Electrodynamometer Deflection Equation for AC

The deflection ($θ$) of the pointer in an electrodynamometer instrument operated with AC depends on the average torque produced by the interaction between the fixed and moving coils.

AC Torque Principle

The instantaneous torque ($T$) is proportional to the product of the instantaneous currents in the fixed ($i_1$) and moving ($i_2$) coils and the rate of change of mutual inductance ($M$) with respect to the deflection angle ($θ$).

When operating with AC, let the currents be sinusoidal, such as $i_1 = I_1 \sin(ωt)$ and $i_2 = I_2 \sin(ωt + Φ)$, where $I_1$ and $I_2$ are the RMS values, and $Φ$ is the phase difference between the currents.

The average torque ($T_{avg}$) over one cycle is proportional to the product of the RMS values of the currents and the cosine of their phase difference:

$T_{avg} \propto I_1 I_2 \cos Φ \frac{dM}{dθ}$

Pointer Deflection Derivation

The pointer deflection ($θ$) is directly proportional to the average torque ($T_{avg}$). This relationship is governed by the instrument's spring control mechanism.

Let $k$ be a constant that includes the spring constant and other factors relating torque to deflection.

Therefore, the deflection is expressed as:

$θ \propto T_{avg}$

$θ = \frac{1}{k} \cdot (I_1 I_2 \cos Φ \frac{dM}{dθ})$

This simplifies to the standard equation:

$θ = \frac{I_1 I_2}{k} \cos Φ \frac{dM}{dθ}$