A Model of the D'Arsonval Meter Derive a model of a D’Arsonval meter in terms of the coil angular displacement θ and the coil current i. The input is the applied voltage vi . Discuss the case where there are n coils around the core.

Question

A Model of the D'Arsonval Meter

Derive a model of a D’Arsonval meter in terms of the coil angular displacement θ and the coil current i. The input is the applied voltage [latex]v_{i}[/latex] . Discuss the case where there are n coils around the core.

Accepted Answer

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
[latex]T = f r = \left(2B \frac{L}{2} i \right) r = (BLr) i[/latex]
If a torsional viscous damping torque [latex]c\dot{θ}[/latex], 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
[latex]I \frac{d^{2}θ}{dt^{2}} + c \frac{d θ}{d t} + k_{T} θ = T = (BLr)i [/latex]      (1)
where I is the inertia of the core/coil unit.
The rotation of the coil induces a voltage [latex]v_{b}[/latex] in the coil that is proportional to the coil’s linear velocity v such that [latex]v_{b} = BL v[/latex]. The linear velocity is related to the coil’s angular velocity [latex]\dot{θ}[/latex] by [latex]v = r \dot{θ}[/latex]. Thus,
[latex]v_{b} = BL v = BL r \frac{dθ}{dt}[/latex]
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
[latex]v_{i} − L \frac{di}{dt} − Ri − v_{b} = 0 [/latex]
or
[latex]L \frac{di}{dt} + Ri + BLr \frac{dθ}{dt} = v_{i} [/latex]    (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 [latex]v_{b} = n B Lr \dot{θ}[/latex]. Thus equations (1) and (2) become
[latex]I \frac{d^{2}θ}{dt^{2}} + c \frac{dθ}{dt} + k_{T} θ = n(BLr)i [/latex]    (3)
[latex]L \frac{d i}{d t} + Ri + nBLr \frac{d θ}{d t} = v_{i} [/latex] (4)

Note that if the applied voltage [latex]v_{i}[/latex] is constant, the system will reach a steady-state in which the pointer comes to rest. At steady-state, [latex]\dot{θ} = di/dt = 0[/latex], and equation (4) gives
[latex]i = \frac{v_{i}}{R} [/latex]
and equation (3) gives
[latex]θ = \frac{nBLri}{k_{T}} = \frac{nBLrvi}{Rk_{T}} [/latex]
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 [latex]v_{i}[/latex] .

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

Related Answered Questions

Differentiation with Op Amps Design an op-amp circuit that differentiates the input voltage. ...

Verified Answer:

Model of a Field-Controlled dc Motor Develop a model of the field-controlled motor shown in Figure 6.4.5. ...

Verified Answer:

An Electromagnetic Speaker Develop a model of the electromagnetic speaker shown in Figure 6.6.2, and obtain the transfer function relating the diaphragm displacement x to the applied voltage v. ...

Verified Answer:

Calculating Motor-Amplifier Requirements The trapezoidal profile requirements for a specific application are given in the following table, along with the load and motor data. Determine the motor and amplifier requirements. ...

Verified Answer:

Verified Answer:

Verified Answer:

Integration with Op Amps Determine the transfer function Vo(s)/Vi(s) of the circuit shown in Figure 6.3.8. ...

Verified Answer:

Verified Answer:

Circuit Analysis Using Impedance For the circuit shown in Figure 6.3.3a, determine the transfer function between the input voltage vs and the output voltage vo. ...

Verified Answer:

Coupled RC Loops Determine the transfer function Vo(s)/Vs (s) of the circuit shown in Figure 6.3.1. ...

Verified Answer: