How Do I Calculate Induced Voltage in Parallel Conductors?

[Steve Wetzel]

Say I have two parallel conductors that are both L long and d distance apart (center to center). I run AC power of frequency f and RMS voltage V through one conductor. That current will create a fluctuating magnetic field which will pass through the the other conductor and generate a voltage. How do I calculate what that voltage is?

We know:

V = RMS voltage of powered conductor

L = length of the two conductors

d = center to center distance of the conductors

f = frequency of the ac current on the powered conductorMy specific case is trying to calculate the induced voltage on the ground check conductor in an SHD-GC medium voltage mining cable. The cable construction is 3 insulated conductor shielded with a copper braid shield (for flexibility). in the interstices of this construction are 2 ground conductors and one ground check. The non-symmetry is causing an induced current on the ground check which is causing problems.What is the formula to calculate this?I originally posted this in April (https://www.physicsforums.com/threads/calculating-induced-voltage-in-two-parallel-conductors.946052/) and thought I was going to get some help there but that help never came. Can anyone assist?

 

[anorlunda]

My specific case is trying to calculate the induced voltage on the ground check conductor in an SHD-GC medium voltage mining cable.

How many conductors are in that cable?

Is it something like this? http://general-cable.dcatalog.com/v/Mining-Cable/#page=42 that has 3 power conductors, 2 grounding conductors and one ground check conductor?

Your question makes it sound like only one power conductor and one ground check conductor.

 

[Hesch]

That current will create a fluctuating magnetic field which will pass through the other conductor and generate a voltage. How do I calculate what that voltage is?

Well, no emf is induced in wire, due to some magneting field is passing through the wire ( except for some Eddy voltages and -currents ).

The formula to be used is:

E = dΨ/dt , Ψ is the magnetic flux through some loop formed by a wire. A straight wire doesn't form a loop.

Now, if you connect a voltmeter to the ends of the straight wire, you will measure some emf, but that's because the wires from the ends of the straight wire and the voltmeter will contribute to creation of a closed loop with a significant area.

 

[Tom.G]

How are each of the Ground conductors and the Ground Check conductor terminated at each end?
What device or circuitry is used to monitor the Ground Check conductor?

A detailed wiring schematic would be helpful even if it is a hand drawn sketch. That way we are all talking about the same thing.

Remember that the Mining field is new to most of us here and many of the details you consider 'normal' are unfamiliar to us. There is, however, a huge knowledge base here of the underlying principles and quirks that could be the underlying cause.

Cheers,
Tom

 

[anorlunda]

The question IMO is more complicated than the OP makes it sound. The application is detection of broken wires during fault (short circuit) conditions. So there are multiple conductors, each of which can be broken or not, or faulted or not. External resistors and circuits are also part of the ground check function. Everything is non-ideal.

Here's a statement of the problem domain.

Certain mining equipments are fed with portable three-phase A.C. power distribution cables comprising generally three helically stranded insulated power conductors and at least two grounding conductors located in the outer interstices formed by the power conductors. It is well known that such cables are subjected to a high degree of abuse due to constant handling, reeling and unreeling. Such cables are often run over by the equipment, hit by stones from the mine blast, dragged over rock and trampled under foot.

Here's some explanation from the https://arlweb.msha.gov/

75.803 Fail Safe Ground Check Circuits on High-Voltage Resistance Grounded Systems
Ground check circuits are required to be designed so as to ensure a safe dependable path for fault current by causing the circuit breaker to open when either of the following occurs:

• 
  1. The ground check wire is broken at any point; or
  2. The grounding conductor is broken at any point.
  

If low-resistance parallel paths are present that prevent the ground check circuit from actuating the ground check relay when the grounding conductor is broken, the ground check circuit shall be acceptable as compliance with this Section if the ground check circuit is designed to cause the circuit breaker to open when the impedance of the grounding circuit increases beyond the amount necessary to cause a 100-volt drop external to the grounding resistor during fault conditions.
The following method may be approved by electrical inspection personnel as an alternate method for ensuring continuity of a safe, dependable path for fault current for resistance-grounded circuits extending to permanently installed, stationary equipment located on the surface:

1. 
   1. Grounding circuit shall originate at the grounded side of the grounding resistor and shall extend along with the power conductors and shall serve as a grounding conductor for the frames of all equipment receiving power from the circuit.
   2. Second grounding circuit shall connect the frames of the stationary equipment to a low resistance ground field located near the utilization location.
   
   3. The resistance of the grounding resistor and the resistance of the ground field shall be maintained in such a manner that not more than 100 volts will appear between the equipment frame and Earth under fault conditions in the event that the grounding conductor should be severed.
   

 

[Tom.G]

After reading several parts of the MSHA.GOV site that @anorlunda referenced, the most likely fault is that a Ground or Ground Check wire is indeed open. The next likely suspect is a piece of mining equipment has a power fault to its frame. Of course both could be true with an equipment fault opening a Ground conductor.

I haven't found any details on the Ground Check circuitry, the following assumes that the Ground Check wire is tied to one or both Ground wires at the far end.
If there are junctions in the power cable (hopefully with connectors) you could try a set of jumpers tying the Ground & Ground Check conductors to help localize a problem. This is of course a generic trouble shooting approach that may or may not be safe or applicable in your particular situation.

Cheers,
Tom

 

[anorlunda]

I'm hoping that we hear more from @Steve Wetzel again. The two parallel conductors scenario in post #1 sound so overly simplified that we can't give a good answer.

How many conductors? Broken or whole? Faulted or not? Terminations and external circuitry? And what is the purpose of the induced EMF question in the first place?

 

[jim hardy]

Seems to me it's more likely capacitive coupling into the ground check wire ?

Is there a published picofarads per foot for that cable ?

 

[Steve Wetzel]

Wow!

Thank you ALL for your responses. I never expected to find such support here. I am impressed. I will try to address your very good questions and comments below:

1) Annorlunda:

The cable has 3 shielded power conductors, 2 bare ground conductors, and one insulated ground check conductor (also called a pilot, its for control). Here is a spec sheet on this cable: http://industrial.southwire.com/en/cablespec/spec/47381/

The reason for the confusion is my fault. I set up my question with the more simple case. I assume I will have to look at the induced component from each of the three phase conductors separately .

2) Hsch and Tom.G.

You are both absolutely correct I loop is needed. I do not have a schematic of the ground check circuit. I will see if I can get one!

3) Anorlunda

Your second post is indeed correct as to the function of the ground check circuit but the application issue I am dealing with is a long run (300 - 4000 m) of large cable (350 and 500 kcmil) which creates a voltage on the ground check conductor which causes the ground fault monitoring system to trip out.

4) Tom.G

No fault is occurring, its simply an induced voltage problem.

5) Anorlunda

Sorry for the delay, I did not check messages over the weekend, I never though I would get such support here!

The situation I am dealing with is 3 conductors, terminated, with a ground fault monitoring system. And again, the purpose of the induced EMF question is the mine would like some guidelines on how long of a specific cable they can run before they have to worry about this issue.

6) jim hardy

You can find the published capacitive reactance of the cable in the datasheet. http://industrial.southwire.com/en/cablespec/spec/47381/And I was able to find a link to a ground-fault ground-check monitor from Littlefuse: http://m.littelfuse.com/~/media/protection-relays/product-manuals/trailing-cable/littelfuse_protectionrelays_se_107_manual.pdf

Thank you all again for your comments and questions. I will be able to respond in a timely manner now.

 

[anorlunda]

I will try to address your very good questions and comments below:

Ah, thanks that is great help. It sounds like you need good troubleshooting help. Other PF members are usually better at that than I am.

But one thing comes to mind. You may have power leakage at the well bottom causing return currents through the Earth and thus Earth voltage gradients. If that is the problem, then induced EMF from other conductors is the wrong question.

One thing I do know about troubleshooting is that prematurely jumping to a conclusion as to the cause is a mistake.

OK all you PF EEs, let's see if we can help Steve.

 

[jim hardy]

not certain about their MΩ*MFT units
and not sure of your length,

I am dealing with is a long run (300 - 4000 m) of large cable (350 and 500 kcmil)

can i just say around ten thousand feet ? Would that make for 0.0034 megohms, 3400 Ω ?

Resistance grounding concepts are described well in IEEE Green Book, standard 142.
Your grounding resistor should be equal or less than the distributed capacitive reactance of your system.
Do you know how many ohms yours is ?
The Littelfuse bulletin addresses that

2.1.2 GF The ground-fault-circuit trip level is 0.5, 2.0, or 4.0 A when current is sensed with a CT200-series current transformer. Since the ground-fault-circuit trip level should not be greater than 20% of the grounding resistor let-through current, these levels are appropriate for use with 5-, 15-, or 25-A grounding resistors. For other applications, the trip level of the ground-fault circuit is 0.25, 1.0, or 2.0% of the primary rating of the 5-A-secondary current transformer

That's first thing i'd check...
Have you an oscilloscope?
In big three phase systems the ground current contains exaggerated amount of third harmonic.
My instinct would be to get a number for voltage across your grounding resistor, just read it with a DMM,
and get some idea what is the harmonic content of that voltage by oscilloscope or spectrum analyzer..
If your supply voltage waveform (phase to neutral not phase to phase) is either peaky or flat topped, you'll be surprised at the voltage across your grounding resistor caused by 3rd harmonic current through all that Xc from two miles of cable.
Remember 3rd harmonic current is additive among the phases, it doesn't sum to zero like fundamental.
And at third harmonic Xc of your cable is 3X less.

Armed with that information you can decide whether your ground detector settings are logical for your particular system.

That's where i'd start. You'll get to know your system's quirks that way.

old jim

 

[jim hardy]

Usually my troubleshooting starts off in the absolute wrong direction.
But if you rule out the dumb things first, sometimes it saves a lot of tail-chasing.

Any variable speed drives out on your loads ?

 

[Steve Wetzel]

Jim,

Thank you for pointing out the units. I hate them also and keep pushing that we use the more correct kft. abbreviation as its really per 1000 ft. Apparently "MFT." is an old industry abbreviation and switching to something that makes sense just caused problems for customers...

I don't know what the grounding resistor is being used but I will try to find out. It may take some time.

As for a scope I can ask them. I work for a cable manufacturer, not the mine. They have asked for our help and I thought it would be fun to dig into this.

And in this instance, there are no VFDs involved, but they can cause big issues in a cable of this construction due to the non-symetiral grounds and ground check in much shorter distances than traditional 60Hz applications.

I see on your profile page you enjoy boats, old Evinrudes, and telescopes. Me too! Grew up boating on Lake Wisconsin and spent more time on the water than on land all summer long when I was in high school. I now live just far enough out of Madison, WI that I have semi-dark Skys so I bough an 8" SCT which is fun. Now the Evenrude connection... I live 4 miles outside of Cambridge, WI where Lake Riply is. Ole Evenrude grew up on that lake and he built a sail boat there. that was, of course, before he invented the outboard motor.

 

[Steve Wetzel]

Jim,
Regarding your comment about capacitive coupling. You think so? The phase conductors are all shielded with a copper/nylon braid shield on top of a semi-con bedding tape on top of an extruded insulation shield. that braid is in contact with the ground wires so that's all at ground potential. You think the 45 mils of insulation and low voltage of the ground check is going to cause a capacitance issue?

 

[Tom.G]

Just got thru the datasheet of the Ground-Fault Ground-Check Monitor. The below from pg 11 of
http://m.littelfuse.com/~/media/protection-relays/product-manuals/trailing-cable/littelfuse_protectionrelays_se_107_manual.pdf

The Ground-Check circuit is a current loop with a 5.6V Zener at the far end. The Induced-ac Withstand voltage is 25VAC.

• 12VDC source with 240Ω series impedance
• Remote terminator of 5.6V ±5% Zener diode
• Max Ground-Check wire resistance of 40±10Ω
• Nominal Ground-Check loop current 25mA
• Nominal Trip Voltage ±1V from nominal (calculated)

Based on the resistance of the 350kcmil cable, the maximum run is about 32 000 feet or 9700 meters to stay within the max loop resistance. However with a 25Amp Ground Fault current you are limited to about 10 000 feet (3048 meters), after which the 25VAC Withstand voltage is exceeded.

With above information of 240Ω source resistance, 25VAC and the cable data from http://industrial.southwire.com/en/cablespec/spec/47381/,
there may be enough to calculate the capacitive induced voltage.

 

[Steve Wetzel]

Thanks for taking a look Jim. How would you go about performing the calculation? As the phase conductors are shielded and in contact with the ground conductors they should be at or very near ground potential. I am trying to understand what the capacitive coupling is that you are talking about. Is it the capacitance between the ground check conductor and the ground shields from the two adjacent phase conductors?

 

[jim hardy]

. I am trying to understand what the capacitive coupling is that you are talking about. Is it the capacitance between the ground check conductor and the ground shields from the two adjacent phase conductors?

Yes, capacitance between the conductors.
I'm pretty sure i saw in the cable spec sheet that the braided shield is 60% coverage . While I've never been around that type cable, plain old circuit theory says a cable thousands of feet long has substantial capacitance . 60% of that is still considerable.
Add to that I've no idea what is the sensitivity of your ground detector circuit and it makes me think 'were this mine i'd want to know those details.'

How would you go about performing the calculation?

Phase to ground voltage / capacitive reactance would give an upper limit for current, but if the shield is 60% effective use 40% of capacitive reactance.
How does result compare to Tom.G's 25 ma?

That's where i'd start,,, then learn how that ground detector works.

@Tom.G seems