Characteristic Impedance vs Frequency

Question

I am reading this paper and I see that characteristic impedance does not take frequency into account. I am surprised by this as I figured different rise time signals would have different frequency content and see different impedances. Z0 is proportional to sqrt(L/C) and I figured those values change at different frequencies.

Are these characteristic impedance equations only valid for certain frequency range or am I wrong in assuming that characteristic impedance will have a frequency dependency?

https://www.polarinstruments.com/support/cits/IPC1999.pdf

Yes, there's a small frequency dependence, because \$\epsilon_r\$ varies a bit with frequency. — Spehro Pefhany, Oct 15 '20 at 21:11
As f increases and dielectric constant decreases , thus Zo increases and load RC affects bandwidth of transmission line. Thus selected substrates for microwave logic (CML) and radio signals may use Polyamide, ceramic or Teflon and a wide range of better dielectrics than just FR4 epoxy/fibreglass with Dk=4.2 and effectively reducing with higher f. I tended to use w/h ratios =1 for most to get near 50 Ohms for 74HC logic and 25 Ohms for 3.6V logic demands thinner insulation layers than normal to raise the w/h >1 or reduce h/w<1 — Tony Stewart EE75, Oct 15 '20 at 21:19

Andy aka · Accepted Answer · 2022-12-13T13:43:37.840

I am reading this paper and I see that characteristic impedance does not take frequency into account.

At frequencies above about 100 kHz to 1 MHz, the characteristic impedance can be said to be \$\sqrt{L/C}\$ but, this doesn't account for lower frequencies where the characteristic impedance is: -

$$Z_0 = \sqrt{\dfrac{R+j\omega L}{G+j\omega C}}$$

_{Where R is the series resistance per unit length and G is the parallel conductance per unit length.}

So, at really low frequencies we can say that \$Z_0\$ is \$\sqrt{R/G}\$.
At mid-audio frequencies we can say that \$Z_0\$ is \$\sqrt{R/j\omega C}\$.

Here's an example of telephone cable that I calculated myself using excel: -

Extracted from my answer here.

Are these characteristic impedance equations only valid for certain frequency range or am I wrong in assuming that characteristic impedance will have a frequency dependency?

It has a frequency dependency for sure but, for track impedance calculations we are normally only interested in frequencies that are significantly above 1 MHz hence, \$Z_0 = \sqrt{L/C}\$.

Characteristic Impedance vs Frequency

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