# Mismatched transmission line at DC

The characteristic impedance equation in transmission line theory is this:

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

Take the case of a twisted pair transmission line, say the cat6 cable represented in this link . Cat6 cable info

Either conductor in the pair is well separated by their insulation and an additional plastic spacer, so G is essentially zero. As for R,L, and C, the link above provides that

• R=8.00 ohms/100meters

• L=0 henries

• C=6nF/100meters

If you let omega go to zero, Z_0 goes to infinity. This makes sense if the transmission has no load attached to it, but when a mismatched load is attached things get a little messed up. If you try to calculate the load reflection coefficient, you'll get "-1" as a result. $$\lim_{Z_0\to\infty} \frac{Z_L - Z_0}{Z_L+Z_0} = -1$$ This implies that zero voltage will be measured at the load $$\Gamma_L = \frac {V_L^-}{V_L^+}=-1$$ $$V_L = V_L^+ + V_L^- = V_L^+(1 +\Gamma_L) = 0$$

Now we all now that DC voltage dividers work and that's exactly the circuit that I've explained. If I haven't made any mistakes, could someone tell me the physical significance of this result?

/product_details.aspx?id=33613 "cable info"

• L is not zero!! Commented Aug 17, 2016 at 5:47
• The spec sheet does not list L. Plus L being non-zero doesn't affect the end result I'm presenting. Commented Aug 17, 2016 at 11:45
• Using $$V_L=V_L^+(1+\Gamma_L )$$ as a steady state solution entails the driver side of the line is terminated to the characteristic impedance. Wherever along the line you look toward the source you see a driver with a finite voltage thru an infinite impedance: output dead. Commented Aug 17, 2016 at 13:01
• My question then is "Why is the line of infinite impedance? " The source will indeed be terminated to the line, some voltage will drop across the source impedance, but the line should act like a resistor spanning between the, say positive pole of the source to the positive pole of the load and also a resistor spanning between the negative poles of the source and load...right? Commented Aug 17, 2016 at 13:49
• Who flagged this as low quality? Low quality means quality of the question, if the OP has something wrong then try to get them to correct it not shut it down. It looks like a great question to me as far as length and content Commented Aug 17, 2016 at 19:30

## 1 Answer

This question makes no sense, since there is no reactance at DC.

Do you expect to measure 377 ohms with an open circuited ohmmeter ?

• I do suggest that the line is loaded. There is some impedance spanning the far terminals of the transmission line. Commented Aug 17, 2016 at 19:25
• Your question makes no sense - there no reactance at DC. You will only measure the load and cable resistance. Transmission line theory is related to signal wavelength which has no meaning at DC. Sorry Commented Aug 17, 2016 at 19:36