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The standing wave pattern

Quelle: Wikipedia[1]

Using complex notation for the voltage amplitudes, for a signal at frequency ν, the actual (real) voltages Vactual as a function of time t are understood to relate to the complex voltages according to:

\[ V_\text{actual} = \Re (e^{i 2 \pi \nu t} V) \] .

Thus taking the real part of the complex quantity inside the parenthesis, the actual voltage consists of a sine wave at frequency ν with a peak amplitude equal to the complex magnitude of V, and with a phase given by the phase of the complex V. Then with the position along a transmission line given by x, with the line ending in a load located at x0, the complex amplitudes of the forward and reverse waves would be written as:

\[\begin{align} V_f(x) &= e^{-i k(x-x_0)} A \\ V_r(x) &= \Gamma e^{i k(x-x_0)} A \end{align}\]

for some complex amplitude A (corresponding to the forward wave at x0). Here k is the wavenumber due to the guided wavelength along the transmission line. Note that some treatments use phasors where the time dependence is according to \(e^{-i 2 \pi \nu t}\) and spatial dependence (for a wave in the +x direction) of \(e^{+i k(x - x_0)}\). Either convention obtains the same result for Vactual.

According to the superposition principle the net voltage present at any point x on the transmission line is equal to the sum of the voltages due to the forward and reflected waves:

\[\begin{align} V_\text{net}(x) &= V_f(x) + V_r(x) \\ &= e^{-i (x - x_0)} \left( 1 + \Gamma e^{i 2k(x - x_0)}\right ) A \end{align}\]

Since we are interested in the variations of the magnitude of Vnet along the line (as a function of x), we shall solve instead for the squared magnitude of that quantity, which simplifies the mathematics. To obtain the squared magnitude we multiply the above quantity by its complex conjugate:

Depending on the phase of the third term, one can see that the maximum and minimum values of \[\begin{align} |V_\text{net}(x)|^2 &= V_\text{net}(x) V^*_\text{net}(x) \\ &= e^{-i (x - x_0)} \left(1 + \Gamma e^{i 2k(x - x_0)}\right) A \, e^{+i (x - x_0)} \left(1 + \Gamma^* e^{-i 2k(x - x_0)}\right) A^* \\ &= \left[1 + |\Gamma|^2 + 2 \Re (\Gamma e^{i 2k(x - x_0)})\right] |A|^2 \end{align}\] Vnet (the square root of the quantity in the equations) are (1 + |Γ|)|A| and (1 − |Γ|)|A| respectively, for a standing wave ratio of:

\[ \text{SWR} = \frac{|V_\text{max}|}{|V_\text{min}|} = \frac{1 + |\Gamma|}{1 - |\Gamma|}\]

as we had earlier asserted. Along the line, the above expression for \( |V_\text{net}(x)|^2\) is seen to oscillate sinusoidally between \(|V_\text{min}|^2\) and \(|V_\text{max}|^2\) with a period of 2π/2k. This is half of the guided wavelength λ = 2π/k for the frequency ν. That can be seen as due to interference between two waves of that frequency which are travelling in opposite directions.


\[\begin{align} |V_\text{net}(x)|^2 = \left[1 + |\Gamma|^2 + 2 \Re (\Gamma e^{i 2k(x - x_0)})\right] |A|^2 \end{align}\] (A: Die der Vorwärtswelle entsprechende komplexe Amplitude am Ort \( x_0 \))

mit \[ \Gamma e^{i 2k(x - x_0)} = \Gamma \Big[ \cos \big( 2k(x - x_0) \big) + i \sin \big( 2k(x - x_0)\big) \Big] \]

folgt für den Realteil des 3. Terms der Ausgangsformel \[ 2 \Re (\Gamma e^{i 2k(x - x_0)}) = 2 \Gamma \cos \big( 2k(x - x_0) \big) \]

setzt man das wieder oben ein, folgt:

\[\begin{align} |V_\text{net}(x)|^2 = \left[1 + |\Gamma|^2 + 2 \Gamma \cos \big( 2k(x - x_0) \big) \right] |A|^2 \end{align}\]

meine Schreibweise:

\[ |U(x)|^2 = P_{vor} \cdot Z_W \left[1 + |r|^2 + 2 |r| \cos \big( 2 \frac {2 \pi (x - x_0)}{\lambda} \big) \right] \]