# Modeling a long single wire in a circuit?

I'm currently trying to make a model (equivalent circuit) of a long single wire with a resistor and inductor in series. Is this a proper way to do it?

What I have done:

1. Calculating the resistance per unit length of a single wire

2. Calculating the self-inductance per unit length of a single wire

Is this a proper way to model up a long single straight wire? If it isn't, what is the proper way to do it?

• depends very much on the frequency Commented Sep 20, 2018 at 23:54

What is the wire used for? Detecting ghost ULF e-fields or UHF microwave? or connecting some signal? ( Paranormal is pseudo-science)

Either way, the physics describing a single wire depends greatly on material and geometry and affects characteristic impedance, resonant frequency or wavelength, Q or amount of ringing in a step response or overshoot.

For an unterminated wire, it is just an antenna resonator at 1/4 wavelength or an inefficient antenna at lower frequencies getting lower signal efficiency at lower frequencies for the same electric field.

or "sometimes a wire is just a wire" ( Einstein joke about his smoking a cigar at a psych conference and reference to phallic symbols)

For a terminated wire, let's consider the lumped element. , for

Short answer: The proper way is to model 2 wires unless you are wanting an unterminated antenna. Then for twisted pair 40 twists/m use 2nH/m, 100pF/m x m$$\\Omega\$$/m and leakage conductance has some very small value or >gigaohm resistance/m. (depends on leakage, creepage)

The inductance is a ratio of L/d or length/diameter ratio with some other factors.

We choose a wire gauge, based on current, impedance etc for a given length, which determines distributed resistance/unit [m$$\\Omega\$$/m]

If we assume each path is different Signal and Return wire then it gets more complicated with shielding and common-mode impedance then ingress from external noise fields causing common-mode to convert to a differential signal. Then we have EMI issues.

But for brevity here is the single wire model. The same model (no citation possible nor needed) is used for the return but may have different values.

The inductance for my rule of thumb is 1nH/m but depending on twists and L/d ratio this can vary from 0.5 to 2nH/m. The lowest inductance is where the width is much greater than the length for connections or multiple insulated strands are used (Litz Wire)

The capacitance [pF/m] depends on the return path, which rises with reducing the gap.

-If none and just earth ground, then the capacitance is inverse to the gap and the air gap is in series with the insulation (PVC) -if the capacitance is low the characteristic impedance rises but also with a low source impedance, the Q also increases and thus ringing is worse for a step pulse. The dielectric constant, (Dk, or $$\ε_R\$$) may be around 4x times air for many plastics, PVC, FR4 with variations.

So capacitance per unit or pF/m can vary from air to straight pair to twisted pair to shielded twisted pair.

Thus a wire becomes a transmission line when the cycle time of the signal energy is shorter than the propagation delay.

The result is shown in Telegrapher's Equation for a distributed RLC wire

When is a wire just a conductor? When it is DC or short AC line current with no transient loads but transients occur on startup and step loads are broad bandwidth.

## Rebuttal Question

When is a busbar not a zero Ohm conductor?
( when it is short-circuited or driving a very low impedance)

Consider a small busbar with 50Khz pulses on it for a DC to AC bridge for an inverter that will have pulses >100A on it. Imagine multilayer strips of flat copper, insulated with screw terminals. How many layers? How much ESL is OK>

It could be an industrial motor, 3 phase inverter using IGBT's in the kW range or a Wireless Power Transfer system to charge e-cars. The DC comes from any source (AC grid to DC, solar PV. You need a 0Ω Ohms but we know that is impossible. So, this depends on the conductor ESR and the inductance, ESL and the geometry again or Length/(width+thickness) ratio.

A failure in the design could cause the IGBT's to self-destruct from ringing or shoot-thru. The cap is the size of two fists with specs shown in my model.

So every value of RLC of each component must be estimated from data sheets or calculations and the results are critical. There is a busbar-cap parallel and series resonant (anti-resonant) effect due to the LCL circuit having 3 reactive components.

It may interest some folks that the same effect occurs with crystals except uW power, inductance near 1H load caps in pF and motional capacitance in fF (femtofarad) as a CLC circuit.

• Hey thanks for responding! The wire is used as the bus line of a CAN Bus "non-transceiver" network Commented Sep 21, 2018 at 3:07
• You "can" but only very slow short datalinks. try to send CMOS low f clock logic on a wire to a scope and see how long the straight wire you can go without ringing from an earth grounded source. Then try a wire pair. Then twisted pair then CAT5 cable with some resistor load like 100 ~200 Ohms Commented Sep 21, 2018 at 3:15
• @ Tony I use 1uH/meter for a single wire in free-space, no nearby return wire or plane; there is a natural-log factor in the math. What is your rule for twisted-pair, per meter? Commented Sep 21, 2018 at 4:46
• 120? Ohms 12 twist/in 15pF / ft as I recall? Coax 30pF/ft 50 Ohms 1uH/m=10nH/cm=1nH/mm for air or PCB tracks, unless reasonably fat or flat wire then 50% of this for rule of thumb not precise Commented Sep 21, 2018 at 6:12
• twisted Magnet wire has more capacitance so much lower impedance. Commented Sep 21, 2018 at 6:19

In the lumped circuit approximation, a single wire can't carry signal or power. There must be a return path for any current to flow in the wire, and the loop area between the wire and return path will be important to determining the effective inductance of the wire. You might also want to model the capacitance between the wire and its surroundings.

Outside the lumped circuit approximation, a single wire carrying a signal without a return path is an antenna. You will need to include radiation effects to model the antenna accurately.