# Does the impedance along a transmission line change when there are standing waves?

I know the characteristic impedance can't change because it's determined by the physical properties of a transmission line. But the impedance seen when there are standing waves changes along the length, is that correct?

Then my SWR meter won't be accurate because the impedance isn't 50 ohms, is that right? So, the higher the SWR the more inaccurate by SWR meter will be?

VSWR is related to the magnitude of the reflection coefficient $$\Gamma$$:

$$\text{VSWR} = {1+|\Gamma| \over 1-|\Gamma|}$$

The reflection coefficient can be calculated from the load impedance $$Z_L$$ and the characteristic impedance $$Z_0$$:

$$\Gamma = {Z_L - Z_0 \over Z_L + Z_0}$$

The reflection coefficient, like the load impedance, a complex number with a magnitude and phase. But only the magnitude of the reflection coefficient is relevant to the calculation of VSWR. Changing the length of the transmission line between the SWR meter and the load alters the phase of the reflected wave, but assuming a lossless line it does not alter the magnitude.

Thus, while the impedance seen looking into the line varies depending on the length of the line, the VSWR remains constant on a lossless line.

In practice transmission lines have some loss, and this has the effect of decreasing the magnitude of the reflection coefficient with increasing line length, which means the VSWR will approach 1:1.

It is difficult to say where the SWR meter is most "accurate" since the VSWR is not a constant but varies along the transmission line. If your transmitter is specified to work into a VSWR of 1.6:1 or below, then placing the SWR meter next to the transmitter is most accurate as that measures the VSWR seen by the transmitter. On the other hand if your objective is to measure the SWR bandwidth of an antenna, placing the meter at the antenna will be more accurate because the transmission line loss would otherwise result in increasing the measured bandwidth. Then again, in many applications the transmission line losses are low enough that the difference is of no practical significance.

Yes, the characteristic impedance of the transmission line $$Z_0$$ does not change (it is a property of the cable itself), but the impedance seen from the end of the cable depends on the length of the cable. But also no, your VSWR meter will work just fine. VSWR is a measurement of how far the measured impedance is from $$50\Omega$$ (or whatever is the impedance of your system).

# VSWR, impedance, and cable length

If you terminate your $$50\Omega$$ coax line with an antenna whose input impedance is something else, like $$75\Omega$$, some of the power transmitted reflects when it hits the antenna. This is because both current (along the transmission line) and the voltage between your outer and inner conductors must remain continuous at all points. The amount (and phase) of reflected signal is called reflection coefficient $$\Gamma$$:

$$\Gamma = {Z_L - Z_O\over Z_L + Z_O}$$

The Voltage Standing Wave Ratio is just an another way of expressing the magnitude (so we don't care about the phase of the reflected signal) of the $$\Gamma$$:

$$VSWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}$$

When the length of your transmission line changes, you don't affect the amount of reflected signal (assuming lossless transmission line), only its phase. The simplest example is openly terminated coaxial cable ($$Z_l = \infty$$): A short open ended coax acts as a capacitor and a bit over quarter wavelength long cable ($$l > \lambda/4$$) acts as an inductor. If you have long enough coax, the losses will dissipate some of the reflected signal, leading to improved VSWR.

If you know the complex propagation constant $$\gamma$$ (so you know both how fast the wave travels and how much there are losses), you can calculate the input impedance $$Z_{in}$$ that is seen from the end of the coaxial cable:

$$Z_{in}(l )=Z_{0}\,{\frac {Z_L+Z_{0}\tanh (\gamma l )}{Z_{0}+Z_L\,\tanh (\gamma l )}}$$

# Standing waves

When ever there is a impedance mismatch in the end of the cable, that is, your antenna is not exactly $$50 \Omega$$, a wave is reflected back. If you look at the animation below (from Wikipedia), you can see the blue wave is transmitted from left, hits a short on the right, and is reflected to left as red wave. The black curve is the sum of these two waves. In some locations the red and blue cancel each other out and a $$\lambda/4$$ away from those points they amplify each others. In this animation the load impedance is $$Z_l=0\Omega$$, but the graph is similar with other impedances as well, except that the minimum voltage is not exactly zero.

This sum wave is called as standing wave and the ratio of the minimum and maximum voltage along this transmission line is called the Voltage Standing Wave Ratio (VSWR). Your VSWR meter measures the amplitudes from red and blue curves for example through a directional coupler. If you had a suitable waveguide where you can move a probe along the transmission line, you could also measure the minimum and maximum amplitude from the waveguide itself. I don't think this measurement method has any practical uses outside of demonstrating the phenomenon.

• so the ability of the swr meter (which is designed to work with a 50 ohm impedance) to measure 1+|T|/1-|T| accurately is not affected if the impedance of the transmission line it's connected to is not 50 ohms ? Sep 7 '19 at 6:01
• @Andrew, that is corret. So the VSWR meter measures how much power is reflected and it does not matter if the reflection is caused by the load (antenna) at the end of the coaxial cable or by the coaxial cable itself. However, if your radio has a higher impedance output and you try to use the SWR meter to know if your antenna is matched, it wont work. Maybe you can describe what you are actually trying to measure? Sep 7 '19 at 6:13
• OH2FXN - i didn't mean that, what i mean is that if the swr isn't perfect, then the impedance seen by the swr meter looking into its antenna socket wont be 50 ohms due to the impedance change away from 50 ohms caused by the standing waves on the line, and so the meter won't be calibrated correctly because it is only accurate when the impedance it's working with is the impedance it's designed to work with eg : 50 ohms. so it will measure the vswr with an error caused by the impedance not being 50 ohms. Sep 7 '19 at 7:16
• Are you looking a graph like en.wikipedia.org/wiki/Standing_wave_ratio#/media/… where the blue wave is transmitted from left, hits a short on the right, and is reflected to left as red wave? And when seeing that the (black) sum of these two waves varies along the wire, it seems that the VSWR value must be different for each location? Even in this case, the answer would be no: the VSWR meter takes in the magnitude from both blue and red curve, it doesn't look at the black curve. Sep 7 '19 at 7:57
• A directional coupler would be one way of building a VSWR meter: take 1% of the forward going wave and 1% of the returning wave and their ratio is the reflection coefficient $\Gamma$. So the impedance changes along the transmission line, but the VSWR does not. Sep 7 '19 at 8:17

In an ideal system, the VSWR is not changed by the coax length. However, in the real world, two factors affect this.

First, the coax is lossy, and high SWR magnifies the loss, causing swr to appear to approach 1:1 at the transmitter. If you are measuring complex swr (i.e., with a network analyzer), and your goal is to determine if the antenna is well matched and your coax is not so long as to absorb too much of the high SWR, then this is not of great consequence and you can still see the mismatch, although perhaps with a smaller magnitude than it is at the antenna.

Second, if you are measuring a balanced antenna connected via coax with no balun or a balun that is unable to block all the common mode current, then the coax becomes part of the antenna, and the SWR measured at the transmitter may actually be raised by the coax, and may actually appear to change randomly if you move the coax with respect to the antenna, as the geometry between the antenna and the coax can change the common mode coupling with the antenna.

This second effect can be minimized with a good balun at the antenna (and perhaps at the transmitter). It can also be minimized by cutting your coax to an electrical multiple of 1/4 wavelength (i.e., consider VF of the coax), which will at least make common mode radiation not add more of the complex component to your SWR -- but realize that your coax is still part of the antenna and may be radiating.

A few antennas intentionally take advantage of this to either soften directionality of the antenna by radiating from the coax, or with careful placement of a un-un on the coax, tune a section of the coax for a band not otherwise supported by the rest of the antenna.