I Inductive heating and eddy currents

[artis]

Without going into the math let me refresh you with some common knowledge. The thing you are forgetting is time.Every waveform unfolds in time, yes a faster/steeper current rise produces stronger induced current which is just logical isn't it? Because you are putting more energy in per given amount of time to create the faster rise, but you can't just look at the energy put in at a given small window because you won't heat your metal with just a single pulse. You need to look at the energy put in over time.

Also take into account that your square waveform after it hits max flattens off , in this flat top section there is no more change of current so current becomes steady and that part is wasted as it produces no further induction, in a sine wave such part doesn't exist.
Also a faster dB/dt change induces stronger eddy current yes but that stronger eddy current produces a stronger back EMF which decreases the strength of the applied field. I would assume with steeper dB/dt your field will not be able to penetrate as deep as with a sine waveform.
You have to take these assumptions also into account while doing the math. So far it seems your just happy to calculate the power through the coil but calculate the power into the actual material when it's inserted.
That complicates stuff fast, and frankly I don't know the result but I suspect the benefit is not what you think it would be

 

[Baluncore]

So far it seems your just happy to calculate the power through the coil but calculate the power into the actual material when it's inserted.

As a transformer with a centre-tapped Cu primary and a single turn resistive Pb secondary, the driver magnetising current will be VAR while the shorted turn will be real VA power. The relative magnitude of those two driver current components will be interesting. I am waiting for the LTspice model.

What are the resistivities of solid and molten lead ?

 

[artis]

The relative magnitude of those two driver current components will be interesting.

I think also the geometry of the setup will be important as would be for any real transformer, given the OP goes with his desired geometry I suspect the B field coupling to the "secondary" aka metal to be melted piece , will be rather poor because
1) the metal is not within the coils but rather outside of them
2) only one coil is active at any given time, so B field only encompasses one side of the metal and eddy currents will set up a backEMF which will further deflect the source B field which will loop around instead.

The only way he can improve this is to use a core but at the frequencies mentioned it would have to be ferrite one.

The way I would do it I would simply make a ceramic oven that serves as a container for the metal , and simply place a larger coil around the oven as is actually done in commercial ovens whether induction or resistance ones. then the metal sits in the very middle of the coil and no fancy cores or multiple coils etc are needed also the coupling is best for what can be achieved without a core.

 

[Baluncore]

2) only one coil is active at any given time, so B field only encompasses one side of the metal and eddy currents will set up a backEMF which will further deflect the source B field which will loop around instead.

Not really, they are both active since in the diagram they both share a core. It is a centre-tapped push-pull transformer driven by a B-class amplifier.
Any deflection of the B field can only be because eddy currents are present. The more deflection there is, the more heating there will be of the Pb slug.

The only way he can improve this is to use a core but at the frequencies mentioned it would have to be ferrite one.

Any core would be defeated by the wide airgap about the Pb slug. Guiding the external magnetic path is really not of great importance.

..., and simply place a larger coil around the oven as is actually done in commercial ovens whether induction or resistance ones. then the metal sits in the very middle of the coil and no fancy cores or multiple coils etc are needed also the coupling is best for what can be achieved without a core.

That is precluded by the production line running through between the coils. On the other hand, two coils in Helmholtz series give the required gap and the even field about the Pb slug, it is in effect one coil with the sample in the middle, passing between the ends.

 

[artis]

That is precluded by the production line running through between the coils.

I seem to have missed something, what production line ?

 

[Baluncore]

I seem to have missed something, what production line ?

Let me set up the application, then I will get to my questions.
Large ceramic disc about 1.5" thick and about 24" in diameter on a rotary. On the top side the disc has machined casting molds. Each mold will get a room temp slug of lead of proper volume. Inductive heating will melt the lead so it can take shape of the mold.

The idea for coils is to have two of them, "iron" core. One will be held directly over the lead slug, and one held directly below the lead slug on the bottom side of the plate.

 

[hutchphd]

What are the resistivities of solid and molten lead ?

The resistance of the metal typically ~doubles at transition to liquid. Lead is typical
I am perplexed about why this "center tapped" excitation is interesting. The theory is linear...why should there be surprises lurking?

.

 

[Baluncore]

I am perplexed about why this "center tapped" excitation is interesting.

Is it interesting? Maybe you can explain what perplexes you.
It is different, mostly because of the way it was originally described in terms of PWM of two separate coils at the opposite ends of the same iron core.

 

[hutchphd]

Is it interesting? Maybe you can explain what perplexes you.

A center-tapped transformer with a resistive secondary (tertiary?) is not a novel system. In fact it can be done analytically I reckon.
What are the exact question(s) under investigation? Are we looking to improve something? Its a pretty standard transformer loaded in a pretty standard way.

 

[artis]

Well now I see why the OP wants to have it this way, because he is using a rotary disc with molds and he wants to quickly heat up each mold and then turn the disc by an increment (servo motor drive possibly) to the next mold so he needs a system that can given the B field but at the same time allow for rotary movement , so the ordinary coil doesn't provide this.

Honestly depending on the size of the Pb in question I start to think that maybe simple resistive heating could accomplish this even faster and with less problems.
@MagneticMagic imagine something like an arc furnace only here you could make one electrode within the mold or change the mold material to something conductive and then apply a second electrode that can be lifted up or down and it touches the metallic sample in the mold.
Depending on the amperage used this will heat the sample faster because DC current heats all cross section of metal evenly at the same time.

I can't find the formula (I had one) that calculates the B field strength within an airgap if the B field strength is known for a core.
Such a wide airgap will reduce the field considerably , the ferrite core will help in this regard to make the total field strength higher

 

[MagneticMagic]

You need to look at the energy put in over time.

I did exactly that.
dB/dt * time = some J

Look at B in cos function
There are 4 distinct quadrants (that last gify shows it)
0-90deg (Bmax to zero)
90-180 (zero to Bmax)
180-270 (Bmax to zero)
270-360 (zero to Bmax)

The zero point is where dB/dt is max, as shown in that last gify.

I evaluated only the 0-90deg in my math above.
0-90 has a dB/dt, or J value from eddy (in my example, J per 25usec)
The overall J in eddy per full one cycle of frequency is the 0-90 J * 4

The push-pull in DC is technically AC.

To get same Bmax you need 100A DC, or 70.7A AC_RMS

My math should be correct.