Aufgaben:Exercise 4.2: Mismatched Line - Revision history

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	[[Category:Linear and Time-Invariant Systems: Exercises\|^4.1 Results of Line Transmission Theory^]]		[[Category:Linear and Time-Invariant Systems: Exercises\|^4.1 Results of Line Transmission Theory^]]
	[[de:Aufgaben:4.~~2_Fehlangepasste_Leitung~~]]		[[de:Aufgaben:Aufgabe 4.2: Fehlangepasste Leitung]]

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	The line is supplied by a voltage source with internal resistance  $R_{\rm 1} = 100 \ \rm \Omega$  and is terminated by resistor  $R_{\rm 2}$.  The input impedance  (German:  "Eingangswiderstand"   ⇒   subscript  "E")  of the line is given by		The line is supplied by a voltage source with internal resistance  $R_{\rm 1} = 100 \ \rm \Omega$  and is terminated by resistor  $R_{\rm 2}$.  The input impedance  (German:  "Eingangswiderstand"   ⇒   subscript  "E")  of the line is given by
	:$$Z_{\rm E}(f) = Z_{\rm W}\cdot \frac {R_2 + Z_{\rm W} \cdot {\rm tanh}(\gamma(f) \cdot l)}{Z_{\rm W}+ R_2 \cdot {\rm tanh}(\gamma(f) \cdot l)}\hspace{0.05cm},\hspace{0.3cm}{\rm tanh}(x)= \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\~~rme~~}^{-x}}\hspace{0.05cm}, \hspace{0.3cm}x \in {\cal C}\hspace{0.05cm}.$$		:$$Z_{\rm E}(f) = Z_{\rm W}\cdot \frac {R_2 + Z_{\rm W} \cdot {\rm tanh}(\gamma(f) \cdot l)}{Z_{\rm W}+ R_2 \cdot {\rm tanh}(\gamma(f) \cdot l)}\hspace{0.05cm},\hspace{0.3cm}{\rm tanh}(x)= \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\rm e}^{-x}}\hspace{0.05cm}, \hspace{0.3cm}x \in {\cal C}\hspace{0.05cm}.$$

	The complex propagation function should – again very simplified – approximated by a real function:		The complex propagation function should – again very simplified – approximated by a real function:
Line 90:		Line 90:

	'''(3)'''  With the terminating resistor  $R_{\rm 2} = 0$   ⇒   short circuit follows from the given equation with real  $x = \gamma (f) \cdot l$:		'''(3)'''  With the terminating resistor  $R_{\rm 2} = 0$   ⇒   short circuit follows from the given equation with real  $x = \gamma (f) \cdot l$:
	:$$\frac{Z_{\rm E}(f)}{Z_{\rm W}} = {\rm tanh}(x)= \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\~~rme~~}^{-x}}= \frac {{\rm e}^{2x}-1}{{\rm e}^{2x}+1}.$$		:$$\frac{Z_{\rm E}(f)}{Z_{\rm W}} = {\rm tanh}(x)= \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\rm e}^{-x}}= \frac {{\rm e}^{2x}-1}{{\rm e}^{2x}+1}.$$
	In particular:		In particular:
	:$${Z_{\rm E}(f)}/{Z_{\rm W}} = 0.99 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}{\rm e}^{2x} = 199\hspace{0.3cm}\Rightarrow \hspace{0.3cm}x ={1}/{2}\cdot {\rm ln}\hspace{0.1cm}(199) \approx 2.65\,{\rm Np}\hspace{0.05cm}.$$		:$${Z_{\rm E}(f)}/{Z_{\rm W}} = 0.99 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}{\rm e}^{2x} = 199\hspace{0.3cm}\Rightarrow \hspace{0.3cm}x ={1}/{2}\cdot {\rm ln}\hspace{0.1cm}(199) \approx 2.65\,{\rm Np}\hspace{0.05cm}.$$
	:$$f_{\rm U} = 10\,\text {MHz:}\hspace{0.2cm}\alpha(f_{\rm U})= 0.5\,{\rm Np/km}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}l_{\rm min}= \frac{2.65\,{\rm Np}}{0.5\,{\~~rmNp~~/km}}\hspace{0.15cm}\underline{= 5.3\,{\rm km}} \hspace{0.05cm},$$		:$$f_{\rm U} = 10\,\text {MHz:}\hspace{0.2cm}\alpha(f_{\rm U})= 0.5\,{\rm Np/km}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}l_{\rm min}= \frac{2.65\,{\rm Np}}{0.5\,{\rm Np/km}}\hspace{0.15cm}\underline{= 5.3\,{\rm km}} \hspace{0.05cm},$$
	:$$ f_{\rm O} = 40\,\text {MHz:}\hspace{0.2cm}\alpha(f_{\rm U})= 1.0\,{\rm Np/km}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}l_{\rm min}= \frac{2.65\,{\rm Np}}{1.0\,{\~~rmNp~~/km}}\hspace{0.15cm}\underline{= 2.65\,{\rm km}} \hspace{0.05cm}.$$		:$$ f_{\rm O} = 40\,\text {MHz:}\hspace{0.2cm}\alpha(f_{\rm U})= 1.0\,{\rm Np/km}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}l_{\rm min}= \frac{2.65\,{\rm Np}}{1.0\,{\rm Np/km}}\hspace{0.15cm}\underline{= 2.65\,{\rm km}} \hspace{0.05cm}.$$
	That means:		That means:
	*At the frequency  $f_{\rm O} = 40\ {\rm MHz}$ , the line length  $l= 2.65 \ \rm km$ is already sufficient to largely suppress reflections.		*At the frequency  $f_{\rm O} = 40\ {\rm MHz}$ , the line length  $l= 2.65 \ \rm km$ is already sufficient to largely suppress reflections.

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	The line is supplied by a voltage source with internal resistance  $R_{\rm 1} = 100 \ \rm \Omega$  and is terminated by resistor  $R_{\rm 2}$.  The input impedance  (German:  "Eingangswiderstand"   ⇒   subscript  "E")  of the line is given by		The line is supplied by a voltage source with internal resistance  $R_{\rm 1} = 100 \ \rm \Omega$  and is terminated by resistor  $R_{\rm 2}$.  The input impedance  (German:  "Eingangswiderstand"   ⇒   subscript  "E")  of the line is given by
	:$$Z_{\rm E}(f) = Z_{\rm W}\cdot \frac {R_2 + Z_{\rm W} \cdot {\rm tanh}(\gamma(f) \cdot l)}		:$$Z_{\rm E}(f) = Z_{\rm W}\cdot \frac {R_2 + Z_{\rm W} \cdot {\rm tanh}(\gamma(f) \cdot l)}{Z_{\rm W}+ R_2 \cdot {\rm tanh}(\gamma(f) \cdot l)}\hspace{0.05cm},\hspace{0.3cm}{\rm tanh}(x)= \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\rme}^{-x}}\hspace{0.05cm}, \hspace{0.3cm}x \in {\cal C}\hspace{0.05cm}.$$
	{Z_{\rm W}+ R_2 \cdot {\rm tanh}(\gamma(f) \cdot l)}
	\hspace{0.05cm},\hspace{0.3cm}{\rm tanh}(x)
	= \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\rm
	e}^{-x}}\hspace{0.05cm}, \hspace{0.3cm}x \in {\cal C}
	\hspace{0.05cm}.$$

	The complex propagation function should – again very simplified – approximated by a real function:		The complex propagation function should – again very simplified – approximated by a real function:
Line 73:		Line 68:
	*Therefore,  $Z_{\rm W}$  is also independent of the line length  $l$  and is determined solely by the primary line parameters $R\hspace{0.05cm}'$, $L\hspace{0.05cm}'$, $G\hspace{0.08cm}'$ and $C\hspace{0.08cm}'$.		*Therefore,  $Z_{\rm W}$  is also independent of the line length  $l$  and is determined solely by the primary line parameters $R\hspace{0.05cm}'$, $L\hspace{0.05cm}'$, $G\hspace{0.08cm}'$ and $C\hspace{0.08cm}'$.
	*The following equation given in the theory section		*The following equation given in the theory section
	:$$Z_{\rm W}(f) = \sqrt{\frac {R\hspace{0.05cm}' + {\rm j} \cdot \omega L\hspace{0.05cm}'}{G\hspace{0.08cm}' + {\rm j} \cdot \omega C\hspace{0.08cm}'}}		:$$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}$$
	\hspace{0.1cm}\bigg \|_{\hspace{0.05cm} \omega \hspace{0.05cm}= \hspace{0.05cm}2\pi f}$$
	makes it clear that the wave impedance does depend on the frequency and is generally also complex-valued.		makes it clear that the wave impedance does depend on the frequency and is generally also complex-valued.

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	'''(2)'''  With the terminating resistor  $Z_{\rm 2}(f) = Z_{\rm W}(f)$  the resistance value transformed to the beginning of the line is also equal to the characteristic impedance, independent of the line length:		'''(2)'''  With the terminating resistor  $Z_{\rm 2}(f) = Z_{\rm W}(f)$  the resistance value transformed to the beginning of the line is also equal to the characteristic impedance, independent of the line length:
	:$$Z_{\rm E}(f) = Z_{\rm W}(f)\cdot \frac {Z_{\rm 2}(f) + Z_{\rm W}(f) \cdot {\rm tanh}(\gamma(f) \cdot l)}		:$$Z_{\rm E}(f) = Z_{\rm W}(f)\cdot \frac {Z_{\rm 2}(f) + Z_{\rm W}(f) \cdot {\rm tanh}(\gamma(f) \cdot l)}{Z_{\rm W}(f)+ Z_{\rm 2}(f) \cdot {\rm tanh}(\gamma(f) \cdot l)}=Z_{\rm W}(f)\cdot \frac {Z_{\rm W}(f) + Z_{\rm W}(f) \cdot {\rm tanh}(\gamma(f) \cdot l)}{Z_{\rm W}(f)+ Z_{\rm W}(f) \cdot {\rm tanh}(\gamma(f) \cdot l)}=Z_{\rm W}(f) \hspace{0.05cm}.$$
	{Z_{\rm W}(f)+ Z_{\rm 2}(f) \cdot {\rm tanh}(\gamma(f) \cdot l)}=
	Z_{\rm W}(f)\cdot \frac {Z_{\rm W}(f) + Z_{\rm W}(f) \cdot {\rm tanh}(\gamma(f) \cdot l)}
	{Z_{\rm W}(f)+ Z_{\rm W}(f) \cdot {\rm tanh}(\gamma(f) \cdot l)}=
	Z_{\rm W}(f) \hspace{0.05cm}.$$
	<u>Solutions 1, 2 and 4</u> are correct:		<u>Solutions 1, 2 and 4</u> are correct:
	*Since  $Z_{\rm W}(f) = Z_{\rm W}$  was assumed to be frequency-independent in the exercise, the input impedance  $Z_{\rm E}(f) = Z_{\rm E}$  is also frequency-independent.		*Since  $Z_{\rm W}(f) = Z_{\rm W}$  was assumed to be frequency-independent in the exercise, the input impedance  $Z_{\rm E}(f) = Z_{\rm E}$  is also frequency-independent.
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	'''(3)'''  With the terminating resistor  $R_{\rm 2} = 0$   ⇒   short circuit follows from the given equation with real  $x = \gamma (f) \cdot l$:		'''(3)'''  With the terminating resistor  $R_{\rm 2} = 0$   ⇒   short circuit follows from the given equation with real  $x = \gamma (f) \cdot l$:
	:$$\frac{Z_{\rm E}(f)}{Z_{\rm W}} = {\rm tanh}(x)		:$$\frac{Z_{\rm E}(f)}{Z_{\rm W}} = {\rm tanh}(x)= \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\rme}^{-x}}= \frac {{\rm e}^{2x}-1}{{\rm e}^{2x}+1}.$$
	= \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\rm
	e}^{-x}}= \frac {{\rm e}^{2x}-1}{{\rm e}^{2x}+1}.$$
	In particular:		In particular:
	:$${Z_{\rm E}(f)}/{Z_{\rm W}} = 0.99 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}		:$${Z_{\rm E}(f)}/{Z_{\rm W}} = 0.99 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}{\rm e}^{2x} = 199\hspace{0.3cm}\Rightarrow \hspace{0.3cm}x ={1}/{2}\cdot {\rm ln}\hspace{0.1cm}(199) \approx 2.65\,{\rm Np}\hspace{0.05cm}.$$
	{\rm e}^{2x} = 199\hspace{0.3cm}\Rightarrow \hspace{0.3cm}		:$$f_{\rm U} = 10\,\text {MHz:}\hspace{0.2cm}\alpha(f_{\rm U})= 0.5\,{\rm Np/km}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}l_{\rm min}= \frac{2.65\,{\rm Np}}{0.5\,{\rmNp/km}}\hspace{0.15cm}\underline{= 5.3\,{\rm km}} \hspace{0.05cm},$$
	x ={1}/{2}\cdot {\rm ln}\hspace{0.1cm}(199) \approx 2.65\,{\rm Np}\hspace{0.05cm}.$$		:$$ f_{\rm O} = 40\,\text {MHz:}\hspace{0.2cm}\alpha(f_{\rm U})= 1.0\,{\rm Np/km}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}l_{\rm min}= \frac{2.65\,{\rm Np}}{1.0\,{\rmNp/km}}\hspace{0.15cm}\underline{= 2.65\,{\rm km}} \hspace{0.05cm}.$$
	:$$f_{\rm U} = 10\,\text {MHz:}\hspace{0.2cm}\alpha(f_{\rm U})= 0.5\,{\rm Np/km}\hspace{0.3cm}
	\Rightarrow \hspace{0.3cm}l_{\rm min}= \frac{2.65\,{\rm Np}}{0.5\,{\rm
	Np/km}}\hspace{0.15cm}\underline{= 5.3\,{\rm km}} \hspace{0.05cm},$$
	:$$ f_{\rm O} = 40\,\text {MHz:}\hspace{0.2cm}\alpha(f_{\rm U})= 1.0\,{\rm Np/km}\hspace{0.3cm}
	\Rightarrow \hspace{0.3cm}l_{\rm min}= \frac{2.65\,{\rm Np}}{1.0\,{\rm
	Np/km}}\hspace{0.15cm}\underline{= 2.65\,{\rm km}} \hspace{0.05cm}.$$
	That means:		That means:
	*At the frequency  $f_{\rm O} = 40\ {\rm MHz}$ , the line length  $l= 2.65 \ \rm km$ is already sufficient to largely suppress reflections.		*At the frequency  $f_{\rm O} = 40\ {\rm MHz}$ , the line length  $l= 2.65 \ \rm km$ is already sufficient to largely suppress reflections.
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	'''(4)'''  Similarly, one obtains for  $R_2 → ∞$   ⇒   idle:		'''(4)'''  Similarly, one obtains for  $R_2 → ∞$   ⇒   idle:
	:$$\frac{Z_{\rm E}(f)}{Z_{\rm W}} = \frac{1}{{\rm tanh}(x)}		:$$\frac{Z_{\rm E}(f)}{Z_{\rm W}} = \frac{1}{{\rm tanh}(x)}= \frac {{\rm e}^{2x}+1}{{\rm e}^{2x}-1}\hspace{0.05cm}.$$
	= \frac {{\rm e}^{2x}+1}{{\rm e}^{2x}-1}\hspace{0.05cm}.$$
	In contrast to the short-circuit case, this now results in the quotient  $Z_{\rm E}/Z_{\rm W} > 1$:		In contrast to the short-circuit case, this now results in the quotient  $Z_{\rm E}/Z_{\rm W} > 1$:
	:$${Z_{\rm E}(f)}/{Z_{\rm W}} = 1.01 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}		:$${Z_{\rm E}(f)}/{Z_{\rm W}} = 1.01 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}{\rm e}^{2x} = 201\hspace{0.3cm}\Rightarrow \hspace{0.3cm}x ={1}/{2}\cdot{\rm ln}\hspace{0.1cm}(201) \approx 2.65\,{\rm Np}\hspace{0.05cm}.$$
	{\rm e}^{2x} = 201\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
	x ={1}/{2}\cdot{\rm ln}\hspace{0.1cm}(201) \approx 2.65\,{\rm Np}\hspace{0.05cm}.$$
	Approximately, the same result is obtained here as in subtask  '''(3)''':		Approximately, the same result is obtained here as in subtask  '''(3)''':
	*At the frequency  $f_{\rm O} = 40\ {\rm MHz}$ , the line length  $l= 2.65 \ \rm km$ is already sufficient to largely suppress reflections.		*At the frequency  $f_{\rm O} = 40\ {\rm MHz}$ , the line length  $l= 2.65 \ \rm km$ is already sufficient to largely suppress reflections.

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	[[Category:Linear and Time-Invariant Systems: Exercises\|^4.1 Results of Line Transmission Theory^]]		[[Category:Linear and Time-Invariant Systems: Exercises\|^4.1 Results of Line Transmission Theory^]]
			[[de:Aufgaben:4.2_Fehlangepasste_Leitung]]

Guenter at 16:24, 11 November 2021

2021-11-11T16:24:30Z

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	* In particular,  it should be examined whether there are reflections.		* In particular,  it should be examined whether there are reflections.
	* To avoid confusion:		* To avoid confusion:
	**$f_{\rm U}$  stands for  "lowest frequency"   ⇒   German:   "unterste Frequenz"   ⇒   subscript "U",		#$f_{\rm U}$  stands for  "lowest frequency"   ⇒   German:   "unterste Frequenz"   ⇒   subscript "U",
	**$f_{\rm O}$  stands for  "highest frequency"  ⇒   German:   "oberste Frequenz"   ⇒   subscript "O".		#$f_{\rm O}$  stands for  "highest frequency"  ⇒   German:   "oberste Frequenz"   ⇒   subscript "O".

Guenter at 16:23, 11 November 2021

2021-11-11T16:23:25Z

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	''Notes:''		''Notes:''
	*The exercise belongs to the chapter  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory\|Some Results from Line Transmission Theory]].		*The exercise belongs to the chapter  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory\|Some Results from Line Transmission Theory]].
	* In particular, it should be examined whether there are reflections.		* In particular,  it should be examined whether there are reflections.
			* To avoid confusion:
			**$f_{\rm U}$  stands for  "lowest frequency"   ⇒   German:   "unterste Frequenz"   ⇒   subscript "U",
			**$f_{\rm O}$  stands for  "highest frequency"  ⇒   German:   "oberste Frequenz"   ⇒   subscript "O".

Hwang at 08:15, 8 November 2021

2021-11-08T08:15:54Z

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	*Since  $Z_{\rm W}(f) = Z_{\rm W}$  was assumed to be frequency-independent in the exercise, the input impedance  $Z_{\rm E}(f) = Z_{\rm E}$  is also frequency-independent.		*Since  $Z_{\rm W}(f) = Z_{\rm W}$  was assumed to be frequency-independent in the exercise, the input impedance  $Z_{\rm E}(f) = Z_{\rm E}$  is also frequency-independent.
	*In contrast, with frequency-dependent wave impedance with real termination, reflections cannot be avoided for all frequencies.		*In contrast, with frequency-dependent wave impedance with real termination, reflections cannot be avoided for all frequencies.
	*~~Die Beschaltung~~  $R_1 = R_2 =Z_{\rm W}$   ⇒   $R_1 =Z_{\rm E}$  is to be aimed at, since then the maximum power is delivered by the source.		*The wiring  $R_1 = R_2 =Z_{\rm W}$   ⇒   $R_1 =Z_{\rm E}$  is to be aimed at, since then the maximum power is delivered by the source.




	'''(3)'''  ~~Mit dem Abschlusswiderstand~~  $R_{\rm 2} = 0$   ⇒   ~~Kurzschluss folgt aus der angegebenen Gleichung mit reellem~~  $x = \gamma (f) \cdot l$:		'''(3)'''  With the terminating resistor  $R_{\rm 2} = 0$   ⇒   short circuit follows from the given equation with real  $x = \gamma (f) \cdot l$:
	:$$\frac{Z_{\rm E}(f)}{Z_{\rm W}} = {\rm tanh}(x)		:$$\frac{Z_{\rm E}(f)}{Z_{\rm W}} = {\rm tanh}(x)
	= \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\rm		= \frac {{\rm e}^{x}-{\rm e}^{-x}}{{\rm e}^{x}+{\rm
	e}^{-x}}= \frac {{\rm e}^{2x}-1}{{\rm e}^{2x}+1}.$$		e}^{-x}}= \frac {{\rm e}^{2x}-1}{{\rm e}^{2x}+1}.$$
	~~Insbesondere gilt~~:		In particular:
	:$${Z_{\rm E}(f)}/{Z_{\rm W}} = 0.99 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}		:$${Z_{\rm E}(f)}/{Z_{\rm W}} = 0.99 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
	{\rm e}^{2x} = 199\hspace{0.3cm}\Rightarrow \hspace{0.3cm}		{\rm e}^{2x} = 199\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
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	\Rightarrow \hspace{0.3cm}l_{\rm min}= \frac{2.65\,{\rm Np}}{1.0\,{\rm		\Rightarrow \hspace{0.3cm}l_{\rm min}= \frac{2.65\,{\rm Np}}{1.0\,{\rm
	Np/km}}\hspace{0.15cm}\underline{= 2.65\,{\rm km}} \hspace{0.05cm}.$$		Np/km}}\hspace{0.15cm}\underline{= 2.65\,{\rm km}} \hspace{0.05cm}.$$
	~~Das heißt~~:		That means:
	*~~Bei der Frequenz~~  $f_{\rm O} = 40\ {\rm MHz}$  ~~genügt bereits die Leitungslänge~~  $l= 2.65 \ \rm km$~~, um Reflexionen weitgehend zu unterdrücken~~.		*At the frequency  $f_{\rm O} = 40\ {\rm MHz}$ , the line length  $l= 2.65 \ \rm km$ is already sufficient to largely suppress reflections.
	*~~Bei niedrigerer Frequenz~~  $f_{\rm U} = 10\ {\rm MHz}$  ~~ist wegen des geringeren Dämpfungsmaßes eine größere Kabellänge erforderlich~~.		*At a lower frequency  $f_{\rm U} = 10\ {\rm MHz}$ , a longer cable length is required due to the lower attenuation function.
	*~~Diese Aussagen beziehen sich natürlich nur auf das Vermeiden von Reflexionen~~.		*Of course, these statements only refer to the avoidance of reflections.
	*~~Insgesamt ist natürlich die niedrigere Signalfrequenz günstiger als die höhere~~.		*Overall, of course, the lower signal frequency is more favorable than the higher one.



	'''(4)'''  ~~In gleicher Weise erhält man für~~  $R_2 → ∞$   ⇒   ~~Leerlauf die Gleichung~~		'''(4)'''  Similarly, one obtains for  $R_2 → ∞$   ⇒   idle:
	:$$\frac{Z_{\rm E}(f)}{Z_{\rm W}} = \frac{1}{{\rm tanh}(x)}		:$$\frac{Z_{\rm E}(f)}{Z_{\rm W}} = \frac{1}{{\rm tanh}(x)}
	= \frac {{\rm e}^{2x}+1}{{\rm e}^{2x}-1}\hspace{0.05cm}.$$		= \frac {{\rm e}^{2x}+1}{{\rm e}^{2x}-1}\hspace{0.05cm}.$$
	~~Im Gegensatz zum Kurzschluss–Fall ergibt sich nun für den Quotienten~~  $Z_{\rm E}/Z_{\rm W} > 1$:		In contrast to the short-circuit case, this now results in the quotient  $Z_{\rm E}/Z_{\rm W} > 1$:
	:$${Z_{\rm E}(f)}/{Z_{\rm W}} = 1.01 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}		:$${Z_{\rm E}(f)}/{Z_{\rm W}} = 1.01 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
	{\rm e}^{2x} = 201\hspace{0.3cm}\Rightarrow \hspace{0.3cm}		{\rm e}^{2x} = 201\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
	x ={1}/{2}\cdot{\rm ln}\hspace{0.1cm}(201) \approx 2.65\,{\rm Np}\hspace{0.05cm}.$$		x ={1}/{2}\cdot{\rm ln}\hspace{0.1cm}(201) \approx 2.65\,{\rm Np}\hspace{0.05cm}.$$
	~~Näherungsweise erhält man hier das gleiche Ergebnis wie bei Teilaufgabe~~  '''(3)''':		Approximately, the same result is obtained here as in subtask  '''(3)''':
	*~~Bei der Frequenz~~  $f_{\rm O} = 40\ {\rm MHz}$  ~~genügt bereits die Leitungslänge~~  $l= 2.65 \ \rm km$~~, um Reflexionen weitgehend zu unterdrücken~~.		*At the frequency  $f_{\rm O} = 40\ {\rm MHz}$ , the line length  $l= 2.65 \ \rm km$ is already sufficient to largely suppress reflections.
	*~~Bei der niedrigeren Frequenz~~  $f_{\rm U} = 10\ {\rm MHz}$  ~~ist wegen des geringeren Dämpfungsmaßes eine größere Kabellänge erforderlich~~.		*At a lower frequency  $f_{\rm U} = 10\ {\rm MHz}$ , a longer cable length is required due to the lower attenuation function.

	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 16:55, 6 November 2021

2021-11-06T16:55:58Z

← Older revision		Revision as of 18:55, 6 November 2021
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	}}		}}

	[[File:P_ID1799__LZI_A_4_2.png\|right\|frame\|~~Message~~ line with wiring]]		[[File:P_ID1799__LZI_A_4_2.png\|right\|frame\|Transmission line with wiring]]
	A transmission system occupies the range from  $f_{\rm U} = 10 \ \rm MHz$  to  $f_{\rm O} = 40 \ \rm MHz$.		A transmission system occupies the range from  $f_{\rm U} = 10 \ \rm MHz$  to  $f_{\rm O} = 40 \ \rm MHz$.

	The transmission line used ~~also~~ has a constant wave impedance   $Z_{\rm W} = 100 \ \rm \Omega$  (real), which does not quite correspond to reality, since the wave impedance usually decreases slightly with frequency and often an imaginary part (usually smaller) must also be taken into account.		The transmission line used has a constant wave impedance   $Z_{\rm W} = 100 \ \rm \Omega$  (real),  which does not quite correspond to reality,  since the wave impedance usually decreases slightly with frequency and often an imaginary part  (usually smaller)  must also be taken into account.

	The line is supplied by a voltage source with internal resistance  $R_{\rm 1} = 100 \ \rm \Omega$  and is terminated by resistor  $R_{\rm 2}$~~ ~~.  The input impedance of the line is given by		The line is supplied by a voltage source with internal resistance  $R_{\rm 1} = 100 \ \rm \Omega$  and is terminated by resistor  $R_{\rm 2}$.  The input impedance  (German:  "Eingangswiderstand"   ⇒   subscript  "E")  of the line is given by
	:$$Z_{\rm E}(f) = Z_{\rm W}\cdot \frac {R_2 + Z_{\rm W} \cdot {\rm tanh}(\gamma(f) \cdot l)}		:$$Z_{\rm E}(f) = Z_{\rm W}\cdot \frac {R_2 + Z_{\rm W} \cdot {\rm tanh}(\gamma(f) \cdot l)}
	{Z_{\rm W}+ R_2 \cdot {\rm tanh}(\gamma(f) \cdot l)}		{Z_{\rm W}+ R_2 \cdot {\rm tanh}(\gamma(f) \cdot l)}

Hwang at 15:02, 5 November 2021

2021-11-05T15:02:57Z

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Guenter at 15:01, 15 October 2021

2021-10-15T15:01:25Z

← Older revision		Revision as of 17:01, 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^]]