Aufgaben:Exercise 4.1Z: Transmission Behavior of Short Cables - Revision history

Guenter at 15:14, 9 November 2021

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Guenter at 16:35, 6 November 2021

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Guenter at 16:23, 6 November 2021

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Hwang at 14:26, 5 November 2021

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Guenter at 15:00, 15 October 2021

2021-10-15T15:00:38Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.1Z: Übertragungsmaß to Aufgaben:Exercise 4.1Z: Transmission Behavior of Short Cables

2021-10-15T14:48:44Z

Guenter moved page Aufgabe 4.1Z: Übertragungsmaß to Exercise 4.1Z: Transmission Behavior of Short Cables

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Javier: Text replacement - "Category:Exercises for Linear and Time-Invariant Systems" to "Category:Linear and Time-Invariant Systems: Exercises"

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Text replacement - "Category:Exercises for Linear and Time-Invariant Systems" to "Category:Linear and Time-Invariant Systems: Exercises"

Javier: Text replacement - "Category:Aufgaben zu Lineare zeitinvariante Systeme" to "Category:Exercises for Linear and Time-Invariant Systems"

2021-03-23T12:27:23Z

Text replacement - "Category:Aufgaben zu Lineare zeitinvariante Systeme" to "Category:Exercises for Linear and Time-Invariant Systems"

@@ Line 3: / Line 3: @@
 }}
-[[File:LZI_Z_4_1.png|right|frame|Short line section]]
+[[File:EN_LZI_Z_4_1.png|right|frame|Short line section]]
 We assume a homogeneous and reflection-free terminated line of length&nbsp; $l$&nbsp; so that the following applies to the spectral function at the output:
 :$$U_2(f)  =  U_1(f) \cdot  {\rm e}^{-\hspace{0.02cm}\gamma(f) \hspace{0.05cm} \cdot \hspace{0.05cm}l}  \hspace{0.05cm}.$$

@@ Line 35: / Line 35: @@
-''Notes:''
+Notes:
 *The exercise belongs to the chapter&nbsp;   [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory|Some Results from Line Transmission Theory]].
@@ Line 97: / Line 97: @@
 The DC signal attenuation becomes relevant,
-*if the useful signal is to be transmitted in the baseband and has a DC component, or
+*if the useful signal is to be transmitted in the baseband and has a DC component,&nbsp; or
-*if the network termination at the participant must be supplied with power from the local exchange (remote power supply).
+*if the network termination at the participant must be supplied with power from the local exchange&nbsp; ("remote power supply").
-'''(2)'''&nbsp; With&nbsp; $f = 10^{5} \ \rm  Hz$&nbsp; and the specified values, the following holds:
+'''(2)'''&nbsp; With&nbsp; $f = 10^{5} \ \rm  Hz$&nbsp; and the specified values,&nbsp; the following holds:
 :$$f \cdot  2\pi  L'  =  10^5\,\frac{1}{ {\rm s}} \cdot 2 \cdot
 ^{-3}\,\frac{\rm \Omega \cdot s}{ {\rm km}}= 200 \,\frac{\rm
@@ Line 119: / Line 119: @@
+'''(3)'''&nbsp; The limit for&nbsp; $f  &#8594; \infty$&nbsp; results if one neglects the second terms in the numerator&nbsp; $R\hspace{0.03cm}'$&nbsp; and in the denominator&nbsp; $G\hspace{0.08cm}'$&nbsp;:
 :$$\lim_{f \rightarrow \infty} \hspace{0.1cm} Z_{\rm W}(f)
   = \lim_{\omega \rightarrow \infty} \hspace{0.1cm}  \sqrt{\frac {R\hspace{0.03cm}' + {\rm j}   \cdot \omega L'}{G' + {\rm j}   \cdot \omega  C\hspace{0.03cm}'}}
   =\sqrt{\frac {2 \pi L\hspace{0.03cm}' }{2 \pi C\hspace{0.03cm}'}}=\sqrt{\frac {2 \cdot 10^{-3}\,{\rm \Omega \cdot s} }
   {2 \cdot 10^{-73}\,{\rm s/\Omega} }} \hspace{0.15cm}\underline{= 100\,{\rm \Omega }}\hspace{0.05cm}.$$
-*The approximation for the attenuation function is more difficult to derive. Starting from
+*The approximation for the attenuation function is more difficult to derive.&nbsp; Starting from
 :$$\alpha(\omega)  =  \sqrt{ {1}/{2}\cdot \left (R\hspace{0.03cm}' G\hspace{0.03cm}' - \omega^2 \cdot L\hspace{0.03cm}'  C\hspace{0.03cm}'\right)+
    {1}/{2}\sqrt{(R\hspace{0.03cm}'\hspace{0.03cm}^2 + \omega^2 \cdot L\hspace{0.03cm}'\hspace{0.03cm}^2) \cdot (G\hspace{0.03cm}'\hspace{0.03cm}^2 + \omega^2 \cdot C\hspace{0.03cm}\hspace{0.03cm}'^2)}}$$
@@ Line 137: / Line 136: @@
   \left [-1 +\sqrt{1 + \frac{R'^2}{ \omega^2 \cdot L'^2}+ \frac{G\hspace{0.03cm}'^2}{ \omega^2 \cdot C\hspace{0.03cm}'^2}} \hspace{0.1cm}
   \right].$$
-*Using the approximation $\sqrt{1 + x}\approx 1+x/2$&nbsp; valid for small&nbsp; $x$&nbsp;, one arrives at the intermediate result for (infinitely) large frequencies:
+*Using the approximation&nbsp; $\sqrt{1 + x}\approx 1+x/2$&nbsp; valid for small&nbsp; $x$,&nbsp; one arrives at the intermediate result for (infinitely) large frequencies:
 :$$2 \cdot \alpha^2(\omega \rightarrow \infty)    =   R\hspace{0.03cm}' G\hspace{0.05cm}' + \omega^2 \cdot L'
   C\hspace{0.05cm}'\cdot
@@ Line 149: / Line 148: @@
     {1}/{2}\cdot \frac{R\hspace{0.03cm}' C\hspace{0.03cm}' + G\hspace{0.03cm}'  L\hspace{0.03cm}'}{\sqrt{ C\hspace{0.03cm}'  L\hspace{0.03cm}' }}=
     {1}/{2}\cdot \left [R\hspace{0.03cm}' \cdot \sqrt{\frac{C\hspace{0.03cm}'}{L\hspace{0.03cm}'}}+G\hspace{0.03cm}' \cdot \sqrt{\frac{L\hspace{0.03cm}'}{C\hspace{0.03cm}'}}\right]\hspace{0.05cm}.$$
-*With the numerical values inserted, we get
+*With the numerical values inserted,&nbsp; we get
 :$$\alpha(f \rightarrow \infty)   =  \alpha(\omega \rightarrow \infty)
   =  {0.5\,{\rm Np/km}}\cdot \left [100 \cdot \sqrt{\frac{2 \cdot 10^{-7}}{2 \cdot10^{-3}}}+10^{-6} \cdot
    \sqrt{\frac{2 \cdot10^{-3}}{2 \cdot10^{-7}}}\right]
 \hspace{0.15cm}\underline{\approx 0.5 \, {\rm Np}/{\rm km}}\hspace{0.05cm}.$$

@@ Line 6: / Line 6: @@
 We assume a homogeneous and reflection-free terminated line of length&nbsp; $l$&nbsp; so that the following applies to the spectral function at the output:
 :$$U_2(f)  =  U_1(f) \cdot  {\rm e}^{-\hspace{0.02cm}\gamma(f) \hspace{0.05cm} \cdot \hspace{0.05cm}l}  \hspace{0.05cm}.$$
-Here &nbsp;$\gamma(f)$&nbsp; describes the complex propagation function of an extremely short line of infinitesimal length &nbsp;$dx$, which can be represented with the parameters &nbsp;$R\hspace{0.05cm}'$, &nbsp;$L\hspace{0.05cm}'$, &nbsp;$G\hspace{0.08cm}'$ and &nbsp;$C\hspace{0.08cm}'$ (see diagram) as follows:
+Here &nbsp;$\gamma(f)$&nbsp; describes the&nbsp; '''complex propagation function'''&nbsp; of an extremely short line of infinitesimal length &nbsp;$dx$,&nbsp; which can be represented with the parameters &nbsp;$R\hspace{0.05cm}'$, &nbsp;$L\hspace{0.05cm}'$, &nbsp;$G\hspace{0.08cm}'$ and &nbsp;$C\hspace{0.08cm}'$ (see diagram) as follows:
 :$$\gamma(f)  =  \sqrt{(R\hspace{0.05cm}' + {\rm j}   \cdot 2\pi f \cdot  L\hspace{0.05cm}')  \cdot  (G\hspace{0.08cm}' + {\rm j}  \cdot  2\pi f \cdot   C\hspace{0.08cm}')} =
   \alpha (f) + {\rm j}   \cdot \beta (f)\hspace{0.05cm}.$$
 The real part of &nbsp;$\gamma(f)$&nbsp; results in
-*the attenuation function per unit length &nbsp;$\alpha(f)$&nbsp;, and
+*The real part of &nbsp;$\gamma(f)$&nbsp; results in the attenuation function&nbsp;$\alpha(f)$&nbsp; (per unit length).
-*the imaginary part of the phase function &nbsp;$\beta(f)$.
+*The imaginary part of &nbsp;$\gamma(f)$&nbsp; results in the phase function &nbsp;$\beta(f)$ (per unit length).
@@ Line 22: / Line 22: @@
    {1}/{2}\cdot \sqrt{(R\hspace{0.05cm}'\hspace{0.05cm}^2 + \omega^2 \cdot L\hspace{0.05cm}'\hspace{0.05cm}^2) \cdot (G\hspace{0.08cm}'\hspace{0.05cm}^2 + \omega^2 \cdot C\hspace{0.08cm}'\hspace{0.05cm}^2)}}
   \hspace{0.1cm}\bigg |_{\hspace{0.05cm}\omega \hspace{0.05cm}= \hspace{0.05cm}2\pi f}.$$
-Another important descriptive quantity besides &nbsp;$\gamma(f)$&nbsp; is the wave impedance &nbsp;$Z_{\rm W}(f)$, which gives the relationship between voltage and current of the two running waves at each location. It holds:
+*For the attenuation function &nbsp;$a(f)$&nbsp; the pseudo unit&nbsp; "Neper"&nbsp; (Np)&nbsp; has to be added additionally and for the phase function &nbsp;$b(f)$&nbsp; "Radian"&nbsp; (rad). &nbsp;
 :$$Z_{\rm W}(f)  =  \sqrt{\frac {R\hspace{0.05cm}' + {\rm j}   \cdot \omega  L\hspace{0.05cm}'}{G\hspace{0.08cm}' + {\rm j}   \cdot \omega  C\hspace{0.08cm}'}}
   \hspace{0.1cm}\bigg |_{\hspace{0.05cm} \omega \hspace{0.05cm}= \hspace{0.05cm}2\pi f}.$$
@@ Line 35: / Line 38: @@
 *The exercise belongs to the chapter&nbsp;   [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory|Some Results from Line Transmission Theory]].
-*Use the following numerical values for the numerical calculations:
+*Use the following values for the numerical calculations:
 :$$R\hspace{0.05cm}' = 100\,\,{\rm \Omega}/{ {\rm km} }\hspace{0.05cm},\hspace{0.3cm}
   G\hspace{0.08cm}' = 1\,\,{\rm &micro; S}/{ {\rm km}}\hspace{0.05cm},\hspace{0.3cm}
@@ Line 48: / Line 51: @@
 <quiz display=simple>
-{Specify &nbsp;$\alpha(f)$, &nbsp;$\beta(f)$ and &nbsp;$Z_{\rm W}(f)$&nbsp; for frequency &nbsp;$f = 0$ (direct current).
+{Specify &nbsp;$\alpha(f)$, &nbsp;$\beta(f)$ and &nbsp;$Z_{\rm W}(f)$&nbsp; for frequency &nbsp;$f = 0$&nbsp; ("direct current").
 |type="{}"}
 $\alpha(f =0) \ =$  { 0.01 3% } $\ \rm Np/km$
@@ Line 55: / Line 58: @@
-{Calculate the attenuation function per unit length &nbsp;$\alpha(f)$&nbsp; for &nbsp;$f = 100\ \rm  kHz$.
+{Calculate the attenuation function &nbsp;$\alpha(f)$&nbsp; (per unit length)&nbsp; for &nbsp;$f = 100\ \rm  kHz$.
 |type="{}"}
 $\alpha(f = 100\ \rm  kHz) \ = \ $  { 0.486 3% } $\ \rm Np/km$
-{Give the approximations of &nbsp;$Z_{\rm W}(f)$&nbsp; and &nbsp;$\alpha(f)$&nbsp; valid for &nbsp;$f &#8594; \infty$&nbsp;.
+{Give the approximations of &nbsp;$Z_{\rm W}(f)$&nbsp; and &nbsp;$\alpha(f)$,&nbsp; valid for &nbsp;$f &#8594; \infty$&nbsp;.
 |type="{}"}
 $ Z_{\rm W}(f &#8594; \infty) \ = \ $  { 100 3% } $\ \rm \Omega$
@@ Line 66: / Line 69: @@
-{Use &nbsp;$\omega L\hspace{0.03cm}' \ll R\hspace{0.05cm}'$&nbsp; and  &nbsp;$\omega C\hspace{0.08cm}' \gg G\hspace{0.08cm}'$&nbsp; to derive an &nbsp;$\alpha(f)$&ndash; approximation for (not too) small frequencies. <br>What is the attenuation function for &nbsp;$ f = 1 \ \rm kHz$&nbsp; and &nbsp;$ f = 4 \ \rm kHz$?
+{Use &nbsp;$\omega L\hspace{0.03cm}' \ll R\hspace{0.05cm}'$&nbsp; and  &nbsp;$\omega C\hspace{0.08cm}' \gg G\hspace{0.08cm}'$&nbsp; to derive an &nbsp;$\alpha(f)$&nbsp;  approximation for&nbsp; (not too)&nbsp; small frequencies. <br>What is the attenuation function per unit length for &nbsp;$ f = 1 \ \rm kHz$&nbsp; and &nbsp;$ f = 4 \ \rm kHz$?
 |type="{}"}
 $\alpha(f = 1\  \rm kHz) \ = \ $  { 0.1 3% } $\ \rm Np/km$
@@ Line 72: / Line 75: @@
-{For the same frequency range, give a suitable approximation for the wave impedance &nbsp;$Z_{\rm W}(f)$&nbsp;. <br>What value results for &nbsp;$ f = 1 \ \rm kHz$?
+{For the same frequency range,&nbsp; give a suitable approximation for the wave impedance &nbsp;$Z_{\rm W}(f)$&nbsp;. <br>What value results for &nbsp;$ f = 1 \ \rm kHz$?
 |type="{}"}
 ${\rm Re}\{Z_{\rm W}(f = 1\ \rm kHz)\} \ =  \ $ { 500 3% } $\ \rm \Omega$

← Older revision		Revision as of 15:00, 15 October 2021
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−	[[Category:Linear and Time-Invariant Systems: Exercises\|^4.1 ~~Einige Ergebnisse der Leitungstheorie~~^]]	+	[[Category:Linear and Time-Invariant Systems: Exercises\|^4.1 Results of Line Transmission Theory^]]

@@ Line 22: / Line 22: @@
    {1}/{2}\cdot \sqrt{(R\hspace{0.05cm}'\hspace{0.05cm}^2 + \omega^2 \cdot L\hspace{0.05cm}'\hspace{0.05cm}^2) \cdot (G\hspace{0.08cm}'\hspace{0.05cm}^2 + \omega^2 \cdot C\hspace{0.08cm}'\hspace{0.05cm}^2)}}
   \hspace{0.1cm}\bigg |_{\hspace{0.05cm}\omega \hspace{0.05cm}= \hspace{0.05cm}2\pi f}.$$
-Bei der Dämpfungsfunktion &nbsp;$a(f)$&nbsp; ist zusätzlich die Pseudoeinheit &bdquo;Neper"&nbsp; (Np) hinzuzufügen und bei der Phasenfunktion &nbsp;$b(f)$&nbsp;  &bdquo;Radian" (rad).&nbsp;  Da die Leitungsbeläge jeweils auf die Leitungslänge bezogen sind, weisen &nbsp;$\alpha(f)$&nbsp; bzw. &nbsp;$\beta(f)$&nbsp; die Einheiten &bdquo;Np/km" bzw. &bdquo;rad/km" auf.
+Bei der Dämpfungsfunktion &nbsp;$a(f)$&nbsp; ist zusätzlich die Pseudoeinheit "Neper"&nbsp; (Np) hinzuzufügen und bei der Phasenfunktion &nbsp;$b(f)$&nbsp;  "Radian" (rad).&nbsp;  Da die Leitungsbeläge jeweils auf die Leitungslänge bezogen sind, weisen &nbsp;$\alpha(f)$&nbsp; bzw. &nbsp;$\beta(f)$&nbsp; die Einheiten "Np/km" bzw. "rad/km" auf.
 Eine weitere wichtige Beschreibungsgröße neben &nbsp;$\gamma(f)$&nbsp; ist der Wellenwiderstand &nbsp;$Z_{\rm W}(f)$, der an jedem Ort den Zusammenhang zwischen Spannung und Strom der beiden laufenden Wellen angibt. Es gilt:
@@ Line 107: / Line 107: @@
 ^{-7}\,\frac{\rm  s}{ {\rm \Omega \cdot km}}= 0.02
 \,\frac{\rm 1 }{ {\rm \Omega \cdot km}} \hspace{0.05cm}.$$
-Damit ergibt sich für das Dämpfungsmaß in &bdquo;Np/km":
+Damit ergibt sich für das Dämpfungsmaß in "Np/km":
 :$$\alpha(f = 100\,{\rm kHz})
 =  \sqrt{ {1}/{2}\cdot \left (100 \cdot 10^{-6} - 200 \cdot 0.02 \right)+

← Older revision		Revision as of 12:40, 23 March 2021
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−	[[Category:~~Exercises for~~ Linear and Time-Invariant Systems\|^4.1 Einige Ergebnisse der Leitungstheorie^]]	+	[[Category:Linear and Time-Invariant Systems: Exercises\|^4.1 Einige Ergebnisse der Leitungstheorie^]]

← Older revision		Revision as of 12:27, 23 March 2021
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−	[[Category:~~Aufgaben zu Lineare zeitinvariante Systeme~~\|^4.1 Einige Ergebnisse der Leitungstheorie^]]	+	[[Category:Exercises for Linear and Time-Invariant Systems\|^4.1 Einige Ergebnisse der Leitungstheorie^]]