So I'm fairly new to calculus, and integrals, and I'm working out an RLC circuit with this formula:
(if it doesn't show up it's here: http://latex.codecogs.com/gif.latex?V_{0}&space;cos(\omega&space;t)=RI+L\frac{dI_{L}}{dt}+\frac{1}{C}\int&space;Idt)
Where V0cos(ωt) is the AC voltage, ω is the angular velocity (2∏f), I is the total current, and L R and C are the respective component values.
*
I know there are better ways to do this, finding voltages across the components and such, but this way intrigues me. And I don't understand it, and I'd like to. That equation looks all fine and all, so the next step is to simplify the integral. I must be missing something here, because when I simplified it I got:
(If it doesn't show up it's here: http://latex.codecogs.com/gif.latex?V_{0}&space;cos(\omega&space;t)=RI+L\frac{dI_{L}}{dt}+\frac{It}{C_{0}}+C)
*
*Note: In the above schematics, ignore the subscript "L" under the I, it should be simply I.
Where C0 is the capacitor value and C is the constant of integration.
The answer in the book on the other hand, shows the answer as this:
(If it doesn't show up it's here: http://latex.codecogs.com/gif.latex?\omega&space;V_{0}&space;cos(\omega&space;t)=L\frac{d^{2}I}{dt^{2}}-R\frac{dI}{dt}+\frac{1}{C}I)
Things that are interesting to me is the extra ω in front and the new stuff being multiplied by R (I thought would still be I but now it's dI/dt).
If anyone can bear with me through all this and help me understand what they did and why, I would be most thankful! Another note by the way, the book says this last function is a "linear second-order nonhomogeneous differential equation" which scares me... Thanks.
(if it doesn't show up it's here: http://latex.codecogs.com/gif.latex?V_{0}&space;cos(\omega&space;t)=RI+L\frac{dI_{L}}{dt}+\frac{1}{C}\int&space;Idt)
Where V0cos(ωt) is the AC voltage, ω is the angular velocity (2∏f), I is the total current, and L R and C are the respective component values.
*
I know there are better ways to do this, finding voltages across the components and such, but this way intrigues me. And I don't understand it, and I'd like to. That equation looks all fine and all, so the next step is to simplify the integral. I must be missing something here, because when I simplified it I got:
(If it doesn't show up it's here: http://latex.codecogs.com/gif.latex?V_{0}&space;cos(\omega&space;t)=RI+L\frac{dI_{L}}{dt}+\frac{It}{C_{0}}+C)
*
*Note: In the above schematics, ignore the subscript "L" under the I, it should be simply I.
Where C0 is the capacitor value and C is the constant of integration.
The answer in the book on the other hand, shows the answer as this:
(If it doesn't show up it's here: http://latex.codecogs.com/gif.latex?\omega&space;V_{0}&space;cos(\omega&space;t)=L\frac{d^{2}I}{dt^{2}}-R\frac{dI}{dt}+\frac{1}{C}I)
Things that are interesting to me is the extra ω in front and the new stuff being multiplied by R (I thought would still be I but now it's dI/dt).
If anyone can bear with me through all this and help me understand what they did and why, I would be most thankful! Another note by the way, the book says this last function is a "linear second-order nonhomogeneous differential equation" which scares me... Thanks.