# Find the natural response of the RLC circuit

1. Nov 21, 2016

### eehelp150

1. The problem statement, all variables and given/known data

Find the natural response
V1 is a squarewave

2. Relevant equations
KCL, KVL

3. The attempt at a solution
How would I find the natural response using differential equations?
Am I doing it right?
by KCL
$$\frac{V_1-V_{in}}{R_1}+\frac{1}{L}\int_{0}^{t}V_1(t)dt + \frac{V_1-V_2}{R_2}=0$$
$$\frac{V_2-V_1}{R_2}+C\dot{V_2}=0$$

$$V_2-V_1+R_2C\dot{V_2}=0$$
$$V_1=R_2C\dot{V_2}$$
substitute to original eq.

$$\frac{V_2}{R_1}+\frac{R_2C\dot{V_2}}{R_1}+\frac{1}{L}\int_{0}^{t}(R_2CV_2+V_2)+C\dot{V_2}=\frac{V_{in}}{R_1}$$
take first derivative of everything
$$\dot{\frac{V_2}{R_1}}+\ddot{\frac{R_2CV_2}{R_1}}+\dot{\frac{1}{L}(R_2CV_2+V_2)}+C\ddot{V_2}=\dot{\frac{V_{in}}{R_1}}$$

Last edited: Nov 21, 2016
2. Nov 22, 2016

### Simon Bridge

Start by holding in your mind the definition of "natural response".
It may help to express voltages across components in terms of the currents through them.

3. Nov 22, 2016

### eehelp150

Is my work thus far correct?

4. Nov 22, 2016

### Simon Bridge

You didn't start by saying what "natural response" means.
You didn't say what reasoning you used to build those equations.
Therefore - I cannot tell.

5. Nov 22, 2016

### eehelp150

The example given by my professor:
$$\ddot{V_C}+\frac{1}{RC}\dot{V_C}+\frac{1}{LC}V_C=\frac{V_{in}}{RC}$$
$$\ddot{V_{natural}}+\frac{1}{RC}\dot{V_{natural}}+\frac{1}{LC}V_{natural}=0$$

6. Nov 22, 2016

### Simon Bridge

Did he not annotate his work? Do you understand what he did and why?
Do you know what "natural frequency" means here?

Good luck.

7. Nov 22, 2016

### eehelp150

No, he did not annotate his work.
His work is laid out something like this:
[Circuit]
By KCL:
Node 1: .....
$$\ddot{V_C}+\frac{1}{RC}\dot{V_C}+\frac{1}{LC}V_C=\frac{V_{in}}{RC}$$

Natural Response:
$$\ddot{V_{natural}}+\frac{1}{RC}\dot{V_{natural}}+\frac{1}{LC}V_{natural}=0$$

I do not understand what you mean by annotating my work. I am simply applying KCL to the two nodes. What is there to annotate other than "by KCL"?

The natural frequency of a RLC circuit is $$\frac{1}{\sqrt{LC}}$$

8. Nov 22, 2016

### Simon Bridge

See post #1 (I don't really care what your prof wrote.)
"By KVL" is only good enough if your notation is self-explanatory. eg. You provided a diagram... but...
I see a $V_1$ on your diagram, but I do not see a $V_{in}$ nor a $V_2$ - but $V_2$ and $V_{in}$ both appear in your equations.
I also see no C or L on the diagram, but they appear in your equations. I could guess $C=C_1$ and $L=L_1$ but how can I be sure given above?
It also means I cannot be sure that the $R_1$ and $R_2$ in your equations refers to the same things on the diagram - since it seems nothing else does.

More generally:
KVL and KCL are usually expressed in terms of the current.
You normally annotate the voltage and current sums by saying which loop/node they belong to - and the diagram is usually labelled with "loop 1" etc or the identity of the loops used is explicit in annotation. You provided no comment at all with your equations, just "here is what I did". OK. And...

On top of that, you have not said anything about how this relates to the thing you have to find.
After being asked twice, you answered by giving me an equation... that is called, "confusing the map for the territory".
ie. If I ask you, "what is 'daytime'" ... you'd (I hope) say that it is the time between sunup and sundown... if pressed you could spell it out as the time when the first limb of the Sun appears above the horizon at dawn to the time the last limb disappears below the horizon at dusk. There is an equation for this, which takes things like position on the earth, elevation, and time of year, as inputs, and outputs the the length of daytime as a time period - but the equation is not what the thing is - it's a map to let you predict things about it.
So, bearing that in mind, what is the natural frequency?

You can still get the answer without understanding it - but then you are not doing physics.

All this is about communicating in a clear way, and showing your understanding. You are in an international forum - people here come from a wide range of cultures and expectations. They do not have the references or context for your work that you do. And then you are asking for free-of-charge assistance... make it easy for people to help you. Note: if you do not understand, this is fine - we've all been there - the first step to wisdom and all that.

This communication thing is quite serious - it is not all that hard to figure out what is true compared with convincing someone else you are right - or just communicating what you've figured out. This is what most of science is about.
If your prof really just wrote a bunch of equations on the board apropos of nothing, then he's a hack and it is no wonder you get puzzled: you've had a bad role model. You cannot hope to understand anything that way. Seek out video lectures on the topic from someone competent, see how they explain stuff. Also look through some recent posts by people on these forums where they explain stuff with maths, and see how they do it. You should be learning the same sort of clarity and discipline. It takes a while to get used to - and there is a bit of an art to deciding how much to explain and when, but it's really no biggie.

9. Nov 22, 2016

### eehelp150

My bad. $V_1$ on the diagram should be $V_{in}$. C should be C1 and L should be L1. I drew the diagram on LTspice differently than what I had on paper (I worked it out on paper before typing it up). This should be the proper diagram:

I should have written Node V1, as I was applying nodal analysis.

The natural frequency is $w_o=\frac{1}{\sqrt{LC}}$
If you are asking for the numerical value of $w_o$, I do not have it. The prompt is to find the differential equation corresponding to the natural response. We are to find the answer in the form of something like: $\ddot{V_{natural}}+\frac{1}{RC}\dot{V_{natural}}+\frac{1}{LC}V_{natural}=0$

I realize that and I am extremely grateful for the help I've been given here. Can you recommend an online resource that involves solving (examples and solutions) RLC circuits using differential equations? The textbook we are supposed to use does not use DEs to solve RLC circuits. Many of the examples I've found online are the very basic equation and not provide any examples beyond the simplest RLC circuit.