Question 15.4: Using magnetic circuit concepts, analyze the toroidal coil s...
Using magnetic circuit concepts, analyze the toroidal coil shown in Figure 15.8 to find an expression for the flux.
As indicated in Figure 15.11, the magnetic circuit of the toroidal coil is analogous to a simple electrical circuit with a resistance connected across a voltage source.
The mean length of the magnetic path is
l = 2π R
The cross section of the core is circular with radius r. Thus, the area of the cross section is
A = πr^2
Substituting into Equation 15.21, we find the reluctance to be
\mathcal{R} = \frac{l}{ μA} (15.21)
\mathcal{R} = \frac{l}{ μA} =\frac{2πR}{μπr^2} = \frac{2R}{μr^2}
The magnetomotive force is
\mathcal{F} = NI
Solving Equation 15.22 for the flux, we have
\mathcal{F} = RΦ (15.22)
\phi =\frac{\mathcal{F}}{\mathcal{R}}
Substituting the expressions for \mathcal{F} and \mathcal{R} found earlier, we get
\phi = \frac{μNr^2I}{2R}
This is the same expression for the flux that we obtained in Examples 15.2 and 15.3 by applying Ampère’s law.
