# What is the relationship between the complex frequecy of a RLC circuit and the half power frequency of it?

First of all, I would like to apologize for possible mistakes in the writing of this text, since English is not my mother tongue and it is my first time in this blog.

Secondly, and more important, is the matter of this question. Taking the following second order equation of the current in a series RLC circuit:

$$\frac{d^2i}{dt^2} + \frac{R}{L}\frac{di}{dt} + \frac{i}{LC} = 0$$

where $$\frac{R}{L}=2\alpha,$$ with $$\alpha$$ being the Neper frequency,

and $$\frac{1}{LC}=\omega_0^2,$$ with $$\omega_0$$ being the resonance frequency.

According to Nilsson's book "Electric Circuits 7th Edition", the solution for the described equation is:

$$A_1*e^{s_1t}+A_2*e^{s_2t}$$

Where $$s_1$$ and $$s_2$$ are called the complex frequencies and are described by the following equations:

$$s_1= -(\frac{R}{2L})+ \sqrt[ ]{(\frac{R}{2L})^2-\frac{1}{LC}}$$

$$s_2= -(\frac{R}{2L})- \sqrt[ ]{(\frac{R}{2L})^2-\frac{1}{LC}}.$$

Well now, analysing the frequency response of the circuit, one can obtain that the resonance frequency is exactly the same:

$$\frac{1}{LC}=\omega_0^2$$

but the half power frequencies change the signs, being:

$$\omega_1=-(\frac{R}{2L})+ \sqrt[ ]{(\frac{R}{2L})^2+\frac{1}{LC}}$$

$$\omega_2=(\frac{R}{2L})+ \sqrt[ ]{(\frac{R}{2L})^2+\frac{1}{LC}}.$$

So now, why does this similarity between $$s_{1,2}$$ and $$\omega_{1,2}$$ exist?

Well, first of all, a complex series RLC circuit can be analyzed using the following method.

The total impedance of the circuit is given by:

$$\underline{\text{Z}}_{\space\text{in}}=\text{R}+\text{j}\omega\text{L}+\frac{1}{\text{j}\omega\text{C}}\tag1$$

The input voltage can be written as:

$$\underline{\text{V}}_{\space\text{in}}=\hat{\text{u}}\exp\left(\varphi\text{j}\right)\tag2$$

The input current can now be written as:

$$\underline{\text{I}}_{\space\text{in}}=\frac{\underline{\text{V}}_{\space\text{in}}}{\underline{\text{Z}}_{\space\text{in}}}\tag3$$

The total complex power is now given by:

$$\underline{\text{S}}=\underline{\text{V}}_{\space\text{in}}\cdot\underline{\text{I}}_{\space\text{in}}=\underline{\text{V}}_{\space\text{in}}\cdot\frac{\underline{\text{V}}_{\space\text{in}}}{\underline{\text{Z}}_{\space\text{in}}}=\frac{\underline{\text{V}}_{\space\text{in}}^2}{\underline{\text{Z}}_{\space\text{in}}}=\frac{\hat{\text{u}}^2\exp\left(2\varphi\text{j}\right)}{\text{R}+\text{j}\omega\text{L}+\frac{1}{\text{j}\omega\text{C}}}\tag4$$

The real power is given by:

$$\text{P}=\left|\underline{\text{S}}\right|\cos\left(\varphi\right)=\frac{\hat{\text{u}}^2}{\sqrt{\text{R}^2+\left(\omega\text{L}-\frac{1}{\omega\text{C}}\right)^2}}\cdot\cos\left(\left|\arg\left(\underline{\text{Z}}_{\space\text{in}}\right)\right|\right)\tag5$$

The apparant power is given by:

$$\text{Q}=\left|\underline{\text{S}}\right|\sin\left(\varphi\right)=\frac{\hat{\text{u}}^2}{\sqrt{\text{R}^2+\left(\omega\text{L}-\frac{1}{\omega\text{C}}\right)^2}}\cdot\sin\left(\left|\arg\left(\underline{\text{Z}}_{\space\text{in}}\right)\right|\right)\tag6$$

Where:

$$\arg\left(\underline{\text{Z}}_{\space\text{in}}\right)=\begin{cases} 0\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\text{if}\space\space\omega\text{L}=\frac{1}{\omega\text{C}}\\ \\ \arctan\left(\frac{\omega\text{L}-\frac{1}{\omega\text{C}}}{\text{R}}\right)\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\text{if}\space\space\omega\text{L}>\frac{1}{\omega\text{C}}\\ \\ \frac{3\pi}{2}+\arctan\left(\frac{\text{R}}{\left|\omega\text{L}-\frac{1}{\omega\text{C}}\right|}\right)\space\space\space\space\space\space\space\space\space\text{if}\space\space\omega\text{L}<\frac{1}{\omega\text{C}} \end{cases}\tag7$$