Consider a device similar to that shown in Figure 11.10 but with a permanent magnet replacing the coil. Suppose that the total reluctance of the magnetic circuit is R(θ ). (Equation (11.23) indicates...

Consider a device similar to that shown in Figure 11.10 but with a permanent magnet replacing the coil. Suppose that the total reluctance of the magnetic circuit is R(θ ). (Equation (11.23) indicates how R(θ ) for the gap might vary for small angles, but R(θ ) would be valid for all θ and would include core reluctance.) The bond graph below shows a 2-port C based on R(θ ) as well as a 1-port C representing the demagnetization curve for the permanent magnet. The relevant parts of the constitutive laws are also sketched. (The M₁
− ϕ sketch can be related to the lower right-hand quadrant of the B –H hysteresis loop shown in Figure 11.5.)

Noting that M₂
= −M₁, the flux ϕ can be eliminated from the constitutive laws so that the “cogging torque” τ can be expressed in terms of the parameters M₀, ϕ₀
and the reluctance R(θ ). Find the law that relates τ to θ.