To find the complete response of a series 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.
\(\displaystyle i(0) = C\frac{dv(0)}{dt}\)
\(\boxed{\frac{dv(0)}{dt} = \frac{i(0)}{C}}\)
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The final (steady state) capacitor voltage \(v(\infty)\).
Use the fact that the capacitor acts like an open circuit to DC at steady state.
\(\boxed{v(\infty) = V_s}\)
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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 KVL around the loop for \(t > 0\),
\(\displaystyle -V_s + R\,i(t) + L\frac{di(t)}{dt} + v(t) = 0\)
But \(\displaystyle i(t) = C\frac{dv(t)}{dt}\),
\(\displaystyle -V_s + RC\frac{dv}{dt} + LC\frac{d^{2}v}{dt^2} + v(t) = 0\)
\(\boxed{\frac{d^{2}v}{dt^2} + \frac{R}{L}\frac{dv}{dt} + \frac{1}{LC}v = \frac{V_s}{LC}}\)
This second-order circuit can also be solved by using Laplace transforms.
Since the characteristic equation for the series RLC circuit is not affected by the presence of the DC source, the roots are the same as the roots obtained for the source-free series RLC circuit.
\(\boxed{\alpha = \frac{R}{2L}}\)
\(\boxed{\omega_{0} = \frac{1}{\sqrt{LC}}}\)
\(\boxed{s_1 = -\alpha + \sqrt{\alpha^2 - \left.\omega_{0}\right.^2}}\)
\(\boxed{s_2 = -\alpha - \sqrt{\alpha^2 - \left.\omega_{0}\right.^2}}\)
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.
The solution (or complete response) to the series RLC circuit with DC source consists of the transient response \(v_{t}(t)\) and the steady-state response \(v_{ss}(t)\).
\(\displaystyle v(t) = v_{t}(t) + v_{ss}(t)\)
The form of the transient response \(v_{t}(t)\) is the same as the form of the solution (or natural response) obtained for the source-free series RLC circuit.
The steady-state response \(v_{ss}(t)\) is the final value of \(v(t)\).
\(\displaystyle v_{ss}(t) = v(\infty) = V_s\)
Overdamped Case (\(\zeta > 1\))
The roots \((s_1 \neq s_2)\) are real and distinct.
The complete response is
\(\boxed{v\left(t\right) = v\left(\infty\right) + A_1 e^{s_1 t} + A_2 e^{s_2 t}}\)
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 complete response.
\(\boxed{v(0) = v(\infty) + A_1 + A_2}\)
The initial value \((t = 0)\) of the derivative of the complete 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 complete response is
\(\boxed{v\left(t\right) = v\left(\infty\right) + e^{-\alpha t}\left(A_1 + A_2 t\right)}\)
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 complete response.
\(\boxed{v(0) = v(\infty) + A_1}\)
The initial value \((t = 0)\) of the derivative of the complete 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 complete response is
\(\boxed{v\left(t\right) = v\left(\infty\right) + e^{-\alpha t} \left[A_1\cos\left(\omega_{d}t\right) + A_2\sin\left(\omega_{d}t\right)\right] = v\left(\infty\right) + B e^{-\alpha t}\cos\left(\omega_{d}t - \theta\right)}\)
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 complete response.
\(\boxed{v(0) = v(\infty) + B\cos(-\theta) = v(\infty) + B\cos(\theta)}\)
\(\boxed{B = \frac{v(0) - v(\infty)}{\cos(\theta)}}\)
The initial value \((t = 0)\) of the derivative of the complete response.
\(\displaystyle \frac{dv(t)}{dt} = -\alpha B e^{-\alpha t}\cos\left(\omega_{d}t - \theta\right) - \omega_d B e^{-\alpha t}\sin\left(\omega_{d}t - \theta\right)\)
\(\boxed{\frac{dv(0)}{dt} = -\alpha B\cos\left(\theta\right) + \omega_d B\sin\left(\theta\right)}\)
\(\displaystyle \frac{dv(0)}{dt} = -\alpha\left[v(0) - v(\infty)\right] + \omega_d \tan\left(\theta\right)\left[v(0) - v(\infty)\right]\)
\(\boxed{\theta = \tan^{-1}\left(\frac{1}{\omega_{d}\left[v(0) - v(\infty)\right]}\left[\alpha\left[v(0) - v(\infty)\right] + \frac{dv(0)}{dt}\right]\right)}\)