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

• 

  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.

• 

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

• 

  The system is analyzed as being differential short pieces. The conductance $G$ 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 (syntax error): {\displaystyle \begin{align*} \frac{\doo v}{dx} &= -L\frac{\doo i}{\doo t} - Ri \\ \frac{\doo i}{dx} &= -C\frac{\doo v}{\doo t} - Gv \end{align*} }

The solutions to the above equations is the sum of forward and backward traveling (reflected) waves: $v(z,t)=v^{+}e^{-\alpha z}e^{\imath (\omega t-\gamma z)}+v^{-}e^{\alpha z}e^{\imath (\omega t+\gamma z)}$ and if we assume that $\alpha =0$ we have the telegraphers equations https://en.wikipedia.org/wiki/Telegrapher's_equations

$v(z,t)=v^{+}e^{\imath (\omega t-\gamma z)}+v^{-}e^{\imath (\omega t+\gamma z)}$ and a similar for $i=i(z,t)$. If we replace $i$ by Ohm law, we get

$i(z,t)={\frac {v^{+}}{Z_{0}}}e^{\imath (\omega t-\gamma z)}+{\frac {v^{-}}{Z_{0}}}e^{\imath (\omega t+\gamma z)}={\frac {v^{+}}{Z_{0}}}e^{\imath (\omega t-\gamma z)}\left(1-{\frac {v^{-}}{v^{+}}}e^{\imath (2\gamma z)}\right)$

The fraction ${\frac {v^{-}}{v^{+}}}$ is called reflection coefficient $\Gamma$

${\frac {v^{-}}{v^{+}}}=\Gamma e^{\imath \phi _{z}}={\frac {Z_{L}-Z_{0}}{Z_{L}+Z_{0}}}$

which gives

$i(z,t)=={\frac {v^{+}}{Z_{0}}}e^{\imath (\omega t-\gamma z)}\left(1-\Gamma e^{\imath (\phi _{z}+2\gamma z)}\right)$

The characteristic impedance is

Thus we have ${\frac {d^{2}v}{dL^{2}}}=\gamma ^{2}v$ and similar for the current. The constant $\gamma ^{2}=(R+\imath \omega L)(G+\imath \omega C)$.

$Z_{0}={\frac {v^{+}}{i^{+}}}=-{\frac {v^{-}}{i^{-}}}={\sqrt {\frac {R'+\imath \omega L'}{G'+\imath \omega C'}}}$

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: 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_L = 0} and 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 = -1} , and a phase reversal of the reflected voltage wave.
• A matched load: 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_L = Z_0} , and 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 = 0} and no reflections.

The voltage standing wave ratio or VSWR

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 \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)