Thus, the $p$–transfer function is:
:$$H_{\rm L}(p)= K \cdot \frac {p - p_{\rm o }}
:$$H_{\rm L}(p)= K \cdot \frac {p - p_{\rm o }}{(p - p_{\rm x 1})(p - p_{\rm x 2})}\hspace{0.05cm} .$$
{(p - p_{\rm x 1})(p - p_{\rm x 2})}
\hspace{0.05cm} .$$
Considering the substitution $p = {\rm j} \cdot 2 \pi f$, the conventional transfer function can be specified, which is also called frequency response:
Considering the substitution $p = {\rm j} \cdot 2 \pi f$, the conventional transfer function can be specified, which is also called "frequency response":
From this equation it can also be seen the relationship between
*the transfer function $H(f)$,
*the transfer function $H(f)$,
*the attenuation function $a(f)$ and
*the attenuation function $a(f)$ and
*the phase function $b(f)$
*the phase function $b(f)$.
can also be seen.
The attenuation and phase values can be determined as follows for a frequency $f$ indirectly specified by the point $p = {\rm j} \cdot 2 \pi f$ :
:$$a(f)\hspace{0.15cm}{\rm in}\hspace{0.15cm}{\rm Np} \hspace{0.25cm} = \hspace{0.2cm} -{\rm ln} \hspace{0.1cm} K
The corresponding magnitudes $|R_{\rm o}|$, $|R_{\rm x1}|$ and $|R_{\rm x1}|$ as well as the angles $\phi_{\rm o}$, $\phi_{\rm x1}$ and $\phi_{\rm x2}$ can be taken from the graph .
The attenuation and phase values can be determined as follows for a frequency $f$ indirectly specified by the point $p = {\rm j} \cdot 2 \pi f$ :
The corresponding magnitudes $|R_{\rm o}|$, $|R_{\rm x1}|$ and $|R_{\rm x2}|$ as well as the angles $\phi_{\rm o}$, $\phi_{\rm x1}$ and $\phi_{\rm x2}$ can be taken from the graph .
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''Please note:''
Please note:
*The exercise belongs to the chapter [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function|Laplace Transform and p-Transfer Function]].
*The exercise belongs to the chapter [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function|Laplace Transform and p-Transfer Function]].
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<quiz display=simple>
<quiz display=simple>
{Berechnen Sie $H(f)$. Wie groß ist dessen Betrag bei sehr großen Frequenzen?
{Compute $H(f)$. What is its magnitude at very large frequencies?
|type="{}"}
|type="{}"}
$|H(f → ∞)| \ = \ $ { 0. }
$|H(f → ∞)| \ = \ $ { 0. }
{Berechnen Sie den Betragsfrequenzgang und den Dämpfungswert für $f → 0$.
{Compute the magnitude frequency response and the attenuation value for $f → 0$.
|type="{}"}
|type="{}"}
$|H(f = 0)| \ = \ $ { 0.278 3% }
$|H(f = 0)| \ = \ $ { 0.278 3% }
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{Berechnen Sie gemäß der beschriebenen Vorgehensweise den Dämpfungswert bei $f =4/(2 \pi)$ in Neper $\rm(Np)$ und Dezibel $\rm(dB)$.
{Compute the attenuation value at $f =4/(2 \pi)$ in neper $\rm(Np)$ and decibel $\rm(dB)$ according to the described approach.
|type="{}"}
|type="{}"}
$a(f = 2/ \pi)\ = \ $ { 0.155 3% } $\ \rm Np$
$a(f = 2/ \pi)\ = \ $ { 0.155 3% } $\ \rm Np$
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{Berechnen Sie gemäß der beschriebenen Vorgehensweise den Phasenwert bei der Frequenz $f = 4/(2 \pi)$.
{Compute the phase value at frequency $f = 4/(2 \pi)$ according to the described approach.
'''(3)''' According to the detailed description in the [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function#Graphical_determination_of_attenuation_and_phase|theory part]], the following holds for the attenuation function:
[[File:P_ID2843__LZI_A_3_4.png|right|frame|Pol–Nullstellen–Diagramm und einige Hilfsgrößen]]
'''(3)''' Entsprechend der detaillierten Beschreibung im [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function#Graphical_determination_of_attenuation_and_phase|Theorieteil]] gilt für die Dämpfungsfunktion:
:$$a(f)= -{\rm ln} \hspace{0.1cm} K
*Furthermore, the additional unit "neper" $\rm (Np)$ must be taken into account.
*The attenuation at $f = 2/\pi$ is searched-for. For this, we set $p = {\rm j} \cdot 2 \pi f = 4$ and determine the following distances:
$$H_{\rm L}(p)= K \cdot \frac {p - p_{\rm o }}{(p - p_{\rm x 1})(p - p_{\rm x 2})}\hspace{0.05cm} .$$
Considering the substitution $p = {\rm j} \cdot 2 \pi f$, the conventional transfer function can be specified, which is also called "frequency response":
The corresponding magnitudes $|R_{\rm o}|$, $|R_{\rm x1}|$ and $|R_{\rm x2}|$ as well as the angles $\phi_{\rm o}$, $\phi_{\rm x1}$ and $\phi_{\rm x2}$ can be taken from the graph .
