Question 6.4.1: A Model of the D'Arsonval Meter Derive a model of a D’Arsonv...

Let the length of one side of the coil be L/2 and its radius be r. Then the torque T acting on both sides of the coil due to the magnetic field B is
T = f r = \left(2B \frac{L}{2} i \right) r = (BLr) i
If a torsional viscous damping torque c\dot{θ}, for example, due to air resistance or damping in the bearings, also acts on the core shaft as it rotates, the equation of motion of the core/coil unit is
I \frac{d^{2}θ}{dt^{2}} + c \frac{d θ}{d t} + k_{T} θ = T = (BLr)i       (1)
where I is the inertia of the core/coil unit.
The rotation of the coil induces a voltage v_{b} in the coil that is proportional to the coil’s linear velocity v such that v_{b} = BL v. The linear velocity is related to the coil’s angular velocity \dot{θ} by v = r \dot{θ}. Thus,
v_{b} = BL v = BL r \frac{dθ}{dt}
The coil circuit is represented in part (b) of Figure 6.4.2, where R represents the resistance of the wire in the coil. Kirchhoff’s voltage law applied to the coil circuit gives
v_{i}  −  L \frac{di}{dt}  −  Ri  −  v_{b} = 0
or
L \frac{di}{dt} + Ri + BLr \frac{dθ}{dt} = v_{i}      (2)
The model consists of equations (1) and (2). Note that the system model is third order.
If there are n coils, the resulting torque expression is T = n(BLr)i and the induced voltage expression is v_{b} = n B Lr \dot{θ}. Thus equations (1) and (2) become
I \frac{d^{2}θ}{dt^{2}} + c \frac{dθ}{dt} + k_{T} θ = n(BLr)i    (3)
L \frac{d i}{d t} + Ri + nBLr \frac{d θ}{d t} = v_{i}       (4)

Note that if the applied voltage v_{i} is constant, the system will reach a steady-state in which the pointer comes to rest. At steady-state, \dot{θ} = di/dt = 0, and equation (4) gives
i = \frac{v_{i}}{R}
and equation (3) gives
θ = \frac{nBLri}{k_{T}} = \frac{nBLrvi}{Rk_{T}}
This equation can be used to calibrate the device by relating the pointer displacement θ to either the measured current i or the measured voltage v_{i} .