# Settling time for an LR circuit with an AC sine source

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

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.

• And to focus on the time aspect, you can consider the system settled after 3-5 tau (depending on how closely you're counting) – W5VO May 21 '16 at 21:55
• But ταυ = L/R becomes infinity if the circuit is pure inductive like in my previews question right? and current will never alternate in theory for a real pure inductive circuit? – user16307 May 21 '16 at 21:59
• @user16307 If the circuit is purely inductive, then it's not a LR circuit (and an impossible theoretical construct at that point). If you set up the initial conditions of your circuit to have a DC current bias as you have done, theoretically that bias will never dissipate without a resistance. – W5VO May 21 '16 at 22:13
• @W5VO I'm asking because in simulation the circuit does settle to steady state for a pure inductive circuit. Does LTspice add series resistance even-though we don't add any? – user16307 May 21 '16 at 22:17
• If $\tau=\infty$ then $e^{-t/\tau}=1$ for all $t$, so there is no transient, just a steady state sinusoid. – Chu May 21 '16 at 22:21

A circuit output settles for a the DC source excitation, the AC source shall never let the output settle as time tends to infinity.