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# LC natural response derivation 2

Starting from the differential equation, we come up with a proposed exponential solution and plug it into the equation. This gives us a characteristic equation. A "natural frequency" emerges. Created by Willy McAllister.

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• this is a tiny thing, but do engineers not use i for imaginary number because i is already used for current so often?
• That is exactly right. We (engineers) use i for current and j for the imaginary unit (sqrt(-1)). The i comes from Ampere himself. When he talked about current in his native French he called it "intensité du courant".
• Hi. I'm sorry, can you explain me why is it Vc+ Vl ? Isn't it Ic=IL and Vc=Vl because they are in parallel? if we do Vc+ Vl, instead of Vc - Vl, it doesn't make sense to me. Hoping for your reply, thank you
(1 vote)
• This is slightly tricky because of the Sign Convention for Passive Components. You are correct that VC = VL = little v. Little v for both is measured from the bottom to the top (+ is at the top).

BUT, the currents are not identical, it is not the case that IL = IC. They are in opposite directions. If we point the current arrow DOWN for the inductor, it points UP for the capacitor. This introduces a negative sign when applying the I-V equation to the capacitor.

v = L di/dt

-i = C dv/dt

This is a weird case for such a super-simple circuit. It introduces a quirk in the I-V equation for one of them.
• Why the K variables in the Superposition of i are K1 and K2? Why not just K?
(1 vote)
• As we solve this problem, we came up with an s^2 term. To solve for s we have to take a square root. Square roots always have two possible solutions, one with a plus sign, and one with a negative sign. This is where the terms s1 and s2, and eventually K1 and K2 come from. We can't choose one of the solutions as the right one, so we carry through the rest of the solution with both alternatives.
• I'm sorry, but how do you get the equation at (eq. with the green color)?
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• The green equation at :56 comes from dividing both sides of the previous equation by L.
I just noticed an error in the green equation, the denominator of the left-hand second derivative term should be dt^2, instead of just plain dt.
• @ s=1/t, and the unit is rad/sec , but if you multiply rad/sec times sec , the unit is rad and you said st has no units
and what is meant by natural frequency , what is special about it? and why is the characteristics equation named that name?
(1 vote)

We use the term "natural" frequency to describe the way something vibrates when there is no external energy applied (when there is no "driving" force).

A good example is a pendulum. When you pull the pendulum's weight to the side you are initializing it with some starting energy. When you let go of the weight the pendulum swings back and forth with some period and some amplitude. Since you are no longer touching the pendulum there is no additional energy coming in from outside, and it will do whatever it does "naturally". In this state we would call its frequency the "natural frequency". If you grab the weight and move it back and forth at your will, that is no longer its natural frequency but rather its driven frequency.
• In the last equation, why is it possible to add the two possible solutions for `s`?
Shouldn't it be that we attempt `i_1` and `i_2` with the respective `s`'s? (And most probably get only one valid answer?)
• Good question. Did you try to solve for i_1 by itself? What did you get?

The characteristic equation produced two roots, (s_1 and s_2). We propose a general solution by tossing both of them into the proposed answer. If one of them is not needed to get a valid result (a function that solves the original differential equation) then the useless s will disappear during the math. If both s terms are needed to get a valid answer they will both hang around to the end.

To review, there are two approaches you can take.
1. Try one root, try the other root, try both roots together. Do as many as needed to get a valid answer.

2. Try both roots together. If only one is needed count on the math to make the other one melt away. If both are needed then you've done the math once.
• What are the other types of frequencies, besides "natural frequency"?
(1 vote)
• We use the term 'natural frequency' when focussed on the few components right in front of us, without regard for any signals coming in from outside. 'Natural frequency' is what the LC (or RC, or RL, or RLC) does 'naturally' when nothing else is driving the circuit.

If we attach a signal source it can introduce its own frequency signal to the circuit. We would use the term 'frequency' to talk about that.
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• At , the square of t in denominator is missing. Is it ok or a mistake?
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• Good eye. I made a notation error by leaving out the ^2 on the bottom of the second derivative term. This is a small flaw, it doesn't harm the derivation.
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• This may be a math question. Why there are two solutions (positive & nagative).
I was not able to see two answers when dealing with imaginary numbers in Sal lectures.
(1 vote)
• Excuse me. do you know which lesson is Energy Storage in LC Circuits and Electromagnetic Oscillations ? I want to learn this lesson.
(1 vote)