#50 Transient Analysis #50 — Complete response of a driven parallel RLC

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Separate natural and full response

   

   Natural response comes only from stored energy

   An external source adds a forced response

   Full response = forced response + natural response

   The forced part says where the circuit will settle

   The natural part says how it gets there

   The source sets the destination; the poles set the path

   Narration transcript

   So far, we studied natural response. That means the circuit was moving only because of stored energy. Full response is different. Now an external source drives the circuit, so the total response becomes the sum of two parts: the forced response and the natural response. The forced part tells us where the circuit wants to settle because of the source. The natural part tells us how the circuit gets there.

2. 2. Parallel RLC driven by a current step

   

   Apply a DC current step I_s to a parallel RLC

   Response variable: i_L(t)

   As t→∞, C behaves as an open circuit

   As t→∞, L behaves as a short circuit

   v(∞)=0 and i_R(∞)=0

   $i_{\mathrm{L}}(\infty )=I_{\mathrm{s}}$

   The final value is the constant forced component

   Narration transcript

   Consider a parallel R L C circuit driven by a D C current step source. We will observe the inductor current i sub L of t. At long time, the capacitor behaves like an open circuit and the inductor behaves like a short circuit. That means the node voltage goes to zero, the resistor current goes to zero, and the entire source current flows through the inductor. So the final value is i sub L of infinity equals I sub s.

3. 3. Write the full-response forms

   

   Start with the common final value x(∞)

   Overdamped: x=x∞+A₁e^s₁t+A₂e^s₂t

   Critical: x=x∞+(A₁+A₂t)e^−αt

   Underdamped: x=x∞+e^−αt[B₁cosω_dt+B₂sinω_dt]

   Pole type selects the transient shape

   The source usually determines x∞

   All transient terms decay to zero

   Narration transcript

   The key idea is simple: keep the same final value, then add the transient shape around it. For an overdamped case, x of t equals x infinity plus two real exponential terms. For the critically damped case, x of t equals x infinity plus a linear term times e to the minus alpha t. For the underdamped case, x of t equals x infinity plus e to the minus alpha t times a sinusoidal combination. So the pole type still controls the transient shape. The source mainly sets the steady final value.

4. 4. Find the two constants from initial conditions

   

   A second-order response needs two independent conditions

   Condition 1: x(0⁺)

   Condition 2: x′(0⁺)

   Continuity laws provide x(0⁺)

   KCL or KVL provides the initial derivative

   x∞ → correct form → two conditions → constants

   Recheck both initial and final values in the result

   Narration transcript

   How do we find the constants? Exactly the same way as before: use the initial conditions. For a second-order response, we need two independent conditions, typically x of zero plus and x prime of zero plus. Those two conditions determine the coefficients in the transient term. So the full response structure is easy: first find x of infinity, then choose the correct transient form, then solve the constants from the initial conditions.

5. 5. Interpret an underdamped full response

   

   Let the current settle to a nonzero i_L(∞)

   Complex poles create a transient oscillation

   $i_{\mathrm{L}}=i_{\mathrm{L}}(\infty )+e^{-\alphat}[B₁\cos \omega_{\mathrm{d}}t+B₂\sin \omega_{\mathrm{d}}t]$

   The waveform rings around the final value

   The ±Ke^−αt envelopes bound the oscillation

   As t→∞, the transient term vanishes

   Result: i_L(t)→i_L(∞)=I_s

   Narration transcript

   Here is one underdamped full-response example. The current settles to a non-zero final value, but it does not go there directly. Instead, it oscillates around the final value while the envelope decays. This picture is the most important intuition for the full second-order step response: the source fixes the destination, and the poles decide the path.

6. 6. Summarize the full second-order response

   

   Full response = forced + natural

   For a DC step, the forced response is often a constant final value

   Poles produce an over-, critical-, or underdamped path

   Two conditions determine the transient constants

   1) x∞ 2) form 3) x(0⁺),x′(0⁺)

   The full response starts at the initial state and ends at the forced state

   Next: series RLC and circuit duality

   Narration transcript

   Let us summarize. Full response equals forced response plus natural response. In a D C step problem, the forced response often becomes the steady final value. The transient part still follows the same second-order cases we already learned: overdamped, critical, or underdamped. To solve any full response, find x infinity, choose the transient form from the poles, and compute the constants from the initial conditions. Next lesson, we connect this idea to the series R L C form and duality.