# Time to Frequency Transform - Where are the Initial Conditions?

In a circuit with sinusoidal source and charged capacitors and inductors we have initial conditions:

$$u_{C}=u(0)+\frac{1}{C} \int_{t_{0}}^{t}i(\tau)d\tau$$

$$i_{L}=i(0)+\frac{1}{L} \int_{t_{0}}^{t}u(\tau)d\tau$$

and when we work in time domain we account for those in the equations.

If instead of phasors we use Laplace transform then we account for those initial conditions too

$$U_{C}(s)=\frac{1}{Cs}I(s)+\frac{1}{s}u(0)$$

$$I_{L}(s)=\frac{1}{Ls}U(s)+\frac{1}{s}i(0)$$

But when we move from time to frequency domain using phasors then I don't understand where those initial conditions appear. In the transformed equations I don't see initial conditions anywhere.

• Did you want to show the frequency domain spectrum as it changes in time? or the steady state transfer function? – Sunnyskyguy EE75 Jan 28 at 21:43
• @SunnyskyguyEE75 I don't want to show something, this is a question it popped in my mind. – Adam Jan 28 at 21:45