Laplace transforms:DC Motor circuit

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Problem

Find the steady state current i(t) through a DC motor represented by a series R-L-Motor circuit. The resistance (R) is from the armature winding. The inductance (L) is the equivalent inductance of the wire coil (which turns by current flowing through the coil in a permanent magnetic field). The motor has input current i(t) and output angular velocity ω(t).

Solution

The torque is proportional to the armature current.

Similarly, relating mechanical (T(t)ω(t)) and electrical (vm(t)i(t)) power, the conservation of energy requires the same proportionality between the voltage across the motor (vm(t)) and the angular velocity (ω(t)).

We want to find the Laplace transfer function of the motor.

Summing the voltages around the series circuit gives us our differential equation.