In power transmission why inductance L increases with increased spacing

[jaus tail]

In power transmission where single phase power is transmitted in 2 wire AC system.
Now inductance is
The equation for inductance is then:
L = 2 X 10^-7 In(De^1/4/_rx)

Why does inductance increase with increase in distance between conductors? When the spacing is increased, the air gap increases between conductors, so reluctance increases, so flux linkages decrease, so inductance must decrease, as L = flux linkages / current.

Why does inductance physically increase with increase in conductor spacing?

 

[jim hardy]

When the spacing is increased, the air gap increases between conductors, so reluctance increases, so flux linkages decrease(?),

i think you need to draw a picture.

what's formula fpr inductance of a loop of wire ?

 

[jaus tail]

Formula for inductance is L = flux linkages / current.
If distance is increased between the two conductors above, why does L increase? I think flux linkages would reduce because mutual flux would reduce.
L = 2 * 10-7 ln ( D / r / 0.7788)

I couldn't find equation for L in rectangular loop in the book.

 

[CraigHB]

When you look at formula for inductance in a wire pair you usually see a caveat that one wire is assumed to be a source and the other a return. In that case flux cancels an amount. The closer the wires the more significantly flux cancels. That's why signal wires are often encased in a twisted pair, to reduce inductance.

For two wires carrying current in the same direction flux does not cancel, but currents are shared between two conductors so the overall value is reduced in the manner of parallel inductance. Proximity of the conductors does not have the same effect. Flux couples rather than cancels as wires become closer together.

 

[jaus tail]

Yes, even in the question in first post, the other conductor is return path for current in first conductor. So current in both conductors are opposite to each other.
but if you look at the center, then the magnetic lines of force are in same director for both conductors. Like both lines of flux are going into the screen like a cross.

so wouldn't the flux lines add each other instead of cancelling each other out?

 

[dlgoff]

but if you look at the center, then the magnetic lines of force are in same director for both conductors.

No. They cancel.

image compliments https://www.quora.com/Why-do-two-wi...h-current-flowing-in-opposite-direction-repel

 

[jaus tail]

I'm sorry but how are they cancelling. The scenario that i asked is just as the first picture. The one with red and blue flux line. the currents in conductors are flowing in opp directions.

And it says that the lines of force add in the region.

 

[dlgoff]

I'm sorry but how are they cancelling. The scenario that i asked is just as the first picture. The one with red and blue flux line. the currents in conductors are flowing in opp directions.

And it says that the lines of force add in the region.

You're right with this image. I misread it when uploading. Sorry.

 

[jaus tail]

So the question remains same. Why does inductance increase with increase in spacing? In fact now if you look at image, it's clear the fluxes of the two lines are adding each other. So as distance between them increases, the flux reduces, so L must reduce.
Why does L increase with spacing b/w conductors?

 

[jim hardy]

In fact now if you look at image, it's clear the fluxes of the two lines are adding each other.

Of course they add. Then why do you insist flux decreases as the wires separate?

That transmission line has a source and a load so it is a loop that encloses a finite area.
Anyplace in between those wires flux is being pushed, in your picture, into the screen by both wires.
The more area the more flux..

Try superposition...
In your mind, double the distance between the wires. Within the area between them, each wire now pushes more not less flux into the screen.

Formula for inductance of a flat loop is not so nice looking as for a solenoid and that's why most authors avoid it.

https://www.eeweb.com/toolbox/rectangle-loop-inductance

old jim

 

[jaus tail]

But if the distance between two wires is increased why would there be more flux at one point? I mean it's like saying that if i increase the air gap of induction motor there'd be more flux linkages.

Flux equals mmf / reluctance. MMF depends directly on the current in the wire. Reluctance is the air gap reluctance. As distance increases the air gap would also increase between two conductors, so there should be less lines of flux. it's like a magnet will have strong force to objects near to it, than to objects away from it.

I saw the formula but i don't understand actually how can it be. if that's the case that inductance increases with distance then each transmission line will have infinite inductance as there are many many transmission lines all around the world. And its possible that some conductor will carry current in opposite direction.

I guess I'm missing some very basic point. Any insight?

 

[jim hardy]

I guess I'm missing some very basic point. Any insight?

reluctance ?

In your drawing, each wire is surrounded by concentric circles of flux. The wider the separation of the wires the more of those circles you can fit between them.
Is this where you found that drawing?
http://www.skm-eleksys.com/2011/03/transmission-line-parameters-resistance.html
I see the same one there.
As they explain, Inductance is figured in (milli?)Henries per unit length , mile or km.

if that's the case that inductance increases with distance then each transmission line will have infinite inductance

Infinite? Only if length is infinite.

 

[256bits]

I guess I'm missing some very basic point. Any insight?

Take another look at the magnetic circuit of the 2-wire electric circuit.

The air gap you mention between the wires - that is not the air gap in a magnetic circuit on which you seem to be basing the reluctance, Jim has discussed reluctance..

The magnetic circuit for a loop of wire is that area for the field flux. That area is the area encompased by the loop, and all of the area outside the loop. The area outside the loop is pretty much constant, so only the area within the loop affects the magnetic circuit. ( In the basic case at least )

Decreasing the area - is moving the two wires together -
Increasing the area - moving the two wires apart -

--------------
as an aside :
A solenoid is just a bunch of 2-loop wire circuits piled on top of one another, with the flux produced from one loop added on to the other. One can look at one wire next to the other and see what happens. That is picture 2 in the dlgoff post. One can look one wire and the wire directly opposte - that is picture 1 in the same post.

Try this post and follow the flux paths for a solenoid. - Replace with just one loop follow the logic.
http://www.tpub.com/neets/book2/2b.htm

 

[jaus tail]

If I try superposition, then the lines of flux one conductor pushes into the screen between the distance between the wires, well these lines of flux won't extend till infinity. I think there would less lines of flux as distance from conductor increases. Isn't that what happens in a magnet?

A magnet pulls objects that are nearer but if you increase the distance between the magnetic material n magnet, the magnet may not be able to pull the material.
Is is because lines of force/flux become sparse as distance between magnet n material increases.

Similarly if I increase distance between conductors, if the flux lines increase, where did extra mmf come from for additional flux lines.

If we take basic magnet, it has n s pole. The magnetic field is strongest near the pole n fades gradually.

In your drawing, each wire is surrounded by concentric circles of flux. The wider the separation of the wires the more of those circles you can fit between them.

I think this is what I'm having trouble understanding. Why would there be more flux lines? The MMF is constant. As distance away from current carrying lines is increased there should be less lines.
Isn't that what happens in case of magnet? Like if magnet is away from a nail, the magnet won't attract the nail. But if magnet is near to nail only then will magnet attract the nail.
I understood the formula but actually where do the extra lines of flux come from? Where does the extra magnetic energy come from?
Just like there's a fire, the heat is strongest near the fire.
Like HV line, the ionization of air is maximum near the line.

 

[CraigHB]

Yes that's right flux intensity falls off with distance, that's why mutual inductance also so falls off with distance. The problem is you are not understanding the manner in which the flux of two wires interacts as distance increases or decreases. For common paths, proximity couples flux increasing inductance to that of a single wire. For opposite paths proximity cancels flux reducing inductance below that of a single wire. You can use the idea of a loop as area goes to zero to get a feeling for it, but still that's the math for a loop not a linear path so it won't make a lot of sense with a zero area. That's something that can't happen realistically.

 

[jaus tail]

That is what I thought that maybe the flux lines cancel each other n so as distance increases the cancelling element reduced.

Buy if you look at dgloffs post. Then the flux lines add in case currents are in reverse direction.

 

[jim hardy]

well these lines of flux won't extend till infinity.

Is is because lines of force/flux become sparse as distance between magnet n material increases.

Why can't they extend to infinity ? They just become very faint.
Look at Biot-Savart. Zero is the limit as r approaches infinity.

I think this is what I'm having trouble understanding. Why would there be more flux lines?

Draw an end view. Each wire is surrounded by concentric circles. The wider the space between the wires the more of those circles traverse that space.
Also realize that they're not really lines but a fluid-like continuum. We just represent the continuous field with lines whose closeness indicates intensity.

Like if magnet is away from a nail, the magnet won't attract the nail.

Add "perceptibly"

 

[CraigHB]

Why can't they extend to infinity ? They just become very faint.

That's a good point. Forces created by some body such as gravity and magnetism never actually go to zero in space, they just become more and more disperse until they can be no longer be measured. The forces are still propagating through space albeit too small to be observed at some far off distance.

 

[jaus tail]

So if they become faint then wouldn't their effect also be less? I mean isn't that what faint implies? Less strength.
I looked up Biot savart's Law... Definition of BIOT-SAVART LAW. : a statement in electromagnetism: the magnetic intensity at any point due to a steady current in an infinitely long straight wire is directly proportional to the current and inversely proportional to the distance from point to wire.

The magnetic intensity is inversely proportional to distance from point to wire. So if distance is increased then intensity would reduce as per Biot Savart's law.

 

[jim hardy]

The magnetic intensity is inversely proportional to distance from point to wire. So if distance is increased then intensity would reduce as per Biot Savart's law.

Intensity at any single point, sure. Flux isn't intensity at any point, it's the integral of flux density over an area. When you add area by separating the wires you have a lot more points to tally up with your integral.

Draw two end views with one having separation twice the other. Do the circles near the wires get farther apart ? Or do you get more of them in the middle ? Why would the circles nearest each wire change any? Have you tried superposition yet ? What's the intensity contribution from each wire midway between them, and what is it 1/10th of the way between them?

 

[jaus tail]

Sorry for late reply.

by the way...i was about to ask a doubt in this question but asking the doubt sort of gave some insight...kindly let me know if this conclusion can be true..

Flux isn't intensity at any point, it's the integral of flux density over an area. When you add area by separating the wires you have a lot more points to tally up with your integral.

Draw two end views with one having separation twice the other. Do the circles near the wires get farther apart ?

I thought that shouldn't the circles near the wire get farther apart. Current in the line is constant. So this current will produce flux. Now the value of this flux won't change unless current changes.

Flux density is flux / area. As I increase area, the flux density would reduce since flux is constant.

Why would flux increase without increase in current?

Is this the reason:
like flux density is like current density of material. if I increase cross section area of material, resistance reduces, current increases, so current density remains same.
Likewise increasing area, increases flux so that flux density must always remain constant.Points against this conclusion:
That's like saying there is water in a pan, if I increase area of pan, there'll be more water...but wouldn't the water just spread out more.

It's like there is a source of fire. The farther i go from source, the less heat i'll get.Any insight would be helpful...

 

[dlgoff]

It's like there is a source of fire. The farther i go from source, the less heat i'll get.

less heat yes. but the burns you get don't go away. you just get more "lesser" burns as you get farther from the source.

 

[jim hardy]

I thought that shouldn't the circles near the wire get farther apart.

why would they?


Lavoisier:
http://web.lemoyne.edu/~giunta/ea/lavprefann.html

"Instead of applying observation to the things we wished to know, we have chosen rather to imagine them. Advancing from one ill founded supposition to another, we have at last bewildered ourselves amidst a multitude of errors. These errors becoming prejudices, are, of course, adopted as principles, and we thus bewilder ourselves more and more. The method, too, by which we conduct our reasonings is as absurd; we abuse words which we do not understand, and call this the art of reasoning.

Vanity gets in our way.

When matters have been brought this length, when errors have been thus accumulated, there is but one remedy by which order can be restored to the faculty of thinking; this is, to forget all that we have learned, to trace back our ideas to their source, to follow the train in which they rise, and, as my Lord Bacon says, to frame the human understanding anew.

Start with Ampere's law.
http://mlg.eng.cam.ac.uk/mchutchon/electromagnetismeqns.pdf

 

[jaus tail]

Thanks for your patience with me and trying to explain me.
I found this link
http://physics503.one-school.net/2008/06/magnetics-effects-of-current-carrying.html
that says:
The strength of the field decreases out as you move further out.

The book from which I'm studying says:
Inductance is flux linkages per unit current.
Flux linkages are due to internal flux, external flux, and flux linkages in parallel current

Now for external flux:

Consider the inductance in left conductor due to external flux linkages from right conductor. I understand that there'd be more flux as the distance is increased but not all of that flux is linking or enclosing or cutting the left conductor. Like in the figure that I've drawn, only the green flux is linking with the left conductor. The remaining fluxes will just pass through the distance between the more conductor. So even though i have more flux, i don't actually get more flux linking with left conductor. So where are the extra flux linkages that increase L?

I'm trying to find theory of inductance of rectangular loops online but all I'm getting are calculators where i have to enter values and will get result.

 

[jim hardy]

not all of that flux is linking or enclosing or cutting the left conductor.

the conducting loop encloses all the flux inside it.

 

[jim hardy]

At last - an end view !

Ever read "Flatland" ? https://www.amazon.com/dp/1508474184/?tag=pfamazon01-20

...Thinking in just two dimensions...
First, walk around any flux circle in your endview. Its intensity is determined by its length and the current it encircles. That's flux density.

Then , walk that orange line from end to end and tally the circles that cross it. That's flux.

 

[jaus tail]

Oh...i thought that as we separate the wires its the flux (concentric circles between wires) that will thin out instead of more circles coming in the picture because that's what happens in case of current through conductor if we increase cross section area.

So ideally the conductors must be as close to each other as possible to reduce L? if we neglect other factors as capacitance and physical limitations?

 

[jim hardy]

Oh...i thought that as we separate the wires its the flux (concentric circles between wires) that will thin out instead of more circles coming in the picture

think about it. They're out there already because they aren't constrained to the area between the wires. Moving the wire fences in more of them. Unlike your current analogy because current is constrained to inside of the conductor.

 

[jim hardy]

So ideally the conductors must be as close to each other as possible to reduce L?

that's why you see so much twisted pair. It minimizes the area encircled by the wires.

They're paying attention in house wiring now to running both conductors together. It minimizes the line frequency magnetic field in the living space.

 

[jaus tail]

Thanks. Sorry for late reply.
Well I have one last question i guess that should put an end to my doubt...

Talking strictly about L in left conductor due to flux from right conductor.
So the lines of flux are always there and as we add more space between the conductors there are more flux lines in the area so the flux linkages increase.
But then if we look at the blue flux. Then even this flux is encircling both the conductors. Why don't we take that into consideration? Like if we increase distance between conductors(move left conductor to the left), then blue flux will now go between them. But then otherwise the blue flux is also encircling the left conductor so why is L because of the blue flux ignored?

Like L because of mutual inductance, why must this L only involve flux that flows only between the conductors?

 

[jim hardy]

so why is L because of the blue flux ignored?

because your blue flux line does not pass through the area enclosed by your current loop.
So it doesn't link the turn(s) of your inductor...

 

[jaus tail]

Okay. Thanks a lot.
The reason i didn't draw end view earlier was cause i didnt know what an end view is. Here we have front view, top view and side view.

 

[jim hardy]

Here we have front view, top view and side view.

Ahhh real drafting ! ok, call it front view...