Source-free 2nd order lin. circuit

[EvLer]

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.

 

[Andrew Mason]

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;

I am not sure I understand what you are asking exactly, but perhaps this will help.

An LC circuit with resistance is simply a damped harmonic oscillator. The differential equation:

$$L\ddot x(t) + R\dot x(t) + \frac{1}{C}x(t) = 0$$

has the general solution:

$$x = A_0e^{-\gamma t}sin(\omega t+\phi)$$

where $\omega^2 = \omega_0^2 - \gamma^2$ and
$\omega_0^2 = 1/LC$ and
$\gamma = R/2L$

The relationship between V and I in an RLC circuit is:

$$V = IZ$$ where is the impedance: $Z^2 = R^2 + (\omega L - 1/\omega C)^2$

AM

 

[thepatientmental]

You are correct in thinking that the equations for Vc(t) and iL(t) are the same. This is because in a source-free circuit, the voltage and current are related by the same differential equation.

In this case, the solution to the differential equation will be the same for both Vc(t) and iL(t). This is because the circuit elements (resistor, inductor, and capacitor) are the same for both variables, so the coefficients in the differential equation will be the same.

Therefore, the roots of the characteristic equation will also be the same for both Vc(t) and iL(t). This means that the solutions for Vc(t) and iL(t) will have the same form, just with different constants. This is why you see the same solution for both variables.

So, in conclusion, you are not wrong. The equations for Vc(t) and iL(t) will be the same in a source-free circuit, and this is because of the relationship between voltage and current in this type of circuit.