Question 5.15.1: Derive the equations of motion for the actuator shown in Fig...

This is a moving magnet actuator and its behavior is governed by rotational motion equations (5.87):

\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),

\frac{dθ(t)}{dt} = ω(t).            (5.297)

To solve Eq. (5.297) we need expressions for

\frac{∂Λ(i,θ)}{∂θ},                (5.298)

L = \frac{∂Λ(i,θ)}{∂i},                (5.299)

and

T(i,θ).                (5.300)

Assumptions: In order to simplify the analysis we make the following assumptions. First, we assume that the core is operating in a linear region of its magnetization curve with a permeability μ >> μ_{0} .

B = μH            (core).                 (5.301)

Thus, the magnetic circuit is linear, and the effects of coil and magnet can be separated,

B = B_{coil} + B_{mag} ,                  (5.302)
Λ(i,θ) = Λ_{coil}(i) + Λ_{mag}(θ) ,              (5.303)

and

T(i,θ) = T_{coil}(i,θ) + T_{core}(θ) ,              (5.304)

where

B_{coil} = field due to the coil
B_{mag} = field due to the magnet
T_{coil}(i,θ) = torque on the magnet due to the coil
T_{core}(θ) = torque on the magnet due to the core
Λ_{mag}(θ) = flux linkage due to the magnet
Λ_{coil}(i) = flux linkage due to the current.      (5.305)

Notice that ∧_{mag}(θ) and T_{core}(θ) are functions of θ only. If the magnetic circuit is nonlinear, then a nonlinear analysis needs to be performed for each time stepped integration of Eq. (5.297). The second assumption is that the magnet has a linear second-quadrant demagnetization curve of the form

B = μ_{0} (H + M_{s}),            (5.306)

where M_{s} is uniform and fixed.

Flux linkage: The flux linkage is given by Eq. (5.303). Therefore, Eqs. (5.298) and (5.299) reduce to

\frac{∂Λ(i,θ)}{∂θ} =\frac{dΛ_{mag}(θ)}{dθ} ,            (5.307)

and

L = \frac{dΛ_{coil}(i)}{di} ,            (5.308)

respectively.
Flux linkage due to the magnet: The flux linkage due to the magnet is determined using FEA,

FEA ⇒ Λ_{mag}(θ) .              (5.309)

Specifically, FEA calculations are performed for a discrete set of angles θ_{k} (k = 1, 2, . . .) that span the anticipated rotation of the magnet. The flux linkage ∧_{mag}(θ_{k}) is computed (with the coil turned off) for each value of θ_{k},

Λ_{mag}(θ_{k})= \underbrace{\int_{s} B_{mag}(θ_{k}) ⋅  ds.}_{integration  over  coil}

Typical FEA calculations are shown in Fig. 5.54. Specifically, flux plots are shown for (a) θ = 0°, (b) θ = 45°, and (c) θ = 90°. Notice that the maximum flux linkage occurs when θ = 0°. A normalized plot of Φ_{ave} (θ) (the average flux through the coil) is shown in (d). From these calculations we obtain the data( θ_{k}, Λ_{mag}(θ_{k})) and then fit it to a polynomial,

Λ_{mag}(θ) =\sum\limits_{k=0}^{N} a_{k}θ^{k}.              (5.310)

From this we obtain

\frac{dΛ_{mag}(θ)}{dθ} =\sum\limits_{k=1}^{N} a_{k} k θ^{(k-1)}              (FEA),           (5.311)

which is the desired analytical expression for Eq. (5.307).

Flux linkage due to the coil: The flux linkage due to the coil can be determined using FEA or magnetic circuit analysis. As the circuit is linear, the flux linkage is proportional to the current and the constant of proportionality C can be computed using FEA,

Λ_{coil}(i) = niC        (FEA).         (5.312)

On the other hand, if the gap g in the circuit is small relative to the dimensions of the cross-sectional area, then the flux Φ_{coil}(i) through each turn of the coil can be determined using magnetic circuit analysis as in Example 3.5.1. This gives

Φ_{coil} (i) = \frac{μ_{0} ni A_{gap}}{g},          (5.313)

where g and A_{gap} are the length and area of the gap. As the coil has n turns, the flux linkage is

Λ_{coil}(i) = n Φ_{coil} (i)

= \frac{μ_{0} n^{2}i A_{gap}}{g}.

In either case, from Eq. (5.308) we have

L = \left\{\begin{matrix} nC \qquad (FEA), \\ \frac{μ_{0} n^{2} A_{gap}}{g} \qquad (circuit  analysis). \end{matrix} \right.             (5.314)

Torque: As noted already, the torque on the magnet can be separated into two components

T(i,θ) = T_{coil}(i,θ)  + T_{core}(θ).          (5.315)

The term T_{coil}(i,θ) is the torque on the magnet due to the current through the coil. This is referred to as the drive torque. The other term, T_{core}(θ) is the torque on the magnet due to the presence of the magnetic circuit (core). It is independent of the current, and is referred to as the reluctance torque.

Drive torque: The drive torque can be computed using the formula for the torque on a bipolar cylinder in an external field (Example 3.4.4),

T_{coil} = πM_{s} h R_{mag}^{2}B_{ext}sin(θ)\hat{z},

where h and R_{mag} are the length and radius of the magnet. In addition,

B_{ext} = \frac{μ_{0} ni}{g}                    (5.316)

is the field in the gap due to the coil (Example 3.5.1). Therefore,

T_{coil} (i,θ)= πM_{s} h R_{mag}^{2} \frac{μ_{0} ni}{g}sin(θ).          (5.317)

This result is valid when magnetic circuit analysis applies. When this is not the case, an analytical expression for T_{coil}(i,θ) can be determined using FEA. Specifically, choose a reference current i_{ref} and perform a single FEA to compute the field components B_{r,coil} and B_{Φ,coil} at a series of points (R_{mag},Φ_{j}) that span the circumference of the magnet,

B_{coil}(R_{mag},Φ_{j}) =  B_{r,coil}(R_{mag},Φ_{j}) \hat{r}+B_{Φ,coil}(R_{mag},Φ_{j}) \hat{Φ} .        (5.318)

This data is then substituted into Eq. (3.134), which in turn can be evaluated numerically for a sequence of angles θ_{k} that span the angular rotation of the magnet. The resulting data (θ_{k},T_{coil}(θ_{k})) is then fit to a finite polynomial

Eq. (3.134): T = \int_{v} ρ_{m}(r×B_{ext})dv × \oint_{s} σ_{m}(r×B_{ext})ds.

T_{coil}(i, θ) = \frac{i}{i_{ref}}\sum\limits_{k=0}^{N} d_{k} θ^{k},            (5.319)

where the multiplier i/i_{ref} reflects the linearity of the circuit.

Reluctance torque: The reluctance torque is computed using FEA

FEA ⇒ T_{core}(θ).           (5.320)

This torque is computed for a set of angular values θ_{k} that span the angular rotation of the magnet. The computed data (θ_{k},T_{core}(θ_{k})) is then fit to a finite polynomial,

T_{core}(θ) = \sum\limits_{k=0}^{N} C_{k} θ^{k}.           (5.321)

A sample plot of T_{core}(θ) (obtained from FEA) is shown in Fig. 5.55; it is negative because it acts to rotate the magnet counterclockwise (clockwise rotation is taken to be positive). Notice that T_{core}(θ) is maximum at θ = 45°.

Calculations: We demonstrate the hybrid approach with some sample calculations. The following parameters are used in the analysis:

R = 3.0 Ω
n = 500 turns
M_{s} = 4.3 × 10^{5}   A/m
R_{mag} = 4.0 mm
g = 10.0 mm
h = 25.0 mm
j_{m} = 6 × 10^{-7} kg⋅m^{2} .

The width of the core is 10 mm. Therefore, A_{gap} = 250  mm^{2}. First, determine Eqs. (5.311), (5.314), (5.317), and (5.321) and then substitute these into the equations of motion (5.297). We apply a step function voltage pulse of the form

V_{s}(t) = \left\{\begin{matrix} V_{0} \qquad (0  ≤  t  ≤  T_{0}) \\ 0 \qquad (T_{0} ≤  t  < ∞),  \end{matrix} \right.

where V_{0} = 12 V and T_{0} = 2.0 ms. The initial conditions are i_{0} = 0 A, θ_{0} = 0.175 radians (10°), and ω_{0} = 0 rad/s. We solve Eq. (5.297) numerically using Euler’s method. A normalized plot of i(t) is shown in Fig. 5.56. Notice that the current begins decaying after T_{0} when the applied voltage becomes zero. Normalized torque profiles are shown in Fig. 5.57. Finally, a plot of θ(t) is shown in Fig. 5.58. As expected, θ(t) reaches a maximum and then returns to its initial value due to the reluctance torque.