Transient Analysis of R-L-C Circuits by Classical Method
Time-domain differential equations, complementary functions, particular integrals, and complete dynamic solutions for first-order and second-order linear circuits.
2.1 & 2.2 First-Order Differential Equations & General Solution Structure
The Classical Complete Solution Principle
In the classical time-domain method, any linear circuit subjected to excitation is modeled by a non-homogeneous differential equation with constant coefficients: \[a_n \frac{d^n x}{dt^n} + \dots + a_1 \frac{dx}{dt} + a_0 x = f(t)\] The complete response \(x(t)\) is strictly the superposition of two components: \[x(t) = x_c(t) + x_p(t)\]
| Component | Mathematical Name | Physical Engineering Meaning | Key Characteristic |
|---|---|---|---|
| \(x_c(t)\) | Complementary Function | Transient / Natural Response: Energy exchange between circuit elements decaying to zero due to resistance. | Depends strictly on internal network structure (\(R, L, C\)), independent of source waveform. |
| \(x_p(t)\) | Particular Integral | Forced / Steady-State Response: Sustained output dictated by the external excitation as \(t \to \infty\). | Takes the same mathematical functional form as the forcing function \(f(t)\). |
Method of Undetermined Coefficients (PI Selection Rules)
| Excitation Type \(f(t)\) | Assumed Form of Particular Integral \(x_p(t)\) |
|---|---|
| Constant DC: \(V_0\) or \(I_0\) | \(K\) (constant) |
| Exponential: \(V_0 e^{-at}\) (where \(-a\) is not a natural root) | \(K e^{-at}\) |
| Exponential: \(V_0 e^{-at}\) (where \(-a\) is a simple natural root) | \(K t e^{-at}\) |
| Sinusoidal: \(V_m \cos(\omega t + \theta)\) or \(\sin(\omega t)\) | \(A \cos(\omega t) + B \sin(\omega t)\) |
2.3 Response of R-L and R-C Circuits (DC, Exponential, Sinusoidal)
1. DC Step Excitation
2. Exponential Excitation
For a series RC circuit excited by \(v(t) = V_0 e^{-at}\), the complete solution derived via classical ODE is: \[i(t) = \frac{V_0}{R(1 - a RC)} \left[ e^{-t/RC} - a RC \, e^{-at} \right]\] (See full derivation in Derivation Sheet Section 5).
3. Sinusoidal AC Excitation
When series RL is excited by \(v(t) = V_m \sin(\omega t + \theta)\) switched on at \(t = 0\): \[L \frac{di}{dt} + R i = V_m \sin(\omega t + \theta)\] Particular integral: \[i_p(t) = \frac{V_m}{Z} \sin(\omega t + \theta - \phi), \quad \text{where } Z = \sqrt{R^2 + (\omega L)^2}, \quad \phi = \tan^{-1}\left(\frac{\omega L}{R}\right)\] Complementary function: \(i_c(t) = A e^{-Rt/L}\). Using initial condition \(i(0^+) = 0\): \[i(0^+) = A + \frac{V_m}{Z} \sin(\theta - \phi) = 0 \implies A = -\frac{V_m}{Z} \sin(\theta - \phi)\] \[i(t) = -\frac{V_m}{Z} \sin(\theta - \phi) e^{-Rt/L} + \frac{V_m}{Z} \sin(\omega t + \theta - \phi)\]
2.4 Response of Series R-L-C Circuits (DC Excitation)
Applying KVL to a series RLC circuit excited by a constant voltage \(V\): \[L \frac{d^2 i}{dt^2} + R \frac{di}{dt} + \frac{1}{C} i = 0\] Characteristic equation: \[s^2 + \frac{R}{L} s + \frac{1}{LC} = 0 \iff s^2 + 2\alpha s + \omega_0^2 = 0\] Roots: \[s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}, \quad \alpha = \frac{R}{2L}, \quad \omega_0 = \frac{1}{\sqrt{LC}}\]
| Damping Regime | Mathematical Criterion | Roots Nature | Current Solution \(i(t)\) |
|---|---|---|---|
| Overdamped | \(\alpha > \omega_0 \iff R > 2\sqrt{\frac{L}{C}}\) | Real, unequal, negative (\(s_1 \ne s_2 < 0\)) | \(i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}\) |
| Critically Damped | \(\alpha = \omega_0 \iff R = 2\sqrt{\frac{L}{C}}\) | Real, equal, negative (\(s_1 = s_2 = -\alpha\)) | \(i(t) = (A_1 + A_2 t) e^{-\alpha t}\) |
| Underdamped | \(\alpha < \omega_0 \iff R < 2\sqrt{\frac{L}{C}}\) | Complex conjugates (\(-\alpha \pm j\omega_d\)) | \(i(t) = e^{-\alpha t}[A_1 \cos(\omega_d t) + A_2 \sin(\omega_d t)]\), \(\omega_d = \sqrt{\omega_0^2 - \alpha^2}\) |
\(R = 2\,\Omega, \quad L = 0.5\text{H}, \quad C = 1\text{F}\). Uncharged initially.
2. Roots: \[s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2} = -2 \pm \sqrt{4 - 2} = -2 \pm \sqrt{2}\] \[s_1 = -2 + \sqrt{2} \approx -0.586, \quad s_2 = -2 - \sqrt{2} \approx -3.414\] General solution: \(i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}\).
3. Initial Conditions: \[i(0^+) = 0 \implies A_1 + A_2 = 0 \implies A_2 = -A_1\] Across inductor at \(t = 0^+\): \[v_L(0^+) = V - R i(0^+) - v_C(0^+) = 100 - 0 - 0 = 100\text{ V}\] \[\frac{di}{dt}(0^+) = \frac{v_L(0^+)}{L} = \frac{100}{0.5} = 200\text{ A/s}\] Differentiating general solution: \[\frac{di}{dt}(0^+) = s_1 A_1 + s_2 A_2 = A_1(s_1 - s_2) = A_1[(-2+\sqrt{2}) - (-2-\sqrt{2})] = A_1(2\sqrt{2}) = 200\] \[A_1 = \frac{200}{2\sqrt{2}} = 50\sqrt{2} \approx 70.71\text{ A}, \quad A_2 = -50\sqrt{2}\text{ A}\]
2.6 & 2.7 Parallel R-L-C Circuits & Comparison with Series
Applying KCL to a parallel RLC circuit fed by a source current \(I_s\): \[\frac{v}{R} + \frac{1}{L}\int v \, dt + C \frac{dv}{dt} = I_s\] Differentiating: \[C \frac{d^2 v}{dt^2} + \frac{1}{R} \frac{dv}{dt} + \frac{1}{L} v = \frac{dI_s}{dt} \implies \frac{d^2 v}{dt^2} + \frac{1}{RC} \frac{dv}{dt} + \frac{1}{LC} v = 0\] Characteristic equation: \(s^2 + 2\alpha s + \omega_0^2 = 0\), where: \[\alpha = \frac{1}{2RC}, \quad \omega_0 = \frac{1}{\sqrt{LC}}\]
Comparison Table: Series RLC vs. Parallel RLC
| Feature / Parameter | Series R-L-C Circuit | Parallel R-L-C Circuit |
|---|---|---|
| Primary Variable | Current \(i(t)\) (common to all elements) | Voltage \(v(t)\) (common across all elements) |
| Damping Factor \(\alpha\) | \(\alpha = \frac{R}{2L}\) | \(\alpha = \frac{1}{2RC}\) |
| Resonant Frequency \(\omega_0\) | \(\omega_0 = \frac{1}{\sqrt{LC}}\) | \(\omega_0 = \frac{1}{\sqrt{LC}}\) (Identical!) |
| Overdamped Condition | \(R > 2\sqrt{\frac{L}{C}}\) | \(R < \frac{1}{2}\sqrt{\frac{L}{C}}\) |
| Critically Damped Condition | \(R = 2\sqrt{\frac{L}{C}}\) | \(R = \frac{1}{2}\sqrt{\frac{L}{C}}\) |
| Underdamped Condition | \(R < 2\sqrt{\frac{L}{C}}\) | \(R > \frac{1}{2}\sqrt{\frac{L}{C}}\) |
| Effect of Increasing \(R\) | Increases damping (suppresses oscillations) | Decreases damping (encourages oscillations) |