Difference between revisions of "Aufgaben:Exercise 4.3: Operational Attenuation"

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{{quiz-Header|Buchseite=Lineare zeitinvariante Systeme/Einige Ergebnisse der Leitungstheorie
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{{quiz-Header|Buchseite=Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory
 
}}
 
}}
  
[[File:P_ID1800__LZI_A_4_3.png|right|frame|Betrachtetes Leitungsmodell]]
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[[File:P_ID1800__LZI_A_4_3.png|right|frame|Considered line model]]
Wird eine Nachrichtenverbindung der Länge  $l$  nicht an beiden Enden mit ihrem Wellenwiderstand  $Z_{\rm W}$  abgeschlossen, so kommt es stets zu Reflexionen.  
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If a communication link of length  $l$  is not terminated at both ends with its wave impedance  $Z_{\rm W}$,  reflections will always occur.
  
Anstelle der Wellendämpfung  $a_{\rm W} = \alpha \cdot l$  muss man in diesem Fall die Betriebsdämpfung  $a_{\rm B}$  betrachten, die hier ohne Frequenzabhängigkeit angegeben wird.  Das heißt:   Wir betrachten hier stets nur eine einzige Frequenz $f_0$:
+
Instead of the wave attenuation  $a_{\rm W} = \alpha \cdot l$  one has to consider in this case the  '''operating attenuation'''  $a_{\rm B}$  (German:  "Betriebsdämpfung"   ⇒   subscipt  "B"),  which is given here without frequency dependence.  That means:  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>
+
 
Die vier Anteile &ndash; alle mit der Pseudoeinheit "Neper"&nbsp;  (Np) &ndash; beschreiben folgende Sachverhalte:
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The four parts&nbsp; &ndash; all with the pseudo unit "Neper"&nbsp;  (Np) &ndash;&nbsp; describe the following facts:
* 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.
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* 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".
* Der zweite Summand gibt die&nbsp; '''senderseitige Stoßdämpfung'''&nbsp; an.&nbsp; Dieser Term berücksichtigt den Leistungsverlust durch Reflexionen am Übergang "Sender &rarr; Leitung":
+
* 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}}}
 
:$${\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}.$$
 
  \hspace{0.05cm}.$$
* In analoger Weise gilt für die&nbsp; '''empfängerseitige Stoßdämpfung'''&nbsp; am Leitungsende &nbsp; &rArr; &nbsp; Übergang "Leitung &rarr; Empfänger":
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* 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}}}
 
:$${\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}.$$
 
  \hspace{0.05cm}.$$
* Die&nbsp; '''Wechselwirkungsdämpfung'''&nbsp; beschreibt die Signaldämpfung durch die Auswirkung einer doppelt reflektierten Welle, die sich dem Nutzsignal konstruktiv oder destruktiv überlagern kann.&nbsp; Für diesen letzten Anteil
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* The&nbsp; '''interaction attenuation&nbsp; (IA)'''&nbsp; describes the signal attenuation due to the effect of a doubly reflected wave,&nbsp; which can be constructively or destructively superimposed on the useful signal.&nbsp; For this last part holds:
:$${ 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}
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:$${ 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}|$$
 
   \gamma \hspace{0.05cm}\cdot\hspace{0.05cm} l}|$$
:verwenden wir in dieser Aufgabe folgende Gleichungen und Nomenklatur:
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:we use the following equations and nomenclature in this exercise:
:$${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}
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:$${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}|
 
   \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}
 
  \hspace{0.05cm},\hspace{0.3cm}r_\alpha = r_1 \cdot r_2\cdot {\rm e}^{-2 \hspace{0.05cm}\cdot\hspace{0.05cm}
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''Hinweise:''
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Notes:  
*Die Aufgabe gehört zum Kapitel&nbsp;  [[Linear_and_Time_Invariant_Systems/Einige_Ergebnisse_der_Leitungstheorie|Einige Ergebnisse der Leitungstheorie]].
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*The exercise belongs to the chapter&nbsp;  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory|Some Results from Line Transmission Theory]].
*Wesentliche Informationen finden Sie auf der Seite&nbsp; [[Linear_and_Time_Invariant_Systems/Einige_Ergebnisse_der_Leitungstheorie#Einfluss_von_Reflexionen_.E2.80.93_Betriebsd.C3.A4mpfung|Einfluss von Reflexionen - Betriebsdämpfung]]
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*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]]
 
   
 
   
*Gehen Sie bei numerischen Berechnungen von folgenden Zahlenwerten aus:
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*Use the following numerical values for numerical calculations:
 
:$$Z_{\rm W} = 100\,{\rm \Omega}\hspace{0.05cm},\hspace{0.3cm}R_1 = 200\,{\rm \Omega}\hspace{0.05cm},\hspace{0.3cm}
 
:$$Z_{\rm W} = 100\,{\rm \Omega}\hspace{0.05cm},\hspace{0.3cm}R_1 = 200\,{\rm \Omega}\hspace{0.05cm},\hspace{0.3cm}
 
  R_2 = 1\,{\rm k\Omega}\hspace{0.05cm},\hspace{0.3cm}l = 2\,{\rm km}\hspace{0.05cm},\hspace{0.3cm}
 
  R_2 = 1\,{\rm k\Omega}\hspace{0.05cm},\hspace{0.3cm}l = 2\,{\rm km}\hspace{0.05cm},\hspace{0.3cm}
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===Fragebogen===
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===Questions===
  
 
<quiz display=simple>
 
<quiz display=simple>
{Welcher Wert ergäbe sich für die Betriebsdämpfung <u>bei Anpassung</u>, wenn es also keine Reflexionen geben würde?
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{What would be the value of the operating attenuation&nbsp; <u>with matching</u>,&nbsp; i.e. if there were no reflections?
 
|type="{}"}
 
|type="{}"}
 
$a_{\rm B} \ = \ $  { 0.2 3% } $\ \rm Np$
 
$a_{\rm B} \ = \ $  { 0.2 3% } $\ \rm Np$
  
  
{Berechnen Sie die beiden Anteile der Stoßdämpfung für &nbsp;$Z_{\rm W} = 100\,{\rm \Omega}$&nbsp; sowie&nbsp; $R_1 = 200\,{\rm \Omega}$&nbsp; und &nbsp;$ R_2 = 1\,{\rm k\Omega}$.
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{Calculate the two parts of the reflection loss for &nbsp;$Z_{\rm W} = 100\,{\rm \Omega}$&nbsp; as well as&nbsp; $R_1 = 200\,{\rm \Omega}$&nbsp; and &nbsp;$ R_2 = 1\,{\rm k\Omega}$.
 
|type="{}"}
 
|type="{}"}
 
$\ln\ |q_1| \ = \ $ { 0.059 3% } $\ \rm Np$
 
$\ln\ |q_1| \ = \ $ { 0.059 3% } $\ \rm Np$
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{Berechnen Sie die Reflexionsfaktoren &nbsp;$r_1$, &nbsp;$r_2$ und &nbsp;$r_\alpha$.
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{Calculate the reflection factors &nbsp;$r_1$, &nbsp;$r_2$ and &nbsp;$r_\alpha$.
 
|type="{}"}
 
|type="{}"}
 
$r_1 \ = \ $ { 0.333 3% }
 
$r_1 \ = \ $ { 0.333 3% }
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$r_\alpha\ = \ $ { 0.183 3% }
 
$r_\alpha\ = \ $ { 0.183 3% }
  
{Welche Bedingung muss die Hilfsgröße &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 $&nbsp; erfüllen, <br>damit es zu konstruktiver bzw. destruktiver Überlagerung bezüglich der Wechselwirkungsdämpfung kommt?
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{Which condition must the auxiliary &nbsp;$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="[]"}
 
|type="[]"}
+ $A$&nbsp; ist minimal &nbsp; &#8658; &nbsp; konstruktive Überlagerung,
+
+ $A$&nbsp; is minimal &nbsp; &#8658; &nbsp; constructive superposition,
+ $A$&nbsp; ist maximal &nbsp; &#8658; &nbsp; destruktive Überlagerung.
+
+ $A$&nbsp; is maximal &nbsp; &#8658; &nbsp; destructive superposition.
  
  
{Geben Sie den kleinstmöglichen Wert &nbsp;$\beta_\text{min}$&nbsp; für das Phasenmaß &nbsp;$\beta(f_0)$&nbsp; an, <br>damit es zu einer konstruktiven Überlagerung kommt.
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{Specify the smallest possible value &nbsp;$\beta_\text{min}$&nbsp; for the phase function (per unit length) &nbsp;$\beta(f_0)$&nbsp;, <br>so that constructive superposition occurs.
 
|type="{}"}
 
|type="{}"}
 
$\beta_\text{min} \ = \ $ { 0.785 3% } $\ \rm rad/km$
 
$\beta_\text{min} \ = \ $ { 0.785 3% } $\ \rm rad/km$
  
  
{Wie groß kann der Wechselwirkungsdämpfungsanteil maximal werden?&nbsp; Welche Voraussetzungen müssen hierfür gelten?
+
{What is the maximum size of the interaction attenuation component?&nbsp; What are the requirements for this?
 
|type="{}"}
 
|type="{}"}
${\rm Max}\ \big [a_\text{WWD}\big ] \ = \ $ { 0.693 3% } $\ \rm Np$
+
${\rm Max}\ \big [a_\text{IA}\big ] \ = \ $ { 0.693 3% } $\ \rm Np$
  
  
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</quiz>
 
</quiz>
  
===Musterlösung===
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===Solution===
 
{{ML-Kopf}}
 
{{ML-Kopf}}
'''(1)'''&nbsp; Bei Widerstandsanpassung&nbsp; $(R_1 = R_2 =Z_{\rm W})$&nbsp; verbleibt von den vier Summanden nur der erste:
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'''(1)'''&nbsp; With resistance matching&nbsp; $(R_1 = R_2 =Z_{\rm W})$&nbsp; only the first of the four summands remains:
:$${a}_{\rm B} =  \alpha \cdot l = 0.1\,{\rm Np/km} \cdot 2\,{\rm km} \hspace{0.15cm}\underline{= 0.2\,{\rm Np}}\hspace{0.05cm}.$$
+
:$${a}_{\rm B} =  {a}_{\rm W} =\alpha \cdot l = 0.1\,{\rm Np/km} \cdot 2\,{\rm km} \hspace{0.15cm}\underline{= 0.2\,{\rm Np}}\hspace{0.05cm}.$$
  
  
 
+
'''(2)'''&nbsp; According to the given equations we obtain:
'''(2)'''&nbsp; Entsprechend den angegebenen Gleichungen ergibt sich:
 
 
:$$q_1  =  \frac {R_1 + Z_{\rm W}}{2 \cdot \sqrt{R_1 \cdot Z_{\rm
 
:$$q_1  =  \frac {R_1 + Z_{\rm W}}{2 \cdot \sqrt{R_1 \cdot Z_{\rm
 
W}}}= \frac {200 + 100}{2 \cdot \sqrt{200 \cdot 100}}= 1.061
 
W}}}= \frac {200 + 100}{2 \cdot \sqrt{200 \cdot 100}}= 1.061
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'''(3)'''&nbsp; With the given circuit resistors we obtain
'''(3)'''&nbsp; Mit den vorgegebenen Beschaltungswiderständen erhält man
 
 
:$$r_1= \frac {200\,{\rm \Omega} - 100\,{\rm \Omega}}{200\,{\rm \Omega} + 100\,{\rm \Omega}} \hspace{0.15cm}\underline{= 0.333}\hspace{0.05cm},$$
 
:$$r_1= \frac {200\,{\rm \Omega} - 100\,{\rm \Omega}}{200\,{\rm \Omega} + 100\,{\rm \Omega}} \hspace{0.15cm}\underline{= 0.333}\hspace{0.05cm},$$
 
:$$r_2=\frac {1000\,{\rm \Omega} - 100\,{\rm \Omega}}{1000\,{\rm \Omega} + 100\,{\rm \Omega}}\hspace{0.15cm}\underline{ =
 
:$$r_2=\frac {1000\,{\rm \Omega} - 100\,{\rm \Omega}}{1000\,{\rm \Omega} + 100\,{\rm \Omega}}\hspace{0.15cm}\underline{ =
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'''(4)'''&nbsp; <u>Both statements are correct</u>:  
'''(4)'''&nbsp; <u>Beide Aussagen sind richtig</u>:  
+
*With constructive superposition is &nbsp; &rArr; &nbsp; ${a}_{\rm IA} = {\rm ln}\hspace{0.1cm}A < 0 \; \Rightarrow \; A < 1$&nbsp; and minimal.  
*Bei konstruktiver Überlagerung ist&nbsp; ${a}_{\rm WWD} = {\rm ln}\hspace{0.1cm}A < 0 \; \Rightarrow \; A < 1$&nbsp; und 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 wave and returning wave.
*Im Gegensatz dazu bewirkt der maximale Wert von&nbsp; $A&nbsp;$&nbsp; (für den  $A > 1$ gilt) &nbsp; eine positive Wechselwirkungsdämpfung, also eine zusätzliche Dämpfung des Nutzsignals aufgrund der destruktiven Überlagerung von hin&ndash; und rücklaufender Welle.
 
 
 
  
  
'''(5)'''&nbsp; In der letzten Teilaufgabe wurde gezeigt, dass konstruktive Überlagerung gleichbedeutend ist mit der Minimierung von
+
'''(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}
 
:$$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
 
   \beta \hspace{0.05cm}\cdot\hspace{0.05cm} l}|=  |1- r_\alpha \cdot \cos(2  \beta  l)+ {\rm j} \cdot \sin(2  \beta
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:$$ \Rightarrow \hspace{0.3cm }A    =  \sqrt {1+ {r_\alpha}^2- 2 \cdot r_\alpha \cdot \cos(2  \beta  l)}
 
:$$ \Rightarrow \hspace{0.3cm }A    =  \sqrt {1+ {r_\alpha}^2- 2 \cdot r_\alpha \cdot \cos(2  \beta  l)}
 
  \hspace{0.05cm}.$$
 
  \hspace{0.05cm}.$$
*Das Minimum ergibt sich für
+
*The minimum results for
 
:$$\cos(2  \beta  l) = +1 \hspace{0.2cm}\Rightarrow \hspace{0.2cm}2  \beta
 
:$$\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{0.2cm}\beta_{\rm min} = {\pi}({2l})\hspace{0.15cm}\underline{= 0.785\,{\rm rad/km}}
 
  \hspace{0.05cm}.$$
 
  \hspace{0.05cm}.$$
*Dagegen kommt es zu destruktiver Überlagerung, falls das Phasenmaß folgende Bedingung erfüllt:
+
*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
 
:$$\cos(2  \beta  l) = -1 \hspace{0.2cm}\Rightarrow \hspace{0.2cm}2  \beta
  l= {\pi}/{2}, {3\pi}/{2}, {5\pi}/{2},\text{ ...}$$
+
  l= {\pi}/{2},\ {3\pi}/{2},\ {5\pi}/{2},\text{ ...}$$
  
  
'''(6)'''&nbsp; Das Argument&nbsp; $A  = \sqrt {1+ {r_\alpha}^2- 2 \cdot r_\alpha \cdot \cos(2  \beta  l)}$&nbsp; kann maximal&nbsp; $A = 2$&nbsp; werden &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; ${\rm Max}\ [a_\text{IA}] \; \underline {= 0.693 \ \rm Np}$.
  
Hierfür müssen  folgende Voraussetzungen erfüllt sein:
+
The following requirements must be met for this:
* nicht abgeschlossene Leitung&nbsp; $(r_1 = r_2  = 1)$,
+
* not terminated line&nbsp; $(r_1 = r_2  = 1)$,
* kurze Kabellänge, so dass der Term&nbsp; $\alpha \cdot l$&nbsp; nicht wirksam ist&nbsp; $(r_\alpha = 1)$,
+
* short cable length,&nbsp; so the term&nbsp; $\alpha \cdot l$&nbsp; is not effective&nbsp; $(r_\alpha = 1)$,
* Phasenverlauf entsprechend der Teilaufgabe&nbsp; '''(5)'''.
+
* phase progression according to the subtask&nbsp; '''(5)'''.
 
{{ML-Fuß}}
 
{{ML-Fuß}}
  
  
  
[[Category:Linear and Time-Invariant Systems: Exercises|^4.1 Einige Ergebnisse der Leitungstheorie^]]
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[[Category:Linear and Time-Invariant Systems: Exercises|^4.1 Results of Line Transmission Theory^]]

Latest revision as of 16:02, 12 November 2021

Considered line model

If a communication link of length  $l$  is not terminated at both ends with its wave impedance  $Z_{\rm W}$,  reflections will always occur.

Instead of the wave attenuation  $a_{\rm W} = \alpha \cdot l$  one has to consider in this case the  operating attenuation  $a_{\rm B}$  (German:  "Betriebsdämpfung"   ⇒   subscipt  "B"),  which is given here without frequency dependence.  That means:  We always only consider a single frequency $f_0$:

$${ a}_{\rm B} = { a}_{\rm B}(f_0)= { a}_{\rm W}+ {\rm ln}\hspace{0.1cm} |q_1|+{\rm ln}\hspace{0.1cm} |q_2|+{ a}_{\rm IA}\hspace{0.05cm}.$$

The four parts  – all with the pseudo unit "Neper"  (Np) –  describe the following facts:

  • The first summand  $a_{\rm W} = \alpha \cdot l$  models the  wave attenuation  of the wave propagating along the line.  Note that attenuations are denoted by  "$a$",  while the attenuation function per unit length is denoted by  "$\alpha$"   ⇒   read: "alpha".
  • The second summand gives the  transmitter-side reflection loss.  This term takes into account the power loss due to reflections at the  "transmitter → line"  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  receiver-side reflection loss  at the end of the line   ⇒   transition  "line → 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  interaction attenuation  (IA)  describes the signal attenuation due to the effect of a doubly reflected wave,  which can be constructively or destructively superimposed on the useful signal.  For this last part holds:
$${ a}_{\rm IA} = {\rm ln}\hspace{0.1cm} |1- r_1 \cdot r_2 \cdot {\rm e}^{- 2\hspace{0.05cm}\cdot\hspace{0.05cm} \gamma \hspace{0.05cm}\cdot\hspace{0.05cm} l}|$$
we use the following equations and nomenclature in this exercise:
$${a}_{\rm IA} = {\rm ln}\hspace{0.1cm}A, \hspace{0.3cm}A = |1- r_\alpha \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \hspace{0.05cm}\cdot\hspace{0.05cm} \beta \hspace{0.05cm}\cdot\hspace{0.05cm} l}| \hspace{0.05cm},\hspace{0.3cm}r_\alpha = r_1 \cdot r_2\cdot {\rm e}^{-2 \hspace{0.05cm}\cdot\hspace{0.05cm} \alpha \hspace{0.05cm}\cdot\hspace{0.05cm} l},\hspace{0.3cm}r_1= \frac {R_1 - Z_{\rm W}}{R_1 + Z_{\rm W}}, \hspace{0.3cm}r_2= \frac {R_2 - Z_{\rm W}}{R_2 + Z_{\rm W}} \hspace{0.05cm}.$$




Notes:

  • Use the following numerical values for numerical calculations:
$$Z_{\rm W} = 100\,{\rm \Omega}\hspace{0.05cm},\hspace{0.3cm}R_1 = 200\,{\rm \Omega}\hspace{0.05cm},\hspace{0.3cm} R_2 = 1\,{\rm k\Omega}\hspace{0.05cm},\hspace{0.3cm}l = 2\,{\rm km}\hspace{0.05cm},\hspace{0.3cm} \alpha = 0.1\,{\rm Np/km} \hspace{0.05cm}.$$


Questions

1

What would be the value of the operating attenuation  with matching,  i.e. if there were no reflections?

$a_{\rm B} \ = \ $

$\ \rm Np$

2

Calculate the two parts of the reflection loss for  $Z_{\rm W} = 100\,{\rm \Omega}$  as well as  $R_1 = 200\,{\rm \Omega}$  and  $ R_2 = 1\,{\rm k\Omega}$.

$\ln\ |q_1| \ = \ $

$\ \rm Np$
$\ln\ |q_2| \ = \ $

$\ \rm Np$

3

Calculate the reflection factors  $r_1$,  $r_2$ and  $r_\alpha$.

$r_1 \ = \ $

$r_2\ = \ $

$r_\alpha\ = \ $

4

Which condition must the auxiliary  $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 $  satisfy,  so that constructive or destructive superposition occurs with respect to the interaction attenuation?

$A$  is minimal   ⇒   constructive superposition,
$A$  is maximal   ⇒   destructive superposition.

5

Specify the smallest possible value  $\beta_\text{min}$  for the phase function (per unit length)  $\beta(f_0)$ ,
so that constructive superposition occurs.

$\beta_\text{min} \ = \ $

$\ \rm rad/km$

6

What is the maximum size of the interaction attenuation component?  What are the requirements for this?

${\rm Max}\ \big [a_\text{IA}\big ] \ = \ $

$\ \rm Np$


Solution

(1)  With resistance matching  $(R_1 = R_2 =Z_{\rm W})$  only the first of the four summands remains:

$${a}_{\rm B} = {a}_{\rm W} =\alpha \cdot l = 0.1\,{\rm Np/km} \cdot 2\,{\rm km} \hspace{0.15cm}\underline{= 0.2\,{\rm Np}}\hspace{0.05cm}.$$


(2)  According to the given equations we obtain:

$$q_1 = \frac {R_1 + Z_{\rm W}}{2 \cdot \sqrt{R_1 \cdot Z_{\rm W}}}= \frac {200 + 100}{2 \cdot \sqrt{200 \cdot 100}}= 1.061 \hspace{0.2cm}\Rightarrow \hspace{0.2cm} {\rm ln}\hspace{0.1cm}|q_1| \hspace{0.15cm}\underline{= 0.059 \,{\rm Np}} \hspace{0.05cm},$$
$$q_2 = \frac {R_2 + Z_{\rm W}}{2 \cdot \sqrt{R_2 \cdot Z_{\rm W}}}= \frac {1000 + 100}{2 \cdot \sqrt{1000 \cdot 100}}= 1.739 \hspace{0.2cm}\Rightarrow \hspace{0.2cm} {\rm ln}\hspace{0.1cm}|q_2| \hspace{0.15cm}\underline{= 0.553 \,{\rm Np}}\hspace{0.05cm}.$$


(3)  With the given circuit resistors we obtain

$$r_1= \frac {200\,{\rm \Omega} - 100\,{\rm \Omega}}{200\,{\rm \Omega} + 100\,{\rm \Omega}} \hspace{0.15cm}\underline{= 0.333}\hspace{0.05cm},$$
$$r_2=\frac {1000\,{\rm \Omega} - 100\,{\rm \Omega}}{1000\,{\rm \Omega} + 100\,{\rm \Omega}}\hspace{0.15cm}\underline{ = 0.818}$$
$$\Rightarrow \hspace{0.3cm} r_\alpha = r_1 \cdot r_2\cdot {\rm e}^{-2 \hspace{0.05cm}\cdot\hspace{0.05cm} \alpha \hspace{0.05cm}\cdot\hspace{0.05cm} l}= 0.333 \cdot 0.818\cdot {\rm e}^{-4}\hspace{0.15cm}\underline{= 0.183} \hspace{0.05cm}.$$


(4)  Both statements are correct:

  • With constructive superposition is   ⇒   ${a}_{\rm IA} = {\rm ln}\hspace{0.1cm}A < 0 \; \Rightarrow \; A < 1$  and minimal.
  • In contrast, the maximum value of  $A $  (for which $A > 1$)   causes positive interaction attenuation, i.e., additional attenuation of the useful signal due to destructive superposition of outgoing wave and returning wave.


(5)  In the last subtask,  it was shown that constructive superposition is equivalent to the minimization of

$$A = |1- r_\alpha \cdot {\rm e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2\hspace{0.05cm}\cdot\hspace{0.05cm} \beta \hspace{0.05cm}\cdot\hspace{0.05cm} l}|= |1- r_\alpha \cdot \cos(2 \beta l)+ {\rm j} \cdot \sin(2 \beta l)| = \sqrt {1- 2 \cdot r_\alpha \cdot \cos(2 \beta l)+ {r_\alpha}^2 \cdot \cos^2(2 \beta l)+ {r_\alpha}^2 \cdot \sin^2(2 \beta l)}$$
$$ \Rightarrow \hspace{0.3cm }A = \sqrt {1+ {r_\alpha}^2- 2 \cdot r_\alpha \cdot \cos(2 \beta l)} \hspace{0.05cm}.$$
  • The minimum results for
$$\cos(2 \beta l) = +1 \hspace{0.2cm}\Rightarrow \hspace{0.2cm}2 \beta l= \pi,\ 2\pi,\ 3\pi, ... \hspace{0.2cm}\Rightarrow \hspace{0.2cm}\beta_{\rm min} = {\pi}({2l})\hspace{0.15cm}\underline{= 0.785\,{\rm rad/km}} \hspace{0.05cm}.$$
  • In contrast, destructive superposition occurs if the phase function (per unit length) satisfies the following condition:
$$\cos(2 \beta l) = -1 \hspace{0.2cm}\Rightarrow \hspace{0.2cm}2 \beta l= {\pi}/{2},\ {3\pi}/{2},\ {5\pi}/{2},\text{ ...}$$


(6)  The argument  $A = \sqrt {1+ {r_\alpha}^2- 2 \cdot r_\alpha \cdot \cos(2 \beta l)}$  can become maximum  $A = 2$    ⇒   ${\rm Max}\ [a_\text{IA}] \; \underline {= 0.693 \ \rm Np}$.

The following requirements must be met for this:

  • not terminated line  $(r_1 = r_2 = 1)$,
  • short cable length,  so the term  $\alpha \cdot l$  is not effective  $(r_\alpha = 1)$,
  • phase progression according to the subtask  (5).