# Effective Inductance Problem

Discussion in 'Homework Help' started by AACIRCUITS, Jul 13, 2011.

1. ### AACIRCUITS Thread Starter New Member

Jul 13, 2011
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I require a core which does not have a uniform cross sectional area.
Ultimately i want to be able to determine the step response of the RL circuit. But i'm very rusty on my circuit theory.
All values are up in the air at the moment, i just need direction in how to design the core, windings etc.
If one can picture a rectangular section core (assume 20.0mm thick x 60.0mm wide radially in a C shape, then have a straight length around 100mm (20x60), this horse shoe section would have the coil winding. Then I want to taper the ends and combine the two ends of the horse shoe, so that there is a complete ring now with a narrow bridge of still 20.0 thick, but now only around 4.0mm wide (i want to achieve saturation in this narrow area).
Can anyone point me in the right direction on how to mathematically draw up this simple circuit? Work out inductance, and therefore step response?
Al

2. ### t_n_k AAC Fanatic!

Mar 6, 2009
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If you draw the picture we can probably work something out.

3. ### AACIRCUITS Thread Starter New Member

Jul 13, 2011
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Thanks for the response, attached is a quick sketch.

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4. ### t_n_k AAC Fanatic!

Mar 6, 2009
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OK I see what you mean now.

The inductance of the "Inductor" will vary of course - depending on the magnetizing current supplied.

In principle the following would occur ......

At low magnetizing currents the inductance will be relatively high with the magnetic material operating in a quasi-linear manner.

As the magnetizing current increases the magnetic material will approach the region of saturation onset and inductance will begin to fall.

As the current increases even further the saturation becomes significant - primarily at the thinnest core cross-section. Inductance decreases even further to a notional minimum value. This minimum value would probably be evaluated in the first instance by treating the thinnest section part of the magnetic circuit as an air gap. Flux fringing and other practical issues would make the predicted inductance subject to error.

One would need to know the magnetic material properties (B-H) to estimate the inductance values at the major points of interest.

A "proper" (rather than approximate) analysis would probably involve the use of finite element techniques.

The burning question - does this concept have a useful purpose?

Last edited: Jul 14, 2011
5. ### AACIRCUITS Thread Starter New Member

Jul 13, 2011
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Does this concept have a useful purpose. Yes.
I've got a B-H curves, and finite element analysis software.
But lack the knowledge to move forward.
A few Questions;
I plan to use a square wave function, if i start off with a linear inductance @ low amps, then there would be pre-determined rise and fall time, as i start to saturate the core, you say that inductance falls, does this mean that the rise and fall times will decrease?
So if I have my B-H curve, i can set up a function for μ.
If i have 0 amps, and know μ value for core, how do i approximate what the effective μ for the loop?
Al

6. ### t_n_k AAC Fanatic!

Mar 6, 2009
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You apparently don't want to share about the purpose. If it's for a design with a possible commercial gain then I understand there might be reluctance on your part to divulge anything. On the same score, it would seem inappropriate to seek help "gratis" without some concession to a forum members' natural inquisitiveness. Not that we ask for or expect payment (or even thanks) for any help provided on the forum.

As to your questions:

Yes, as inductance decreases the rise and fall times will decrease. If you are driving the circuit through a wide dynamic range including the saturation region, the resulting excitation current waveform will be more complex than a simple first order exponential. Remember also that there will be loss factors (e.g eddy current) associated with the rate of change of flux in the core.

The material (incremental) permeability value is relevant to the analysis. Again yes, this value is available from the B-H curve. You actually don't want the effective μ for the loop - rather you need the circuit reluctance (itself a dynamic value with saturation) which is related to the overall magnetic circuit geometry. For a non-uniform geometry the reluctance is normally derived as a summation of the reluctances of the discrete geometric segments. An analytical problem arises when the various parts of the magnetic circuit are at different states of flux density and therefore possibly at different incremental permeability values. Again the degree of flux fringing off the core at high saturation points would be difficult to model.

7. ### ErnieM AAC Fanatic!

Apr 24, 2011
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This sounds like something I worked on years back, not my design, for a light dimmer used on the B1. The device looked like a conventional AC/DC full wave rectifier, but had a common core choke in the input power leads. The inductance was set high so normally the choke acted as a high Z blocking current from the output. The choke also had a gazillion turns on it so a low current could saturate it and thus allow current to flow to the load.

Whole thing was wound on a standard toroidal core a few inches around and an inch or two in height.

8. ### AACIRCUITS Thread Starter New Member

Jul 13, 2011
6
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t_n_k
I understand where you're coming from, and although i don't see how it impacts the problem mentioned at the start of this thread, I'd be happy to share the purpose, but i believe you would be reluctant to provide any further assistance, or just plain not believe me, but in the interest of moving from this sticking point. This is only one component of a bigger machine. So explaining it's final purpose does not "reveal too much". Also I'm not after anyone to do all my calculations for me, just educate/enlighten me so i may keep working on the details myself. I'm designing a type of electric motor/generator to be exact. The analysis i have done on Maxwell 3D for the last 2 years gives me great confidence in it's viability (Using standard everyday theory). If you would be so kind as to humor me on this, and not laugh me out of this forum. Let's assume i'm working on Perpetual motion...[insert crickets here]...Please refrain from non productive comments.

Last edited: Jul 14, 2011
9. ### AACIRCUITS Thread Starter New Member

Jul 13, 2011
6
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Back to the "non-perpetual motion" problem at hand.
Is there some literature on how to approximate the magnetic circuit?
If i start with the zero current instance, where μ is constant throughout the material, is it a matter of charting the cross sectional area of the closed loop and working out an equivalent area of equal length (l), so i can use;
L=N^2/(l/μA), or do i need to work out reluctance (l/μA) of each section as you say?
So i can work the reluctance R(0A),
if i break up the model into 4 sections, horse shoe shape, each taper, thin bridge.
So i end up with R1, R2, R3 & R4 (which is just related to the change in area) which i can total up. If i divide my winding turns by this, i get my inductance at i0?
for 0<i<min saturation current, μ is constant, so the above may still apply.
for min saturation current<i, i will have to get specific values for μ for each section and follow above steps?

Last edited: Jul 14, 2011
10. ### t_n_k AAC Fanatic!

Mar 6, 2009
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If you assume the flux density in the thinnest section is below the saturation onset then you can assume a constant permeability in the material around the magnetic circuit.

Calculation of the individual part reluctances can then be done with the same μ value. These would then be added to give the total reluctance. The inductance would then be deduced from the resulting flux linkages per amp of current excitation in the coil.

I guess simple symmetrically tapered sections will have logarithmic expressions for their reluctance values - again based on the known geometry.

11. ### AACIRCUITS Thread Starter New Member

Jul 13, 2011
6
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Thanks mate, will see how i go.