# Designing an RLC circuit with specific transfer function

I have a transfer function below.

$$H(s)=(s^2+(1/2)s+(1/2))/(s^2+s+1)$$

Using this, I have to design an RLC circuit with that transfer function.

At first hand, I did some inverse Laplace Transform, but it didn't seem to be helpful.

So I'm stuck in here not knowing how to implement that circuit only with a Transfer Function...

Any small hints or clues would be appreciated.

Thanks in advance.

## 1 Answer

The first thing you should do is try to find out what a RLC circuit even looks like with the variables you want.

$$\frac{q(t)}{C}+L q''(t)+R q'(t)=u(t)$$

Then the Laplace $$\mathcal{L}(Q)\to \frac{\text{C} \left(\text{L} q'(0)+\text{L} q(0) s+q(0) R+\mathcal{L}(U)\right)}{\text{C} \text{L} s^2+\text{C} R s+1}$$

Solving this gives you a general function of:

$$q(t)\to c_1 e^{\frac{1}{2} t \left(-\frac{\sqrt{C R^2-4 L}}{\sqrt{C} L}-\frac{R}{L}\right)}+c_2 e^{\frac{1}{2} t \left(\frac{\sqrt{C R^2-4 L}}{\sqrt{C} L}-\frac{R}{L}\right)}$$

Comparing to the solution of the Transfer function and its solution:

$$H(s)=\frac{s^2+\frac{s}{2}+\frac{1}{2}}{s^2+s+1}$$

$$\delta (t)+\frac{1}{12} e^{-\frac{1}{2} \left(1+i \sqrt{3}\right) t} \left(i \left(\sqrt{3}+3 i\right) e^{i \sqrt{3} t}-i \sqrt{3}-3\right)$$

You may be able to get yourself closer to your circuit design.

If you have access to mathematica and System modeler (or matlab and simulink) you could also just design exactly the circuit to your specific functions. Keep in mind your transferfunction which is in the form $$H(s)=\frac{U(s)}{Y(s)}$$ has an input..which is why the results don't look the same, and where that $$\delta (t)$$ shows up and other bits.

At this article you can read more about the differential equations, and how they relate that to your design specifics.