Aufgaben:Exercise 3.3: p-Transfer Function - Revision history

Guenter at 13:16, 13 October 2021

2021-10-13T13:16:49Z

Guenter at 13:06, 13 October 2021

2021-10-13T13:06:17Z

Oezer at 22:46, 11 October 2021

2021-10-11T22:46:33Z

Oezer at 22:45, 11 October 2021

2021-10-11T22:45:33Z

Oezer at 22:44, 11 October 2021

2021-10-11T22:44:11Z

Oezer at 22:43, 11 October 2021

2021-10-11T22:43:08Z

Oezer at 22:42, 11 October 2021

2021-10-11T22:42:31Z

Oezer at 22:41, 11 October 2021

2021-10-11T22:41:41Z

Oezer at 22:39, 11 October 2021

2021-10-11T22:39:59Z

Oezer at 22:39, 11 October 2021

2021-10-11T22:39:44Z

@@ Line 121: / Line 121: @@
-'''(2)'''&nbsp; <u>Suggested solution 2</u> is correct:
+'''(2)'''&nbsp; <u>Suggested solution 2</u>&nbsp; is correct:
 *Replacing&nbsp; $p$&nbsp; by&nbsp; ${\rm j } \cdot 2\pi f$ the following is obtained:
 :$$H(f)= \frac {1 - (2\pi f)^2 \cdot LC}
@@ Line 127: / Line 127: @@
   \hspace{0.05cm} .$$
-*So, there is always a frequency at which the numerator is zero, namely the resonance frequency of&nbsp; $L$&nbsp; and&nbsp; $C$.
+*So,&nbsp; there is always a frequency at which the numerator is zero, namely the resonance frequency of&nbsp; $L$&nbsp; and&nbsp; $C$.
 *For this frequency &nbsp;$f_0 = 1 \ \rm MHz/2\pi$&nbsp; the series connection of &nbsp;$L$&nbsp; and &nbsp;$C$&nbsp; acts like a short circuit.
-*From this it follows:&nbsp; Regardless of the values of &nbsp;$R$, &nbsp;$L$ and &nbsp;$C$ it is a&nbsp; $\rm band&ndash;stop \:filter$.
+*From this it follows:&nbsp; Regardless of the values of &nbsp;$R$, &nbsp;$L$&nbsp; and &nbsp;$C$ it is a&nbsp; $\rm band&ndash;stop \:filter$.
@@ Line 158: / Line 158: @@
   s}}  \hspace{0.05cm}.$$
-*The normalization of the variable &nbsp;$p$&nbsp; and all poles and zeros to the unit &nbsp;$(  \rm 1/&micro; s)$&nbsp; simplifies the numerical evaluation, especially in the time domain.
+*The normalization of the variable &nbsp;$p$&nbsp; and all poles and zeros to the unit &nbsp;$(  \rm 1/&micro; s)$&nbsp; simplifies the numerical evaluation,&nbsp; especially in the time domain.
-*If the unit is dispensed with altogether, all&nbsp; $t$&ndash;values are obtained in microseconds.
+*If the unit is dispensed with altogether,&nbsp; all&nbsp; $t$&ndash;values are obtained in microseconds.
-'''(6)'''&nbsp; If the denominator polynomial is set &nbsp;$N(p) = 0$, the following conditional equation arises as a result:
+'''(6)'''&nbsp; If the denominator polynomial is set &nbsp;$N(p) = 0$,&nbsp; the following conditional equation arises as a result:
 :$$p^2 + 2A \cdot p + B^2 = 0 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
   p_{\rm x1,\hspace{0.05cm}2}= -A \pm \sqrt{A^2 - B^2}
@@ Line 183: / Line 183: @@
-'''(7)'''&nbsp; <u>Suggested solution 1</u> is correct:
+'''(7)'''&nbsp; <u>Suggested solution 1</u>&nbsp; is correct:
 *Since only one of the components is to be changed, &nbsp;$L$&nbsp; and &nbsp;$C$&nbsp; must remain the same because otherwise the zeros would also be shifted.
 * The resistance value &nbsp;$R$&nbsp; must be changed.
@@ Line 190: / Line 190: @@
 '''(8)'''&nbsp; According to the result in subtask&nbsp; '''(7)'''&nbsp; there is a double pole for &nbsp;$\underline {A = B = 2 \cdot 10^{-6} \cdot \rm  1/s}$.
-*To do this, the ohmic resistance must be reduced from &nbsp;$50 \ \rm \Omega$&nbsp; to &nbsp;$40 \ \rm \Omega$&nbsp;.
+*To do this,&nbsp; the ohmic resistance must be reduced from &nbsp;$50 \ \rm \Omega$&nbsp; to &nbsp;$40 \ \rm \Omega$&nbsp;.
-*Then, the double pole is at &nbsp;${-2 \cdot 10^{6} \cdot \rm  1/s}$.
+*Then,&nbsp; the double pole is at &nbsp;${-2 \cdot 10^{6} \cdot \rm  1/s}$.
 *Or with another normalization at &nbsp;${-2 \cdot \rm  (1/&micro; s)}$.
 {{ML-Fuß}}

@@ Line 4: / Line 4: @@
 [[File:P_ID1765__LZI_A_3_3.png|right|frame|Considered two-port network]]
-Any linear time-invariant system that can be realized by a circuit of discrete time-constant components (resistances &nbsp;$R$, capacitances &nbsp;$C$, inductances &nbsp;$L$, amplifier elements, etc.) is causal and also has a fractional&ndash;rational&nbsp; $p$&ndash;transfer function of the form
+Any linear time-invariant system that can be realized by a circuit of discrete time-constant components&nbsp; (resistances &nbsp;$R$,&nbsp; capacitances &nbsp;$C$,&nbsp; inductances &nbsp;$L$,&nbsp; amplifier elements, etc.)&nbsp; is causal and has a fractional&ndash;rational&nbsp; $p$&ndash;transfer function of the form
 :$$H_{\rm L}(p)= \frac {A_Z \cdot p^Z +\text{ ...} + A_1 \cdot p + A_0}
   {B_N \cdot p^N + \text{ ...} + B_1 \cdot p + B_0}= \frac {Z(p)}{N(p)}
@@ Line 21: / Line 21: @@
 The&nbsp; $Z + N + 1$&nbsp; parameters mean:
-* $K = A_Z/B_n$&nbsp; is a constant factor. If &nbsp;$Z = N$ applies, then this is dimensionless.
+* $K = A_Z/B_n$&nbsp; is a constant factor.&nbsp; If &nbsp;$Z = N$ applies, then this is dimensionless.
-* The solutions of the equation &nbsp;$Z(p) = 0$&nbsp; yield the &nbsp;$Z$ zeros &nbsp;$p_{{\rm o}1}$, ... , $p_{{\rm o}N}$&nbsp; of &nbsp;$H_{\rm L}(p)$.
+* The solutions of the equation &nbsp;$Z(p) = 0$&nbsp; yield the &nbsp;$Z$ zeros &nbsp;$p_{{\rm o}1}$, ... , $p_{{\rm o}Z}$&nbsp; of &nbsp;$H_{\rm L}(p)$.
 * The zeros of the denominator polynomial &nbsp;$N(p)$&nbsp; yield the &nbsp;$N$ &nbsp;poles &nbsp;$p_{{\rm x}1}$, ... , $p_{{\rm x}N}$&nbsp; of the transfer function.
@@ Line 37: / Line 37: @@
-''Please note:''
+Please note:
 *The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function|Laplace Transform and p-Transfer Function]].

@@ Line 158: / Line 158: @@
   s}}  \hspace{0.05cm}.$$
-*The normalization of the variables &nbsp;$p$&nbsp; and all poles and zeros to the unit &nbsp;$(  \rm 1/&micro; s)$&nbsp; simplifies the numerical evaluation, especially in the time domain.
+*The normalization of the variable &nbsp;$p$&nbsp; and all poles and zeros to the unit &nbsp;$(  \rm 1/&micro; s)$&nbsp; simplifies the numerical evaluation, especially in the time domain.
 *If the unit is dispensed with altogether, all&nbsp; $t$&ndash;values are obtained in microseconds.

@@ Line 129: / Line 129: @@
 *So, there is always a frequency at which the numerator is zero, namely the resonance frequency of&nbsp; $L$&nbsp; and&nbsp; $C$.
 *For this frequency &nbsp;$f_0 = 1 \ \rm MHz/2\pi$&nbsp; the series connection of &nbsp;$L$&nbsp; and &nbsp;$C$&nbsp; acts like a short circuit.
-*From this it follows:&nbsp; Regardless of the values of &nbsp;$R$, &nbsp;$L$ and &nbsp;$C$ it is a&nbsp; $\rm band-stop \:filter$.
+*From this it follows:&nbsp; Regardless of the values of &nbsp;$R$, &nbsp;$L$ and &nbsp;$C$ it is a&nbsp; $\rm band&ndash;stop \:filter$.

@@ Line 129: / Line 129: @@
 *So, there is always a frequency at which the numerator is zero, namely the resonance frequency of&nbsp; $L$&nbsp; and&nbsp; $C$.
 *For this frequency &nbsp;$f_0 = 1 \ \rm MHz/2\pi$&nbsp; the series connection of &nbsp;$L$&nbsp; and &nbsp;$C$&nbsp; acts like a short circuit.
-*From this it follows:&nbsp; Regardless of the values of &nbsp;$R$, &nbsp;$L$ and &nbsp;$C$ it is a&nbsp; $\rm band-stop &nbsp;filter$.
+*From this it follows:&nbsp; Regardless of the values of &nbsp;$R$, &nbsp;$L$ and &nbsp;$C$ it is a&nbsp; $\rm band-stop \:filter$.

@@ Line 66: / Line 66: @@
-{Express &nbsp;$H_{\rm L}(p)$&nbsp; in pole&ndash;zero& form.&nbsp; How many zeros&nbsp; $(Z)$&nbsp; and poles&nbsp; $(N)$&nbsp; are there?&nbsp; What is the constant factor &nbsp;$K$?
+{Express &nbsp;$H_{\rm L}(p)$&nbsp; in pole&ndash;zero form.&nbsp; How many zeros&nbsp; $(Z)$&nbsp; and poles&nbsp; $(N)$&nbsp; are there?&nbsp; What is the constant factor &nbsp;$K$?
 |type="{}"}
 $Z \ = \ $  { 2 3% }