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Hysteresis losses in an inductor  resistance proportional to frequency? 
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#1
Apr1512, 09:25 AM

P: 130

1. The problem statement, all variables and given/known data
Consider an inductor, with a core of ferromagnetic material, connected to an a.c. power supply. Explain why the contribution of the hysteresis losses to the resistance of the coil is theoretically proportional to the frequency of the oscillator. 3. The attempt at a solution The energy lost per second, i.e. the power, is simply the amount of times the hysteresis loop of the material is 'completed'. So the power used is proportional to the frequency. However, how do I then show from this that the resistance is also proportional to the frequency? It feels like I'm oversimplifying it, but is it simply because I^2 = P/R and since I is not proportional to the frequency, this means R is? I've looked hard, and it may be simple. A text book of mine says to consider representing the hysteresis losses as a resistor in parallel with the inductor  why parallel? I would have assumed the correct approach would be to consider it in series  after all, in a circuit diagram we consider the resistance of an inductor as being placed in series with the inductor. 


#2
Apr1512, 03:33 PM

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You got started off right  the loss is proportional to frequency since frequency = no. of times the loop is traversed per second. The energy dissipated per cycle is proportional to the area within the loop. The additional increase in R with frequency, on top of the above loss mechanism, is due to skin effect, eddy currents, etc.
As for R in series vs. in parallel with L  it can be either way. In either case you get, for a given frequency, a real component plus a positive reactive component. With series it's R + jwL, with parallel it's wL/√(1 + w^{2}L^{2}/R^{2})exp{j(π/2  tan^{1}wL/R)}. The argument in the exponent is always positive. 


#3
Apr1612, 04:15 PM

P: 130

Also, doesn't the skin effect only effect the currents in the core, but not the circuit? I know that the skin effect cancels out the field in the interior of the core, and thus lowers the inductance. 


#4
Apr1612, 08:31 PM

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Hysteresis losses in an inductor  resistance proportional to frequency?
For eddy currents see http://en.wikipedia.org/wiki/Eddy_current For the skin effect see http://en.wikipedia.org/wiki/Skin_effect. The skin thickness δ is inversely proportional to the square root of the frequency. This means the resistance is inversely proportional to δ. Power inductor and transformer cores are constructed in laminations isolated electrically from each other to minimize eddy currents. Skin effect is in the wires, not the core, far as I know, but I cold well be wrong  anybody? Lowsignal inductors sometimes use powdered iron to effect the same thing. Other core materials are inherently electrically nonconducting. 


#5
Apr1712, 12:46 PM

P: 130

Thanks for the help Rude_man. I apologise but there is one thing that is not intuitively obvious to me;
"At higher frequency the more effectively do the induced currents cancel the field in the interior of the core. This leads, at high freq., to the skin effect where the field and eddy current are confined to a surface layer whose thickness is δ. The variation of the inductance is associated with the skin depth". Although it does not state where this skin effect occurs, the core or the circuit, it specifically mentions that the eddy currents, which form in the core, witness the skin effect as opposed to the currents in the circuit. I have here no mention that the skin effect has any effect on the resistance of the inductor, only its inductance. Although if the skin effect DOES have an effect on the current of the circuit then it WOULD make sense to me how the resistance of the circuit/inductor would be affected. Later it goes on to mention that the variation of resistance is associated with energy dissipation (with no mention of the skin effect). 


#6
Apr1712, 08:58 PM

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However, this does assume constant current. If you assume constant voltage, then H drops with frequency (since L goes up and so does reactance wL) and the entire hysteresis curve takes on a new shape. It should be easy to be convinced that if you reduce the crosssectional area of a wire, that the resistance increases! http://en.wikipedia.org/wiki/Skin_effect P.S. resistance is always associated with energy dissipation. Pretty hard to get around i^{2}R! 


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