Inductor Current Analysis: Voltage Source with Linear Ramp

To determine the current in the inductor at t=0, we need to analyze the behavior of the inductor during the two different time intervals: 0 to T/2 and T/2 to T.

During the time interval from 0 to T/2:

• The voltage across the inductor is decreasing linearly from V0 to 0.
• Since the voltage is changing, the inductor will resist the change, resulting in an induced voltage that opposes the decreasing current.
• The rate of change of current in the inductor is given by: di/dt = -(V0 / L), where L is the inductance of the inductor.
• Integrating this equation, we get: i0 - i(0) = -(V0 / L) * (T/2), where i(0) is the current in the inductor at t=0.

During the time interval from T/2 to T:

• The voltage across the inductor is increasing linearly from 0 to V0.
• Similar to the previous interval, the inductor will resist the change in voltage, resulting in an induced voltage that opposes the increasing current.
• The rate of change of current in the inductor is given by: di/dt = (V0 / L).
• Integrating this equation, we get: i(T) - i0 = (V0 / L) * (T/2).

Combining the two equations, we can solve for i(0):

i0 - i(0) = -(V0 / L) * (T/2) i(T) - i0 = (V0 / L) * (T/2)

Rearranging the equations and solving for i(0):

i(0) = i0 + (V0 / L) * (T/2) - (V0 / L) * (T/2) i(0) = i0

Therefore, the current in the inductor at t=0 is equal to i0, which means it remains unchanged from its value at t=T.