Differential Equation of Series RL Circuit

One of the properties of any LTI system, such as the RL circuit you describe, is that the application of a sinusoidal source will give a sinusoidal result that is altered in magnitude and phase. This portion is apparent in your steady state response where the output is a sinusoid of magnitude and phase determined by the system paramaters (W,L, and R).

The transient response, in your example, is an applied voltage, that is super imposed (added to) the steady state response, but its effects decay with time, exponentially according to the time constant (-R/L*t).

initial charge up or inrush into the storage elements, or the transition from the devices from the at rest condition. A simpler example, is to think of the step response of the system. If you apply a constant voltage, at t0, the current will exponentially ramp up to the steady state of V/R. The relative amounts of inductance and resistance will determine the rate (time constant) at which the current ramps (the storage elements charge).

Look at it this way: the first half-cycle of the applied sine wave is all positive, so it has a DC component. After that, all the successive full cycles average zero. So there's an initial asymmetric, DC kick shot in at startup. After many cycles, the circuit sort of forgets about that initial insult and settles down.

If r=0, meaning you apply a sine wave to a pure inductor starting at t=0, the dc component never dies away.

Something like that.

John