Aufgaben:Exercise 4.1: Attenuation Function - Revision history

Guenter at 15:38, 6 November 2021

2021-11-06T15:38:40Z

Guenter at 15:27, 6 November 2021

2021-11-06T15:27:11Z

Guenter at 15:25, 6 November 2021

2021-11-06T15:25:02Z

Guenter at 15:23, 6 November 2021

2021-11-06T15:23:13Z

Hwang at 13:30, 5 November 2021

2021-11-05T13:30:03Z

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Guenter: Guenter moved page Aufgaben:Exercise 4.1: Attenuation Constant to Aufgaben:Exercise 4.1: Attenuation Function

2021-10-29T14:42:40Z

Guenter moved page Exercise 4.1: Attenuation Constant to Exercise 4.1: Attenuation Function

Guenter at 15:00, 15 October 2021

2021-10-15T15:00:11Z

Guenter at 14:59, 15 October 2021

2021-10-15T14:59:07Z

Guenter at 14:56, 15 October 2021

2021-10-15T14:56:40Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.1: Dämpfungsmaß to Aufgaben:Exercise 4.1: Attenuation Constant

2021-10-15T14:32:35Z

Guenter moved page Aufgabe 4.1: Dämpfungsmaß to Exercise 4.1: Attenuation Constant

@@ Line 81: / Line 81: @@
   L' = 1.8\,\,{\rm mH}/{ {\rm km}} \hspace{0.05cm},\hspace{0.3cm}
   G\hspace{0.03cm}' = 0.5\,\,{\rm &micro; S}/{ {\rm km}} \hspace{0.05cm},\hspace{0.3cm}
-  C\hspace{0.03cm}' = 6.7\,\,{\rm nF}/{ {\rm km}}
+  C\hspace{0.03cm}' = 6.7\,\,{\rm nF}/{ {\rm km}}$:
 :$$\alpha_{\rm I}(f)   =  1/2 \cdot
   \left [2.2 \cdot \sqrt{\frac{6.7 \cdot 10^{-9}}{ 1.8 \cdot 10^{-3}} }
@@ Line 93: / Line 92: @@
+'''(2)'''&nbsp; The bound &nbsp;$α_{\rm I}(f)$&nbsp; calculated in subtask&nbsp; '''(1)'''&nbsp; is valid only for &nbsp;$f \gg f_∗$,&nbsp; while the bound &nbsp;$α_{\rm II}(f)$&nbsp; is valid for &nbsp;$f \ll f_∗$.
 *The characteristic frequency is obtained as the intersection of the two approximations:
 :$$\alpha_{\rm II}(f = f_{\star})  =   \sqrt{1/2 \cdot  \omega_{\star}  \cdot R' \cdot C' }\hspace{0.1cm}
   \bigg |_{\omega_{\star} \hspace{0.05cm}= \hspace{0.05cm}2\pi f_{\star}} = \alpha_{\rm I}(f = f_{\star})$$
-*For the copper cable with&nbsp; $\text{0.6 mm}$&nbsp; diameter, the following equation holds:
+*For the copper cable with&nbsp; $\text{0.6 mm}$&nbsp; diameter,&nbsp; the following equation holds:
 :$$f_{\star}  =   \frac {{\alpha^2_{_{\rm I}}(f = f_{\star})}}{\pi \cdot R' \cdot C'}=
      \frac {0.496^2 \, {\rm 1/km^2}}{\pi \cdot 130\,{\rm \Omega/km} \cdot 35 \cdot 10^{-9}\,{\rm s/(\Omega \cdot km)}}
 \hspace{0.15cm}\underline{= 17.2\,{\rm kHz}}\hspace{0.05cm}.$$
-*In contrast, for the bronze line with diameter &nbsp; $\text{5 mm}$&nbsp;:
+*In contrast,&nbsp; for the bronze line with diameter&nbsp; $\text{5 mm}$&nbsp;:
 :$$f_{\star}  =
      \frac {(2.25 \cdot 10^{-3})^2 }{\pi \cdot 2.2 \cdot 6.7 \cdot 10^{-9}}\,{\rm kHz}
 \hspace{0.15cm}\underline{= 0.109\,{\rm kHz}}\hspace{0.05cm}.$$
 '''(3)'''&nbsp; For the copper cable &nbsp;$f_0 \ll f_∗$ holds.
-*Therefore, the approximation is &nbsp;$α_{\rm II}(f)$ &nbsp; &rArr; &nbsp; "strong attenuation" should be used:
+*Therefore,&nbsp; the approximation is &nbsp;$α_{\rm II}(f)$ &nbsp; &rArr; &nbsp; "strong attenuation"&nbsp; should be used:
 :$$\alpha(f = f_0)  \approx \sqrt{\pi \cdot f_0 \cdot R' \cdot C'}= \sqrt{\pi \cdot 2 \cdot 10^{3} \cdot 130 \cdot 35 \cdot 10^{-9}}
   \hspace{0.1cm}{\rm Np}/{ {\rm km} }
 \hspace{0.15cm}\underline{ = 0.17 \hspace{0.1cm}{\rm Np}/{ {\rm km} }}
   \hspace{0.05cm}.$$
-*For the bronze line, because of &nbsp;$f_0 \gg f_∗$&nbsp; the approximation is &nbsp;$α_{\rm I}(f)$ &nbsp; &rArr; &nbsp; "weak attenuation" is more suitable, see subtask&nbsp; '''(1)''':
+*For the bronze line,&nbsp; because of &nbsp;$f_0 \gg f_∗$&nbsp; the approximation is &nbsp;$α_{\rm I}(f)$ &nbsp; &rArr; &nbsp; "weak attenuation"&nbsp; is more suitable,&nbsp; see subtask&nbsp; '''(1)''':
 :$$\alpha(f = f_0)
 \hspace{0.15cm}\underline{= 0.0023\hspace{0.1cm}{\rm Np}/{ {\rm km} }}

← Older revision		Revision as of 15:27, 6 November 2021
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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]].

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1797__LZI_A_4_1.png|right|frame|Attenuation function and two bounds]]
+[[File:P_ID1797__LZI_A_4_1.png|right|frame|Attenuation function per unit length &nbsp; &rArr; &nbsp; $\alpha(f)$&nbsp; and two bounds]]
 The attenuation function per unit length &nbsp; &rArr; &nbsp; $\alpha(f)$&nbsp; &ndash; pronounced&nbsp; "alpha"&nbsp; &ndash; of a line indicates the attenuation related to the line length.&nbsp; This quantity is determined by the primary line parameters &nbsp;$R\hspace{0.05cm}'$, &nbsp;$L\hspace{0.05cm}'$, &nbsp;$G\hspace{0.08cm}'$ and &nbsp;$C\hspace{0.08cm}'$&nbsp; where the exact equation is somewhat complicated.&nbsp;&nbsp; Therefore, two more manageable approximations have been developed:
 :$$\frac{\alpha_{_{\rm I}}(f)}{\rm Np}  = {1}/{2} \cdot \left [R\hspace{0.05cm}' \cdot \sqrt{{C\hspace{0.08cm}'}/{ L\hspace{0.05cm}'} } + G\hspace{0.08cm}' \cdot \sqrt{{L\hspace{0.05cm}'}/{ C\hspace{0.08cm}'} }\hspace{0.05cm}\right ]

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1797__LZI_A_4_1.png|right|frame|Attenuation function and bounds]]
+[[File:P_ID1797__LZI_A_4_1.png|right|frame|Attenuation function and two bounds]]
-The attenuation function per unit length &nbsp;$\alpha(f)$ &ndash; pronounced "alpha" &ndash; of a line indicates the attenuation related to the line length.&nbsp; This quantity is determined by the primary line parameters &nbsp;$R\hspace{0.05cm}'$, &nbsp;$L\hspace{0.05cm}'$, &nbsp;$G\hspace{0.08cm}'$ and &nbsp;$C\hspace{0.08cm}'$&nbsp; where the exact equation is somewhat complicated. Therefore, two more manageable approximations have been developed:
+The attenuation function per unit length &nbsp; &rArr; &nbsp; $\alpha(f)$&nbsp; &ndash; pronounced&nbsp; "alpha"&nbsp; &ndash; of a line indicates the attenuation related to the line length.&nbsp; This quantity is determined by the primary line parameters &nbsp;$R\hspace{0.05cm}'$, &nbsp;$L\hspace{0.05cm}'$, &nbsp;$G\hspace{0.08cm}'$ and &nbsp;$C\hspace{0.08cm}'$&nbsp; where the exact equation is somewhat complicated.&nbsp;&nbsp; Therefore, two more manageable approximations have been developed:
 :$$\frac{\alpha_{_{\rm I}}(f)}{\rm Np}  = {1}/{2} \cdot \left [R\hspace{0.05cm}' \cdot \sqrt{{C\hspace{0.08cm}'}/{ L\hspace{0.05cm}'} } + G\hspace{0.08cm}' \cdot \sqrt{{L\hspace{0.05cm}'}/{ C\hspace{0.08cm}'} }\hspace{0.05cm}\right ]
   \hspace{0.05cm},$$
 :$$\frac{\alpha_{_{\rm II}}(f)}{\rm Np}  =  \sqrt{1/2 \cdot \omega  \cdot {R\hspace{0.05cm}' \cdot C\hspace{0.08cm}'} }\hspace{0.1cm}
   \bigg |_{\omega \hspace{0.05cm}= \hspace{0.05cm}2\pi f}\hspace{0.05cm}.$$
-These two approximations are shown in the graph together with the actual &nbsp;$\alpha(f)$&nbsp; curve. The intersection of&nbsp; $\alpha_{\rm I}(f)$&nbsp; and&nbsp; $\alpha_{\rm II}(f)$&nbsp; gives the characteristic frequency&nbsp; $f_∗$&nbsp; with the following meaning:
+These two approximations are shown in the graph together with the actual &nbsp;$\alpha(f)$&nbsp; curve.&nbsp; The intersection of&nbsp; $\alpha_{\rm I}(f)$&nbsp; and&nbsp; $\alpha_{\rm II}(f)$&nbsp; gives the characteristic frequency&nbsp; $f_∗$&nbsp; with the following meaning:
 *For &nbsp;$f \gg f_∗$&nbsp; holds &nbsp;$α(f) ≈ α_{\rm I}(f)$.
 *For &nbsp;$f \ll f_∗$&nbsp; holds &nbsp;$α(f) ≈ α_{\rm II}(f)$.
-These approximations are used to determine the attenuation function &nbsp;$\alpha(f)$&nbsp; for a message signal of frequency &nbsp;$f_0 = 2 \ \rm kHz$&nbsp;, whereby the following transmission media are to be considered:
+These approximations are used to determine the function &nbsp;$\alpha(f)$&nbsp; for a message signal of frequency &nbsp;$f_0 = 2 \ \rm kHz$,&nbsp; whereby the following transmission media are to be considered:
 *a copper cable with &nbsp;$0.6 \ \rm mm$&nbsp; diameter:
@@ Line 34: / Line 34: @@
 ''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]].
+*The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory|Some Results from Line Transmission Theory]].
-*The reference unit "Neper" (Np) in the above equations for &nbsp;$α_{\rm I}(f)$&nbsp; and &nbsp;$α_{\rm II}(f)$&nbsp; and thus also for the total attenuation function &nbsp;$α(f)$&nbsp; results from the fact that the magnitude frequency response is defined as &nbsp;$|H(f)| = {\rm e}^{-a}$&nbsp;.
+*The reference unit in the above equations for &nbsp;$α_{\rm I}(f)$&nbsp; and &nbsp;$α_{\rm II}(f)$&nbsp; and thus also for the total attenuation function (per unit length) &nbsp;$α(f)$&nbsp; results from the fact that the magnitude frequency response is defined as &nbsp;$|H(f)| = {\rm e}^{-a}$&nbsp;.
 *From this follows for the attenuation &nbsp;$ a = - {\rm ln} \; |H(f)|$, where the relationship via the natural logarithm is denoted by "Neper" (Np).
-*The unit of the attenuation function per unit length &nbsp;$α = a/l$&nbsp; is thus "Np/km".
+*The unit of the attenuation function per unit length &nbsp;$α = a/l$&nbsp; is thus&nbsp; "Np/km".
@@ Line 56: / Line 56: @@
-{Using the two approximations, give the attenuation function for frequency &nbsp;$f_0 = 2 \ \rm kHz$&nbsp;.
+{Using the two approximations,&nbsp; give the attenuation function for frequency &nbsp;$f_0 = 2 \ \rm kHz$&nbsp;.
 |type="{}"}
 ${\rm Copper}\hspace{-0.1cm}: \hspace{0.2cm} \alpha (f = f_0) \ = \ $  { 0.17 3% } $\ \rm Np/km$

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

← Older revision	Revision as of 14:42, 29 October 2021
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← Older revision		Revision as of 14:59, 15 October 2021
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−	[[Category:Linear and Time-Invariant Systems: Exercises\|^4.1 Some Results of the ~~Conduction~~ Theory^]]	+	[[Category:Linear and Time-Invariant Systems: Exercises\|^4.1 Some Results of the Line Transmission Theory^]]

← Older revision		Revision as of 14:56, 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 Some Results of the Conduction Theory^]]

← Older revision	Revision as of 14:32, 15 October 2021
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