Why is the intrinsic impedance of a lossless transmission line an ohmic resistance?

[Jasper Chow]

It is generally agreed that a lossless tranmission line is made up of countless equivalent ideal capacitors and inductors. Since ideal capacitors and inductor consume no power, the equivalent net should have an equivalent impedance consistent with that property. But the caculating result is (L/C)^0.5, a pure resistor. How can this be? A resistorless net turns out to have a pure ohmic impedance and dissipate power?

 

[WimRFP]

Your statement about a resistorless network showing resistance is true. So there can be two things: all energy you supply is stored, or dissipated, radiated, converted, etc.

You "see" Zo = (L/C)^0.5 only, when you connect/terminate the transmission line with a resistor equal Zo. So the generator that supplies the energy is absorbed/dissipated by the resistor at the end of the transmission line.

You can also give the line an infinite length, then the energy you supplied is in fact stored in the line, as it never reaches the end.

If you have a line with finite length without resistor and you apply a step (going from 0 to 5V for example), first the source experience Zo. The leading edge traveling down the line and reaches the end. As there is no resistor present, the leading edge bounces back and travels towards the load. Once that reflection reaches the source, the source sees infinite impedance.

During the short time the source sees Zo, it supplies energy. This is stored in the transmission line.

 

[godfreyl]

But the caculating result is (L/C)^0.5, a pure resistor. How can this be? A resistorless net turns out to have a pure ohmic impedance and dissipate power?

An ideal, lossless transmision line does not have any resistance and it does not dissipate any power.

The value you calculate with that formula is the "characteristic impedance". If a resistor of that value is connected as a load at the end of the transmission line, then any electrical power travelling down the transmission line will be completely absorbed by the load.

If the load impedance at the end of the transmission line is any other value of resistance, then some energy is absorbed by the load, and some is reflected back up the transmission line to the source.

If the load impedance at the end of the transmission line is a capacitor, an inductor, an open circuit or a short circuit, then all of the energy is reflected back up the transmission line to the source.

When some or all of the energy sent down the transmission line is reflected back to the source, it arives back after a certain delay, due to the time it takes to travel down the line and back up again.

Sometimes when you make an international phone call, you can hear an echo of your own voice. This happens when the line carrying your phone call is not properly terminated with the correct impedance.

Another problem can arise when a transmission line is used to carry an RF signal, since the length of even a fairly short transmission line may be a significant fraction of a wavelength, or even more than one wavelength. If the load impedance at the end of the line is not equal to the characteristic impedance of the line, then the input impedance, looking into the line will vary with frequency.

 

[chuckey]

Resistance is the ratio of voltage divided by the current. Look at it that way.
Frank