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Sourcefree 2nd order lin. circuit 
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#1
Mar2605, 10:19 AM

P: 460

RLC second order linear network question:
So, we are given this equation which is the same for Vc(t) and iL(t) expressed as x(t): 2nd deriv of x(t) + R/L(1st deriv of x(t)) + 1/(LC)(x(t)) = 0; And in one of the problems it asks to find both equation for the Vc(t) and iL(t) for t < 0, and now I am confused, it seems to me that they are the same, since the solution is the same for both of them: aS^2 + bS + c = 0; because the coefficients are the same from differential equation, so there are the same roots for Vc(t) and iL(t), and roots are w/t imaginary part, just reals. Am I wrong? Thanks a lot. 


#2
Mar2805, 12:55 PM

Sci Advisor
HW Helper
P: 6,679

An LC circuit with resistance is simply a damped harmonic oscillator. The differential equation: [tex]L\ddot x(t) + R\dot x(t) + \frac{1}{C}x(t) = 0[/tex] has the general solution: [tex]x = A_0e^{\gamma t}sin(\omega t+\phi)[/tex] where [itex]\omega^2 = \omega_0^2  \gamma^2[/itex] and [itex]\omega_0^2 = 1/LC[/itex] and [itex]\gamma = R/2L[/itex] The relationship between V and I in an RLC circuit is: [tex]V = IZ[/tex] where is the impedance: [itex]Z^2 = R^2 + (\omega L  1/\omega C)^2[/itex] AM 


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