High frequency transmission lines

Introduction

Error creating thumbnail: convert-im6.q16: unrecognized color `context-fill' @ warning/color.c/GetColorCompliance/1057. convert-im6.q16: non-conforming drawing primitive definition `fill' @ error/draw.c/RenderMVGContent/4414.

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, general case

• 

  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 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} }

and these two is easy to combine, and it gives the second degree dy

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{\partial^2 v}{dx^2} -LC\frac{\partial^2 v}{\partial t^2} = (RC+GL)\frac{\partial v}{\partial t} + GRv }

and similar equation to i. Lets solve those.

Lossless transmission

If wire resistance and insulation conductance can be neglected (R=G=0), the model depends only on L and C elements. Thus, we have two similar wave equations for v and i (only for v is shown)

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{\partial^2 v}{\partial^2 t^2} - \hat v^2 \frac{\partial v}{\partial x^2} = 0 }

where 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 \hat v = (LC)^{-1/2}} . These reduce to one-dimensional Helmholtz equations, and the result is

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} v(x) &= v^+ e^{-\imath kx} + v^- e^{+\imath kx} \\ i(x) &= \frac{v^+}{Z_0} e^{-\imath kx} + \frac{v^-}{Z_0} e^{+\imath kx} \end{align} }

where k is the wave number 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 k=\omega\sqrt{LC}} 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 Z_0} is the characteristic impedance, which for the lossles transmission line is

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 = \sqrt{ \frac LZ } }

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

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 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 R' = G' = 0} and for distortionless line 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 R'/L' = G'/C'} . The voltage 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 \Gamma = \frac{v^-}{v^+} = - \frac{i^-}{i^+} = \frac{Z_L - Z_0}{Z_L + Z_0} } where 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} is the characteristic impedance of transient line, 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 Z_L} is the impedance of load (antenna). If 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} , then the line is perfectly matched, and there is no mismatch loss and all power is transferred to the load (antenna).

• An open 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 = \infty} 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} .
• 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)