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I want to derive the velocity factor \$v_f\$ $$ v_f=\frac{1}{c\sqrt{L'C'}}\tag 1 $$ and this relationship with the phase velocity \$v_p\$ $$ v_f=\frac{v_p}{c}. \tag 2 $$ \$L'\$ is the inductance per metre and \$C'\$ is the capacitance per metre.

I know the following:

The phase velocity \$v_p\$:

I guess \$\epsilon_r=\mu_r=1\$ here? So \$\epsilon=\epsilon_0\$ and \$\mu=\mu_0\$? $$ v_p=\frac{\omega}{\beta}=\frac{1}{\sqrt{L'C'}}, \tag 3 $$ where \$\lambda=\frac{2\pi}{\beta}\$ and \$\beta=\omega\sqrt{L'C'}\$. And \$c=\frac{1}{\sqrt{\epsilon_0\mu_0}}\$, so $$ v_p=\frac{\omega}{\frac{2\pi}{\lambda}}=\frac{\lambda}{2\pi}\omega=\lambda f =c \tag 4 $$

The velocity factor \$v_f\$:

I guess \$\epsilon_r\neq1\$, \$\mu_r\neq 1\$ here? $$ v_p=\frac{1}{\sqrt{\epsilon\mu}}=\frac{1}{\sqrt{\epsilon_0\mu_0}}\frac{1}{\sqrt{\epsilon_r\mu_r}}=\frac{c}{\sqrt{\epsilon_r\mu_r}} \tag 5 $$

I don't know how to proceed from \$(3)-(5)\$ to find \$(1)\$ and \$(2)\$.

Thanks!

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You have a happy mix of equations for wave quantities in different mediums. You can sometimes accidentally get some valid looking results by combining those equations regardless their different validity ranges, but usually the results are nonsense.

You wanted (1). It's the speed of wave in TEM waveform transmission line divided by the speed of light in vacuum. It's actually the definition of the velocity factor for usual cables, so it cannot be derived more deeply than by dividing your formula (3) by c. Formula (3) is the wave propagation speed in lossless TEM waveform transmission lines and it can be proven by forming the famous telegrapher's equation and comparing it to the general wave equation. Vf gives the speed of waves as a percentage of the speed of the light in vacuum. In TEM waveform transmission lines (coaxial cable, twisted pair etc...) the wave propagates straigtforward along the transmission line axis.

Equation (2) is a more general definition for the velocity factor. It's valid for TEM waveform transmission lines and also for wavequides. In wavequides the waves do not propagate straightforward, but by reflecting inside the wavequide. The reflections result extremely complex total (=sum, interference pattern) electromagnetic field that seems to propagate along the wavequide, The apparent propagation speed is called phase velocity. The real propagation is like zig-zag, not directly along the wavequide. The apparent speed of the interference patterm can be even more than the speed of the light (Vf >1) For example this is the case for the most common metallic wavequides.

The phase velocity or the velocity factor * c must be used for calculating the wavelength when one needs common wavelength dependent pieces of transmission lines such as quarter wave transformers or resonators.

Eq. (5) gives the propagation speed for single electromagnetic wave in lossless homogenic isotropic non-dispersive linear medium. Many insulators such as air and plastics obey this formula usefully exactly.

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