# RLC Circuit with a forced AC voltage source. Solving solution by complexifying the input function

So, in class we studied the rlc circuit with a force ac voltage across the circuit. The teacher assumed a lot of things and we ended up with a sinusoidal function for current with some equations to calculate the voltages across parts of the circuit.

However, I'm trying to solve the differential equation from scratch. So I decided to complexifying the forced AC voltage using Euler's identity and take only the real part of the solution when I finished. $$L\ddot{y}+R\dot{y}+\frac{1}{C}y= E\cos(\omega t)$$ $$L\ddot{y}+R\dot{y}+\frac{1}{C}y= E\ e^{\omega it} .$$

I managed to find the complimentary solution for the differential equation, however I'm having a hard time finding the particular solution.

So far I assumed that the complementary solution looks like this.

$$y_p = Ae^{\omega it}$$

And I got A as..

$$A = \frac{E}{(-\omega^2L+\frac{1}{C})+\omega Ri}$$

I multiplied A with 1 but in fraction form with both the numerator and the denominator as the complex conjugate of the denominator of A.

$$A = \frac{(-\omega^2L+\frac{1}{C})-\omega Ri}{(-\omega^2L+\frac{1}{C})-\omega Ri} \cdot \frac{E}{(-\omega^2L+\frac{1}{C})+\omega Ri}$$

From here I have trouble simplifying the equation to form a simple sinusoidal equation. So if anyone can show me how you simplify this particular solution I would appreciate it.

Also, I'm confused on how my complementary solution has a exponential decay function. Since I thought the circuit should always have current flowing even after a long time.

As you have already done, set $y_p = Ae^{i\omega t}$, and solve for $A$. Now that you have found $A$ to be complex, it would be easier to immediately write it as $A = re^{i\theta}$. Recall for a complex number $z = x + iy$, $|z|^2 = x^2 + y^2$ and $\theta = \arctan(y/x)$. This should help you simplify your particular solution since $A$ can be expressed as $A = \frac{E}{r}e^{-i\theta}$, so $y_p = \frac{E}{r}e^{i(\omega t - \theta) }$, and note that if you did it correct, you will see a resonance at $\omega^2 = \frac{1}{LC}$.
It seems that you have not solved for your homogeneous (or complementary) solution as yet. The homogeneous equation has a resistor (or damping) term, which will provide the exponential decay. This will cause this solution to die away and only the particular solution will remain for long $t$.