Previous posts in this series:

If you’re unsure about how capacitors and/or inductors work, check out:

In part 4 we looked at the natural response and step response of RL and RC circuits, which means circuits with resistors together with inductors and capacitors, respectively.

This time we will take a look at **RLC circuits**, which contain both resistor(s), inductor(s) and capacitor(s), and examine their **natural response** and **step response**. We’ll not go into too much detail regarding the theory behind this material, but rather show you quite directly how to find expressions for the responses of RLC circuits.

Since this is a theory-heavy blog post, it is most suited to electrical engineering students as a supplement to other learning material. Others might find this interesting as well, especially those who are curious about how capacitors and inductors can influence a circuit.

The equations and examples in this blog post are taken from the book Electric Circuits by Nilsson and Riedel.

## Intro to RLC Circuits

We need to separate between **parallel** RLC circuits and **series** RLC circuits. Below you will find simplified theoretical circuits to illustrate different concepts in the following order:

- Natural response of a parallel RLC circuit
- Step response of a parallel RLC circuit
- Natural response of a series RLC circuit
- Step response of a series RLC circuit

We will use a couple of these in examples later in this blog post.

#### Damping

As opposed to RL and RC circuits, RLC circuits can be **overdamped**, **underdamped** or **critically damped**. The behavior of these are not the same as we talked about in this post. Here, both overdamped and critically damped circuits can overshoot the final value. However, they will not oscillate beyond that. Underdamped circuits will oscillate beyond the initial overshoot.

The combination of *R*, *L* and *C* values decides what kind of damping the circuit has:

The value of *α* (aka. the **neper frequency**) depends if the RLC circuit is parallel or series:

The value of *ω _{0}* (aka. the

**resonant radian frequency**) is the same regardless of type of circuit:

Both *α* and *ω _{0}* have the unit of rad/s (radians per second).

#### Algorithm

What we want to do is to find an expression of either voltage or current as a function of time: *x(t)*. There are **six different equations** for this, where the one you want to use is chosen based on whether you’re looking for the natural response or step response, and whether the system is overdamped, underdamped or critically damped. Each equation has a set of **coefficients** which need to be found. To find these coefficients you will need the initial value ** x(0)**, the initial time derivative of

*x*

**and sometimes the**

*dx(0)/dt***roots of the characteristic equation**. We’ll go into detail on how to do this later in this post.

To summarize:

- Natural response or step response?
- Find out if the circuit is overdamped, underdamped or critically damped.
- Choose the proper response equation based on 1. and 2.
- Find
*x(0)*and*dx/dt(0)*. - Find the roots of the of the characteristic equation if necessary.
- Find the response equation’s coefficients using what you found in 4.
- Use the coefficients found in 6. (and the roots found in 5. if applicable) to complete the expression for the response using the equation chosen in 3.

#### The Characteristic Equation and Its Roots

For both parallel and series RLC circuits, the so called characteristic equation is

We need *s* in the **overdamped** response equations, and since the characteristic equation is a quadratic equation we will get two different values of *s,* aka. the **roots** of the characteristic equation. Use this equation to find the roots *s _{1}* and

*s*:

_{2}

Natural Response Equations

In the following equations, the variable *x* represent either voltage or current, depending on what we want to calculate. With all natural response equations, we expect the *x* value to start at an initial value *x(0)* and converge towards zero in some way.

#### Overdamped Natural Response

In this scenario, the expected behavior of *x(t)* is a non-oscillating convergence from *x(0)* to zero. It might overshoot the final value.

To find the coefficients *A _{1}* and

*A*, you have to have two different equations. These ones are used in this specific scenario:

_{2}Note the usage of the roots *s _{1}* and

*s*.

_{2}#### Underdamped Natural Response

In this scenario, the expected behavior of *x(t)* is an approach from *x(0)* towards zero. It will dip below zero before oscillating and converging towards its final value.

where

aka. the **damped radian frequency**.

Coefficient equations:

#### Critically Damped Natural Response

In this scenario, the expected behavior of *x(t)* is a similar to the overdamped scenario, except that it is on the verge of oscillating.

Coefficient equations:

## Step Response Equations

These step response equations are quite similar to the natural response equations excepts they go from an initial value *x(0)* to a final value *X _{f}*.

#### Overdamped Step Response

Main equation:

Coefficient equations:

#### Underdamped Step Response

Main equation:

where

Coefficient equations:

#### Critically Damped Step Response

Main equation:

Coefficient equations:

## Examples

Let’s run through a few examples.

#### Overdamped Natural Response of a Parallel RLC Circuit

In this example, we’ll let

*C*= 0.2 μF*L*= 50 mH*R*= 200 Ω

We need to know some initial conditions to get going. In this case we know that

*v(0)*= 12 V*i*= 30 mA_{L}(0)

Let’s double check that this is overdamped:

*α*= 12500 rad/s*ω*= 10000 rad/s_{0}

*α* > *ω _{0}*, thus overdamped.

Since we already know *v(0)*, the first thing we need to do is to find *dv/dt(0)*. We can use the following equation for this, which is a fundamental equation for capacitors:

Ohm’s law gives us *i _{R}* = 60 mA. Kirchoff’s current law then gives us

*i*= -90 mA.

_{C}Now we have all we need: *dv(0)/dt* = -450 kV/s

Next, we need to find the roots. Use the equations mentioned earlier in this blog post.

*s*= -5000 rad/s_{1}*s*= -20000 rad/s_{2}

Now we can dive into the coefficient equations for overdamped natural response. Use both equations to find *A _{1}* and

*A*.

_{2}*A*= -14 V_{1}*A*= 26 V_{2}

Finally, we’re ready for the main overdamped natural response equation. Let’s plot it!

#### Underdamped Step Response of a Series RLC Circuit

Let’s switch all parameters and look at how it turns out.

*V*= 48 V*R*= 280 Ω*L*= 0.1 H*C*= 0.4 μF

Our aim will be to find the voltage across the capacitor *v _{c}*, with the + sign above the capacitor.

Let’s double check if this is underdamped or not:

*α*= 1400 rad/s*ω*= 5000 rad/s_{0}

*ω _{0}* >

*α,*thus underdamped.

Since no energy is stored in the circuits at *t*=0, we have that

*v*= 0_{c}(0)*dv*= 0_{c}(0)/dt

We don’t need the roots for this, so we can jump directly to the coefficients:

*B’*= -48 V_{1}*B’*= -14 V_{2}

After some time, no current will flow through the circuit because of the capacitor. Thus *X _{f}* =

*V*= 48 V.

We now have everything we need for a sweet plot.

## Final Thoughts

That was a lot of theory and equations, but if you took your time to go through this, this was probably what you was looking for in the first place.

We recommend that you read a proper textbook to fully understand these concepts. Electric Circuits by Nilsson and Riedel is an excellent choice.

Another source of knowledge and wisdom is this **free** online course where they among other things look at “time and frequency behavior of second-order circuits containing energy storage elements (capacitors and inductors)”, which is what we’ve briefly looked at here.

This blog post is only meant as a supplement to any material you already have.