# Equivalent circuit to use when calculating the impedance of an electrical cable from its datasheet parameters

I'm trying to work out the impedance of a cable (specifically a single twisted pair of Lapp Kabel 0035836). The datasheet quotes the following information:

Mutual capacitance:
C/C: approx. 120 nF/km
C/S: approx. 160 nF/km

Inductivity:
Approx. 0.50 mH/km


Can I assume the equivalent circuit is something like this:

simulate this circuit – Schematic created using CircuitLab

and can I assume the impedance follows the following equation? $$Z_{total} = Z_{R} + Z_{L} + Z_{C}$$

where

$$Z_{R} = R$$ is the resistance (calculated using the resistivity of copper, the conductor material); $$Z_{L} = i \omega L$$ is the "inductivity" above multiplied by the length; $$Z_{C} = \frac{1}{i \omega C}$$ is the "C/C" value which I assume is the "core to core" capacitance, multiplied by the length.

Is my logic correct, or have I misinterpreted the datasheet parameters and/or the equivalent circuit to use?

It depends on what you're trying to do. If you're trying to do a compartmental analysis, you need to decide how many compartments you need to see what you're trying to see.

Let's say you have 1 meter of cable. Convert your numbers to units/meter. If you then want to split it into 1000 compartments, calculate units per mm, and then put 1000 of those elements in series.

• I'm not sure what a compartmental analysis is. Essentially the purpose of this is to work out the voltage drop in my transmission line to a capacitive load attached at the other end of this cable. The main thing I wanted to know is whether my equivalent circuit is correct. – Sean Mar 20 '17 at 21:11

If your operating at DC maybe, if you ever plan on switching the signal, then no. Why? because transmission lines have wave effects.

Especially if your cable is a long enough to be a transmission line, here is how you'll know:

A cable becomes a transmission line when it has a length greater than λ/8 at the operating frequency where:

λ = 300/fMHz

For example, the wavelength of a 433-MHz frequency is:

λ = 300/fMHz = 300/433 = 0.7 meters or 27.5 inches

So find the highest frequency that will travel through the cable and "plug and chug"

Really what you need to do is use a transmission line model, the power source, cable and load all need to be matched, otherwise you will have:
1) Not optimal power transfer
2) Reflections and other problems.

So, Find the characteristic impedance of your cable and match

If your cable is short, then you could probably approximate the cable with this model (kind of like yours, but don't forget G)

Note: I think the problem is you need to also look at the load, its just easier that way. I simulated this with ltspice and the lumped element model but also the resistance and inductance of the return current. I am graphing the combined cable and load impedance. I see frequency effects above 10kHz with a load of 1Ω (green line), for 100Ω (red line) your good to ~1MHz, 10kΩ (purple line), you start to see frequency effects show up at 5MHz.

• I'm not exactly after a DC impedance, but not RF either. I'm interested in the audio band, up to around 10 kHz. Do I still need to take account of the "G" term in that case? When I calculate the impedance using the equation I put in my question (just adding each term), I get an unexpectedly large impedance at low frequencies. At 1 Hz, for example, I calculate the impedance to be 1.0200e-01 - 2.2105e+08i ohms, which has an absolute value of 2e8 ohms... I must have gone wrong somewhere but I can't tell where. – Sean Mar 21 '17 at 9:20
• In the audio band, you care about none of this, @Sean. Plug 0.01MHz into the equation laptop2d provided, and figure out how long your cable would need to be before you need to worry about it. – Scott Seidman Mar 21 '17 at 13:35
• @ScottSeidman, if I put a current limiting resistor in series with the cable, then there is definitely a "low pass" effect due to the RLC filter created by the resistor and cable's inductance and stray capacitance. I'm trying to work out what the impedance of this series would be. Maybe I should have mentioned in my post that there will be a big resistor in series. – Sean Mar 21 '17 at 17:38
• @Sean It really depends on what your source impedance is, I'm assuming that it will be low because its an audio application. G is usually relatively high >1e6 Ohms. I was looking at the cable, this isn't really a transmission line cable. How long do you plan on running the cable? Even though this is twisted pair, I think mutual inductance would be my biggest worry. Make sure you take advantage of the pairs and run the return current from the load with the current to the load on the same pair. – Voltage Spike Mar 21 '17 at 18:01
• @laptop2d it'll be about 6m, very short. The source impedance is 47k. A capacitive load of ~1nF will be attached on the other end. I'm just trying to work out the voltage drop at the load as a function of frequency, given the datasheet parameters for the cable. – Sean Mar 21 '17 at 19:55