# Why is the relation between voltage ratio and frequency a linear one in a lumped transition line?

In an experiment the cutoff frequency of a lumped transmission line is determined by measuring the ratio of the output/input voltages. The lumped transmission line is a ladder network of 40 capacitors and 40 inductors,

This is connected to an oscilloscope which generates sinusoidal waves. Reflection occurs at the BNC input point because the $$\Z_{transmission-line}>Z_{coaxial-cable}\$$, $$\frac{V_{reflected}}{V_{in}}=\frac{Z_b(\omega)-Z_a}{Z_b(\omega)+Z_a}$$ where $$\Z_a=Z_{coaxial}, Z_b=Z_{transmission-line}\$$ $$Z_b(\omega)=\sqrt{(L/C)(1/(1-\omega^2 LC/4)))}$$ which is the characteristic impedance of the ladder network.

Using $$V_{transmitted}/V_{in}=\frac{2Z_a}{Z_b(\omega)+Z_a}$$, here I assume $$\V_{transmitted }\$$ is proportional to the voltage measured by the probe of the oscilloscope, so the voltage ratio vs $$\\omega\$$ cannot be linear due to $$\Z_b=Z_b(\omega)\$$.

However, my experimental data suggests a linear relation

whereas I expected something (red curve) like due to the last equation.
Why is my interpretation incorrect?

• You need to be more careful around the 71.5 kHz area to see the red curve replicated correctly Feb 7, 2022 at 16:36
• Chern Simons - Hi, You asked the same question yesterday at Physics.SE. No answers yet, As linked in the SE FAQ, duplicating the same question on multiple SE sites (especially without disclosure) is generally discouraged. Lots of advice in that linked topic. Perhaps pick just one site for a while? Then move to another, with a refined question, if unsuccessful. Feb 7, 2022 at 16:58
• @Andy aka So the experimental data plot should replicate the analytic curve? because I see no remblence betwen the two whatsoever (except the minor change in gradient around 71.5kHz) Is my assumption that $V_{transmitted-into-lumped-line} \propto V_{measured-by-probe}$ correct though? Feb 7, 2022 at 17:27
• @ SamGibson♦ will do that in abit! Feb 7, 2022 at 17:27
• @ChernSimons - Just FYI the MathJax delimiters vary across Stack Exchange sites. Here (unlike Physics.SE) the inline MathJax delimiters are \$ not just $. That is why your MathJax in the comment a couple above this one, was not rendered as expected (also in the question itself). (The block MathJax delimiter  is the same in both sites.) Feb 7, 2022 at 17:34

Measuring the properties of a lumped-element transmission line

http://aaronscher.com/RF_wireless_EM/experiment_3/lumped_element_TL.html

The lumped element transmission line is a low pass filter with a cut off frequency ωc =2/√ LC

Type in google search box: 2/sqrt(0.000000015 x 0.000330)/6.28

Result = 143142.22133 Hz

Independent of the reflection/transmission ratio and the other loss factors the measurements in your graph seem to be in the linear range below the cutoff frequency which by my calculation is rounded to 143 kHz. It is not clear to me how you would modify the experiment to see the cutoff frequency since the attenuation due to other factors seems to be dominating the frequency cutoff.

• sorry but I think my question is more on the linear (expected to be non-linear) relation between voltage ratio and angular frequency, rather than the cut-off frequency itself which I understand is independent of other factors. Feb 8, 2022 at 0:29
• I could not find models that predict a linear attenuation factor for signals that are below the cutoff frequency. There are models that describe attenuation factors for the transmission line or lumped circuit TL "analog" with actual resistance, impedance mismatch, and frequency characteristics. But I did not see any formulas that clearly anticipate linear attenuation over frequency range as shown in your results. Feb 8, 2022 at 17:44