In previous tutorial on Moving Iron Instrument Operation construction & working principle was discussed. In this post

**moving iron instrument torque equation**will be derived.###
__Torque Equation of Moving Iron Instruments__

READ HEREMoving Iron Instrument Working Operation CLICK HERE

Consider a small increment in current supplied to the coil of the instrument. due to this current let dθ be the deflection under the deflecting torque Td. Due to such deflection, some mechanical work will be done.

Mechanical Work = Td .dθ

There will be a change in the energy stored i the magnetic field due to the change in inductance. This is because the vane tries to occupy the position of minimum reluctance. The inductance is inversely proportional to the reluctance of the magnetic circuit of coil.

Let I = initial current

L = instrument inductance

θ = deflection

dI = increase in current

dθ = change in deflection

dL = change in inductance

In order to effect an increment dL in the current, there must be an increase in the applied voltage given by,

e = d(L*I)/dt

= I * dL/dt + T * dI/dt as both I and L are changing.

The electrical energy supplied is given by

eIdt = { I * dL/dt + T * dI/dt }Idt

=I²dL + ILdI

The stored energy increases from 1/2*(LI²) to 1/2*[(L+dL)(I+dI)²]

Hence the change in stored energy is given by,

1/2*[(L+dL)(I+dI)²] - 1/2*(LI²)

Neglecting higher order terms,this becomes ILdI + 1/2 * I² dL

The energy supplied in nothing but increase in stored energy plus the energy required for mechanical work done.

I²dL + ILdI = ILdI + 1/2*(I²)dL +Td.dθ

Td.dθ = 1/2( I².dL )

Td = 1/2 I²dL/dθ

While the controlling torque is given by,

Tc = Kθ

where K = spring constant

Kθ = 1/2 I²dL/dθ

θ = 1/2 I²dL/dθ * 1/Kunder equilibrium

Thus the deflection is proportional to the square of the current through the coil. And the instrument gives square law response.