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# RLC natural response - variations

The RLC natural response falls into three categories: overdamped, critically damped, and underdamped. Written by Willy McAllister.

## Introduction

The natural response of a resistor-inductor-capacitor circuit left parenthesis, start text, R, L, C, right parenthesis, end text can take on three different forms, depending on the specific component values.
In two prior articles, we covered an intuitive description of how the start text, R, L, C, end text behaves, and did a formal derivation where we modeled the circuit with a 2nd-order differential equation and solved a specific example circuit. In this article, we look closely at the characteristic equation and give names to the various solutions.
The circuit for the start text, R, L, C, end text natural response.

### What we're building to

The start text, R, L, C, end text characteristic equation is:
s, squared, plus, start fraction, start text, R, end text, divided by, start text, L, end text, end fraction, s, plus, start fraction, 1, divided by, start text, L, C, end text, end fraction, equals, 0
We will solve for the roots of the characteristic equation using the quadratic formula:
s, equals, start fraction, minus, start text, R, end text, plus minus, square root of, start text, R, end text, squared, minus, 4, start text, L, end text, slash, start text, C, end text, end square root, divided by, 2, start text, L, end text, end fraction
By substituting variables alpha and omega, start subscript, o, end subscript we can write s a little simpler as:
s, equals, minus, alpha, plus minus, square root of, alpha, squared, minus, omega, start subscript, o, end subscript, squared, end square root
where,
alpha, equals, start fraction, start text, R, end text, divided by, 2, start text, L, end text, end fraction, comma omega, start subscript, o, end subscript, equals, start fraction, 1, divided by, square root of, start text, L, C, end text, end square root, end fraction
alpha is called the damping factor and omega, start subscript, o, end subscript is the resonant frequency.
Depending on the relative size of alpha and omega, start subscript, o, end subscript, there will be three different forms of the solution for i, left parenthesis, t, right parenthesis,
• overdamped, alpha, is greater than, omega, start subscript, 0, end subscript, leads to the sum of two decaying exponentials
• critically damped, alpha, equals, omega, start subscript, 0, end subscript, gives t times a decaying exponential
• underdamped, alpha, is less than, omega, start subscript, 0, end subscript, leads to a decaying sine

### Modeling and solving the circuit - review

In a previous article we created and solved a 2nd-order differential equation modeling the start text, R, L, C, end text circuit. That equation looks like this:
start text, L, end text, start fraction, d, squared, i, divided by, d, t, squared, end fraction, plus, start text, R, end text, start fraction, d, i, divided by, d, t, end fraction, plus, start fraction, 1, divided by, start text, C, end text, end fraction, i, equals, 0
We proposed a solution with an exponential form (which worked out really nicely for us), and came up with what is called the characteristic equation with this form:
s, squared, plus, start fraction, start text, R, end text, divided by, start text, L, end text, end fraction, s, plus, start fraction, 1, divided by, start text, L, C, end text, end fraction, equals, 0
We solved for s, the roots of the start text, R, L, C, end text characteristic equation, using the quadratic formula,
s, equals, start fraction, minus, start text, R, end text, plus minus, square root of, start text, R, end text, squared, minus, 4, start text, L, end text, slash, start text, C, end text, end square root, divided by, 2, start text, L, end text, end fraction
By substituting variables alpha and omega, start subscript, o, end subscript we wrote s a little simpler as:
s, equals, minus, alpha, plus minus, square root of, alpha, squared, minus, omega, start subscript, o, end subscript, squared, end square root
where alpha, equals, start fraction, start text, R, end text, divided by, 2, start text, L, end text, end fraction, comma and omega, start subscript, o, end subscript, equals, start fraction, 1, divided by, square root of, start text, L, C, end text, end square root, end fraction
alpha is called the damping factor, and omega, start subscript, o, end subscript is called the resonant frequency.
We revised our proposed solution to have this form,
i, equals, K, start subscript, 1, end subscript, e, start superscript, s, start subscript, 1, end subscript, t, end superscript, plus, K, start subscript, 2, end subscript, e, start superscript, s, start subscript, 2, end subscript, t, end superscript
We now take a close look at the expression for s, the roots of the start text, R, L, C, end text characteristic equation, and the impact it has on the solution for i.

## Exact solution

If we want an exact answer for particular values of start text, R, end text, start text, L, end text, and start text, C, end text, we perform a computation like the one we did in the previous article for the example circuit. Alternatively, we can enter the circuit into a circuit simulator to help us find a result.

### Overdamped, critically damped, underdamped

We can get an impression of the full richness of the natural response by looking three possible outcomes in a qualitative sense.
The solution for s depends on the sign of the subtraction that happens under the square root term in the equation:
s, equals, minus, alpha, plus minus, square root of, alpha, squared, minus, omega, start subscript, o, end subscript, squared, end square root
How the roots turn out:
relationsign of alpha, squared, minus, omega, squarednicknames
alpha, is greater than, omega, start subscript, o, end subscriptplusoverdamped2 real roots
alpha, equals, omega, start subscript, o, end subscript0critically damped2 repeated roots
alpha, is less than, omega, start subscript, o, end subscriptminusunderdamped2 complex roots

How the response turns out, i, left parenthesis, t, right parenthesis:
relationsign of alpha, squared, minus, omega, squarednicknamei, left parenthesis, t, right parenthesis
alpha, is greater than, omega, start subscript, o, end subscriptplusoverdamped2 decaying exponentials
alpha, equals, omega, start subscript, o, end subscript0critically dampedt, dot decaying exponential
alpha, is less than, omega, start subscript, o, end subscriptminusunderdampeddecaying sine

If your engineering studies take you into the area of Control Theory, these terms are used to describe how dynamic systems act. For example, the motion of a robot's arm can be described by a second-order differential equation. If your ask your robot to quickly reach for an object, you can describe how its hand moves using these words.
Let's take a look at the three possible outcomes in a bit more detail.

### $\alpha^2 - \omega^2 > 0\quad$alpha, squared, minus, omega, squared, is greater than, 0 overdamped

Under this condition, the omega, start subscript, o, end subscript, squared term is small relative to alpha, squared, so we know the expression inside the square root will be positive. We also know the square root expression will be smaller than alpha. This means s will be two real numbers, both negative.
s, start subscript, 1, comma, 2, end subscript, equals, minus, alpha, plus minus, square root of, alpha, squared, minus, omega, start subscript, o, end subscript, squared, end square root
s, start subscript, 1, end subscript, equals, minus, start text, r, e, a, l, space, n, u, m, b, e, r, end text, start subscript, 1, end subscript and s, start subscript, 2, end subscript, equals, minus, start text, r, e, a, l, space, n, u, m, b, e, r, end text, start subscript, 2, end subscript
(Convince yourself that s, start subscript, 1, end subscript and s, start subscript, 2, end subscript will both be negative.)
The current will be the superposition of two real exponentials that both decay to zero.
i, equals, K, start subscript, 1, end subscript, e, start superscript, minus, start text, r, e, a, l, end text, start subscript, 1, end subscript, t, end superscript, plus, K, start subscript, 2, end subscript, e, start superscript, minus, start text, r, e, a, l, end text, start subscript, 2, end subscript, t, end superscript
The circuit is said to be overdamped because two superimposed exponentials are both driving the the current to zero. A circuit will be overdamped if the resistance is high relative to the resonant frequency.

### $\alpha^2 - \omega^2 = 0 \quad$alpha, squared, minus, omega, squared, equals, 0 critically damped

The boundary between underdamped and overdamped is when alpha, equals, omega, start subscript, o, end subscript. The damping factor and the resonant frequency are in balance, and the terms under the square root subtract to 0. The roots of the characteristic equation, s, are two identical real numbers, called repeated roots:
s, start subscript, 1, comma, 2, end subscript, equals, minus, alpha, plus minus, start cancel, square root of, alpha, squared, minus, omega, start subscript, o, end subscript, squared, end square root, end cancel, start superscript, 0, end superscript
s, start subscript, 1, comma, 2, end subscript, equals, minus, alpha
Solving a 2nd-order differential equation with repeated roots is a bit tricky. I'm not going to do the derivation here, but instead I refer you to a great video Sal did on solving repeated roots. Welcome back ... With repeated roots, the answer is an exponential term multiplied by t.
i, equals, start fraction, start text, V, end text, start subscript, 0, end subscript, divided by, start text, L, end text, end fraction, t, e, start superscript, minus, alpha, t, end superscript
This response is said to be critically damped.

### $\alpha^2 - \omega^2 <0\quad$alpha, squared, minus, omega, squared, is less than, 0 underdamped

When alpha is smaller than omega, start subscript, o, end subscript, the square root term has a negative number inside, and s comes out as two complex conjugate numbers, with real and imaginary parts. The example circuit we worked out in the RLC natural response derivation article is an underdamped system.
The current looks like a sine wave that diminishes over time. Think of the sound a bell makes when you strike it. The bell's note rings out and fades over time. That is an underdamped second-order mechanical system. For second-order electrical circuits, we borrow the term and say the underdamped system "rings" at a frequency of approximately omega, start subscript, o, end subscript, equals, start fraction, 1, divided by, square root of, start text, L, C, end text, end square root, end fraction, point
If we let the resistance get really small and eventually go to 0, then alpha, equals, start text, R, end text, slash, 2, start text, L, end text goes to zero and s, start subscript, 1, comma, 2, end subscript becomes omega, start subscript, o, end subscript. The circuit becomes a pure start text, L, C, end text configuration. When we analyzed the natural response of the LC circuit, we came up with a sine wave that lasted forever. (In real life, start text, R, end text is never really 0, so there is always some energy lost. A bell does not ring forever.)
The first example circuit we worked through earlier in this article had start text, R, end text, equals, 2, \Omega, comma, start text, L, end text, equals, 1, start text, H, end text, comma and start text, C, end text, equals, 1, slash, 5, start text, F, end text, point
We're not going to repeat the solution, but here are a few observations using the alpha and omega, start subscript, o, end subscript notation.
The damping factor alpha is
alpha, equals, start fraction, start text, R, end text, divided by, 2, start text, L, end text, end fraction, equals, start fraction, 2, divided by, 2, dot, 1, end fraction, equals, 1
The resonant frequency, omega, start subscript, o, end subscript is
omega, start subscript, o, end subscript, equals, start fraction, 1, divided by, square root of, start text, L, C, end text, end square root, end fraction, equals, start fraction, 1, divided by, square root of, 1, dot, 1, slash, 5, end square root, end fraction, equals, square root of, 5, end square root
Looking at the terms under the square root:
alpha, squared, minus, omega, squared, equals, 1, squared, minus, square root of, 5, end square root, squared, equals, minus, 4, equals a negative number, which we saw led to a decaying sine solution. Therefore, we would describe the example circuit as an underdamped system.

## Summary

The start text, R, L, C, end text circuit is the electronic equivalent of a swinging pendulum with friction. The circuit can be modeled by this 2nd-order linear differential equation:
start text, L, end text, start fraction, d, squared, i, divided by, d, t, squared, end fraction, plus, start text, R, end text, start fraction, d, i, divided by, d, t, end fraction, plus, start fraction, 1, divided by, start text, C, end text, end fraction, i, equals, 0
The resulting characteristic equation is:
s, squared, plus, start fraction, start text, R, end text, divided by, start text, L, end text, end fraction, s, plus, start fraction, 1, divided by, start text, L, C, end text, end fraction, equals, 0
We solved for the roots of the characteristic equation using the quadratic formula:
s, equals, start fraction, minus, start text, R, end text, plus minus, square root of, start text, R, end text, squared, minus, 4, start text, L, end text, slash, start text, C, end text, end square root, divided by, 2, start text, L, end text, end fraction
By substituting variables alpha and omega, start subscript, o, end subscript we wrote s a little simpler as:
s, equals, minus, alpha, plus minus, square root of, alpha, squared, minus, omega, start subscript, o, end subscript, squared, end square root
where alpha, equals, start fraction, start text, R, end text, divided by, 2, start text, L, end text, end fraction, comma and omega, start subscript, o, end subscript, equals, start fraction, 1, divided by, square root of, start text, L, C, end text, end square root, end fraction
Depending on the relative size of alpha and omega, start subscript, o, end subscript, we came up with three different forms of the solution:
• overdamped, alpha, is greater than, omega, start subscript, 0, end subscript, leads to the sum of two decaying exponentials
• critically damped, alpha, equals, omega, start subscript, 0, end subscript, leads to t times decaying exponential
• underdamped, alpha, is less than, omega, start subscript, 0, end subscript, leads to a decaying sine

## Want to join the conversation?

• how to get the equation for i=
​V0/W0*e−αt*t in crirically damped?
• From the author:Jonathan is correct. The expression for current should be i = V0/L t e^(-at). This has been corrected in the article. The article still just states the result, without derivation.

Inspired by Jonathan's solution in this clarification, I wrote up this more complete derivation of the Critically Damped case:
http://spinningnumbers.org/a/rlc-natural-response-variations.html#critically-damped
This website is a continuation of the EE contributions I made at KA.
• why do you get barely any questions, tips, or thanks on your electrical engineering section?
on other sections, i see over 100 questions but on this section, i hardly ever see 10
• If I add a SPDT switch in the circuit, I can charge the capacitor. But if there is one way to charge without the switch. (For exmaple, I add a battery and it will work and form same-frequncy wave until battery does not have any energy)
(1 vote)
• Providing an initial charge to a capacitor or initial current to an inductor is a tricky bit of work in a circuit simulator. It is difficult to implement a mechanical switch. They are usually not part of the simulator's primitive elements.

It's possible to mimic a mechanical switch using very large transistors. I show an example of this on my web site, spinningnumbers.org. This is a follow-on to my work at KA where all the articles are updated. If you navigate to the Natural and Forced Response topic, find the article on RLC Variations. The simulation models have giant MOSFETS acting like mechanical switches to initialize the voltage. Remove the spaces and try this URL,

https : //spinningnumbers . org/a/rlc-natural-response-variations.html#under-damped

(Scroll down to the end of the Under Damped chapter to find the simulation model.)
• The article begins by telling:
( α > ωo ) implies overdamped circuit. Or,
( R/2L > ωo ) implies overdamped circuit..................... (1)

But at a later point, the article also tells:
"A circuit will be overdamped if the resistance is high relative to the resonant frequency."
This means,
( R > ωo ) implies overdamped circuit..................... (2)

If ( R/2L > ωo ) and ( R > ωo ), both implies an overdamped circuit, this means both are equivalent statements.

How can ( R/2L > ωo ) means the same thing as ( R > ωo )?
(1 vote)
• Equation (1) is correct.
For equation (2) you are taking a very qualitative statement and trying to turn it into an equation, but this doesn't work. The units are all wrong... ohms > radians/sec. I'm just trying to say you end up with an over damped circuit if the resistance is high. In technical terms, a circuit is over damped when equation (1) is true.
• In the hidden section about the underdamped natural frequency approximation, it states that we can only approximate the frequency as w0 because the damping factor under the radical shifts the frequency over time (the wavelength changes).

I think this is misleading and not true. The natural frequency does not change; only the envelope changes.
lets begin with the end solution for the differential eqn:
i = <some function of t> * <a sinusoidal>. in our RLC case:
i = 5e^-t * sin2t.
We can clearly see that the argument of the sine function depends linearly on t. Nothing else. The multiplier (or Eigenvalue) acts only as an envelope function which does nothing in term of frequency.

The natural frequency is determined by the roots of the differential equation, which in turn are characterized by the "damping factor" alpha and the "frequency" omega. But these names are only naming convention after the case that the damping factor equals 0 (the only case we can state that w is the natural frequency, as it is the only factor which determines the roots). Otherwise, the frequency is a "combination" of alpha and w. And it stays constant! The name "damping factor" arise from the fact that when it is not zero, the sinusoidal decays.

The only way to change the system's FIXED natural frequency is by changing the boundary conditions.
(1 vote)
• I used a program to graph the overdamped current solution, and I am pretty sure it never goes up to two amperes as it is shown. The equation 1.25*(exp(-x)-exp(-9x)) has a zero derivative at (-ln(1/9))/8=0.2747 which makes the function value 0.8443, never rises above this.
(1 vote)
• Suppose a parallel RLC circuit is critically damped at the natural frequency of 100 Hz. I have values for R and L. How do I calculate a value of C such that it is critically damped. I know it can be calculated using 1/2RC = 1/sqrt(LC) but how do I incorporate an element of 100 Hz from this formula? This formula doesn't give me the critical damped circuit with the natural frequency of 100 Hz. Please tell me.
(1 vote)
• I don't really have a good feel for what the i-t curves would look like for these three variations of the RLC natural response. Why don't you give specific examples with their graphs?

A more general criticism of the entire EE course -- there are not nearly enough problems to try to solve to test the reader's understanding. Please provide problems for each module!!
(1 vote)