Digital Signal Processing Reference
In-Depth Information
Common oscillating circuit in steady state
0,0045
0,0040
0,0035
0,0030
0,0025
0,0020
0,0015
0,0010
0,0005
0,0000
200
150
100
50
0
-50
- 100
- 150
- 200
ˆ
R
Uf
()
u
R
u
L
u
C
R
L
C
u
uuuu
=++
R
L
C
di
uiRL
1
=+
+
³
i t
()
ϕ
f
dt
C
Starting with complex voltage and complex current
ˆ
j
ω
t
ˆ
j
ϕ
jt
ω
ˆ
jt
ω
ˆ
j
ϕ
jt
ω
u
e
=
=
ee
i
=
Ie
=
Iee
u
i
1
ˆ
j
ω
t
ˆ
j
ω
t
ˆ
j
ω
t
ˆ
j
ω
t
j
ω
t
U
e
=
I
e
R
+
j
ω
L
Ie
+
I
e
ª º
÷
¬ ¼
e
jC
ω
75
100
125
150
f
(
)
§
1
·
§
·
Resonance frequency
Hz
ˆ
ˆ
R
es
UI
=
Rj
+
ω
L
C
−
¨
¨
¸
¸
ω
©
¹
©
¹
Amplitude and phase at the resistor R
NB:
•
Integral equation mesh rule above (sum of all
voltages is zero).
ˆ
ˆ
j
ϕ
U e
u
(
)
§
1
·
j
ϕϕ
−
u
i
Z
=
=
=
Ze
=
R
+
j
ω
L
−
¨
¸
ˆ
ˆ
j
ϕ
I
Ie
ω
C
i
©
¹
•
Ohm´s law, law of inductance, law of capacitance
are used for u
R
, u
L
and u
C
.The result is a differential
equation.
1
ω
L
−
2
1
§
·
ω
C
2
Z
=
R
+
ω
L
−
and
tan
ϕ
=
¨
¸
•
By using complex calculus, the differential
equation easily turns into an algebraic one, because
e
j
ω
t
can simply be differentiated and integrated.
ω
C
R
©
¹
1
ω
L
−
ω
C
•
In electro technology, j instead of i is used for an
imaginary quantity, i is used for current.
ϕϕϕ
=− =
n
u
i
R
U
•
Complex amplitude
U
: it includes two pieces of
information (value and angle).
1
1
§
·
Resonance:
j
ω
L
−
=
0
wL
.
ω
=
•
The frequency dependent complex resistor is called
impedance
Z
.
¨
¸
ω
C
ω
C
©
¹
•
1
1
1
1
Z
decreases most if the imaginary value of
Z
(in the
brackets) is zero. This way, the total current is
highest. This leads to the formula for resonance
frequency f
Res
ω
2
=
ω
=±
bzw f
.
=±
Res
LC
LC
2
π
LC
GAUSSian plane
GAUSSian plane
GAUSSian plane
j Im
j Im
j Im
j
ω
L
j
ω
L
j
ω
L
R
Z=R
Z
=R
R
Real
Real
Real
1
1
−
j
1
−
j
−
j
ω
C
ω
C
ω
C
f
>
f
f
<
f
f
=
f
Re
s
Re
s
Re
s
Illustration 309:
Frequency dependence of an oscillatory circuit with forced oscillation
The highest reaction or deflection (here in the form of a brownout at the resistor R) is in the resonance
frequency. If the oscillating circuit is only initiated once, it oscillates in the so- called natural frequency. It
is slightly lower than resonance frequency
.
