RLC Circuit Second Order Differential Equation

AI Thread Summary
The discussion revolves around solving a second-order differential equation for an RLC circuit, specifically L C d²i/dt² + (L/R) di/dt + i = i_s(t). The original poster is struggling with the physics behind the equation and how to manipulate it to demonstrate its validity. Participants suggest applying Kirchhoff's Current Law and correctly identifying the relationships between voltage and current across the circuit components. They also guide the poster on how to derive the necessary expressions for the resistor, inductor, and capacitor to arrive at the correct form of the equation. The conversation emphasizes understanding both the mathematical and physical principles involved in RLC circuits.
SALAAH_BEDDIAF
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Homework Statement



Hi there guys I am new to this forum and i have a problem with a bit of cw. It's regarding an RLC circuit. I've come up with a picture (attached) that denotes the equation.

Homework Equations



I know the equation is L C \frac{d^2 i}{d t^2} + \frac{L}{R} \frac{di}{dt} + i =i_s (t) , t≥<br /> 0

The Attempt at a Solution



I do not really know where to start to show that, I've been given the equation and i was able to draw out the circuit but from there I do not really know where to go as I do not know physics of a circuit very well
 

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SALAAH_BEDDIAF said:

Homework Statement



Hi there guys I am new to this forum and i have a problem with a bit of cw. It's regarding an RLC circuit. I've come up with a picture (attached) that denotes the equation.

Homework Equations



I know the equation is L C \frac{d^2 i}{d t^2} + \frac{L}{R} \frac{di}{dt} + i =i_s (t) , t≥<br /> 0

The Attempt at a Solution



I do not really know where to start to show that, I've been given the equation and i was able to draw out the circuit but from there I do not really know where to go as I do not know physics of a circuit very well
attachment.php?attachmentid=56909&d=1363800929.png
Hello SALAAH_BEDDIAF. Welcome to PF !

(If your question is principally in regards to solving the differential equation, then you have posted this in the correct place. On the other hand, if it's principally in regards to understanding the physics leading to the differential equation, then we need to ask a moderator to move this thread to Introductory Physics.)

Is the quantity on the right hand side of the equation, the driving current? -- usually a sinusoidal such as \displaystyle \ i_s(t)=I_0\sin(\omega t)\ or \displaystyle \ i_s(t)=I_0\cos(\omega t)\
 
SammyS said:
attachment.php?attachmentid=56909&d=1363800929.png



Hello SALAAH_BEDDIAF. Welcome to PF !

(If your question is principally in regards to solving the differential equation, then you have posted this in the correct place. On the other hand, if it's principally in regards to understanding the physics leading to the differential equation, then we need to ask a moderator to move this thread to Introductory Physics.)

Is the quantity on the right hand side of the equation, the driving current? -- usually a sinusoidal such as \displaystyle \ i_s(t)=I_0\sin(\omega t)\ or \displaystyle \ i_s(t)=I_0\cos(\omega t)\

no I'm able to solve the equation, it's just that I'm ahving difficulty showing that the equation equals what i have given
 
SALAAH_BEDDIAF said:
no I'm able to solve the equation, it's just that I'm ahving difficulty showing that the equation equals what i have given

Start with
\displaystyle i_s(t)=i_1(t)+i(t)+i_2(t)\ .​
This comes from applying Kirchhoff's Current Law.

Then since the inductor, the resistor and the capacitor are all in parallel, the instantaneous voltage across anyone of these devices is equal to the instantaneous voltage across any other one of these devices.
 
I think i may have got it? here is what i have
Current through resistor: \frac{1}{R}

Current through inductor: \frac{1}{L} \int i\,dt

Current through Capacitor: C\frac{dv}{dt}

Applying kirchhoffs law i got

\frac{1}{R} + \frac{1}{L} \int i\,dt + C\frac{dv}{dt} = 0

Differentiating wrt t i got

\frac{1}{R} \frac{di}{dt} + \frac{i}{L} + C \frac{d^2 i}{dt^2} = 0

Rearranging to get the equation

LC \frac{d^2 i}{dt^2} + \frac{L}{R} \frac{di}{dt} + i = i_s (t)

I'm pretty sure i missed something here, anyone ought to help me out?
 
SALAAH_BEDDIAF said:
I think i may have got it? here is what i have
Current through resistor: \frac{1}{R}

Current through inductor: \frac{1}{L} \int i\,dt

Current through Capacitor: C\frac{dv}{dt}

Applying kirchhoffs law i got

\frac{1}{R} + \frac{1}{L} \int i\,dt + C\frac{dv}{dt} = 0

Differentiating wrt t i got

\frac{1}{R} \frac{di}{dt} + \frac{i}{L} + C \frac{d^2 i}{dt^2} = 0

Rearranging to get the equation

LC \frac{d^2 i}{dt^2} + \frac{L}{R} \frac{di}{dt} + i = i_s (t)

I'm pretty sure i missed something here, anyone ought to help me out?
Some of your expressions are not quite right.

Let \displaystyle \ v(t)\ be the instantaneous voltage across each component.

For the resistor, \displaystyle \ v(t)=i_1 R\ .\ \ Notice that in the figure, the current through R is labeled, i1 .

For the inductor, \displaystyle \ v(t)=L\frac{di}{dt}\ .\ \

For the capacitor, \displaystyle \ i_2=C\frac{dv}{dt}\ .\ \

Use these to find how i, i1, and i2 are related.
 
okay i got i_1 + i_2 + i = i_s (t)
using the equalities v = iR = L\frac{di}{dt} = \frac{Q}{C}
I got i_1 R = L \frac{di}{dt} → i_1 = \frac{L}{R}\frac{di}{dt}
not too sure how to transform C\frac{dv}{dt} → LC\frac{d^2 i}{dt^2}
 
SALAAH_BEDDIAF said:
okay i got i_1 + i_2 + i = i_s (t)
using the equalities v = iR = L\frac{di}{dt} = \frac{Q}{C}
I got i_1 R = L \frac{di}{dt} → i_1 = \frac{L}{R}\frac{di}{dt}
not too sure how to transform C\frac{dv}{dt} → LC\frac{d^2 i}{dt^2}


Use \displaystyle \ v(t)=L\frac{di}{dt}\ \ to find the derivative of \displaystyle \ v(t)\ .\ \

Then plug that into \displaystyle \ i_2=C\frac{dv}{dt}\ .\ \
 
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