refactor: impedanztransformation
This commit is contained in:
Matthias Jung
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</svg>
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After Width: | Height: | Size: 33 KiB |
@@ -0,0 +1,61 @@
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% 2026 Prof. Dr. Matthias Jung (DL9MJ)
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\begin{tikzpicture}
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\draw(0,0) coordinate(start)
|
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to [sinusoidal voltage source] (0,2)
|
||||
to [short] ++(1,0)
|
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to [short, -o] ++(0,-0.5) coordinate(l1);
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\draw(start)
|
||||
to [short] ++(1,0)
|
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to [short, -o] ++(0,+0.5) coordinate(l2);
|
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%
|
||||
\draw[very thick](l1) to [short, o-o] ++(5,0) coordinate(l3);
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\draw[very thick](l2) to [short, o-o] ++(5,0) coordinate(l4);
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%
|
||||
\draw[short, DARCgreen] (l3) to [short, l_={$R_\mathrm{a}=0$}] (l4);
|
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%
|
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\draw(l1) to [open] ++(0,-2.5) coordinate(s1);
|
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%
|
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\draw[-Triangle] (s1) -- ++(5.5,0);
|
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%
|
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\def\x{0.5}
|
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%
|
||||
\draw[DARCred] (s1)
|
||||
sin ++(+1,+1*\x) coordinate(sinus)
|
||||
cos ++(+1,-1*\x)
|
||||
sin ++(+1,-1*\x)
|
||||
cos ++(+1,+1*\x)
|
||||
sin ++(+1,+1*\x)
|
||||
;
|
||||
\draw[DARCblue] (s1) ++(0,-1*\x)
|
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cos ++(+1,+1*\x)
|
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sin ++(+1,+1*\x) coordinate(cosius)
|
||||
cos ++(+1,-1*\x)
|
||||
sin ++(+1,-1*\x)
|
||||
cos ++(+1,+1*\x)
|
||||
;
|
||||
%
|
||||
\draw[DARCred] (sinus) -- ++(0.5*\x,0.5*\x) node[right] {$I$};
|
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\draw[DARCblue] (cosius) -- ++(0.5*\x,0.5*\x) node[right] {$U$};
|
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%
|
||||
\draw(s1)
|
||||
++(0,-1) node[](){$R=\infty$}
|
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++(+1,0) node[](){$R=0$}
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++(+1,0) node[](){$R=\infty$}
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++(+1,0) node[](){$R=0$}
|
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++(+1,0) node[](){$R=\infty$}
|
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++(+1,0) node[](){$R=0$};
|
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%
|
||||
\draw(s1)
|
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++(0,-1.5) coordinate(s2)
|
||||
++(1,0) coordinate(s3)
|
||||
++(1,0) coordinate(s4)
|
||||
++(1,0) coordinate(s5)
|
||||
++(1,0) coordinate(s6)
|
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++(1,0) coordinate(s7);
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% 1:[color], 2:[top], 3:start, 4:end, 5:shift, 6:rotate, 7:text
|
||||
\DARCmeasure{s2}{s3}{0,0}{0}{$\frac{\lambda}{4}$}
|
||||
\DARCmeasure{s3}{s4}{0,0}{0}{$\frac{\lambda}{4}$}
|
||||
\DARCmeasure{s4}{s5}{0,0}{0}{$\frac{\lambda}{4}$}
|
||||
\DARCmeasure{s5}{s6}{0,0}{0}{$\frac{\lambda}{4}$}
|
||||
\DARCmeasure{s6}{s7}{0,0}{0}{$\frac{\lambda}{4}$}
|
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\end{tikzpicture}
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@@ -0,0 +1,218 @@
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<?xml version="1.0" encoding="UTF-8"?>
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After Width: | Height: | Size: 33 KiB |
@@ -0,0 +1,61 @@
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% 2026 Prof. Dr. Matthias Jung (DL9MJ)
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\begin{tikzpicture}
|
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\draw(0,0) coordinate(start)
|
||||
to [sinusoidal voltage source] (0,2)
|
||||
to [short] ++(1,0)
|
||||
to [short, -o] ++(0,-0.5) coordinate(l1);
|
||||
\draw(start)
|
||||
to [short] ++(1,0)
|
||||
to [short, -o] ++(0,+0.5) coordinate(l2);
|
||||
%
|
||||
\draw[very thick](l1) to [short, o-o] ++(5,0) coordinate(l3);
|
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\draw[very thick](l2) to [short, o-o] ++(5,0) coordinate(l4);
|
||||
%
|
||||
\draw[DARCgreen] (l3) to [open, l_={$R_\mathrm{a}=\infty$}] (l4);
|
||||
%
|
||||
\draw(l1) to [open] ++(0,-2.5) coordinate(s1);
|
||||
%
|
||||
\draw[-Triangle] (s1) -- ++(5.5,0);
|
||||
%
|
||||
\def\x{0.5}
|
||||
%
|
||||
\draw[DARCblue] (s1)
|
||||
sin ++(+1,+1*\x) coordinate(sinus)
|
||||
cos ++(+1,-1*\x)
|
||||
sin ++(+1,-1*\x)
|
||||
cos ++(+1,+1*\x)
|
||||
sin ++(+1,+1*\x)
|
||||
;
|
||||
\draw[DARCred] (s1) ++(0,+1*\x)
|
||||
cos ++(+1,-1*\x)
|
||||
sin ++(+1,-1*\x)
|
||||
cos ++(+1,+1*\x)
|
||||
sin ++(+1,+1*\x) coordinate(cosius)
|
||||
cos ++(+1,-1*\x)
|
||||
;
|
||||
%
|
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\draw[DARCblue] (sinus) -- ++(0.5*\x,0.5*\x) node[right] {$U$};
|
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\draw[DARCred] (cosius) -- ++(0.5*\x,0.5*\x) node[right] {$I$};
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%
|
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\draw(s1)
|
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++(0,-1) node[](){$R=0$}
|
||||
++(+1,0) node[](){$R=\infty$}
|
||||
++(+1,0) node[](){$R=0$}
|
||||
++(+1,0) node[](){$R=\infty$}
|
||||
++(+1,0) node[](){$R=0$}
|
||||
++(+1,0) node[](){$R=\infty$};
|
||||
%
|
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\draw(s1)
|
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++(0,-1.5) coordinate(s2)
|
||||
++(1,0) coordinate(s3)
|
||||
++(1,0) coordinate(s4)
|
||||
++(1,0) coordinate(s5)
|
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++(1,0) coordinate(s6)
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++(1,0) coordinate(s7);
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% 1:[color], 2:[top], 3:start, 4:end, 5:shift, 6:rotate, 7:text
|
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\DARCmeasure{s2}{s3}{0,0}{0}{$\frac{\lambda}{4}$}
|
||||
\DARCmeasure{s3}{s4}{0,0}{0}{$\frac{\lambda}{4}$}
|
||||
\DARCmeasure{s4}{s5}{0,0}{0}{$\frac{\lambda}{4}$}
|
||||
\DARCmeasure{s5}{s6}{0,0}{0}{$\frac{\lambda}{4}$}
|
||||
\DARCmeasure{s6}{s7}{0,0}{0}{$\frac{\lambda}{4}$}
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\end{tikzpicture}
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+57
-68
@@ -1,89 +1,78 @@
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<?xml version="1.0" encoding="UTF-8"?>
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<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="112.931pt" height="38.841pt" viewBox="0 0 112.931 38.841">
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<defs>
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<g>
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<symbol overflow="visible" id="glyph0-0">
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</symbol>
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<symbol overflow="visible" id="glyph0-1">
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</symbol>
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<symbol overflow="visible" id="glyph1-0">
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Before Width: | Height: | Size: 13 KiB After Width: | Height: | Size: 11 KiB |
@@ -1,9 +1,18 @@
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||||
Ist der Wellenwiderstand eines Speisekabels nicht identisch mit dem Widerstand der Last, so kann neben der auftretenden Stehwelle auch eine Impedanztransformation erzielt werden. Das heißt, dass eine Signalquelle an einem Ende des Kabels einen anderen Widerstand "sieht" als am anderen Ende des Kabels angeschlossen ist.
|
||||
Nicht jede Antenne besitzt an ihrem Speisepunkt genau die Impedanz, die für den Anschluss an eine bestimmte Speiseleitung oder einen Sender benötigt wird. Weicht die Impedanz beispielsweise von den üblichen $\qty{50}{\ohm}$ ab, muss sie entsprechend angepasst werden, damit die HF-Leistung möglichst verlustarm übertragen werden kann. Dazu wird die vorhandene Impedanz in eine andere, gewünschte Impedanz *transformiert*. Dies bezeichnet man als *Impedanztransformation* beziehungsweise *Impedanzanpassung*.
|
||||
|
||||
Insbesondere zwei Fälle sind von Bedeutung: Die $\lambda/4$-Leitung zur gezielten Impedanztransformation sowie $\lambda/2$-Leitungen und Vielfache davon, die unabhängig vom Wellenwiderstand keine Impedanztransformation vornehmen.
|
||||
Für die Impedanzanpassung bzw. Impedanztransformation gibt es verschiedene Möglichkeiten. Häufig verwendet werden beispielsweise:
|
||||
|
||||
* Transformatoren,
|
||||
* $\frac{\lambda}{4}$-Leitungen oder
|
||||
* Anpassnetzwerke aus Spulen und Kondensatoren.
|
||||
|
||||
Transformatoren haben wir bereits bei der endgespeisten Antenne mit 1:49 Unun kennengelernt. Im Folgenden betrachten wir deshalb zwei weitere Möglichkeiten genauer: die Impedanztransformation mit $\frac{\lambda}{4}$-Leitungen aus dem Vorherigen Abschnitt und die Anpassung mit LC-Netzwerken. Schauen wir zunächst nochmal auf die Transformationsleitungen, hier ist es übrigens egal ob wir eine symmetrische Speiseleitung oder eine unsymmetrische Koaxialleitung verwenden, die Transformation funktioniert in beiden Fällen:
|
||||
|
||||
Bei einer Leitung, deren elektrische Länge $\lambda/4$ beträgt, werden Wirkwiderstände, die kleiner als der Wellenwiderstand der Leitung sind, zu Widerständen die größer als der Wellenwiderstand der Leitung sind. Umgekehrt werden Wirkwiderstände, die größer als der Wellenwiderstand der Leitung sind, zu Widerständen die kleiner als der Wellenwiderstand sind. Diesen Umstand macht man sich z.B. zunutze, um hochohmige Antennen auf ein niederohmiges System ($\qty{50}{\ohm}$) anzupassen.
|
||||
|
||||
[question:AG410]
|
||||
[question:AG409]
|
||||
|
||||
Bei einer Leitungslänge von $\lambda/2$ hebt sich der Effekt jedoch wieder auf, so dass keine Impedanztransformation auftritt.
|
||||
|
||||
[question:AG412]
|
||||
|
||||
@@ -1,13 +1,42 @@
|
||||
Einen interessanten Sonderfall der Impedanztransformation mittels Übertragungsleitungen stellt die sogenannte Lecherleitung dar. Dies ist eine am Ende offene Zweidrahtleitung. Bei Frequenzen, für die die Lecherleitung eine elektrische Länge von $\lambda/4$ aufweist, wird das offene Ende ($\qty{\infty}{\ohm}$) in einen Kurzschluss ($\qty{0}{\ohm}$) transformiert. Die Lecherleitung ist in diesem Resonanzfalle dann am anderen Ende leitend, obwohl die beiden Drähte gar nicht miteinander verbunden sind.
|
||||
Eine sogenannte *Lecherleitung* besteht aus zwei parallelen Leitern, auf denen sich durch die Überlagerung von hin- und rücklaufender Welle stehende HF-Wellen ausbilden. Sie kann am Ende entweder offen (vgl. Abbildung [ref:a_lecherleitung_offen]) oder kurzgeschlossen (vgl. Abbildung [ref:a_lecherleitung_kurzgeschlossen]) sein. In beiden Fällen entstehen charakteristische Strom- und Spannungsverteilungen, die beispielsweise zur Bestimmung der Wellenlänge genutzt werden können.
|
||||
|
||||
Umgekehrt gilt, dass eine am Ende kurzgeschlossene Leitung im Resonanzfalle sehr hochohmig wird.
|
||||
<margin>
|
||||
[picture:1112:a_lecherleitung_offen:Lecherleitung mit offenem Ende]
|
||||
</margin>
|
||||
|
||||
% TODO: Bild der Lecherleitung
|
||||
Bei einer am Ende *offenen* Lecherleitung kann am Leitungsende kein Strom fließen. Dort befindet sich daher ein Stromminimum und gleichzeitig ein Spannungsmaximum. Aus dem Ohmschen Gesetz
|
||||
|
||||
$R=\frac{U}{I}$
|
||||
|
||||
folgt, dass das Verhältnis von Spannung zu Strom an dieser Stelle idealisiert unendlich groß wird, also $R=\infty$. Strom und Spannung sind entlang der Leitung räumlich um $\frac{\lambda}{4}$ gegeneinander verschoben. Im Abstand von $\frac{\lambda}{4}$ vom offenen Leitungsende befindet sich daher ein Strommaximum und gleichzeitig ein Spannungsminimum. Da dort die Spannung idealisiert gegen null geht, ergibt sich nach dem Ohmschen Gesetz entsprechend $R=0$. Nach jeweils weiteren $\frac{\lambda}{4}$ wechseln sich Strommaximum und Spannungsmaximum ab. Nach einer Strecke von $\frac{\lambda}{2}$ liegt wieder die gleiche Strom- und Spannungsverteilung vor, wie in Abbildung [ref:a_lecherleitung_offen] dargestellt.
|
||||
|
||||
---
|
||||
|
||||
Bei einer am Ende *kurzgeschlossenen* Lecherleitung können die beiden Leiter am Leitungsende keine unterschiedliche Spannung besitzen. Dort gilt daher $U=0$. Es befindet sich dort ein Spannungsminimum und gleichzeitig ein Strommaximum. Mit $R=\frac{U}{I}$ folgt damit idealisiert $R=0$. Im Abstand von $\frac{\lambda}{4}$ vom Kurzschluss befindet sich dagegen ein Spannungsmaximum und gleichzeitig ein Stromminimum. Da der Strom dort idealisiert gegen null geht, wird das Verhältnis $\frac{U}{I}$ sehr groß und es gilt idealisiert $R=\infty$. Auch bei der kurzgeschlossenen Lecherleitung wechseln sich Strom- und Spannungsmaxima nach jeweils $\frac{\lambda}{4}$ ab. Nach einer Strecke von $\frac{\lambda}{2}$ liegt wieder die gleiche Strom- und Spannungsverteilung vor, wie in Abbildung [ref:a_lecherleitung_kurzgeschlossen] dargestellt.
|
||||
|
||||
|
||||
<margin>
|
||||
[picture:1111:a_lecherleitung_kurzgeschlossen:Lecherleitung mit kurzgeschlossenem Ende]
|
||||
</margin>
|
||||
|
||||
Welche Frequenz sich auf einer Lecherleitung als Resonanz ausbildet, hängt im Wesentlichen von ihrer Länge ab. Ändert sich die Leitungslänge, ändert sich damit auch die Resonanzfrequenz.
|
||||
|
||||
[question:AG320]
|
||||
|
||||
Bei einer Leitungslänge von $\frac{\lambda}{2}$ wiederholt sich die Strom- und Spannungsverteilung vollständig. Eine Lastimpedanz am Leitungsende erscheint daher am Leitungseingang wieder mit demselben Wert.
|
||||
|
||||
Ein besonders wichtiger Spezialfall ergibt sich, wenn die Lecherleitung bei der betrachteten Frequenz eine elektrische Länge von genau $\frac{\lambda}{4}$ besitzt. Wie wir an den Strom- und Spannungsverteilungen gesehen haben, vertauschen sich über eine Strecke von $\frac{\lambda}{4}$ ein Strommaximum und ein Spannungsmaximum miteinander. Dadurch wird auch eine hohe Impedanz in eine niedrige Impedanz und umgekehrt transformiert.
|
||||
|
||||
Bei einer am Ende *offenen* $\frac{\lambda}{4}$-Leitung ist die Impedanz am Leitungsende idealisiert unendlich groß. Nach einer Strecke von $\frac{\lambda}{4}$ befindet sich am Leitungseingang dagegen ein Spannungsminimum und ein Strommaximum. Die Eingangsimpedanz ist daher nahezu null ($Z_\mathrm{in} \approx \qty{0}{\ohm}$). Ein offenes Leitungsende wird durch eine $\frac{\lambda}{4}$ lange Leitung also näherungsweise in einen Kurzschluss transformiert.
|
||||
|
||||
---
|
||||
|
||||
[question:AG411]
|
||||
|
||||
Einen ähnlichen Effekt können wir mit Koaxialkabeln erreichen.
|
||||
Umgekehrt besitzt eine am Ende *kurzgeschlossene* Leitung am Leitungsende die Impedanz $\qty{0}{\ohm}$. Nach einer Strecke von $\frac{\lambda}{4}$ liegt am Leitungseingang ein Spannungsmaximum und ein Stromminimum vor. Die Eingangsimpedanz wird dort daher sehr groß ($Z_\mathrm{in} \rightarrow \infty$). Ein Kurzschluss am Leitungsende wird durch eine $\frac{\lambda}{4}$ lange Leitung somit näherungsweise in einen Leerlauf transformiert.
|
||||
|
||||
[question:AG410]
|
||||
[question:AG409]
|
||||
<indepth>
|
||||
Das Verhalten einer $\frac{\lambda}{4}$-Leitung lässt sich auch mit Schwingkreisen vergleichen. Eine offene $\frac{\lambda}{4}$-Leitung besitzt am Eingang eine sehr kleine Impedanz und verhält sich dort ähnlich wie ein *Reihenschwingkreis in Resonanz*. Eine kurzgeschlossene $\frac{\lambda}{4}$-Leitung besitzt am Eingang dagegen eine sehr große Impedanz und verhält sich ähnlich wie ein *Parallelschwingkreis in Resonanz*.
|
||||
</indepth>
|
||||
|
||||
Im nächsten Abschnitt betrachten wir, wie sich mit Hilfe von $\frac{\lambda}{4}$-Leitungen gezielt Impedanztransformationen durchführen lassen.
|
||||
+4
-4
@@ -1070,15 +1070,15 @@
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Impedanztransformation",
|
||||
"ident": "impedanztransformation",
|
||||
"title": "Lecherleitung",
|
||||
"ident": "lecherleitung",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Lecherleitung",
|
||||
"ident": "lecherleitung",
|
||||
"title": "Impedanztransformation",
|
||||
"ident": "impedanztransformation",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
|
||||
+4
-4
@@ -1723,15 +1723,15 @@
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Impedanztransformation",
|
||||
"ident": "impedanztransformation",
|
||||
"title": "Lecherleitung",
|
||||
"ident": "lecherleitung",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Lecherleitung",
|
||||
"ident": "lecherleitung",
|
||||
"title": "Impedanztransformation",
|
||||
"ident": "impedanztransformation",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
|
||||
+4
-4
@@ -2512,15 +2512,15 @@
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Impedanztransformation",
|
||||
"ident": "impedanztransformation",
|
||||
"title": "Lecherleitung",
|
||||
"ident": "lecherleitung",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
},
|
||||
{
|
||||
"title": "Lecherleitung",
|
||||
"ident": "lecherleitung",
|
||||
"title": "Impedanztransformation",
|
||||
"ident": "impedanztransformation",
|
||||
"status": "alpha",
|
||||
"class": "A",
|
||||
"video_url": null
|
||||
|
||||
Reference in New Issue
Block a user