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 impedance at any location along a feed line
Equation for the reflection coefficient
Equation for VSWR

To calculate the impedance at any location along a feed line, the following formula may be used. Note that this is the formula for a loss less feed line.

Zin = Z0 (ZL + jZ0tan(ℬl) / Z0 - jZLtan(ℬl))

where ℬ is the phase constant, which can be expressed as 2π/ƛ
where ƛ or wavelength is the speed of light / frequency
where l is the length of the feed line

To calculate the reflection coefficient, the following formula can be used.
𝜞 = ZL - Z0 / ZL + Z0

To calculate VSWR we can use the following formula.

VSWR = 1 + |𝜞| / 1 - |𝜞|

Now we have everything necessary to calculate the impedance at the end of a feed line, the reflection coefficient at that point and the VSWR at that point.

Fortunately, we don't need to run a series of impedance values for a number of different lengths of coax for our data points. Better that we use a visualization tool like SimNEC in order to "see" what is happening.

We're dealing with three separate impedances and/or characteristic impedances, the antenna impedance at a particular frequency (load), the feed line characteristic impedance and finally the impedance of our radio (source). By using a Smith Chart and centering the chart (normalizing) on the characteristic impedance of the feed line, we can see that regardless of length, 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 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. The following charts illustrate the idea.

Smith Chart centered on 450 ohms

In this example, the load impedance is 250 ohms with 100 ohms of capacitive reactance. The impedance for the load can be seen as the bright red circle left of center and below the center horizontal line. The feed line length is represented by the green circle(s) starting at the load impedance and moving clockwise around the Smith Chart. The slight variance in circle diameter is due to loss in the feed line. If this were a pure lossless line, the circles would overlay each other perfectly. Note that the distance to the center point of the Smith Chart (450 + j0) is exactly the same regardless of feed line length. By using our formulas provided above, we can calculate the reflection coefficient, with regard to the source (radio) ,at any point on the feed line to see that the reflection coefficient remains constant. If we have a constant reflection coefficient for any length of feed line, we will have a constant SWR between the feed line and radio as well. Note the black segmented circle in the illustration. This is the 2.0:1 SWR circle. All SWR readings for this solution, regardless of feed line length, remain the same.

Smith Chart centered on 50 ohms

For illustration purposes, the characteristic impedance of the feed line has been changed to 125 ohms and the Smith Chart has been recentered at 50 ohms, since 50 ohms (radio impedance) is our goal for eliminating SWR. We can see that relative to the impedance that we want to hit (50 ohms) in order to have a 1.0:1 SWR, the change in impedance caused by varying the feed line length has a dramatic effect on the reflection coefficient (and thus the SWR) seen by the radio. In this case, using the impedance of the radio and the impedance at any point on the feed line in the reflection coefficient equation (above) results in vastly different values and thus vastly different SWR readings.

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.

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.

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