Question 6.5.1: Derive a model of a D’Arsonval meter in terms of the coil an......

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

If a torsional viscous damping torque c{\dot{\theta}}, 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}\theta}{d t^{2}}+c\frac{d\theta}{d t}+k_{T}\theta=T=(B L r)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} = BLv. The linear velocity is related to the coil’s angular velocity \dot{{\theta}} by v = r\dot{{\theta}}. Thus,

The coil circuit is represented in part (b) of Figure 6.5.2, where R represents the resistance of the wire in the coil. Kirchhoff’s voltage law applied to the coil circuit gives

or

L{\frac{d i}{d t}}+R i+B L r{\frac{d\theta}{d t}}=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 L r{\dot{\theta}}. Thus equations (1) and (2) become

I\frac{d^{2}\theta}{d t^{2}}+c\frac{d\theta}{d t}+k_{T}\theta=n(B L r)i               (3)

L{\frac{d i}{d t}}+R i+n B L r{\frac{d\theta}{d t}}=v_{i}                (4)

Note that if the applied voltage \mathbf{}v_{i} is constant, the system will reach a steady-state in which the pointer comes to rest. At steady-state, \dot{{\theta}} = di/dt = 0, and equation (4) gives

and equation (3) gives

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 \mathbf{}v_{i}.