How to Derive the Equation for an RLC Circuit?

[Ai En]

how to get this equation:
Vout/Vin = [RL]/[-[ω][/2]+jω(R/L)+1/(LC)]

 

[a-tom-ic]

- Show us the picture of the RLC Circuit your using.
- Tell us how Vout is defined.
- Tell us what [omega][/2] stands for (use the boards LaTeX syntax).

 

[Ai En]

http://upload.wikimedia.org/wikipedia/en/1/14/RLC_series_band-pass.svg

 

[a-tom-ic]

Hi Ai En,

as a solution you could use the voltage divider formula to get a first equation:
$$\frac{V_{out}}{V_{in}} = \frac{R_L}{R_L + j \omega L + \frac{1}{j \omega C}}$$

afterwards if you think that's useful you can try to manipulate this formula until you get the one given to you. Since i can not interpret your given equation (as i said use the LaTeX encoding to be more clear about your formula) i can only guess and did for example multiply the whole formula with $\frac{j \omega L}{j \omega L}$ and the denominator again with $\frac{L}{L}$ that results in:
$$\frac{V_{out}}{V_{in}} = \frac{j \omega R_L}{L \cdot \left( - \omega^2 + j \omega \frac{R_L}{L} + \frac{1}{C L}\right)}$$

 

[pmacias]

To derive the equation for an RLC circuit, we first need to understand the components involved and their properties. An RLC circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series. Each of these components has a specific impedance, which affects the overall behavior of the circuit.

The impedance of a resistor is given by Ohm's law: R = V/I, where V is the voltage across the resistor and I is the current flowing through it.

The impedance of an inductor is given by the equation: XL = jωL, where j is the imaginary unit, ω is the angular frequency, and L is the inductance.

The impedance of a capacitor is given by the equation: XC = -j/(ωC), where j is the imaginary unit, ω is the angular frequency, and C is the capacitance.

To find the overall impedance of the RLC circuit, we need to add the individual impedances of each component. This can be done using Kirchhoff's voltage law, which states that the sum of all voltages in a closed loop is equal to zero.

Applying Kirchhoff's voltage law to the RLC circuit, we get the following equation:

Vin - Vout - IR - IX - IC = 0

Where Vin is the input voltage, Vout is the output voltage, IR is the voltage across the resistor, IX is the voltage across the inductor, and IC is the voltage across the capacitor.

Substituting the impedance equations for each component, we get:

Vin - Vout - I(R + jωL) - I(-j/(ωC)) = 0

Rearranging the equation and dividing both sides by Vin, we get:

Vout/Vin = [RL]/[-[ω][/2]+jω(R/L)+1/(LC)]

This is the derived equation for an RLC circuit, which relates the output voltage to the input voltage and the circuit parameters of resistance, inductance, and capacitance. This equation can be used to analyze the behavior of an RLC circuit and determine its resonance frequency, bandwidth, and other important characteristics.