I'm currently wanting to model an RLC circuit that is driven by a toroid transformer with a driving current, I(t). See image below.(adsbygoogle = window.adsbygoogle || []).push({});

In physics class, I've done similar problems with 'ideal' toroids and a simple R circuit. What I'm struggling with is how to model a 'real' toroid transformer.

Normally I would do something like this:

[itex]\mu_0 I_{enclosed} = \oint B \cdot dl[/itex]

[itex]\Rightarrow B(t) = \frac{mu_0 I(t) N}{2 \pi r}[/itex]

where N is the number of turns.

The flux is equal to:

[itex]\Phi = \int B \cdot da = \frac{mu_0 I(t) N}{2 \pi}\int \int dr dx= \frac{mu_0 I(t) N L}{2 \pi} ln\left(\frac{r_{out}}{r_{in}}\right )[/itex]

where L is the width of the toroid, and r is the inner and outer radius.

The EMF is:

[itex]\varepsilon = -\frac{d\Phi}{dt} = - \frac{mu_0 N L}{2 \pi} ln\left(\frac{r_{out}}{r_{in}}\right ) \frac{dI(t)}{dt} [/itex]

Using Kirchoff's voltage law:

[itex]\varepsilon = V_R + V_L + V_C = iR +\frac{di(t)}{dt}L + \frac{1}{C}\int_{-\inf}^{t}i(t)dt[/itex]

Differentiating and diving by L gives the ODE:

[itex]\frac{d^2 i(t)}{d x^2}+\frac{R}{L}\frac{di(t)}{dt}+\frac{i(t)}{LC} = \frac{d \varepsilon (t)}{dt} \frac{1}{L}[/itex]

Which can then be solved for i(t), the induced current in the RLC circuit. Etc. Etc.

My Question Is:

Is this method ok for a 'real' toroid? I feel this is wrong because I need to include the permeability of the toroid, but I don't know how to do this. It was my understanding that the presence of a 'real' toroid strengthened the magnetic field (magnitude of 1000 or so) and subsequently increased the induced current. I will appreciate any feedback. Thanks!

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# RLC circuit driven by a 'Real' toroid

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