How to calculate the speed of AC signal in different mediums?

Most wires will have a speed of travel for AC current of 60 to 70 percent of the speed of light, or about 195 million meters per second.

I also read here that the propagation speed depends on the wire material and its surrounding. What is the mathematical equation that allows me precisely calculate the speed of an AC signal given the physical parameters of the wire and its environment?

For example:

Printed wire using silver paste on plastic, surrounded by air.

In general you would have to do a simulation with the physical arrangement to get an accurate answer.

For specific geometries such as a very conductive trace separated from a perfect ground plane by an infinite expanse of dielectric, the propagation delay is just a factor of the dielectric constant $\epsilon_r$ of the medium.

$t_{pd} = 85\sqrt{0.475\epsilon_r+0.67}$ (picoseconds per inch)

Significant resistance will add something else to the equation because in general it will become frequency dependent.

If you are trying to do something where this would actually matter such as designing a screen printed antenna, you may have to do more than just this kind of approximation.

• Basically I want to know what is the frequency range of which after it, my trace will start acting as a transmission line. For this, I have to compare the wavelength of the signal passing through the trace to the size (length) of my trace. However, I have to consider the speed of the signal in the situation I have which is less than 3x10^8 in order to have a valid comparison. Will the mentioned equation be helpful in this case? Sep 6 '16 at 9:37
• To get a general idea, ~60% of the speed of light is probably close enough. The equation shows with typical FR4 er of ~4 the speed will be ~62% of C. Sep 6 '16 at 9:46

This is so called telegrapher's equation. The propagation speed is $u=\dfrac{1}{\sqrt{LC}}$, where L and C are inductance (Henries) and capacitance (Farads) of your transmission line.

• Is there a way to calculate the speed as a function of the physical properties (conductivity, permittivity) and geometry of the wire? Sep 6 '16 at 8:57
• Yes, you calculate the L and C as a function of the physical properties and geometry, then you calculate the speed via the Telegrapher's Equation. Sep 6 '16 at 9:06
• @Marko Buršič I just read in the links you provided that velocity factor can be calculated using the dielectric of the surrounding material of the wire (If I got that correctly). Does that mean the wire material doesn't have as much importance (copper, silver, aluminum ..etc) ? Sep 6 '16 at 9:14

Does that mean the wire material doesn't have as much importance (copper, silver, aluminum ..etc) ?

The velocity of propagation of electromagnetic radiation in any medium is given by

$v = \frac {1} { \sqrt {\varepsilon _0 \varepsilon _r \mu _0 \mu _r}}$

Where:

$\varepsilon _0$ = permittivity of free space = 8.854 pF/metre

$\varepsilon _r$ = relative permittivity of the medium (the dielectric)

$\mu _0$ = permeability of free space = 4$\pi$ $\cdot 10^{-7}$ Henries / metre

$\mu _r$ the relative permeability of the medium (the conductor).

As the relative permeability and permittivity for free space are 1, then the velocity of electromagnetic propagation in free space is:

$c = \frac {1} { \sqrt {\varepsilon _0 \mu _0 }}$

Many metallic conductors have a relative permeability of close to unity and therefore $\mu _0 \mu _r$ reduces to $\mu _0$; i.e. the permeability of the conductor is of no consequence to the signal velocity. Copper, for instance has $\mu _r$ = 0.999994 (6ppm different from free space and therefore inconsequential in most cases).

If you are using a ferromagnetic conductor that has a high relative permeability, such as Nickel then it does have a significant impact on signal propagation velocity.

• Thanks for the contribution. What if my conductor is surrounded by two dielectrics; printed on a certain substrate and exposed to air from the top side? Speed will be in between the values corresponding to the two different dielectric constants, but will the average of the speeds be a good approximation? Sep 6 '16 at 11:15
• @HaneenSu: much depends on the thickness (and relative thickness)of the dielectric layers. If they are quite thin then the average would be a decent starting point. Sep 6 '16 at 11:22
• makes sense. Thickness of the substrate is 1mm while the air layer is basically the whole environment. So speed should be more towards the air value probably. Sep 6 '16 at 11:25
• I would be inclined to go closer to the dielectrics; the effective dielectric value for the surface layer of FR-4 is around 3.5 (not 1). A good starting point would be the average permittivity of the dielectric layers. Sep 6 '16 at 12:27