RLC circuit

Basically, RLC circuits contain a resistor, inductor, and capacitor. RLC circuits have many intersting properties. Depending on where you measure the output voltage, RLC circuits can act as many different types of filters.

Here's what the typical RLC circuit looks like:

                                   L
                 R             _   _   _
                              / \ / \ / \
      -----/\/\/\/\/\---------\_/-\_/-\_/------
      |                                       |
      |                                       |
      |                                       |
      |                                       |        
   :  |                                     -----        
      |                                            C
      |                                     -----
      |                                       |  
      |                                       |
      -----------------------------------------
                         |
                       -----  Ground
                        ---
                         -

In the circuit, the resistor of resistance R, inductor of inductance L, and capacitor of capacitance C are all hooked up in series. With RLC circuits, it is perfectly possible to stick a voltage or current source anywhere along the line, or to start the capacitor off with a charge across it. The circuit above is the most basic RLC circuit.

No matter where you measure the voltage across, you will have to use a voltage divider relation. Because you use that voltage divider, your system function's denominator will always be equal to the sum of the impedances of the three elements. That denominator will always be: R + LS + (1/ SC). Multiply through by SC, and you get RCS + LCS² + 1. S = j * omega. Thus the resonant frequency will always be 1 / sqrt (L * C). This is the value that will minimize the denominator.

An RLC circuit will act as a band pass filter if the voltage is measured across the resistor. If we do that, H(S), the system function, will be R / (R + LS + 1/SC). Multiplying through, we get H(S) = RCS / LCS² + RS + 1. For small omega, we get a small magnitude, and we get small magnitudes for large values of omega as well. For values around the resonant frequency, though, the magnitude is relatively high. This is the defining characteristic of a band pass filter. In a similar manner, we can find that measuring the voltage across the capacitor will produce a low pass filter. Measuring voltage across the inductor produces a high pass filter.

Despite the fact that they are all very similar, RLC circuits are very versatile. By changing the values of the parameters R, L, and C, as well as any voltages or currents introduced to the system, one can make an RLC circuit do almost anything.