Question 5.11.1: Set up the equations of motion for the rotary bias actuator ...

This is a moving magnet actuator and is governed by the equations for rotational motion (5.87), which we repeat here for convenience:

\frac{di(t)}{dt} = \frac{1}{L}[V_{s}(t) – i(t) (R + R_{coil}) – \frac{∂Λ(i,θ)}{∂θ} ω(t)]

\frac{dω(t)}{dt} = \frac{1}{j_{m}} T(i,θ, ω)

\frac{dθ(t)}{dt} = ω(t).              (5.207)

These equations contain nonlinear terms and can be solved numerically once expressions for L, ∂Λ(i,θ)/∂θ, and T(i,θ, ω) have been determined. We have added an dependence in T(i,θ, ω) to account for the effects of eddy currents in the conductive shell.

Inductance: The inductance of the coil can be estimated using the result of Eq. (3.87) for two parallel wires (Example 3.2.10). Taking into account the fact that there are n turns, the inductance is approximated by

Eq. (3.87): L ≈ \frac{μ_{0}}{π} ln(\frac{s}{a}) (H/m).

L ≈ \frac{μ_{0}n^{2}l_{c}}{π} ln(\frac{s}{r_{w}}),             (5.208)

where s is the mean separation of the coils, l_{c} is the length of each side of the coil (into the page), and r_{w} is the radius of the wire.

Induced voltage: A voltage V_{ind} is induced in the coil as the magnet rotates. It is given by

V_{ind} (t) = \frac{∂Λ(i,θ)}{∂θ} ω(t).         (5.209)

The flux linkage Λ(i,θ) can be written as the sum of two terms,

Λ(i,θ) = Λ_{coil}(i) + Λ_{mag}(θ).

The first term Λ_{coil}(i) is the flux linkage due to the current through the coil itself. The second term Λ_{mag}(θ) is the flux linkage due to the magnet. Notice that only the second term has an angular dependence and, therefore,

\frac{∂Λ(i,θ)}{∂θ} = \frac{∂Λ_{mag}(θ)}{∂θ}.

To determine this term we need an expression for the field of the magnet. For this actuator, the length of the magnet is much greater than its diameter, and a two-dimensional analysis suffices (Fig. 5.27). A two-dimensional field solution for the magnet was derived in Example 3.6.3. The vector potential was found to be

A_{mag}(r,Φ) = \frac{μ_{0} M_{s}}{2} \frac{a^{2}}{r} sin (Φ) \hat{z}          (a < r).       (5.210)

Recall that Eq. (5.210) applies when the magnet is positioned with its north pole aligned with Φ = 0 as defined by Eq. (3.264). If the magnet is rotated by an angle θ then Eq. (5.210) is modified by substituting Φ → Φ — θ, that is,

Eq. (3.264): M = M_{s} [cos(Φ)\hat{r} – sin(Φ)\hat{Φ}] .

A_{mag}(r,Φ,θ) = \frac{μ_{0} M_{s}}{2} \frac{a^{2}}{r} sin (Φ – θ) \hat{z}.              (5.211)

The flux linkage due to the magnet is

Λ_{mag}(θ) = n \int_{S}B_{mag}(r,Φ,θ) ⋅ ds

= n \oint_{coil}A_{mag}(r,Φ,θ) ⋅ dl.

Therefore,

\frac{∂Λ_{mag}(θ)}{∂θ} = n \frac{∂}{∂θ}\oint_{coil}A_{mag}(r,Φ,θ) ⋅ dl

= n \frac{∂}{∂θ} \{ \int_{0}^{-l_{c}}  A_{mag}(R_{c},0,θ)dz + \int_{-l_{c}}^{0}A_{mag}(R_{c},π,θ)dz \}

= n μ_{0}M_{s} l_{c} \frac{a^{2}}{R_{c}} cos(θ),       (5.212)

where θ is the angular position of the magnet relative to the x-axis, and R_{c} is the radial position of the sides of the coil. The coil has n side elements (into the page) that are positioned at r = R_{c}, and Φ = 0 and π, respectively. We substitute Eq. (5.212) into Eq. (5.209) and obtain

V_{ind}(t) = K_{e}(θ(t))ω(t),            (5.213)

where K_{e} is the electrical constant

K_{e}(θ(t)) = n μ_{0} M_{s}l_{c}\frac{a^{2}}{R_{c}} cos(θ(t))                (5.214)

Torque: The torque on the magnet can be written as a superposition of two terms,

T(i, θ, ω) = = T_{coil}(i,θ)+ T_{eddy}(ω).          (5.215)

The terms T_{coil}(i,θ) and T_{eddy}(ω) are the torques due to the coil and eddy currents induced in the surrounding conductive shell, respectively.

Drive torque: We consider the torque imparted to the coil by the magnet. This is given by Eq. (5.79),

T = i \int_{coil} r × (dl × B_{ext}),           (5.216)

where r is the distance from the current element dl to the axis of rotation and integration is in the direction of current flow. Here, r = R_{c} \hat{r}, and dl = dz\hat{z}. Therefore,

r × (dl × B_{ext}) = R_{c} B_{r} dz \hat{z},

where B_{r} is given by Eq. (3.274),

Eq. (3.274): B_{r}(r,Φ) = μ_{0}\frac{M_{s}}{2}\frac{a^{2}}{r^{2}} cos(Φ)    (a < r),

B_{r}(r,Φ,θ) = \frac{μ_{0} M_{s}}{2} \frac{a^{2}}{r^{2}} cos(Φ – θ).              (5.217)

In Eq. (5.217), (r, Φ) is the observation point and θ is the rotational position of the magnet with respect to the x-axis. The coil has n pairs of wires positioned as described in the foregoing. Therefore, the axial component of the torque is

T_{z}(θ) = ni(t) \frac{μ_{0} M_{s}}{2} \frac{a^{2}}{R_{c}} \{ cos(0 – θ) \int_{0}^{-l_{c}} dz+cos(π – θ) \int_{-l_{c}}^{0} dz \}

= – ni(t) μ_{0} M_{s} l_{c}\frac{a^{2}}{R_{c}} cos(θ(t)).                (5.218)

This is the torque imparted to the coil by the magnet. The torque on the magnet is of equal magnitude but in the opposite direction. Therefore, the torque imparted to the magnet by the coil is

T_{coil}(i(t),θ(t)) = ni(t) μ_{0} M_{s}l_{c}\frac{a^{2}}{R_{c}}cos(θ(t)).                (5.219)

By comparing Eq. (5.214) with Eq. (5.219) we find that

T_{coil}(i(t),θ(t)) = K_{t}(θ(t))i(t),            (5.220)

where K_{t}(θ(t)) is the torque constant and K_{t} (θ(t)) = K_{e} (θ(t)).

Eddy currents: As the magnet rotates it induces eddy currents in the conductive shell that give rise to a drag torque. To analyze this, we choose a reference frame that is at rest with respect to the shell, and with its origin on the axis of the magnet. We assume that the shell is thinner than the skin depth. associated with the desired field reversal time. For example, δ ≈ 25 mm for a field reversal of 10 ms. Based on these assumptions, the magnetic field induced in the shell is negligible compared to the magnet’s field, which is obtained from the vector potential equation (5.211). Therefore, the electric field induced at a position (r, Φ) in the shell is

E(r, Φ, θ(t)) = – \frac{∂A_{mag}(r, Φ, θ(t))}{∂t}

= – \frac{∂A_{mag}(r, Φ, θ(t))}{∂θ} ω(t).            (5.221)

This gives rise to an induced current density in the shell,

J(r, Φ, θ(t)) = σE(r, Φ, θ(t)),           (5.222)

where σ is the conductivity of the shell. Substituting Eqs. (5.211) and (5.221) into Eq. (5.222) gives

J_{z}(r, Φ, θ(t)) = σ\frac{μ_{0} M_{s}}{2} \frac{a^{2}}{r} cos(Φ – θ(t))ω(t).            (5.223)

Thus, the current through an infinitesimal cross section r dr dΦ of the shell is

di(t) = J_{z}(r, Φ, θ(t)) r dr dΦ.          (5.224)

The drag torque is obtained by combining Eqs. (5.216), (5.217) and (5.224) and then integrating over the cross-sectional area of the shell (we assume that the shell is as long as the magnet). This gives

T_{eddy}(ω(t)) = σha^{4}(\frac{μ_{0} M_{s}}{2})^{2}ω(t) \int_{0}^{2π} \int_{R_{1}}^{R_{2}} \frac{1}{r} cos^{2}(Φ – θ(t)) dr dΦ

= σπha^{4}(\frac{μ_{0} M_{s}}{2})^{2} ln (\frac{R_{2}}{R_{1}})ω(t).            (5.225)

We write this as

T_{eddy}(ω(t)) = – K_{eddy}ω(t),                  (5.226)

where

K_{eddy} = Sgn(ω(t)) σπha^{4}(\frac{μ_{0} M_{s}}{2})^{2} ln (\frac{R_{2}}{R_{1}}),                    (5.227)

and

Sgn(ω(t)) = \left\{\begin{matrix} 1 \qquad ω(t)  ≥  0\\ -1 \qquad ω(t)  < 0 \end{matrix}. \right.

The term Sgn(ω(t)) takes into account the fact that the torque due to the eddy current acts to oppose the rotation. Notice that we have ignored the current induced in the conductive shell by the drive coil, and the voltage induced in the coil by the eddy currents within the shell. These effects are second order [17].

Equations of motion: Finally, substitute Eqs. (5.213), (5.220), and (5.226) into Eq. (5.207) and obtain the equation of motion for the actuator,

\frac{di(t)}{dt} = \frac{1}{L}[V_{s}(t) – i(t) (R + R_{coil}) – K_{e}(θ(t)) ω(t)]

\frac{dω(t)}{dt} = \frac{1}{j_{m}} [K_{t}(θ(t))i(t) – K_{eddy}ω(t)]              (5.228)

\frac{dθ(t)}{dt} = ω(t),

where L is given by Eq. (5.208).

Calculations: To demonstrate the theory we consider an M-O system that requires a bias field in the range 0.02—0.04 T along a radial length of 100 mm, and a field-reversal time of 12 ms. A magnet for this system was designed in Example 4.10.1. Specifically, we use a sintered NdFeB magnet (M_{s} = 7.2 × 10^{5} A/m) with a radius R = 1 mm, and a length h = 108 mm. This magnet needs to be rotated θ = ±85° in 12 ms.

Preliminary analysis: Before we design the actuator we need to estimate nominal values for several key parameters. First, we determine the moment of inertia of the magnet. This is given by j_{m}=1/2ρπhR^{4}, where ρ = 7.52 × 10^{3}  kg/m^{3} is the density of sintered NdFeB, and R and h are as given in the preceding text. Using these values we obtain j_{m} = 1.275×10^{-9} kg/m^{3}. Second, we estimate the number of amp turns required to reverse the field in 12 ms. This is a three-step process: First, we estimate the angular acceleration α = 2θ/t^{2} required to rotate the magnet through an angle of θ = π in less than half of the reversal time, t = 5 × 10^{-3} s. We use this reduced time because the acceleration must be great enough to compensate for the drag due to the conductive shell. We calculate α = 2.5 × 10^{5} rad/s^{2}. Next, we estimate the drive torque T_{coil} = j_{m} α, which yields T_{coil} = 3.18 ×10^{-4} Nm. Finally, we estimate the amp turns using Eq. (5.219). We use the following nominal values: θ(t) = π/3, l_{c} = h, and R_{c} = 3 × 10^{-3} m. This value of R_{c} (radial position of the coil) places the coil far enough away from the magnet to accommodate the conductive shell. Using these values we obtain

ni = \frac{T_{coil}R_{c}}{μ_{0} M_{s} h a^{2} cos(π/3)}

= 20.            (5.229)

If we choose a nominal value of current i = 200 ma, then n = 100 turns. We are finally ready to perform the actuator design.

Actuator design: Our goal is to determine the radial thickness of the conductive shell that renders the desired field reversal time. We integrate Eq. (5.228) using the fourth-order Runge-Kutta method with the inner radius of the shell set to R_{in} = 2.0 mm (1.0 mm from the surface of the magnet) and the outer radius set to R_{out} = 2.5, 3.0 and 3.5 mm, respectively. Further, we assume that the shell is made from aluminum, which has a conductivity σ = 3.82 × 10^{7} Ωm^{-1}. For these calculations, the radial position of the coil is 0.5 mm beyond the outer surface of the shell, that is, R_{c} = 3.0, 3.5 and 4.0 mm, respectively. The excitation voltage is held constant at V_{s} = 12 V and 36 gauge wire is used with n = 100 turns. The response of the actuator is computed subject to the following initial conditions: i(0) = 0 A, θ(0) =  -85°, ω(0) = 0 rad/s. The resistance and inductance of the coil compute to 32.6 Ω and 2.0 mH, respectively. A parametric plot of θ(t) vs t with R_{out} = 2.5, 3.0 and 3.5 mm is shown in Fig. 5.30. Notice that as R_{out} increases, the eddy current damping increases and the magnitude of the oscillations about the equilibrium position of 90° decrease. When R_{out} = 3.5 mm, the magnet stabilizes at 90° within the desired reversal time of 12 ms. Thus a shell with R_{in} = 2.0 mm and R_{out} = 3.5 mm will suffice for this application. Additional plots of i(t) and V_{emf} (t) = K(θ(t)) ω(t) for R_{out} = 3.0 mm are shown in Fig. 5.31. Last, we note that in practice, the equilibrium position of the magnet is offset from 90° (e.g., 85°) so that it can be rotated by the coil. This offset is usually accomplished using magnetic detent in which a soft-magnetic element is fixed in proximity to the magnet. The presence of this element gives rise to second-order effects, and will not alter the analysis appreciably.