Countless online discussions have online experts describing their various preferred methods for estimating the characteristic impedance of a transmission line… often without really testing whether their simple results are realistic, ie believable. Of course, being social media, it would be unsocial for another participant to question the results, so the unchallenged becomes part of ham lore.
Apparent gross failures are often wrongly attributed to factors like manufacturing tolerances, polluted line surface, other esoteric factors etc that might imply a knowledgeable author… but that is social media, an unreliable source of information.
Let's explore an estimate using measurements with a nanoVNA using the popular eighth wavelength (λ/8) method.
λ/8 method
The λ/8 method relies upon the property of a lossless line terminated in an open circuit that differential impedance \(Z_d=\jmath X=- \jmath \left| Z_0 \right| cot \left(\pi/4\right)=- \jmath\left| Z_0 \right|\). So, if you measure the reactance looking into the λ/8 (\(\frac{\piᶜ}{4} \:or\: 45°\)), you can estimate Zo as equal to the magnitude of the reactance.
A similar expression can be written for the case of a short circuit termination and it leads to the same result that you can estimate Zo as equal to the magnitude of the reactance (an exercise for the reader).
The fact that the two cases lead to the same result can be used to verify that the line length is in fact λ/8 (they will not be equal if the length is a little different to λ/8)… though writeups rarely mention this, or perform the test.
So, the method depends critically on:
- whether the line length is λ/8;
- whether it is sufficiently low loss; and
- whether the differential impedance measurement is valid.
Most online articles do not include details of the measurement setup, perhaps thinking that it not all that relevant. Of course, one of the greatest failings in experiments is to ignore some factor that is in fact relevant.
An NEC-4.2 model
The nanoVNA is such a limited device without a computer attached, lets model a scenario that might well be used by the naive.
Above is a graphic of the scenario. It comprises a vertical section of the modelled transmission line from height 2 to 7m, so it is 5m in length and comprises 1mm diameter copper conductors spaced 20mm, dielectric is a vacuum. Impedance is measured in the horizontal segment bonding both sides of the transmission line.
We might expect the quarter wave resonance of this part of the scenario alone would be approximately 15MHz, but the NEC model gives a slightly lower frequency of 14.97MHz and therefore λ/8 is 7.485MHz.
Also included is a connection from one side of the transmission line to real ground to represent the ground connection of the nanoVNA via its USB cable to a computer used to capture the measurement results.
Above is the impedance plot from the model, it looks well behaved and we might suggest that we would expect that the nanoVNA would measure Zin=Rin+jXin=92.5-j195Ω or |Z|=215Ω.
If you not into the ‘j value' stuff, you might then say that Zo=215Ω. If you were a little more savvy, you might say that Zo=|X|=195Ω. Not a whole lot of difference… but the difference should be concerning. In fact, the relatively large value of R in Z should sound a warning that the naive might overlook.
What should have been expected
Let's look at a simple model of the transmission line using TWLLC.
Parameters | |
Conductivity | 5.800e+7 S/m |
Rel permeability | 1.000 |
Diameter | 0.001000 m |
Spacing | 0.020000 m |
Velocity factor | 1.000 |
Loss tangent | 0.000e+0 |
Frequency | 7.485 MHz |
Twist rate | 0 t/m |
Length | 0.125 wl |
Zload | 1.000e+100+j0.000e+0 Ω |
Yload | 0.000000+j0.000000 S |
Results | |
Zo | 444.04-j1.48 Ω |
Velocity Factor | 1.0000 |
Length | 45.000 °, 0.125000 λ, 5.006554 m, 1.670e+4 ps |
Line Loss (matched) | 2.28e-2 dB |
Line Loss | >100 dB |
Efficiency | ~0 % |
Zin | 8.486e-1-j4.418e+2 Ω |
Above is an extract of the output.
Key calculated results are:
- Zo=Ro+jXo=444.04-j1.48Ω; and
- Zin=Rin+jXin=0.849-j442Ω.
Note that:
- Xin≅Ro, which reconciles with the theory of the λ/8 method; and
- Rin is relatively quite small as expected of low loss line.
Reconciliation
By comparison with the TLLC prediction, a key point of reconciliation failure is that in the measurement Rin is relatively quite large and quite inconsistent with even a low loss line, and so the premise for applying the λ/8 method vanishes.
What's the problem?
Let's model a better DUT,
Above is a graphic of the scenario. It comprises a vertical section of the modelled transmission line from height 2 to 7m, so it is 5m in length and comprises 1mm diameter copper conductors spaced 20mm, dielectric is a vacuum. Impedance is measured in the horizontal segment bonding both sides of the transmission line.
We might expect the quarter wave resonance of this part of the scenario alone would be approximately 15MHz, but the NEC model gives a slightly lower frequency of 14.97MHz and therefore λ/8 is 7.485MHz.
There is no connection to ground, the instrument and the transmission line section are relatively isolated from ground.
Above is the impedance plot from the model, it looks well behaved and we might suggest that we would expect that the nanoVNA would measure Zin=Zoc=Rin+jXin=0.879-j458Ω.
As a simple check, Rin is relatively small, and we might accept that Zo=|Xin|=458Ω… but this value is sensitive to the electrical length.
Running the same model with a short circuit termination, Zsc=3.98234+j463.85Ω.
We can calculate Zo=Ro+jXo from Zoc and Zsc.
As you see, Ro lies between the |Xoc| and |Xsc|.
This calculation if of higher accuracy than basing Zo on |Xoc| or |Xsc| alone of an assumed λ/8 section.
A simpler approximation for low loss line is \(Z_o\approx \sqrt{-X_{oc} X_{sc}}=460.8\) which reduces sensitivity to actual length.
Calculated Zo based on the second NEC model does not reconcile exactly with the TWLLC calculation as they use different methods of modelling, but the results are reasonably close.
What's the problem was asked. The problem is that the λ/8 method depends on a valid differential mode impedance measurement, and that did not happen in the first example, the common mode current path prevented pure differential mode excitation of the DUT.
Conclusions
- Measurement fixtures / scenarios which permit or encourage common mode excitation of the transmission line under test are likely to produce invalid results.
- The measurement fixture / scenario is critical to valid results.
- Naive application λ/8 method can lead to worthless results.