High frequency transmission lines

Teoriaa: Systeemi

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Dipole antennas and transmission coefficient. See the svg file

Antenn system.

Lähetin, johto, antenni, jne. https://www.antenna-theory.com/tutorial/txline/transmissionline.php

https://www.worldradiohistory.com/BOOKSHELF-ARH/Technology/Rider-Books/R-F%20Transmission%20Lines%20-%20Alexander%20Schure.pdf

Transmission line

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  The (long) transmission line is modeled as Z₀. If the frequency (wavelength) of the source is too large (small) compared to dimensions of the system, it need to be considered in more detailed. See also the svg file.

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  The physical realizations of the transmission lines are usually coaxial cables, twisted cables or twin lead cables.

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  The system is analyzed as being differential short pieces. The conductance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is the conductance between the two wires, which exists because of the high frequency.

If a transmission line has a length greater than about 10% of a wavelength, then the line length will noticeably affect the circuit's impedance. The equation in the above image can be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align*} \frac{\partial v}{dx} &= -L\frac{\partial i}{\partial t} - Ri \\ \frac{\partial i}{dx} &= -C\frac{\partial v}{\partial t} - Gv \end{align*} }

The solutions to the above equations is the sum of forward and backward traveling (reflected) waves: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(z,t) = v^+ e^{-\alpha z} e^{\imath( \omega t - \gamma z )} + v^- e^{\alpha z} e^{\imath (\omega t +\gamma z) } } and if we assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=0} we have the telegraphers equations https://en.wikipedia.org/wiki/Telegrapher's_equations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(z,t) = v^+ e^{\imath (\omega t - \gamma z)} + v^- e^{\imath (\omega t +\gamma z)} } and a similar for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=i(z,t)} . If we replace $i$ by Ohm law, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i(z,t) = \frac{v^+}{Z_0} e^{\imath (\omega t - \gamma z) } + \frac{v^-}{Z_0} e^{\imath (\omega t +\gamma z)} = \frac{v^+}{Z_0} e^{\imath (\omega t - \gamma z)} \left( 1 - \frac{v^-}{v^+}e^{\imath(2\gamma z)} \right) }

The fraction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{v^-}{v^+}} is called reflection coefficient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{v^-}{v^+} = \Gamma e^{\imath \phi_z} = \frac{Z_L - Z_0}{Z_L + Z_0}}

which gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i(z,t) = = \frac{v^+}{Z_0} e^{\imath (\omega t - \gamma z)} \left( 1 - \Gamma e^{\imath(\phi_z +2\gamma z)} \right) }

The characteristic impedance is

Thus we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2 v}{dL^2} = \gamma^2 v } and similar for the current. The constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma^2 = (R+\imath \omega L)(G+\imath \omega C)} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_0 = \frac{v^+}{i^+} = - \frac{v^-}{i^-} = \sqrt{ \frac{R' + \imath \omega L'}{G' + \imath \omega C'}} }

For lossless line $R'=G'=0$ and for distortionless line $R'/L'=G'/C'$. The voltage reflection coefficient $\Gamma$

$\Gamma ={\frac {v^{-}}{v^{+}}}=-{\frac {i^{-}}{i^{+}}}={\frac {Z_{L}-Z_{0}}{Z_{L}+Z_{0}}}$ where $Z_{0}$ is the characteristic impedance of transient line, and $Z_{L}$ is the impedance of load (antenna). If $Z_{L}=Z_{0}$, then the line is perfectly matched, and there is no mismatch loss and all power is transferred to the load (antenna).

• An open circuit: $Z_{L}=\infty$ and $\Gamma =+1$.
• A short circuit: $Z_{L}=0$ and $\Gamma =-1$, and a phase reversal of the reflected voltage wave.
• A matched load: $Z_{L}=Z_{0}$, and $\Gamma =0$ and no reflections.

The voltage standing wave ratio or VSWR

${\text{VSWR}}={\frac {|V|_{\text{max}}}{||V|_{\text{min}}|}}={\frac {1+|\Gamma |}{1-|\Gamma |}}$

Siirtolinja (transmission line). Impedanssi. Koaksaalikaapelin impedanssi muodostuu sen kapasitiivisestä rakenteesta. Ei juuri resistiivistä häviötä (impedanssia) https://electronics.stackexchange.com/questions/543100/derivation-of-resistance-of-coaxial-cable. Koaksaalikaapelin ε^r

• 76.7 Ω
• 30 Ω
• The impedance of a centre-fed dipole antenna in free space is 73 Ω, so 75 Ω coax is commonly used for connecting shortwave antennas to receivers.
• Sometimes 300 Ω folded dipole antenna => 4:1 balun transformer is used.

twin-lead transmission lines: the characteristic impedance of is roughly 300 Ω.

Feeding length.

Some transmission lines are

• Coaxial cable
• Two-wire cable
• Microstrip line
• . . .

Skin Effect

The skin effect Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} . The higher the frequency, the more the currents are confined to the surface.

Velocity factor

Caption text

Velocity factor	Line type
0.95	Ladder line
0.82	Twin-lead
0.79	coaxial cable (foam dielectric)
0.75	RG-6 and RG-8 coaxial (thick)
0.66	RG-58 and RG-59 coaxial (thin)