To find the natural response of a parallel RLC circuit, the following is required:
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The initial capacitor voltage \(v(0)\).
Since the capacitor voltage cannot change instantaneously,
\(\boxed{v\left(0^-\right) = v\left(0^+\right) = v\left(0\right) = V_0}\)
where \(v\left(0^-\right)\) is the voltage across the capacitor just before switching and \(v\left(0^+\right)\) is its voltage immediately after switching.
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The initial inductor current \(i(0)\).
Since the inductor current cannot change instantaneously,
\(\boxed{i\left(0^-\right) = i\left(0^+\right) = i\left(0\right) = I_0}\)
where \(i\left(0^-\right)\) is the current through the inductor just before switching and \(i\left(0^+\right)\) is its current immediately after switching.
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The initial value of the derivative of capacitor voltage.
Applying KCL at the top node of parallel RLC circuit,
\(\displaystyle \frac{v(0)}{R} + i(0) + C\frac{dv(0)}{dt} = 0\)
\(\boxed{\frac{dv(0)}{dt} = -\frac{1}{RC}\left[R\,i(0) + v(0)\right]}\)
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The damping ratio \(\zeta\).
The damping ratio \(\zeta\) for an RLC circuit is
\(\boxed{\zeta = \frac{\alpha}{\omega_0}}\)
where \(\omega_0\) is the undamped natural frequency and \(\alpha\) is the neper frequency.
The circuit is being excited by the energy initially stored in the capacitor and inductor.
Applying KCL at the top node,
\(\displaystyle \frac{v(t)}{R} + i(t) + C\frac{dv(t)}{dt} = 0\)
\(\displaystyle \frac{v}{R} + \frac{1}{L} \int_{-\infty}^{t} v\left(\tau\right)\,d\tau + C\frac{dv}{dt} = 0\)
To eliminate the integral, differentiate with respect to \(t\).
\(\displaystyle \frac{d}{dt} \left[\frac{v}{R} + \frac{1}{L} \int_{-\infty}^{t} v\left(\tau\right)\,d\tau + C\frac{dv}{dt} = 0\right]\)
\(\displaystyle \frac{1}{R}\frac{dv}{dt} + \frac{1}{L}v + C\frac{d^{2}v}{dt^2} = 0\)
\(\boxed{\frac{d^{2}v}{dt^2} + \frac{1}{RC}\frac{dv}{dt} + \frac{1}{LC}v = 0}\)
This second-order circuit can also be solved by using Laplace transforms.
To solve the parallel RLC second-order differential equation, replace the derivatives by \(s\) to obtain the characteristic equation.
\(\displaystyle s^2 + \frac{1}{RC} s + \frac{1}{LC} = 0\)
\(\displaystyle s = -\frac{1}{2RC} \pm \sqrt{\left(\frac{1}{2RC}\right)^2 - \frac{1}{LC}}\)
The two roots are
\(\boxed{s_1 = -\frac{1}{2RC} + \sqrt{\left(\frac{1}{2RC}\right)^2 - \frac{1}{LC}}}\)
\(\boxed{s_2 = -\frac{1}{2RC} - \sqrt{\left(\frac{1}{2RC}\right)^2 - \frac{1}{LC}}}\)
A more compact way of expressing the roots.
\(\boxed{\alpha = \frac{1}{2RC}}\)
\(\boxed{\omega_{0} = \frac{1}{\sqrt{LC}}}\)
\(\boxed{s_1 = -\alpha + \sqrt{\alpha^2 - \left.\omega_{0}\right.^2}}\)
\(\boxed{s_2 = -\alpha - \sqrt{\alpha^2 - \left.\omega_{0}\right.^2}}\)
where roots \(s_1\) and \(s_2\) are called natural frequencies, measured in nepers per second \((\textrm{Np}/\textrm{s})\); \(\omega_0\) is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second \((\textrm{rad}/\textrm{s})\); \(\alpha\) is the neper frequency or the damping factor, expressed in nepers per second \((\textrm{Np}/\textrm{s})\); \(R\) is the equivalent resistance; \(L\) is the equivalent inductance; and \(C\) is the equivalent capacitance.
Overdamped Case (\(\zeta > 1\))
The roots \((s_1 \neq s_2)\) are real and distinct.
The natural response is
\(\boxed{v\left(t\right) = A_1 e^{s_1 t} + A_2 e^{s_2 t}}\)
where \(A_1\) and \(A_2\) are constants to be determined from the initial conditions.
Determine the value of \(A_1\) and \(A_2\) constants.
The initial value \((t = 0)\) of the natural response.
\(\boxed{v(0) = A_1 + A_2}\)
The initial value \((t = 0)\) of the derivative of the natural response.
\(\displaystyle \frac{dv(t)}{dt} = A_1 s_1 e^{s_1 t} + A_2 s_2 e^{s_2 t}\)
\(\boxed{\frac{dv(0)}{dt} = A_1 s_1 + A_2 s_2}\)
Critically Damped Case (\(\zeta = 1\))
The roots \((s_1 = s_2)\) are real and repeated.
The natural response is
\(\boxed{v\left(t\right) = e^{-\alpha t}\left(A_1 + A_2 t\right)}\)
where \(A_1\) and \(A_2\) are constants to be determined from the initial conditions.
Determine the value of \(A_1\) and \(A_2\) constants.
The initial value \((t = 0)\) of the natural response.
\(\boxed{v(0) = A_1}\)
The initial value \((t = 0)\) of the derivative of the natural response.
\(\displaystyle \frac{dv(t)}{dt} = -\alpha e^{-\alpha t}\left(A_1 + A_2 t\right) + A_2 e^{-\alpha t}\)
\(\boxed{\frac{dv(0)}{dt} = -\alpha A_1 + A_2}\)
Underdamped Case (\(\zeta < 1\))
The roots are complex.
\(\displaystyle s_1 = -\alpha + \sqrt{-\left(\left.\omega_{0}\right.^2 - \alpha^2\right)} = -\alpha + j\omega_{d}\)
\(\displaystyle s_2 = -\alpha - \sqrt{-\left(\left.\omega_{0}\right.^2 - \alpha^2\right)} = -\alpha - j\omega_{d}\)
The natural response is
\(\boxed{v\left(t\right) = e^{-\alpha t} \left[A_1\cos\left(\omega_{d}t\right) + A_2\sin\left(\omega_{d}t\right)\right] = B e^{-\alpha t}\cos\left(\omega_{d}t - \theta\right)}\)
where \(A_1\), \(A_2\), and \(B\) are constants to be determined from the initial conditions, \(\theta\) is the phase angle, and \(\omega_d\) is the damped natural frequency, expressed in radians per second \((\textrm{rad}/\textrm{s})\).
\(\boxed{A_1 = B\cos(\theta),\quad A_2 = B\sin(\theta)}\)
\(\boxed{B = \sqrt{\left.A_{1}\right.^2 + \left.A_{2}\right.^2},\quad \theta = \tan^{-1}\left(\frac{A_2}{A_1}\right)}\)
\(\boxed{\omega_d = \sqrt{\left.\omega_{0}\right.^2 - \alpha^2}}\)
Determine the value of \(B\) and \(\theta\) constants.
The initial value \((t = 0)\) of the natural response.
\(\boxed{v(0) = B\cos(-\theta) = B\cos(\theta)}\)
\(\boxed{B = \frac{v(0)}{\cos(\theta)}}\)
The initial value \((t = 0)\) of the derivative of the natural response.
\(\displaystyle \frac{dv(t)}{dt} = -\alpha B e^{-\alpha t}\cos\left(\omega_{d}t - \theta\right) - \omega_d B e^{-\alpha t}\sin\left(\omega_{d}t - \theta\right)\)
\(\boxed{\frac{dv(0)}{dt} = -\alpha B\cos\left(\theta\right) + \omega_d B\sin\left(\theta\right)}\)
\(\displaystyle \frac{dv(0)}{dt} = -\alpha\,v(0) + \omega_d \tan\left(\theta\right)\,v(0)\)
\(\boxed{\theta = \tan^{-1}\left(\frac{1}{\omega_{d}\,v(0)}\left[\alpha\,v(0) + \frac{dv(0)}{dt}\right]\right)}\)