# The “Impedance” Equation

In "Right the first time" by Lee W. Ritchey, he says "The impedance will be the same whether the frequency of the EM energy is 1 MHz or 1 GHz. This holds true so long as none of the three variables changes with frequency."

But clearly that equation has a dependence on $\omega$ ? What's going on here?

• I see four possible variables (R0, L0, G0, C0), of which two influence the frequency-dependent terms. Why does he say "three"? What is the context? – CL. Aug 27 '17 at 14:43
• I've updated the picture to include more of what he says. I think you can access a pdf of the book here thehighspeeddesignbook.com/index.php – VanGo Aug 27 '17 at 14:51
• @VanGo - What page of the book? – Kevin White Aug 27 '17 at 15:05

The book is downloadable from here: Right First Time and the relevant section of the book is pages 119 to 122.

In this book the author is specifically talking about PCB design and is making the reasonable assumption that $R_0$ and $G_0$ and negligible in comparison with the inductive and capacitive effects for frequencies of a few 100kHz and above.

This fits with my personal experience of PCB design and several authors have made the same assumption.

The statement $Z = \sqrt{\dfrac{R_0 + \text{j} \omega L_0}{G_0 + \text{j} \omega C_0}}$ is false but at these frequencies the impedance varies very little with frequency and $Z = \sqrt{\dfrac{L_0}{C_0}}$ is a very close approximation.

Take a look at fairly standard cable used for telephones. It has a characteristic impedance that varies quite a lot in the audio range: -

And, this effect is as a direct result of the standard characteristic impedance equation for a transmission line. As frequency drops into the audio range, R starts to dominate over jwl and the impedance becomes: -

$\sqrt{\dfrac{R}{j\omega C}}$

As frequency lowers more the impedance tends to become: -

$\sqrt{\dfrac{R}{G}}$

And, at much higher frequencies, the characteristic impedance is the standard formula used by RF guys: -

$\sqrt{\dfrac{L}{C}}$

It's also noteworthy that many sources refer to a transmission line as "distortionless" if the ratio R to G is the same as L to C. When this happens, the impedance is constant from DC to any frequency.