In first year, you might’ve worked with RC or RL circuits in the transient state. Furthermore, you might’ve learned about using RC circuits to create low and high-pass filters. Both RC and RL circuits are referred to as first-order circuits, as they are modeled as first-order differential equations.
Although useful, these circuits are ultimately limited by the fact that you’re missing either a capacitor or inductor. If we attempt to use all 3 components (resistor, capacitor, inductor), we have created an RLC circuit. If we tried solving an RLC circuit, we would notice that it requires the solution of a second-order differential equation. Here’s a sample RLC circuit with the second-order DE derived.
By leveraging RLC circuits, we can create many interesting things. Specifically, the rest of this document will focus on the creation of RLC Second-Order Filters.
As an aside, the number of energy storage devices directly correlates to the order of our differential equation/circuit. In RL and RC circuits, there was only 1 energy storage device (capacitor or inductor), so everything was of the first order. Now that we have resistors and capacitors, the circuits become second-order. Even without the resistor, an LC circuit is still of second-order.
To put it simply, second-order filters tend to be a lot better than first-order. When describing filters, we often use a term called selectivity. Selectivity reflects an electrical component’1s ability to respond to a certain range of frequencies, without being interfered with by other ones. For example, with our custom transceiver, we configure our transceiver to roughly 433 MHz. Our ability to resist signals outside of this 433 MHz value is the selectivity of the transceiver.
To describe selectivity in filters, we often use the term called roll-off rate. This refers to how quickly the gain of a signal decreases when moving away from a target frequency. Going back to the custom transceiver example, as we move away from 433 MHz, how well can we expect to filter out the new frequencies?
Roll-off rate is expressed in dB/Decade, where a decade represents a 10x increase in frequency. Thus, we can describe the roll-off rate as how much gain the system loses when the frequency increases 10x.