Question 5.10.1: Develop a model for the design of the axial-field resonant a...
Develop a model for the design of the axial-field resonant actuator shown in Fig. 5.23.
The behavior of this device is governed by the equations for an axial-field actuator as derived in Example 5.9.1. We repeat them here for convenience,
\frac{di(t)}{dt} = \frac{1}{L} [V_{s}(t) – i(t)(R+ R_{coil}) – n B_{ext} (R_{2}^{2} – R_{1}^{2}) ω(t)]
\frac{dω(t)}{dt} = \frac{1}{j_{m}} [i(t) n B_{ext} (R_{2}^{2} – R_{1}^{2}) + T_{mech}(θ)]
\frac{dθ(t)}{dt} = ω(t). (5.204)
Mechanical torque: The restoring torque is provided by a torsional pivot mechanism
T_{mech}(θ) = – K_{p}θ (5.205)
Here, K_{p} is the spring constant of the pivot. This simple linear relation is typically only valid for limited rotations, for example, -12° ≤ θ ≤ 12°. The resonant frequency f for this mechanism is
f = \frac{1}{2π} \sqrt{\frac{K_{p}}{j_{m}}} (5.206)
Calculations: We demonstrate the use of Eq. (5.204) with some sample calculations. We numerically integrate these equations using the fourth-order Runge-Kutta method (Appendix C). The following parameters are used for the analysis:
B_{ext} = 0.55 T
R + R_{coil} = 6 Ω
n = 100 turns
R_{1} = 3.2 mm
R_{2} = 11.5 mm
L = 0.4 mH
j_{m} = 0.3 × 10^{-6} kg⋅m^{2}
f = 100 Hz
The field B_{ext} = 0.55 T can be achieved using sintered NdFeB magnets above and below the coil with soft-magnetic flux plates attached to the outer surface of each magnet. The flux plates enhance the field across the coil. We apply a periodic step function voltage V_{s}(t) with a magnitude of ±3 V and a period of 10 ms as shown in Fig. 5.24. It only requires two cycles to bring the system to resonance after which the circuit is assumed to be open (i(t) = 0 for t > 20 ms). The actuator response is computed with the following initial conditions: i(0) = 0 A, θ(0) = 0°, and ω(0) = 0 rad/s. The circuit current is plotted in Fig. 5.25. The rotation angle θ(t) and back voltage V_{emf}(t) are plotted in Fig. 5.26. The back voltage is the voltage induced in the coil as it moves. It is given by
V_{emf}(t) =n B_{ext} (R_{2}^{2} – R_{1}^{2}) ω(t).
Note that V_{emf}(t) passes through zero as the angular displacement peaks ( ω(t) = 0), as it should. Because there are no damping terms, the actuator continues in a resonant mode with an angular oscillation of θ = ± 12°. This type of performance is consistent with many such systems where the only appreciable dissipation results from windage and eddy current losses.
