Difference between revisions of "Aufgaben:Exercise 1.2: Coaxial Cable"

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[[File: P_ID787__LZI_A_1_2.png | Various coaxial cables|right|frame]]
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[[File: P_ID787__LZI_A_1_2.png | Various coaxial cable types|right|frame]]
The frequency response of a normal coaxial cable of length  $l$  (with a diameter of   $2.6 \ \text{mm}$  of the inner conductor and  an external diameter of $9.5 \ \text{mm}$)  is for frequencies  $f > 0$:  
+
The frequency response of a normal coaxial cable of length  $l$  (with a diameter of $2.6 \ \text{mm}$  of the inner conductor and  an external diameter of $9.5 \ \text{mm}$)  is for frequencies  $f > 0$:  
 
:$$H(f) = {\rm e}^{-\alpha_{0\hspace{0.02cm}} \hspace{0.05cm} \cdot \hspace{0.05cm} l}
 
:$$H(f) = {\rm e}^{-\alpha_{0\hspace{0.02cm}} \hspace{0.05cm} \cdot \hspace{0.05cm} l}
 
\cdot {\rm e}^{-(\alpha_1  + {\rm j}\hspace{0.05cm} \cdot
 
\cdot {\rm e}^{-(\alpha_1  + {\rm j}\hspace{0.05cm} \cdot
Line 12: Line 12:
 
\hspace{0.05cm} \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm} l}.$$
 
\hspace{0.05cm} \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm} l}.$$
  
*Der erste, von den Ohmschen Verlusten herrührende Term in dieser Gleichung wird durch die so genannte kilometrische Dämpfung  $α_0 = 0.00162\, \text{Np/km}$  beschrieben.  
+
*The first term in this equation, arising from the ohmic losses, is described by the  "kilometric damping"  $α_0 = 0.00162\, \text{Np/km}$.  
  
  
*Der frequenzproportionale Dämpfungsanteil   ⇒   $α_1 · f · l$  mit  $α_1 = 0.000435 \,\text{Np/(km · MHz)}$  geht auf die Querverluste zurück. Dieser macht sich erst bei sehr hohen Frequenzen bemerkbar und wird im Folgenden vernachlässigt.  
+
*The frequency-proportional damping component   ⇒   $α_1 · f · l$  mit  $α_1 = 0.000435 \,\text{Np/(km · MHz)}$  is due to the lateral losses. This only becomes noticeable at very high frequencies and is neglected in the following.  
  
  
*Auch die frequenzproportionale Phase  $β_1 · f · l$  mit  $β_1 = 21.78 \,\text{rad/(km · MHz)}$  wird außer Acht gelassen werden, da diese nur eine für alle Frequenzen gleiche Laufzeit zur Folge hat.  
+
*Also, the frequency-proportional phase  $β_1 · f · l$  with  $β_1 = 21.78 \,\text{rad/(km · MHz)}$  is left out of consideration because this only leads to an equal transit time for all frequencies.  
  
  
Der Koaxialkabel–Frequenzgang wird deshalb für Frequenzen zwischen  $200 \ \text{kHz}$  und  $400 \ \text{MHz}$  vorwiegend durch den Einfluss
+
Hence, for frequencies between  $200 \ \text{kHz}$  and  $400 \ \text{MHz}$  the frequency response of the coaxial cable is predominantly determined by the influence of
*der Dämpfungskonstanten  $α_2 = 0.2722 \,\text{Np/(km · MHz}^{0.5})$, und
+
*the damping constant  $α_2 = 0.2722 \,\text{Np/(km · MHz}^{0.5})$, and
*der Phasenkonstanten  $β_2 = 0.2722 \,\text{rad/(km · MHz}^{0.5})$  
+
*the phase constant  $β_2 = 0.2722 \,\text{rad/(km · MHz}^{0.5})$,
  
  
bestimmt, die auf den so genannten Skineffekt zurückzuführen sind. Für positive Frequenzen gilt:  
+
which are due to the so-called  "skin effect".  For positive frequencies the following applies:  
 
:$$H(f) = K \cdot {\rm e}^{-(\alpha_2  + {\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \beta_2)\hspace{0.05cm} \cdot \hspace{0.05cm} \sqrt{f} \hspace{0.05cm} \cdot \hspace{0.05cm} l}.$$
 
:$$H(f) = K \cdot {\rm e}^{-(\alpha_2  + {\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \beta_2)\hspace{0.05cm} \cdot \hspace{0.05cm} \sqrt{f} \hspace{0.05cm} \cdot \hspace{0.05cm} l}.$$
Aufgrund der gleichen Zahlenwerte von  $α_2$  und  $β_2$  kann hierfür auch geschrieben werden:  
+
Because of the same numerical values of  $α_2$  and  $β_2$  this can be expressed as follows:  
 
:$$H(f) = K \cdot {\rm e}^{- \sqrt{2\hspace{0.05cm} \cdot \hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} f/f_0} },$$
 
:$$H(f) = K \cdot {\rm e}^{- \sqrt{2\hspace{0.05cm} \cdot \hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} f/f_0} },$$
wobei der Parameter  $f_0$  die beiden Konstanten  $α_2$  und  $β_2$  sowie die Kabellänge  $l$  gleichermaßen berücksichtigt.  
+
where the parameter  $f_0$  equally accounts for the two constants  $α_2$  and  $β_2$  and the cable length  $l$ .  
  
  
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''Please note:''  
 
''Please note:''  
*This task belongs to the chapter  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain | System Description in Frequency Domain]].
+
*This exercise belongs to the chapter  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain | System Description in Frequency Domain]].
 
   
 
   
  
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<quiz display=simple>
 
<quiz display=simple>
  
{Wie groß ist die Frequenzgang–Konstante&nbsp; $K$&nbsp; für die Kabellänge&nbsp; $l = 5\,\text{ km}$?  
+
{What is the frequency response constant&nbsp; $K$&nbsp; for a cable length of&nbsp; $l = 5\,\text{ km}$?  
 
|type="{}"}
 
|type="{}"}
 
$K  \ = \ $ { 0.992 5%  }
 
$K  \ = \ $ { 0.992 5%  }
  
  
{Welche Länge&nbsp; $l_{\rm max}$&nbsp; könnte ein Kabel besitzen, damit ein Gleichsignal um nicht mehr als&nbsp; $3\%$&nbsp; gedämpft wird?  
+
{What length&nbsp; $l_{\rm max}$&nbsp; could a cable have such that a direct&nbsp; (DC)&nbsp; signal is not damped by no more than&nbsp; $3\%$&nbsp;?  
 
|type="{}"}
 
|type="{}"}
 
$l_{\rm max} \ = \ $ { 18.8 5%  } &nbsp;$\text{ km}$
 
$l_{\rm max} \ = \ $ { 18.8 5%  } &nbsp;$\text{ km}$
  
  
{Wie groß ist die charakteristische Frequenz&nbsp; $f_0$&nbsp; für die Kabellänge&nbsp; $l = 5\,\text{ km}$. Berücksichtigen Sie die Beziehung&nbsp; $\rm \sqrt{2j} = 1 + j$.  
+
{What is the characteristic frequency&nbsp; $f_0$&nbsp; for a cable length of&nbsp; $l = 5\,\text{ km}$.&nbsp; Consider the relation&nbsp; $\rm \sqrt{2j} = 1 + j$.  
 
|type="{}"}
 
|type="{}"}
 
$f_0\ = \ $ { 0.54 5%  } &nbsp;$\text{ MHz}$
 
$f_0\ = \ $ { 0.54 5%  } &nbsp;$\text{ MHz}$
  
  
{Am Kabeleingang liegt ein Cosinussignal der Frequenz&nbsp; $f_x = f_0$&nbsp; mit Leistung&nbsp; $P_x = \,\text{1 W}$ an. Wie groß ist die Ausgangsleistung&nbsp; $P_y$?  
+
{A cosine signal of frequency&nbsp; $f_x = f_0$&nbsp; is applied to the cable input with a power of&nbsp; $P_x = \,\text{1 W}$.&nbsp; What is the output power&nbsp; $P_y$?  
 
|type="{}"}
 
|type="{}"}
 
$P_y \ = \ $ { 135 5%  } &nbsp;$\text{ mW}$
 
$P_y \ = \ $ { 135 5%  } &nbsp;$\text{ mW}$
  
  
{Welche Ausgangsleistung erhält man mit der Signalfrequenz&nbsp; $f_x = 10 \ \rm MHz$?
+
{What output power is obtained with the signal frequency&nbsp; $f_x = 10 \ \rm MHz$?
 
|type="{}"}
 
|type="{}"}
 
$P_y \ = \ $ { 0.184 5%  } &nbsp;$\text{ mW}$
 
$P_y \ = \ $ { 0.184 5%  } &nbsp;$\text{ mW}$
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</quiz>
 
</quiz>
  
===Sample solution===
+
===Solution===
 
{{ML-Kopf}}
 
{{ML-Kopf}}
'''(1)'''&nbsp; Für den Gleichsignalübertragungsfaktor gilt:  
+
'''(1)'''&nbsp; For the direct signal transmission factor the following holds:  
 
:$$K = H(f=0) = {\rm e}^{-\alpha_0 \hspace{0.05cm}\cdot
 
:$$K = H(f=0) = {\rm e}^{-\alpha_0 \hspace{0.05cm}\cdot
 
\hspace{0.05cm} l} = {\rm e}^{-0.00162 \hspace{0.05cm} \cdot
 
\hspace{0.05cm} l} = {\rm e}^{-0.00162 \hspace{0.05cm} \cdot
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'''(2)'''&nbsp; Mit&nbsp; ${\rm a_0 } = α_0 · l$&nbsp; müsste folgende Gleichung erfüllt sein:
+
'''(2)'''&nbsp; With&nbsp; ${\rm a_0 } = α_0 · l$&nbsp; the following equation should be satisfied:
 
:$${\rm e}^{\rm -a_0 } \ge 0.97
 
:$${\rm e}^{\rm -a_0 } \ge 0.97
 
\hspace{0.2cm} \Rightarrow \hspace{0.2cm} {\rm a_0 } < \ln \frac{1}{0.97
 
\hspace{0.2cm} \Rightarrow \hspace{0.2cm} {\rm a_0 } < \ln \frac{1}{0.97
 
} \approx 0.0305\,{\rm Np}.$$
 
} \approx 0.0305\,{\rm Np}.$$
*Damit erhält man für die maximale Länge&nbsp; $l_{\rm max} = 0.0305 \ \text{Np}/0.00162 \ \text{Np/km} \rm \underline{\: ≈ \: 18.8 \: km}$.  
+
*Thus, the maximum length is &nbsp; $l_{\rm max} = 0.0305 \ \text{Np}/0.00162 \ \text{Np/km} \rm \underline{\: ≈ \: 18.8 \: km}$.  
  
  
  
'''(3)'''&nbsp; Wegen&nbsp; $β_2 = α_2$&nbsp; und der angegebenen Beziehung&nbsp; $\rm 1 + j = \sqrt{2j}$&nbsp; kann für den Frequenzgang auch geschrieben werden:
+
'''(3)'''&nbsp; Because of&nbsp; $β_2 = α_2$&nbsp; and  the given relation&nbsp; $\rm 1 + j = \sqrt{2j}$&nbsp; the frequency response can be formulated as follows:
 
:$$H(f) = K \cdot {\rm e}^{- \sqrt{2\hspace{0.05cm} \cdot
 
:$$H(f) = K \cdot {\rm e}^{- \sqrt{2\hspace{0.05cm} \cdot
 
\hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot
 
\hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot
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\sqrt{2\hspace{0.05cm} \cdot
 
\sqrt{2\hspace{0.05cm} \cdot
 
\hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} f/f_0} }.$$
 
\hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} f/f_0} }.$$
*Durch Koeffizientenvergleich mit der vorne angegebenen Gleichung erhält man:  
+
*By comparing the coefficients with the equation given above the following is obtained:  
 
:$${1}/{f_0} = \alpha_2^2 \hspace{0.05cm} \cdot \hspace{0.05cm} l^2 = \big ( \frac { {\rm 0.272} }{\rm km \hspace{0.05cm} \cdot \hspace{0.05cm} \sqrt{MHz} }\big )^2 \cdot ({\rm 5 \hspace{0.05cm} km})^2 =  \frac{1.855}{ {\rm MHz} }\hspace{0.2cm} \Rightarrow \hspace{0.2cm} f_0 \hspace{0.15cm}\rm \underline{= 0.540 \: MHz}.$$
 
:$${1}/{f_0} = \alpha_2^2 \hspace{0.05cm} \cdot \hspace{0.05cm} l^2 = \big ( \frac { {\rm 0.272} }{\rm km \hspace{0.05cm} \cdot \hspace{0.05cm} \sqrt{MHz} }\big )^2 \cdot ({\rm 5 \hspace{0.05cm} km})^2 =  \frac{1.855}{ {\rm MHz} }\hspace{0.2cm} \Rightarrow \hspace{0.2cm} f_0 \hspace{0.15cm}\rm \underline{= 0.540 \: MHz}.$$
  
  
  
'''(4)'''&nbsp; Für den Frequenzgang gilt:  
+
'''(4)'''&nbsp; For the frequency response it holds:  
 
:$$\begin{align*}H(f) & = K \cdot {\rm e}^{- \sqrt{2\hspace{0.05cm} \cdot
 
:$$\begin{align*}H(f) & = K \cdot {\rm e}^{- \sqrt{2\hspace{0.05cm} \cdot
 
\hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot
 
\hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot
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{\rm e}^{- 2\hspace{0.05cm} \cdot
 
{\rm e}^{- 2\hspace{0.05cm} \cdot
 
\hspace{0.05cm}\sqrt{ f/f_0} }.\end{align*}$$
 
\hspace{0.05cm}\sqrt{ f/f_0} }.\end{align*}$$
*Für&nbsp; $f = f_0$&nbsp; erhält man hierfür&nbsp; $K^2 \cdot \rm e^{–2} ≈ 0.135$. Daraus folgt weiter:  
+
*For this,&nbsp; $K^2 \cdot \rm e^{–2} ≈ 0.135$ is obtained for&nbsp; $f = f_0$&nbsp;. From this it follows that:  
 
:$$P_y = P_x \cdot |H(f = f_0)|^2 \hspace{0.15cm}\underline{\approx135\hspace{0.05cm}{\rm mW}}.$$
 
:$$P_y = P_x \cdot |H(f = f_0)|^2 \hspace{0.15cm}\underline{\approx135\hspace{0.05cm}{\rm mW}}.$$
  
  
  
'''(5)'''&nbsp; Mit der höheren Frequenz&nbsp; $f_x = 10\:\text{MHz}$&nbsp; ist die Ausgangsleistung gegenüber&nbsp; $f_x = 0.54\:\text{MHz}$&nbsp; signifikant kleiner:  
+
'''(5)'''&nbsp; With the higher frequency&nbsp; $f_x = 10\:\text{MHz}$&nbsp; the output power is significantly smaller compared to&nbsp; $f_x = 0.54\:\text{MHz}$&nbsp;:  
 
:$$P_y = P_x \cdot {\rm e}^{- 2\hspace{0.05cm} \cdot
 
:$$P_y = P_x \cdot {\rm e}^{- 2\hspace{0.05cm} \cdot
 
\hspace{0.05cm}\sqrt{ 10/0.54} }\approx P_x \cdot {\rm e}^{- 8.6 } \hspace{0.15cm}\underline{\approx 0.184 \hspace{0.1cm}{\rm mW}}.$$
 
\hspace{0.05cm}\sqrt{ 10/0.54} }\approx P_x \cdot {\rm e}^{- 8.6 } \hspace{0.15cm}\underline{\approx 0.184 \hspace{0.1cm}{\rm mW}}.$$

Latest revision as of 16:40, 11 July 2021


Various coaxial cable types

The frequency response of a normal coaxial cable of length  $l$  (with a diameter of $2.6 \ \text{mm}$  of the inner conductor and  an external diameter of $9.5 \ \text{mm}$)  is for frequencies  $f > 0$:

$$H(f) = {\rm e}^{-\alpha_{0\hspace{0.02cm}} \hspace{0.05cm} \cdot \hspace{0.05cm} l} \cdot {\rm e}^{-(\alpha_1 + {\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \beta_1)\hspace{0.05cm} \cdot \hspace{0.05cm} f \cdot \hspace{0.05cm} l}\hspace{0.05cm}\cdot\hspace{0.05cm} {\rm e}^{-(\alpha_2 + {\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}\beta_2) \hspace{0.05cm} \cdot \hspace{0.05cm} \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm} l}.$$
  • The first term in this equation, arising from the ohmic losses, is described by the  "kilometric damping"  $α_0 = 0.00162\, \text{Np/km}$.


  • The frequency-proportional damping component   ⇒   $α_1 · f · l$  mit  $α_1 = 0.000435 \,\text{Np/(km · MHz)}$  is due to the lateral losses. This only becomes noticeable at very high frequencies and is neglected in the following.


  • Also, the frequency-proportional phase  $β_1 · f · l$  with  $β_1 = 21.78 \,\text{rad/(km · MHz)}$  is left out of consideration because this only leads to an equal transit time for all frequencies.


Hence, for frequencies between  $200 \ \text{kHz}$  and  $400 \ \text{MHz}$  the frequency response of the coaxial cable is predominantly determined by the influence of

  • the damping constant  $α_2 = 0.2722 \,\text{Np/(km · MHz}^{0.5})$, and
  • the phase constant  $β_2 = 0.2722 \,\text{rad/(km · MHz}^{0.5})$,


which are due to the so-called  "skin effect".  For positive frequencies the following applies:

$$H(f) = K \cdot {\rm e}^{-(\alpha_2 + {\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \beta_2)\hspace{0.05cm} \cdot \hspace{0.05cm} \sqrt{f} \hspace{0.05cm} \cdot \hspace{0.05cm} l}.$$

Because of the same numerical values of  $α_2$  and  $β_2$  this can be expressed as follows:

$$H(f) = K \cdot {\rm e}^{- \sqrt{2\hspace{0.05cm} \cdot \hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} f/f_0} },$$

where the parameter  $f_0$  equally accounts for the two constants  $α_2$  and  $β_2$  and the cable length  $l$ .




Please note:



Questions

1

What is the frequency response constant  $K$  for a cable length of  $l = 5\,\text{ km}$?

$K \ = \ $

2

What length  $l_{\rm max}$  could a cable have such that a direct  (DC)  signal is not damped by no more than  $3\%$ ?

$l_{\rm max} \ = \ $

 $\text{ km}$

3

What is the characteristic frequency  $f_0$  for a cable length of  $l = 5\,\text{ km}$.  Consider the relation  $\rm \sqrt{2j} = 1 + j$.

$f_0\ = \ $

 $\text{ MHz}$

4

A cosine signal of frequency  $f_x = f_0$  is applied to the cable input with a power of  $P_x = \,\text{1 W}$.  What is the output power  $P_y$?

$P_y \ = \ $

 $\text{ mW}$

5

What output power is obtained with the signal frequency  $f_x = 10 \ \rm MHz$?

$P_y \ = \ $

 $\text{ mW}$


Solution

(1)  For the direct signal transmission factor the following holds:

$$K = H(f=0) = {\rm e}^{-\alpha_0 \hspace{0.05cm}\cdot \hspace{0.05cm} l} = {\rm e}^{-0.00162 \hspace{0.05cm} \cdot \hspace{0.05cm} 5}\hspace{0.15cm}\underline{ \approx 0.992}.$$


(2)  With  ${\rm a_0 } = α_0 · l$  the following equation should be satisfied:

$${\rm e}^{\rm -a_0 } \ge 0.97 \hspace{0.2cm} \Rightarrow \hspace{0.2cm} {\rm a_0 } < \ln \frac{1}{0.97 } \approx 0.0305\,{\rm Np}.$$
  • Thus, the maximum length is   $l_{\rm max} = 0.0305 \ \text{Np}/0.00162 \ \text{Np/km} \rm \underline{\: ≈ \: 18.8 \: km}$.


(3)  Because of  $β_2 = α_2$  and the given relation  $\rm 1 + j = \sqrt{2j}$  the frequency response can be formulated as follows:

$$H(f) = K \cdot {\rm e}^{- \sqrt{2\hspace{0.05cm} \cdot \hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} f \hspace{0.05cm} \cdot \hspace{0.05cm} {\alpha_2}^2 \hspace{0.05cm} \cdot \hspace{0.05cm} l^2} }= K \cdot {\rm e}^{- \sqrt{2\hspace{0.05cm} \cdot \hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} f/f_0} }.$$
  • By comparing the coefficients with the equation given above the following is obtained:
$${1}/{f_0} = \alpha_2^2 \hspace{0.05cm} \cdot \hspace{0.05cm} l^2 = \big ( \frac { {\rm 0.272} }{\rm km \hspace{0.05cm} \cdot \hspace{0.05cm} \sqrt{MHz} }\big )^2 \cdot ({\rm 5 \hspace{0.05cm} km})^2 = \frac{1.855}{ {\rm MHz} }\hspace{0.2cm} \Rightarrow \hspace{0.2cm} f_0 \hspace{0.15cm}\rm \underline{= 0.540 \: MHz}.$$


(4)  For the frequency response it holds:

$$\begin{align*}H(f) & = K \cdot {\rm e}^{- \sqrt{2\hspace{0.05cm} \cdot \hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} f/f_0} } = K \cdot {\rm e}^{- \sqrt{ f/f_0} } \cdot {\rm e}^{- {\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}\sqrt{ f/f_0} } \hspace{0.05 cm} \Rightarrow \hspace{0.05 cm} |H(f)|^2 = K^2 \cdot {\rm e}^{- 2\hspace{0.05cm} \cdot \hspace{0.05cm}\sqrt{ f/f_0} }.\end{align*}$$
  • For this,  $K^2 \cdot \rm e^{–2} ≈ 0.135$ is obtained for  $f = f_0$ . From this it follows that:
$$P_y = P_x \cdot |H(f = f_0)|^2 \hspace{0.15cm}\underline{\approx135\hspace{0.05cm}{\rm mW}}.$$


(5)  With the higher frequency  $f_x = 10\:\text{MHz}$  the output power is significantly smaller compared to  $f_x = 0.54\:\text{MHz}$ :

$$P_y = P_x \cdot {\rm e}^{- 2\hspace{0.05cm} \cdot \hspace{0.05cm}\sqrt{ 10/0.54} }\approx P_x \cdot {\rm e}^{- 8.6 } \hspace{0.15cm}\underline{\approx 0.184 \hspace{0.1cm}{\rm mW}}.$$