# Wavelength, Physical Length and Relative Permittivity

*Created Saturday 08 May 2021*

This page will explain some of the relationships between physical length and relative permittivity and especially some techniques to determine these parameters while taking advantage of half-wavelength structures.

## Video Tutorial

The contents of this page are described in the following video:

*Coming soon!*

## A Note about Velocity Factor

The Velocity Factor (VF) is a property of transmission lines, specifically used with coaxial cables. Since any dielectric permittivity greater than 1 (air/vacuum) will have the effect of slowing the phase velocity of a wave, the VF is given as a percentage representing that reduced phase velocity.

## Relationship between Er and Wavelength

We want to be able to relate the permittivity and the physical length of the line. To begin with, we take two equations from Page 7 of Pozar's Microwave Engineering [1].

(1)

(2)

The first equation states that the phase velocity is equal to frequency times wavelength. The second relates the phase velocity to the dielectric constant using the speed of light. Putting them together gives us either:

(3)

or:

(4)

## Adding in Physical Length

We can use transmission line properties to use these equations to our advantage. If we have a half-wavelength line, the properties would be as follows:

(5)

If you have some known properties and a half-wavelength line, it is possible to determine other parameters. For example, if you know the relative permittivity and know the frequency of a half-wavelength line, it's possible to put Equations (4) and (5) together to determine the physical length. Similarly, if you know the physical length and frequency of half-wavelength, you can determine the effective relative permittivity. This could come in handy for evaluating unknown cable types.

## Using Technique in Practice

Based on the equations above, it is necessary to determine the half-wavelength frequency of the cable/transmission line in question. One of the quickest methods is to simply take a one-port VNA measurement of one end of the cable with the other end open. This would give a wrapped phase plot as shown below:

The first point where the phase crosses zero is the frequency at which the line's physical length is half-lambda. We can zoom in on that point for more accuracy. In this example, the phase plot crosses zero at exactly 20.86 MHz.

One nice thing about this method is that it works even when the cable's characteristic impedance does not match the port impedance of the VNA. In that case, the phase will simply be non-linear but will still cross zero at the correct location as shown in the two plots above.

A downside of this method is that longer cables require lower frequency measurement. For a PTFE dielectric (Dk=2) if your VNA goes down to 1 MHz, you could measure a cable length of about 212m (700ft) or so. In addition to this, you must be careful with long lines that the first zero-crossing isn't masked by large sample spacing in the S-Parameter measurement or simulation. For example, if you're looking at a point under 1 MHz, and you sample a range from 0 to 1 GHz linearly with 101 points, each point is 10 MHz spaced apart and you'll never see the intended zero-crossing. Some intuition must be used to determine an appropriate frequency sweep setting.

### Example 1: Determine Unknown εr

Assume that you have a cable of known length (1m in this example) but need to know the velocity factor. Taking the phase(S11) measurement gives the following plot:

The first zero-crossing on the phase plot is at 92.853 MHz. Using Equation (3) above, the resultant relative permittivity is 2.61, indicating a velocity factor of 62%.

### Example 2: Determine Cable Length

In this case, assume you have a cable coiled up which is listed as T10 Drop Cable (75Ω CATV coax). This cable has a VF of 85%, meaning εr is 1.38. You take a phase measurement of the cable and see the following result:

The first zero-crossing of the phase is at 21.86 MHz. Using Equation (4) above, the resultant lambda is 12.24. Since the cable length is half of this, the cable length is approximately 6.12 m (20.08 ft).

# References

[1] Microwave Engineering, David Pozar

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