VSWR Change With Feed Line Length

It's the age old question that seems to continue to circulate on the Amateur Radio Forums of various social media platforms.
The answer to this question is: It depends....

In order to understand what's going on, we need some tools to help us visualize what is going on between our antenna, feed line and radio. To that end, we're going to need the mathematical formulas for the following items:

• Equation for the reflection coefficient

• Equation for VSWR

To calculate the reflection coefficient, the following formula can be used.  The symbol 𝜞 is referred to as Gamma.
𝜞  = ZL - Z0 / ZL + Z0

Note that these calculations require the use of complex numbers. As an example, consider a ZL of 100 + j25 ohms  and Z0 of 50 + j0 ohms (your radio) When adding these values together we'll get (100 + j25) + (50 + j0) = (150 + j25) and if we subtract them we get (50 + j25)

To calculate VSWR we can use the following formula.

VSWR = 1 + |𝜞| / 1 - |𝜞|

Note that in order to calculate the VSWR, we need to use the magnitude of Gamma which uses the symbol |𝜞|. In order to compute the magnitude of gamma we'll use the following formula: Magnitude = √(R*R + X*X). 

Now we have everything necessary to calculate the  reflection coefficient at that point and the VSWR at that point.

Note the reflection coefficient formula uses 2 different impedance values for performing the calculation. A measurement of the reflection coefficient is always done at a natural boundary or discontinuity in the antenna system, like the boundary between the feed line and antenna or the boundary between the radio and the transmission line. If we use the boundary between the radio and the transmission line as the point of interest, ZL (impedance of the load) will be the impedance seen looking into the transmission line and Z0 will be the characteristic impedance of the radio (50 ohms).  

We need to generate 2 sets of reflection coefficient values. For the first set assume an antenna impedance of 100 ohms, a transmission line characteristic impedance of 50 ohms and a radio impedance of 50 ohms. For the second set of values, assume an antenna impedance of 100 ohms, a transmission line characteristic of 450 ohms and a radio impedance of 50 ohms. Fortunately we don't have to manually create a list of reflection coefficients, we can use a visualization tool like SimNEC in order to "see" what is happening as feed line length is changed relative  to  either the characteristic impedance of the feed line or that of the radio. In the end, it's the impedance of the radio that we're most interested in.

In the  first case,  the Smith Chart is normalized  to 50 ohms (the radio impedance), which is  the characteristic impedance of both the transmission line and the radio. In the  second case  we'll leave the Smith Chart centered on 50 ohms (the radio)  but change the  transmission line to 450 ohms  (window line). 

For the first case, the reflection coefficient remains exactly the same as we make our way around the circle of impedances defined by the transformation of the load impedance by the characteristic impedance of the feed line (50 ohms)  for a series of different feed line lengths. This holds for any feed line characteristic impedance. We can see this in SimNEC fairly easily by setting the source impedance to that of the feed line, that is, set the source impedance to be centered on the characteristic impedance of the feed line.  For the second case, the transmission line characteristic impedance is changed to 450 ohms (window line) and the calculations are done again using the impedance seen at the end of the feed line (now transformed through a 450 ohm characteristic impedance) and the impedance of the radio.

What we will see is that when the characteristic impedance of the transmission line matches that of the radio, the circle of impedances (and thus reflection coefficients) form a neat circles around the center point of the Smith Chart, which has been normalized to 50 ohms. In the second case, the Smith Chart remains centered on 50 ohms (the impedance of the radio), but now the circle of impedances has been shifted to a different center point on the Smith Chart, well away from the perfect circle that was previously made around the 50 ohm center point. If we now calculate our reflection coefficient relevant to the impedance at the end of the feed line at any point along the new circle relative to the 50 ohm impednce of the radio, we will see that the reflection coefficient (and thus VSWR) changes wildly as the transmission line length changes.

The following charts illustrate the idea.

So, does the length of the feed line change the SWR along the line? Relative to the feed line characteristic impedance the answer is NO. But, relative to the impedance of the radio, which has an impedance of 50 ohms, the answer is decidedly YES if the feed line characteristic impedance of the feed line does not match that of the radio.

The main takeaway from this is the calculation for the reflection coefficient. If we calculate the reflection coefficient relative to the load and the impedance of the feed line at the end of the feed line, we will find that the reflection coefficient remains constant regardless of feed line length (less losses). But, if we calculate the reflection coefficient relative to the feed line impedance and the radio impedance, we can see that the reflection coefficient varies wildly and thus the SWR at the junction between the radio and the feed line varies.