Dear Jang, 

Thanks for the nice question 

The inductance, L, is defined in terms of the current, I, flowing through the system, and Wm, the magnetic energy density is proportional to B2. The magnetic field, B, exists within and outside of the wire. Within the wire, it increases linearly outwards from the centerline, and outside of the wire it falls off as 1/r, where r is the radius. So, to compute the magnetic field and the magnetic energy distributions we build a model based on Maxwell’s equations, in this case Ampère’s Law. A current source is applied to one end of the wire while the other end is grounded. The tangential magnetic flux density is constrained to zero on the exterior boundaries of the model.

Have a look into the following link for ur reference 

https://www.comsol.co.in/blogs/computing-the-inductance-of-a-wire/

Dear Jang,

Must you calculate? Ever considered injecting a step current with a known di/dt, and then measuring the potential rise from the point of injection to the ground? You can estimate the wire's inductance. To reduce inductance, many approaches can be taken. For example, copper strips or braids may have lower inductance than a single wire.

This may be helpful

http://www.nptel.ac.in/courses/108104051/chapter_1/1_4.html

In practice take in account that the wire is not in free space (proximity of conducting objects)

For rough estimation look at this calculator as well

http://chemandy.com/calculators/round-wire-inductance-calculator.htm

Rule of thumb: For straight round conductor ~0.5 mm L = 10nH/cm 

You can find the answer e.g. in the following publication:

Carson J.R. Wave propagation in overhead wires with earth return, Bell System Technical Journal, 1926, No 5, pp. 539 -554.

Best regards, Wojciech Machczyński.

Thanks for answering this wired question. It looks like two parallel wire model must be considered even for the ground wire since there is really no current along open circuit. The current must be returned somehow. Just..I don't see what is return current path for current flowing through ground. For example, when lighting strikes the sharp metal pin on the top of the building. It is going to earth then...what is its return path? Even lighting must have return path.

Dear Jang, 

this attached paper may have very closed answer for your question

Clearly any wire needs a return path. A grounding wire is carrying all asymmetric return currents of e.g. a three-phase system of power supply in high voltage, but also in low voltage. For high frequency signals, a low inductive grounding is essential. Because of the frequency dependence, a single grounding wire presents a high resistance, if the inductivity is large (omega x L). Therefore, mostly broad copper foils are used for grounding of high frequency signal lines. For power-frequency currents, single wires also work.    

Thanks E. Dullni.

Yes I need good ground for high frequency. Could you tell me how I can use copper foil in this case? Just replacing single wire to the foil or putting foil on the floor widely so that the foil plays as common ground plane?

A ballpark answer is 1 micro henry per meter for single conductors.  Parallel conductors are lower as would be the case for two inductors in parallel.

More complicated answers are addressed in the papers already listed.  In a High Voltage lab, you don't generally have high currents unless there is a flashover.  In all cases, common sense is a great asset to figuring this out without complicated evaluations.

Donggyu Jang.

Your question is very interesting. I believe you should study the concepts of partial and loop inductance. I suggest you to read ''Inductance: Loop and Partial'' and ''Analysis of Multiconductor Transmission Lines'' of C. Paul. Also some EMC books as ''Circuit Modeling for Electromagnetic Compatibility'' of Darney, and ''Electromagnetic Compatibility'' of Ott. By reading this books you should conclude that to compute loop inductance is always neccesary to include the return path (which may be the ground). On the other hand the concept of partial inductance allows you to compute the inductance of a single wire leaving aside the return conductor. All this concepts are very useful and rich, as they take us back again to Maxwell equations and the physical meaning of inductance.

In electrical engineer we mainly use loop inductances. For example, if you check how EMTP programs compute impedance matrix you will find that the self inductance of a conductor is nothing else than the loop inductance considering ground as the return path. For instance, see Carson's formula.

In ''Analysis of Faulted Power Systems'', of Anderson, even he does not explicity say it, partial inductance is used to compute impedance matrix and derive sequence impedances.

 I will be busy In the next few weeks, and I may not have time to study this issue in detail, but I hope this answer helps you.

Best Regards. Ignacio Avila

Formula for inductance of a single wire is available.....see vaedrah.angelfire.com/inductors.htm...formulae for both dc and ac are given..

In your case is it not possible to measure it?

I suspect that there is more to your question.  Please provide details.  In a DC situation, di/dt is zero and inductance should not be an issue.  Inductance becomes important when current is NOT time-invariant, i.e., current varies with time in some fashion (sinusoidal steady state, impulse, ramp, square wave, steep front wave, other).  What tests are you performing and what is the frequency content of the current you are dealing with?  Is it a flashover test?  Is it an impulse test? Are you concerned about energization transients?  Are you using a wire or a wide flat braided strip?  What are the details of the grounding system in the laboratory?

Please provide more details.  My direct e-mail is [email protected]

Regards, GG

Please  refer to the Equation given in Page  87 of Chapter 1 of my Book, " Basic Concepts of Electrical Engineering ", uploaded by me to R.G.

L = N^2 / (l / a *μ0*μr)  Henry....( 1.18.6 )

This Equation is derived on assuming unit Current flowing through the coil - Generally for a circuit element.

For a Single Wire N = 1.

l = length oh the Magnetic path = length of the Wire instead of length of the coil in Meters.

a is the area of cross-section of the magnetic ring in square meters.

Yours being a case of Ground Wire with Overhead Line carrying Current, you have to consider the Image Conductors also. As you are interested in Magnetic Field, unlike Capacitance of the line to Ground (where you consider Image Conductor with negative charge to have Zero Potential at Ground),now you have to Consider Image Conductor to have the current flow the same as in the Original conductor to have Zero Magnetic Field at Ground.

Thus it works out to be a case of Two Parallel Wires (Original and Image Conductor)  with Equal currents flowing in the same direction.

The Magnetic field linking the Ground wire is Zero and hence the Inductance of the Ground conductor is Zero.Hence there is no need to calculate the Inductance of the Ground conductor.

P.S.

Dear Pisupati Sadasiva Subramanyam.

Two parallel wires consisted of forward and return current path thus your description that curent are on the same direction on both wires (or one real wire and one image) doesn't make sense for me as every current must have their own return path. (closed loop circuit should be made.)

And I don't understand 'magnetic field linking the ground wire is zero'. If inductance of the ground is zero. Even narrow wire can work for grounding but as you know, The ground wire is normally very think in comparison to circuit wire thus inductance should not be zero. right?

I don't think there is ideal wire with having zero inductance in the real world.

Calculations for Inductor of a single wire is available in many text book but to make earthing for high voltage lab you have to apply some basic considerations of physics and a bit of common sense. For low frequency current the problem is simpler but for high frequency or transient current as you may expect during flash over or High Voltage breakdown with large fault current, happening frequently in a high voltage lab.

    During high frequency or transient current with high di/dt the eddy current within the conductor will be high and the impedance may be high enough for huge voltage drop during a breakdown. To reduce eddy current a simple solution is to use thin copper strip or foils in parallel or many conductors in parallel. A pipe may be used for main grounding. And it is also to be remembered that we may practically go on reducing the inductance by efficient geometric design but for phenomenological reason we can not make inductance zero.

Hi!

There is not a closed solution for the "Carson integral". It should be numerically solved. If you are looking for approximate solution see  the book: Sunde E.D. "Earth conduction effects in transmission systems", D.Van Nostrand Comppany, Inc. ,New York 1949 - chapter V : Propagation characteristics of earth-return conductors.

Mr.Donggyu Jang,

I hpoe you are Familiar with some of these Basic Concepts.

In the case of Calculation of Line to Ground Capacitance using Image Conductor Theory,the image conductor is assumed to carry Equal and Opposite Current.But there also the Image Conductor does not provide the Return path. For both Line and Image Conductors Ground provides the Return path and it is so to have the Ground Potential Zero. .

Similarly in this Case also to have Zero Magnetic Field Intensity at Ground,the Image conductor is assumed to have Equal Current in both Line and Image Conductors. Here also Ground Provides Return path for both and Line and Image Conductors.

Also, in Field Theory You may be Familiar with Calculation of Magnetic Field Intensity ay a Point due to an infinitely long Filament Conductor. There you assume Return path to be at Infinity.

Also you may be Familiar with calculation of Force between Two parallel Conductors carrying Equal/Equal and Opposite Currents,or Calculation of Field due to such parallel conductors, In the case of two conductors carrying equal currents you assume return paths at infinity for currents in both the Conductors.

The Magnetic Field intensity at a Mid Point between Two Parallel Conductors Carrying Equal Currents is Zero. This Can be Visualized considering the two conductors to be Infinitely Long Filament Conductors carrying Equal Currents.

Refer to any text book on Power Systems and Electro-Magnetic Field Theory.

P.S.

The Image Conductor Theory assumes unifomly Zero Potentials at all Points on the Ground along the ground conductor as in the case of using Grounding Grid.

The performance of earthing systems under transient conditions is different from power frequency behaviour because of more significant influence on inductive and capacitive effects. In contrast to the 50 Hz response, at high frequency the inductance of a small earthing system, such a rod, has a significant effect and the effective length of such systems can be very small.

For Accuracy refer to 

Accurate evaluation of the ground impedance of real earthing systems

UNIVERSITÀ DEGLI STUDI DÌ PADOVA

Facoltà di Ingegneria

Corso di Laurea Magistrale in Ingegneria Elettrica

Accurate evaluation of the ground impedance of real earthing systems

Relatore: PROF. ROBERTO TURRI

Dipartimento di Ingegneria Elettrica

Laureando : Leggio Massimiliano

2.2 Fall of potential Method

Link : http://tesi.cab.unipd.it/39557/1/tesi.pdf

P.S.

The issue has been solved in detail and many papers etc are available today. We call

the inductance of a single wire Lp "Partial inductance" to make sure that we

understand that the value of a single conductor alone has to be treated carefully.  

We would call the value for example Lp_11.  This value goes to infinity if the

length goes to infinity.  To make a closed loop, you have

to use as an approximation two wires where the closing parts are ignored from an

inductance point of view.  So, the inductance of two wires in parallel where we

assume that they are somehow connected at the end is

L = Lp_11 + Lp_22 - 2 Lp_12

We wrote a paper in 1972 on the inductance calculation.  A recent publication is:

Holloway, Kuster, Ruehli, Antonini, "Partial and internal inductance: Two of Clayton

R. Paul's Many Passions"  IEEE Transactions on Electromagnetic Compatibility,

Vol 55, No. 4, pp 598-623, Aug. 2013

Also, Clayton Paul wrote a book: Inductance Loop and Partial, Wiley, 2010

Also, we have a book which is almost finished on the complete equivalent 

circuits  which also include resistance and capacitance besides the partial

inductances. It is also from Wiley and it should be out soon. 

Dear colleagues,

I have recently derived a formula for the inductance per unit length and capacitance per unit length of a single wire. They are respectively,  L_0=R/c and C_0=1/Rc, where c is the speed of light. They give values in the experimental limits. Any comments are welcomed.

Arbab

Hi

Formula for approximative inductance of a single wire (buried horizontal electrode)  is available in many documents. See the book of Sunde:  E. D. Sunde, "Earth conduction effects in transmission systems". Bell Telephone Laboratories incorporated, New York, 1968.

This formula is the self-inductance of the conductor, you should study the mutual coupling inter segment (As you said for the totale inductance). This phenomenon is well detailed (expressions of Grover) in : N. D. Hatziargyriou and M. I. Lorentzou, “Investigation of the effect of mutual coupling of grounding conductors,” National Technical University of Athens Department of Electrical and Computer Engineering, Electric Power Division 42. Athens, Greece, 1998.

Good Luck,

Omar

Since you mentioned, you are in a lab, the inductance is important if you are doing chopped wave impulse tests.  Here the issue is that this ground inductance slows down the chopping frequency and may make it easier on the tested equipment.  In my past experience, the ground mesh in the buried lab helps considerably rather than just a single wire.  Also a flat ground conductor, perhaps 100mm wide will have a lower impedance and be a more realistic test.  Multiple grounds are always a good idea where impulse testing is to be done.

Dear Omar,

Thank you very much for your nice reply and the information you supplied.

Good luck.

Arbab

Dear Omar,

an approximation 1 micro henry per meter for single conductors.But it depends on the shape of conductor (circle, rectangle or square) and geometric configuration. For two wires with circle shape with small mutual distance d and diameter D

L=(u0/pi)ln(2d/D)

for single wire above grounded surface

L=(u0/2pi)arccosh(2d/D).

With regards

Bystrík

The inductance of a single wire is a partial inductance Lp.  The partial inductance of a single wire does not mean much and a single wire alone has an infinite inductance if it is infinitely long.  As soon as we have an idea about the return current path which forms a closed loop, then we get a meaningful value for the inductance.  The inductance of a square loop with 4 sides is:  L = Lp_11 + Lp_22 + Lp_33 + Lp_44 -2 Lp_13 - 2 Lp_24. So, the key contribution to the inductance is the enclosed loop.  So, if we don't know the return current path, then we cannot say what the inductance will be. Now, for an antenna, the return path will be closed by capacitances in PEEC circuits.  Also, a wire under ground must have some resistive return currents, besides the capacitances for higher frequencies, else in does not do anything. So, please think in terms of closed loops. Also, partial inductances must be taken as building blocks rather  than final values of inductances.  It is important to understand the details of a problem to get meaningful values.

Best,  Albert

in the Lab Environment, the wire between the source and the test object is only part of the circuit as there is always a return circuit. Many of the larger labs have a mesh of wire in the flooring that is where grounds are connected to. This "ground plane" is lower inductance because there is many conductors/inductors in parallel and wires in parallel have lower inductance because of the distance between them. In most cases, the rule of thumb estimate of inductance is of sufficient accuracy for test purposes.

In the case of DC, usually inductance is not significant except when transients are present. On can estimate the inductance if the capacitance of the circuit is known by the frequency of the oscillations in these transients events. If the resistance of the circuit is high, there will be no oscillations hence will not yield the circuit inductance so rules of thumb for inductance may be the best that one can do. In longer DC applications where there is no visible return wire, likely the return path is through the earth as ground rods usually are pounded into the earth at each end.

I believe that there are some test meters that will help determine inductance. Others will measure capacitance only but that can help if this is unknown.

The question can lead to confusions since inductance is not a trivial concept. The best source about careful answers on these issues can

be found in the Wiley book, 2017, Ruehli, Antonini, Jiang, "Circuit oriented

electromagnetic modeling using the PEEC techniques". It would be hard

to give a simple answer.