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RLC circuit driven by a 'Real' toroid 
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#1
Dec2012, 03:18 PM

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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.
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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