This article presents a simple practical test of the concepts laid out at Some useful equivalences of very short very mismatched transmission lines.
Above is the DUT, it is a short circuit at the end of 102mm of two wire transmission line with VF=1, conductor diameter 0.47mm and 5mm spacing.
The transmission line is not perfectly uniform, but sufficiently good for this demonstration.
We are going to use port extension or e-delay to adjust the reference place to the short circuited end of the transmission line.
Let’s calculate Z0 and propagation time and phase length at 10MHz of the nominal transmission line. It is chosen for this example because it has a VF very close to 1, and is easily physically measured and parameters calculated.
RF Two Wire Transmission Line Loss Calculator
Parameters | |
Conductivity | 5.800e+7 S/m |
Rel permeability | 1.000 |
Diameter | 0.000470 m |
Spacing | 0.005000 m |
Velocity factor | 1.000 |
Loss tangent | 0.000e+0 |
Frequency | 10.000 MHz |
Twist rate | 0 t/m |
Length | 0.102 m |
Zload | 1.000e-99+j0.000e+0 Ω |
Yload | 1.000e+99+j0.000e+0 S |
Results | |
Zo | 369.31-j2.78 Ω |
Velocity Factor | 1.0000 |
Length | 1.225 °, 0.021 ᶜ, 0.003402 λ, 0.102000 m, 3.402e+2 ps |
Line Loss (matched) | 1.41e-3 dB |
Line Loss | >100 dB |
Efficiency | ~0 % |
Zin | 1.198e-1+j7.953e+0 Ω |
Yin | 0.00189307-j0.12570620 S |
VSWR(50)in, RL(50)in, MML(50)in | 428.02, 0.041 dB 20.314 dB |
So, Z0=369Ω, and its propagation time is 340ps one way, two way is 680ps, βl=0.021ᶜ.
Using the theory set out in Some useful equivalences of very short very mismatched transmission lines, we can calculate the e-delay that should refer the reference plane from Port 1 connector to the end of the transmission line section.
\(\frac{\beta l Z_0}{50}=0.155\), so this is just above the limit of 0.1 radians for best accuracy, but the results should still be good enough visually.
\(edelay_{50}=edelay_{Z_0} \frac{Z_0}{50}=680 \frac{370}{50} \text{ ps}=5.03 \text{ ns}\\\)Let’s estimate Z0 from the impedance of the short circuit stub.
\(Z_0=\frac{X}{\tan (\frac{2 \pi l f}{C_0})}=\frac{8.0}{\tan (\frac{2 \pi \cdot 0.102 \cdot 1e7}{299792458})}=374\), that measurement based value reconciles well with the calculation above.
Let’s plot the phase of s11 SOL calibrated at the Port 1 connector.
Above, phase looks fairly linear, but in fact it departs a little from linearity above about 6MHz, more so as frequency increases.
Above is application of e-delay adjusted to obtain an almost flat phase response to 10MHz. Never mind the phase wrap, but note there is a very slight upward slope in the upper part of the trace (obvious when the marker is swept over the area).
Above is the sweep from 1-11 MHz with e-delay adjusted to 5.05ns. The trace looks very flat, sliding the marker around shows it to be pretty good. This reconciles well with the calculated e-delay of 5.03ns.
Conclusions
Port extension within the limits discussed at at Some useful equivalences of very short very mismatched transmission lines. can provide an accurate, convenient and useful technique for referral of the reference plane without doing a full SOL calibration at that point.