I am trying to calculate the total impedance of two parallel copper wires under the following conditions:
- The copper wires are thick
- I want to include all relevant effects, including (but not limited to):
- Skin effect
- Proximity effect
- Internal self inductance
- External self inductance
- Mutual inductance
- Any capacitive reactances
- For now we can ignore wavelength as it pertains to cable length as my specific question pertains to frequencies less than 100kHz and wire lengths less than 100m so these effects can be ignored.
- The copper wires are parallel to each other, like zip cord.
- The copper wires form a return circuit, i.e. current in conductor A is the same as the current in conductor B but in opposite direction.
- Assume an AC voltage source at one end of this circuit with a frequency of f, and f being no more than 100kHz.
- Assume no load on the other end of the circuit – I merely want to model the impedance due to the wire, not due to the load.
- Assume solid cylindrical copper conductors in a PVC isolator just like zip cord, and the two conductors being very close together (approximately 2 – 6mm).
- Assume the wire is between AWG10 and AWG28
I have done tons of research and know how to calculate AC resistance due to skin effect, I think I understand how to calculate the internal self inductance due to skin effect, and I have some other calculations for external self inductance. What I do not have is a complete picture of
- which of above mentioned phenomena I need to consider
- which of those are duplicates, i.e. different names for the same phenomena
- which formulas to use for the rest of those phenomena
In the end I want to consider ALL phenomena that would have an effect of at least 0.001% in the given frequency range and wire specifications above. So I want to solve:
Zac = Rac + j(Xl – Xc)
where Rac includes all resistances, Xl all inductances and Xc all capacitances.
Ignore the environment for now (i.e. any dielectrics / objects in the vicinity except for these two conductors).
I have reviewed these papers:
http://g3ynh.info/zdocs/refs/NBS/Rosa1908.pdf
http://g3ynh.info/zdocs/comps/Zint.pdf
And many others but get confused as per above.
Best Answer
Inductance:
The above is the formula for the inductance of parallel wires where s is distance between wires, L is the overall length and d is wire diameter and here is a calculator that uses the formula.
Capacitance between two parallel wires is:
For series resistance you will find many references on google for this and please note that it is ambient temperature dependant. There are plenty of references for skin effect also but not so many for the formula for proximity effect but hopefully this reference from wiki might prove useful.
Finally, once you have the resistance, inductance and capacitance you can use the standard formula for characteristic impedance (applies to any length of wire of course and is still valid down to DC): -
You haven't considered "G" but this is usually a very large resistance and its effect is negligible.
So now you have the impedance looking into an infitely long line of cable. However, you want the impedance looking into 100 metres of such cable so you will then have to work with reflection coefficients and the load impedance (stated as unloaded in the question) BUT YOU CANNOT ignore these load effects (and reflections) if you want a proper answer.
Reflections will occur with any incorrectly terminated cable: -
And these can effect results significantly even at moderate frequencies like 100 kHz. At 100 kHz the wavelength is 3 km but this is probably going to be more like 2 km. Major things happen at a quarter of this wavelength i.e. an open circuit can look like a short circuit and a short circuit can look like an open circuit so 500 metres is still pretty close for creating major errors when considering a cable that is 100 metres long.