Determining conductance / capacitance and inductance per unit length

Question

How do you determine conductance / capacitance and inductance per unit for a cable experimentally?

Is there a way to determine them with only a oscilloscope and a pulse generator?

Answer 1

Given the only tools are a pulse gen and oscilloscope it makes sense to try something involving risetimes of RC and RL circuits.

The top circuit measures the capacitance of the cable. The far end must be open circuited. As the pulse is applied, Vo will start at zero and rise exponentially to what even the output of the pulse gen is. The time constant of the exponential will be R*C. Given typical cables are around 100pF/m and assuming 2m of cable, 5k will give a time constant of about Tau=1us (5000 * 100p * 2). This should be a reasonable to measure time constant. If a slower rise is desired increase R2 or lengthen the wire under test. Note technically R is the total resistance which should include the output resistance of the pulse gen, normally 50Ohm.

For inductance, the cable must be short circuited at the far end. As the pulse is applied the Vo will jump to "full scale" and decay. The decay time constant will give the inductance Tau=L/R. R is technically R1 || R2, but with R2=1 the 50ohm won't change it much. Once again if the decay is too fast lengthen the cable or decrease R2.

Note "full scale" above refers to the attentuated voltage (Vo = R2/(50+R2)). The test could be done without R2 but the decay will be ~50x faster so could be tricky to measure.

Ideally to measure the time constant of each circuit the decay should be captured on a DSO and have an ideal exponential fitted to it mathematically. The simple method is to find the point where the voltage has transition to 63% of the final value.

Traps:

The rise time of the pulse gen must be much shorter than the measure rise time of the circuit. At least 10 fold 100 would be better.

The length of the test wire can be made longer to make measurements easier. But the length of wire must not be so long that the propagation time for a wave to travel to the end and back is anywhere near the measured time constant. Otherwise messy transmission line effects will mess up the exponentials.

Of course divide by the length of cable test to get the L & C per unit length

Answer 2

If the cable contains no magnetic material, then you can compute both L and C per unit length from the impedance and the speed of light in the cable.

If you have a long enough length of cable for the speed of your pulse generator an oscilloscope, short the cable far end and time the length of the pulse you get from putting in a step. This is twice the electrical length of the cable. Now terminate the far end, and see what value resistor completely removes the reflection. This is the impedance of the cable.

These measurements are likely to be no more accurate than looking up the cable type in a catalogue.

Answer 3

Is there a way to determine them with only a osciloscope and a pulse generator?

There is a range of frequencies for which the phase velocity and characteristic impedance are relatively constant. 1MHz or 10MHz is probably within that range for most cables.

The phase velocity in that range is given by

\$v_\phi = \frac{1}{\sqrt{LC}}\$

and the characteristic impedance in that range is

\$Z_0 = \sqrt{\frac{L}{C}}\$

With proper techniques, both of these can be measured with a scope and a signal source. (OK, for phase velocity, you will also need an accurate length measurement.) From those measurements, you can calculate L and C

R can be measured at DC values, but tends to increase with frequency due to the skin effect. If you can determine the attenuation \$\alpha\$ (in nepers) per unit length, which should be easier for higher frequencies and/or longer transmission line lengths, one can calculate

\$RC+GL = 2\alpha v_{\phi}\$

(This formula is valid even outside the range where the phase velocity and impedance are relatively constant).

Knowing C and L, one can determine whether it is reasonable to neglect the GL term, if so, one has a value for RC, and hence R at a test frequency.

Answer 4

I might consider measuring L/2 with both conductors shorted at each end and measuring the $$ \tau = L/R$$ rise time asymptote per unit length.

Impedance can be measured by matching the load with a sig.gen. on DSO square edge at 20MHz BW to compute C . The pulse amplitude also attenuates with a matched load relative to the 50 Ohm source and Zo = Load.