How can we mathematically relate the settling time for an LR circuit with an AC sine source? By settling time I mean the time taken for the circuit reaching to steady state from transient state. This is related to the previous question: Miscalculation of current for a pure inductive circuit in LTspice
Answer
Using superposition you can treat the transient response (due to initial conditions in your simulation) entirely separately from the ac steady state response to the AC sine source.
In any introductory circuit theory textbook or even in Wikipedia, you can find the basic formula for the zero-input response:
$$i(t) = i(0)\exp(-t/\tau)$$
where \$\tau=\frac{L}{R}\$.
Here the trick is knowing what to plug in for \$i(0)\$. Your simulator sets the initial current through the inductor to 0. But this is the sum of the current due to the steady state response, and due to the transient. You know that the steady state response should give a current of about -2.5 A at \$t=0\$. So that means the part of the initial current due to the zero input response is +2.5 A, and that's the value you should use in the above equation to get the transient part of the simulation result.
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