusoidal, measurable values should be used as voltage and current.
Therefore the calculation of the oscillating circuit is checked using the following values:
1
1
ˆ
ˆ
ˆ
j t
ω
−
j t
ω
j t
ω
−
j t
ω
u t
( )
=
U
cos(
ω
t
)
=
U
e
(
+
e
)
respectively
( )
i t
=
I
e
(
+
e
)
2
2
These values are now used in the differential equation:
di
uu u u iRL
1
³
=++=+
+
i t
R
L
C
dt
C
ˆ
ˆ
ˆ
11
U
R
I
I
§
1
·
(
)
(
)
(
)
ˆ
jt
ω
−
jt
ω
jt
ω
−
jt
ω
jt
ω
−
jt
ω
jt
ω
−
jt
ω
u
=
e
+
e
=
I
e
+
e
+
L
j
ω
e
−
j
ω
e
+
e
−
e
¨
¸
2
2
2
Cj
2
ω
j
ω
©
¹
ª
1
º
(
)
(
)
(
)
(
)
ˆ
ˆ
jt
ω
−
jt
ω
jt
ω
−
jt
ω
jt
ω
−
jt
ω
jt
ω
−
jt
ω
U
e
+
e
=
I
Re
+
e
+
j Le
ω
−
e
+
e
−
e
=
«
»
jC
ω
¬
¼
-
½
ª
§
1
º
ª
1
º
·
§
·
(
)
ˆ
ˆ
jt
ω
−
jt
ω
jt
ω
−
jt
ω
U
e
+
e
=
I
R j
+
ω
L
−
e
+−
Rj
ω
L
−
e
®
¾
¨
¸
¨
¸
«
»
«
»
ω
C
ω
C
©
¹
©
¹
¬
¼
¬
¼
¯
¿
jt
ω
Thus, this is the solution for the left- turning vector
e
as well as the
−
jt
ω
right- turning vector
e
.
1
§
·
In the literature, the impedance
Z
for the oscillating
circuit is
Z
Rj L
=+
ω
−
¨
¸
ω
C
©
¹
1
§
·
∗
Thus, the
complex conjugated
impedance is
Z
=−
R
j
ω
L
−
¨
¸
ω
C
©
¹
Hence:
ˆ
U
(
)
{
}
(
) (
)
ˆ
ˆ
j
ω
t
−
j
ω
t
j
ω
t
−
j
ω
t
j
ω
t
−
j
ω
t
j
ω
t
−
j
ω
t
U
e
+
e
=
I
Z
e
+
Z
∗
e
bzw.
e
+
e
=
Z
e
+
Z
∗
e
ˆ
I
The frequency dependency of the amplitudes and the angles of both opposed rotating vec-
tors can clearly be seen. This is given by the imaginary values. Here the imaginary values
are physically existent and measurable! The already familiar
locus curve
results in a graph
(Illustrations 110 to Illustration112).
The locus curve of the oscillating circuit or the band- pass is traversed (contrariwise) twice
over the entire positive
and
negative frequency domain (Illustration 309).
