Aufgaben:Exercise 4.3: Operational Attenuation - Revision history

Guenter at 15:02, 12 November 2021

2021-11-12T15:02:35Z

Guenter at 14:51, 12 November 2021

2021-11-12T14:51:47Z

Hwang at 09:06, 8 November 2021

2021-11-08T09:06:19Z

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Guenter: Guenter moved page Aufgaben:Exercise 4.3: Effective Attenuation to Aufgaben:Exercise 4.3: Operational Attenuation

2021-10-29T14:46:04Z

Guenter moved page Exercise 4.3: Effective Attenuation to Exercise 4.3: Operational Attenuation

Guenter at 15:01, 15 October 2021

2021-10-15T15:01:51Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.3: Betriebsdämpfung to Aufgaben:Exercise 4.3: Effective Attenuation

2021-10-15T14:54:44Z

Guenter moved page Aufgabe 4.3: Betriebsdämpfung to Exercise 4.3: Effective Attenuation

Javier: Text replacement - "„" to """

2021-05-28T14:48:47Z

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2021-05-28T14:33:03Z

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

2021-03-23T12:40:22Z

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"

2021-03-23T12:27:25Z

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

@@ Line 86: / Line 86: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; With resistance matching&nbsp; $(R_1 = R_2 =Z_{\rm W})$&nbsp; only the first of the four summands remains:
-:$${a}_{\rm B} =  \alpha \cdot l = 0.1\,{\rm Np/km} \cdot 2\,{\rm km} \hspace{0.15cm}\underline{= 0.2\,{\rm Np}}\hspace{0.05cm}.$$
+:$${a}_{\rm B} =  {a}_{\rm W} =\alpha \cdot l = 0.1\,{\rm Np/km} \cdot 2\,{\rm km} \hspace{0.15cm}\underline{= 0.2\,{\rm Np}}\hspace{0.05cm}.$$
@@ Line 99: / Line 98: @@
 \hspace{0.2cm}\Rightarrow \hspace{0.2cm} {\rm
 ln}\hspace{0.1cm}|q_2| \hspace{0.15cm}\underline{= 0.553 \,{\rm Np}}\hspace{0.05cm}.$$
@@ Line 110: / Line 108: @@
    e}^{-4}\hspace{0.15cm}\underline{= 0.183}
   \hspace{0.05cm}.$$
 '''(4)'''&nbsp; <u>Both statements are correct</u>:
-*With constructive superposition is&nbsp; ${a}_{\rm WWD} = {\rm ln}\hspace{0.1cm}A < 0 \; \Rightarrow \; A < 1$&nbsp; and minimal.
+*With constructive superposition is &nbsp; &rArr; &nbsp; ${a}_{\rm IA} = {\rm ln}\hspace{0.1cm}A < 0 \; \Rightarrow \; A < 1$&nbsp; and minimal.
-*In contrast, the maximum value of&nbsp; $A&nbsp;$&nbsp; (for which $A > 1$) &nbsp; causes positive interaction attenuation, i.e., additional attenuation of the useful signal due to destructive superposition of outgoing&ndash; and returning wave.
+*In contrast, the maximum value of&nbsp; $A&nbsp;$&nbsp; (for which $A > 1$) &nbsp; causes positive interaction attenuation, i.e., additional attenuation of the useful signal due to destructive superposition of outgoing wave and returning wave.
-'''(5)'''&nbsp; In the last subtask, it was shown that constructive superposition is equivalent to the minimization of
+'''(5)'''&nbsp; In the last subtask,&nbsp; it was shown that constructive superposition is equivalent to the minimization of
 :$$A   =  |1- r_\alpha \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2\hspace{0.05cm}\cdot\hspace{0.05cm}
    \beta \hspace{0.05cm}\cdot\hspace{0.05cm} l}|=  |1- r_\alpha \cdot \cos(2  \beta  l)+ {\rm j} \cdot \sin(2  \beta
@@ Line 128: / Line 124: @@
 *The minimum results for
 :$$\cos(2  \beta  l) = +1 \hspace{0.2cm}\Rightarrow \hspace{0.2cm}2  \beta
-  l= \pi, 2\pi, 3\pi, ... \hspace{0.2cm}\Rightarrow
+  l= \pi,\ 2\pi,\ 3\pi, ... \hspace{0.2cm}\Rightarrow
   \hspace{0.2cm}\beta_{\rm min} = {\pi}({2l})\hspace{0.15cm}\underline{= 0.785\,{\rm rad/km}}
   \hspace{0.05cm}.$$
 *In contrast, destructive superposition occurs if the phase function (per unit length) satisfies the following condition:
 :$$\cos(2  \beta  l) = -1 \hspace{0.2cm}\Rightarrow \hspace{0.2cm}2  \beta
-  l= {\pi}/{2}, {3\pi}/{2}, {5\pi}/{2},\text{ ...}$$
+  l= {\pi}/{2},\ {3\pi}/{2},\ {5\pi}/{2},\text{ ...}$$
-'''(6)'''&nbsp; The argument&nbsp; $A  = \sqrt {1+ {r_\alpha}^2- 2 \cdot r_\alpha \cdot \cos(2  \beta  l)}$&nbsp; can become maximal&nbsp; $A = 2$&nbsp; &nbsp; &rArr; &nbsp; ${\rm Max}\ [a_\text{WWD}] \; \underline {= 0.693 \ \rm Np}$.
+'''(6)'''&nbsp; The argument&nbsp; $A  = \sqrt {1+ {r_\alpha}^2- 2 \cdot r_\alpha \cdot \cos(2  \beta  l)}$&nbsp; can become maximum&nbsp; $A = 2$&nbsp; &nbsp; &rArr; &nbsp; ${\rm Max}\ [a_\text{IA}] \; \underline {= 0.693 \ \rm Np}$.
 The following requirements must be met for this:
 * not terminated line&nbsp; $(r_1 = r_2  = 1)$,
-* short cable length, so the term&nbsp; $\alpha \cdot l$&nbsp; is not effective&nbsp; $(r_\alpha = 1)$,
+* short cable length,&nbsp; so the term&nbsp; $\alpha \cdot l$&nbsp; is not effective&nbsp; $(r_\alpha = 1)$,
 * phase progression according to the subtask&nbsp; '''(5)'''.
 {{ML-Fuß}}

@@ Line 4: / Line 4: @@
 [[File:P_ID1800__LZI_A_4_3.png|right|frame|Considered line model]]
-If a communication link of length &nbsp;$l$&nbsp; is not terminated at both ends with its wave impedance &nbsp;$Z_{\rm W}$&nbsp;, reflections will always occur.
+If a communication link of length &nbsp;$l$&nbsp; is not terminated at both ends with its wave impedance &nbsp;$Z_{\rm W}$,&nbsp; reflections will always occur.
-Instead of the wave attenuation &nbsp;$a_{\rm W} = \alpha \cdot l$&nbsp; one has to consider in this case the operating attenuation &nbsp;$a_{\rm B}$&nbsp;, which is given here without frequency dependence.&nbsp; That means: We always only consider a single frequency $f_0$:
+Instead of the wave attenuation &nbsp;$a_{\rm W} = \alpha \cdot l$&nbsp; one has to consider in this case the&nbsp; '''operating attenuation''' &nbsp;$a_{\rm B}$&nbsp; (German:&nbsp; "Betriebsdämpfung" &nbsp; &rArr; &nbsp; subscipt&nbsp; "B"),&nbsp; which is given here without frequency dependence.&nbsp; That means:&nbsp; We always only consider a single frequency $f_0$:
-:$${ a}_{\rm B} = { a}_{\rm B}(f_0)= { a}_{\rm W}+ {\rm ln}\hspace{0.1cm} |q_1|+{\rm ln}\hspace{0.1cm} |q_2|+{ a}_{\rm WWD}\hspace{0.05cm}.$$
+:$${ a}_{\rm B} = { a}_{\rm B}(f_0)= { a}_{\rm W}+ {\rm ln}\hspace{0.1cm} |q_1|+{\rm ln}\hspace{0.1cm} |q_2|+{ a}_{\rm IA}\hspace{0.05cm}.$$
-<br clear=all>
-The four parts &ndash; all with the pseudo unit "Neper"&nbsp;  (Np) &ndash; describe the following facts:
+The four parts&nbsp; &ndash; all with the pseudo unit "Neper"&nbsp;  (Np) &ndash;&nbsp; describe the following facts:
-* The first summand &nbsp;$a_{\rm W} = \alpha \cdot l$&nbsp; models the&nbsp; '''wave attenuation'''&nbsp; of the wave propagating along the line.&nbsp; Note that attenuations are denoted by "$a$", while the attenuation function (kilometric attenuation) is denoted by "$\alpha$" &nbsp; &rArr; &nbsp; read: "alpha".
+* The first summand &nbsp;$a_{\rm W} = \alpha \cdot l$&nbsp; models the&nbsp; '''wave attenuation'''&nbsp; of the wave propagating along the line.&nbsp; Note that attenuations are denoted by&nbsp; "$a$",&nbsp; while the attenuation function per unit length is denoted by&nbsp; "$\alpha$" &nbsp; &rArr; &nbsp; read: "alpha".
-* The second summand gives the&nbsp; '''transmitter-side reflection loss'''&nbsp; This term takes into account the power loss due to reflections at the "transmitter &rarr; line" transition:
+* The second summand gives the&nbsp; '''transmitter-side reflection loss'''.&nbsp; This term takes into account the power loss due to reflections at the&nbsp; "transmitter &rarr; line"&nbsp; transition:
 :$${\rm ln}\hspace{0.1cm} |q_1|= {\rm ln}\hspace{0.1cm}\frac {R_1 + Z_{\rm W}}{2 \cdot \sqrt{R_1 \cdot Z_{\rm W}}}
   \hspace{0.05cm}.$$
-* In an analogous way applies to the&nbsp; '''receiver-side reflection loss'''&nbsp; at the end of the line &nbsp; &rArr; &nbsp; transition "line &rarr; receiver":
+* In an analogous way applies to the&nbsp; '''receiver-side reflection loss'''&nbsp; at the end of the line &nbsp; &rArr; &nbsp; transition&nbsp; "line &rarr; receiver":
 :$${\rm ln}\hspace{0.1cm} |q_2|= {\rm ln}\hspace{0.1cm}\frac {R_2 + Z_{\rm W}}{2 \cdot \sqrt{R_2 \cdot Z_{\rm W}}}
   \hspace{0.05cm}.$$
-* The&nbsp; '''interaction attenuation'''&nbsp; describes the signal attenuation due to the effect of a doubly reflected wave, which can be constructively or destructively superimposed on the useful signal.&nbsp; For this last part
+* The&nbsp; '''interaction attenuation&nbsp; (IA)'''&nbsp; describes the signal attenuation due to the effect of a doubly reflected wave,&nbsp; which can be constructively or destructively superimposed on the useful signal.&nbsp; For this last part holds:
-:$${ a}_{\rm WWD} = {\rm ln}\hspace{0.1cm} |1- r_1 \cdot r_2 \cdot {\rm e}^{- 2\hspace{0.05cm}\cdot\hspace{0.05cm}
+:$${ a}_{\rm IA} = {\rm ln}\hspace{0.1cm} |1- r_1 \cdot r_2 \cdot {\rm e}^{- 2\hspace{0.05cm}\cdot\hspace{0.05cm}
    \gamma \hspace{0.05cm}\cdot\hspace{0.05cm} l}|$$
-:we use the following equations and nomenclature in this task:
+:we use the following equations and nomenclature in this exercise:
-:$${a}_{\rm WWD} = {\rm ln}\hspace{0.1cm}A, \hspace{0.3cm}A = |1- r_\alpha \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \hspace{0.05cm}\cdot\hspace{0.05cm}
+:$${a}_{\rm IA} = {\rm ln}\hspace{0.1cm}A, \hspace{0.3cm}A = |1- r_\alpha \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \hspace{0.05cm}\cdot\hspace{0.05cm}
    \beta \hspace{0.05cm}\cdot\hspace{0.05cm} l}|
   \hspace{0.05cm},\hspace{0.3cm}r_\alpha = r_1 \cdot r_2\cdot {\rm e}^{-2 \hspace{0.05cm}\cdot\hspace{0.05cm}
@@ Line 34: / Line 34: @@
-''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]].
 *You can find essential information on the page&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory#Influence_of_reflections_-_operational_attenuation|Influence of reflections - operational attenuation]]
@@ Line 47: / Line 47: @@
 <quiz display=simple>
-{What would be the value of the operating attenuation <u>with matching</u>, i.e. if there were no reflections?
+{What would be the value of the operating attenuation&nbsp; <u>with matching</u>,&nbsp; i.e. if there were no reflections?
 |type="{}"}
 $a_{\rm B} \ = \ $  { 0.2 3% } $\ \rm Np$
@@ Line 64: / Line 64: @@
 $r_\alpha\ = \ $ { 0.183 3% }
-{Which condition must the auxiliary &nbsp;$A = \vert 1- r_\alpha \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \hspace{0.05cm}\cdot\hspace{0.05cm}  \beta \hspace{0.05cm}\cdot\hspace{0.05cm} l}\vert $&nbsp; satisfy, <br>so that constructive or destructive superposition occurs with respect to the interaction attenuation?
+{Which condition must the auxiliary &nbsp;$A = \vert 1- r_\alpha \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \hspace{0.05cm}\cdot\hspace{0.05cm}  \beta \hspace{0.05cm}\cdot\hspace{0.05cm} l}\vert $&nbsp; satisfy,&nbsp; so that constructive or destructive superposition occurs with respect to the interaction attenuation?
 |type="[]"}
 + $A$&nbsp; is minimal &nbsp; &#8658; &nbsp; constructive superposition,
@@ Line 77: / Line 77: @@
 {What is the maximum size of the interaction attenuation component?&nbsp; What are the requirements for this?
 |type="{}"}
-${\rm Max}\ \big [a_\text{WWD}\big ] \ = \ $ { 0.693 3% } $\ \rm Np$
+${\rm Max}\ \big [a_\text{IA}\big ] \ = \ $ { 0.693 3% } $\ \rm Np$

← Older revision		Revision as of 15:01, 15 October 2021
Line 146:		Line 146:


−	[[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 9: / Line 9: @@
 :$${ a}_{\rm B} = { a}_{\rm B}(f_0)= { a}_{\rm W}+ {\rm ln}\hspace{0.1cm} |q_1|+{\rm ln}\hspace{0.1cm} |q_2|+{ a}_{\rm WWD}\hspace{0.05cm}.$$
 <br clear=all>
-Die vier Anteile &ndash; alle mit der Pseudoeinheit &bdquo;Neper"&nbsp;  (Np) &ndash; beschreiben folgende Sachverhalte:
+Die vier Anteile &ndash; alle mit der Pseudoeinheit "Neper"&nbsp;  (Np) &ndash; beschreiben folgende Sachverhalte:
-* Der erste Summand &nbsp;$a_{\rm W} = \alpha \cdot l$&nbsp; modelliert die&nbsp; '''Wellendämpfung'''&nbsp; der sich entlang der Leitung ausbreitenden Welle.&nbsp; Beachten Sie, dass Dämpfungen mit &bdquo;$a$" bezeichnet werden, während das Dämpfungsmaß (kilometrische Dämpfung) mit &bdquo;$\alpha$" &nbsp; &rArr; &nbsp; sprich: &bdquo;alpha" gekennzeichnet ist.
+* Der erste Summand &nbsp;$a_{\rm W} = \alpha \cdot l$&nbsp; modelliert die&nbsp; '''Wellendämpfung'''&nbsp; der sich entlang der Leitung ausbreitenden Welle.&nbsp; Beachten Sie, dass Dämpfungen mit "$a$" bezeichnet werden, während das Dämpfungsmaß (kilometrische Dämpfung) mit "$\alpha$" &nbsp; &rArr; &nbsp; sprich: "alpha" gekennzeichnet ist.
-* Der zweite Summand gibt die&nbsp; '''senderseitige Stoßdämpfung'''&nbsp; an.&nbsp; Dieser Term berücksichtigt den Leistungsverlust durch Reflexionen am Übergang &bdquo;Sender &rarr; Leitung":
+* Der zweite Summand gibt die&nbsp; '''senderseitige Stoßdämpfung'''&nbsp; an.&nbsp; Dieser Term berücksichtigt den Leistungsverlust durch Reflexionen am Übergang "Sender &rarr; Leitung":
 :$${\rm ln}\hspace{0.1cm} |q_1|= {\rm ln}\hspace{0.1cm}\frac {R_1 + Z_{\rm W}}{2 \cdot \sqrt{R_1 \cdot Z_{\rm W}}}
   \hspace{0.05cm}.$$
-* In analoger Weise gilt für die&nbsp; '''empfängerseitige Stoßdämpfung'''&nbsp; am Leitungsende &nbsp; &rArr; &nbsp; Übergang &bdquo;Leitung &rarr; Empfänger":
+* In analoger Weise gilt für die&nbsp; '''empfängerseitige Stoßdämpfung'''&nbsp; am Leitungsende &nbsp; &rArr; &nbsp; Übergang "Leitung &rarr; Empfänger":
 :$${\rm ln}\hspace{0.1cm} |q_2|= {\rm ln}\hspace{0.1cm}\frac {R_2 + Z_{\rm W}}{2 \cdot \sqrt{R_2 \cdot Z_{\rm W}}}
   \hspace{0.05cm}.$$

@@ Line 9: / Line 9: @@
 :$${ a}_{\rm B} = { a}_{\rm B}(f_0)= { a}_{\rm W}+ {\rm ln}\hspace{0.1cm} |q_1|+{\rm ln}\hspace{0.1cm} |q_2|+{ a}_{\rm WWD}\hspace{0.05cm}.$$
 <br clear=all>
-Die vier Anteile &ndash; alle mit der Pseudoeinheit &bdquo;Neper&rdquo;&nbsp;  (Np) &ndash; beschreiben folgende Sachverhalte:
+Die vier Anteile &ndash; alle mit der Pseudoeinheit &bdquo;Neper"&nbsp;  (Np) &ndash; beschreiben folgende Sachverhalte:
-* Der erste Summand &nbsp;$a_{\rm W} = \alpha \cdot l$&nbsp; modelliert die&nbsp; '''Wellendämpfung'''&nbsp; der sich entlang der Leitung ausbreitenden Welle.&nbsp; Beachten Sie, dass Dämpfungen mit &bdquo;$a$&rdquo; bezeichnet werden, während das Dämpfungsmaß (kilometrische Dämpfung) mit &bdquo;$\alpha$&rdquo; &nbsp; &rArr; &nbsp; sprich: &bdquo;alpha&rdquo; gekennzeichnet ist.
+* Der erste Summand &nbsp;$a_{\rm W} = \alpha \cdot l$&nbsp; modelliert die&nbsp; '''Wellendämpfung'''&nbsp; der sich entlang der Leitung ausbreitenden Welle.&nbsp; Beachten Sie, dass Dämpfungen mit &bdquo;$a$" bezeichnet werden, während das Dämpfungsmaß (kilometrische Dämpfung) mit &bdquo;$\alpha$" &nbsp; &rArr; &nbsp; sprich: &bdquo;alpha" gekennzeichnet ist.
-* Der zweite Summand gibt die&nbsp; '''senderseitige Stoßdämpfung'''&nbsp; an.&nbsp; Dieser Term berücksichtigt den Leistungsverlust durch Reflexionen am Übergang &bdquo;Sender &rarr; Leitung&rdquo;:
+* Der zweite Summand gibt die&nbsp; '''senderseitige Stoßdämpfung'''&nbsp; an.&nbsp; Dieser Term berücksichtigt den Leistungsverlust durch Reflexionen am Übergang &bdquo;Sender &rarr; Leitung":
 :$${\rm ln}\hspace{0.1cm} |q_1|= {\rm ln}\hspace{0.1cm}\frac {R_1 + Z_{\rm W}}{2 \cdot \sqrt{R_1 \cdot Z_{\rm W}}}
   \hspace{0.05cm}.$$
-* In analoger Weise gilt für die&nbsp; '''empfängerseitige Stoßdämpfung'''&nbsp; am Leitungsende &nbsp; &rArr; &nbsp; Übergang &bdquo;Leitung &rarr; Empfänger&rdquo;:
+* In analoger Weise gilt für die&nbsp; '''empfängerseitige Stoßdämpfung'''&nbsp; am Leitungsende &nbsp; &rArr; &nbsp; Übergang &bdquo;Leitung &rarr; Empfänger":
 :$${\rm ln}\hspace{0.1cm} |q_2|= {\rm ln}\hspace{0.1cm}\frac {R_2 + Z_{\rm W}}{2 \cdot \sqrt{R_2 \cdot Z_{\rm W}}}
   \hspace{0.05cm}.$$

← 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^]]