Torque Equation Of Moving Iron Instruments

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

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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.
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/K      under equilibrium 

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