# R-L-C Series Circuits

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• 0:05 What is an RLC series circuit?
• 1:43 Equations
• 3:27 Example calculation
• 6:12 Lesson summary

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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this lesson, you will be able to explain what an RLC series circuit is and use related equations to solve simple problems. A short quiz will follow.

## What is an RLC Series Circuit?

On a basic level, the definition of an RLC series circuit is extremely simple. An RLC series circuit is a circuit where a battery, resistor (with resistance R), an inductor (with inductance L) and a capacitor (with capacitance C), RLC, are all connected in one complete loop (a series circuit). An RLC series circuit looks something like this. They're often used as tuning circuits in analogue radios and as low-pass, high-pass or band-pass filters when recording audio in a studio.

For an RLC circuit to operate, it also has to be an alternating current (or AC) circuit, which is where, instead of current flowing just one way around a circuit, it switches direction super-fast. This 'switching' happens 60 times per second in a standard household circuit. This variation in current forms a sine curve when plotted on a graph, like this one for example.

While the basic structure of an RLC circuit is simple, completing calculations isn't always so simple. An RLC circuit is an example of a resonant circuit, one where the capacitor and inductor fight each other to increase and decrease the resistance (or 'impedance') of the circuit. At resonance, the two cancel each other out; in which case, the total impedance of the circuit is just equal to the resistance of the resistor. To fully interpret RLC circuits you often end up using an equation with complex numbers and all kinds of involved mathematics. But in this lesson, we're just going to look at simple series RLC circuits that are at resonance. This will make things easier for us.

## Equations

There are several important equations that help us understand RLC circuits. We said in another lesson that the resistance (or impedance) of a capacitor and inductor (XC and XL) are given by these two equations, where f is the frequency of the AC power source measured in Hertz, C is the capacitance of the capacitor measured in farads and L is the inductance of the inductor measured in Henrys. The resistance of the resistor is just R, like in any circuit.

For a series RLC circuit that's in resonance, XC and XL will be equal to each other and cancel each other out. Therefore, R becomes the total resistance of the circuit. That only happens when this third equation is true. Again, in this equation f is the frequency of the power source, C is the capacitance of the capacitor and L is the inductance of the inductor.

The only other equation you might need (also discussed in another lesson) is Ohm's Law, which says that the current (I) measured in amps is equal to the voltage (V) measured in volts, divided by the resistance(R) measured in Ohms. This can be used for individual components, like the resistor, or the capacitor or inductor, or it can be used for the totals... the total current of the circuit, total voltage and total resistance. Doing these two calculations using Ohm's Law separately can help you solve problems.

Remember that when you use Ohm's Law for a capacitor, instead of R, you use the symbol XC. And when you use Ohm's Law for an inductor, instead of R, you use the symbol XL. If this is confusing, I recommend you watch the lesson on impedance first, and then come back to this later.

## Example Calculation

Maybe this would be easier if we went through an example problem. Let's say you have a resonant RLC circuit, connected to a 12 V AC power supply. The capacitance of the capacitor is 2 * 10^-6 farads, the inductance of the inductor is 2 Henrys, and the resistance of the resistor is 5 Ohms. You're asked to calculate:

A) The frequency of the power supply, f

B) The current flowing in the circuit, I

C) The voltage across the capacitor, VC.

First of all, let's write out what we know. We know that the voltage of the power supply, V, is 12 V. We know that C = 2 * 10^-6 farads, we know that L is equal to 2 Henrys and we know that R is equal to 5 Ohms. Part A asks us to find f. Taking a look at our equations, we'll see that we know everything except f in this equation. So, that's the one to use. Plug in L and C, type it into a calculator, and we get 79.6 Hz.

Part B asks us to figure out the current flowing in the circuit. Ohm's Law is the only equation in this lesson that contains current. Since this is a series circuit, the current is the same all the way around the loop. So we could use Ohm's Law for the whole circuit, or the resistor, or the capacitor, or the inductor; either way, I, will come out the same. But we just don't have enough information about any of the individual components to do a calculation for them, at least not without using more than one equation.

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