Question

The working of a PMMC (Permanent magnet moving coil) meter is described by a second order differential equation

\(J\frac{{{d^2}\theta }}{{d{t^2}}} + D\frac{{d\theta }}{{dt}} + S'\theta = T\)

Where

J = Moment of inertia of the system,

D = Damping coefficient,

S^’ = Spring constant,

θ = Angular deflection and

T = Activating torque,

Assuming D = 0, an undamped natural angular frequency is

1. \(\sqrt {\frac{S}{J}} \)
2. \(\sqrt {\frac{J}{S}} \)
3. \(\frac{1}{{\sqrt {JS} }}\)
4. \(\frac{1}{{2\mu \sqrt {JS} }}\)

Answer 1

Correct Answer - Option 1 : \(\sqrt {\frac{S}{J}} \)

PMMC works on the electromagnetic effect. A permanent magnet is used to produce flux and there is a coil (which attached to an instrument) carry the current to be measured moves in the flux. A pointer is connected to that coil is deflected in the proportion to the current in the coil.

Second-order differential equation of PMMC is,

\(I\frac{{{d^2}\theta }}{{d{t^2}}} + D\frac{{d\theta }}{{dt}} + S\theta = T\)      …1)

Where

J = Moment of inertia of the system,

D = Damping coefficient,

S^’ = Spring constant,

θ = Angular deflection and

T = Activating torque.

Now, assuming D = 0, an undamped frequency-

ω_n = undamped frequency / natural frequency

Take the Laplace transform of equation 1)

Is² θ(s) + Ds θ(s) + s θ(s) = T

At D = 0,

\(\theta \left( s \right) = \frac{T}{{I{s^2} + s}}\)

\(= \frac{{\frac{T}{J}}}{{{s^2} + \frac{S}{J}}}\)

Compare the denominator with the standard equation,

S² + 2ξ ω_n S + ω²_n = 0

\( \Rightarrow {\omega _n} = \sqrt {\frac{S}{J}} \;,\;\xi = 0\)