Question

Measurement and Instrumentation - Deflecting torque of M.I. Instruments in Terms of Self-Inductance

Answer 1

In the context of measurement and instrumentation, specifically when discussing moving coil instruments (M.I. instruments), the concept of deflecting torque and its relation to self-inductance is important. Let's break down these terms:

Deflecting Torque: In a moving coil instrument, the deflecting torque is the torque that is applied to the moving coil of wire in response to a current passing through it. This torque causes the coil to rotate, which is then converted into a pointer movement to indicate a value on the instrument's scale. Deflecting torque is usually proportional to the current passing through the coil.

Self-Inductance: Self-inductance (often denoted as L) is a property of a coil or conductor that describes its ability to induce an electromotive force (EMF) in itself when the current passing through it changes. In simpler terms, it's a measure of how much a coil resists changes in the current flowing through it. Self-inductance is measured in henries (H).

Now, let's establish the relationship between deflecting torque (T) and self-inductance (L) in a moving coil instrument:

The deflecting torque (T) in a moving coil instrument is directly proportional to the product of the current (I) passing through the coil and the magnetic field strength (B) in which the coil is placed:

T ∝ I * B

However, the magnetic field strength (B) is also related to the self-inductance (L) of the coil and the rate of change of current (di/dt):

B = μ₀ * (N/L) * I

Where:

μ₀ is the permeability of free space.
N is the number of turns in the coil.
L is the self-inductance of the coil.

Substituting the expression for B back into the deflecting torque equation:

T ∝ I * (μ₀ * N / L) * I

Simplifying:

T ∝ (μ₀ * N * I²) / L

In a typical moving coil instrument, the controlling torque (typically provided by springs) is balanced by the deflecting torque:

T_deflecting = T_controlling

Therefore:

(μ₀ * N * I²) / L = K * θ

Where:

K is a constant of proportionality.
θ is the angular deflection of the pointer.

This equation shows the relationship between deflecting torque, self-inductance, current, and the angular deflection of the pointer in a moving coil instrument.

It's important to note that this explanation assumes a simplified scenario and doesn't consider factors like the shape of the coil, the magnetic circuit, and other practical considerations that might affect the instrument's behavior.