#43 Transient Analysis #43 — Step response and RC charging

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

1. 1. Natural, forced, and complete response

   

   Apply a constant source at t=0

   The natural response comes from initially stored energy

   The forced response is the source-driven steady state

   Complete response = natural + forced response

   The natural component decays exponentially

   The forced component establishes the final value

   A first-order circuit has one time constant

   The pre-switch condition determines the start

   Long-time analysis determines the final value

   Narration transcript

   So far we have studied the natural response, where a circuit releases its stored energy with no external source. Now we introduce the step response. What happens when we suddenly apply a constant voltage or current source to a circuit at time t equals zero? The complete response has two parts. The natural response is the familiar exponential decay driven by initial stored energy. The forced response is the steady state that the source drives the circuit toward. The complete response equals the natural response plus the forced response. This is the most important idea in first order circuit analysis.

2. 2. General formula and three-step method

   

   $x(t)=x(\infty )+[x(0⁺)-x(\infty )]e^{(}-t/\tau)$

   x(0⁺): value immediately after switching

   x(∞): long-time steady-state value

   τ: time constant that controls transient speed

   1) Find x(0⁺)

   2) Find x(∞)

   3) Find τ

   For RC, τ=R_eq C

   For RL, τ=L/R_eq

   Narration transcript

   Here is the general formula for any first order circuit with a step input. x of t equals x of infinity plus the quantity x of zero minus x of infinity times e to the negative t over tau. Let me explain each term. x of infinity is the final steady state value, the value the circuit settles to after a long time. x of zero is the initial value at the moment of switching. And tau is the time constant, R C for capacitor circuits or L over R for inductor circuits. The three step method to solve any first order step response is: step one, find x of zero from the initial conditions. Step two, find x of infinity by analyzing the circuit at steady state, where capacitors are open circuits and inductors are short circuits. Step three, find tau by computing the equivalent resistance seen by the storage element. Then plug all three values into the general formula. That is it.

3. 3. The three values for an RC charge

   

   The switch closes at t=0

   The capacitor starts uncharged

   $v_{\mathrm{C}}(0⁺)=0 V$

   At long time the capacitor is an open circuit

   Current falls to zero

   $v_{\mathrm{C}}(\infty )=V_{\mathrm{S}}$

   The resistance seen by the capacitor is R

   τ=RC

   All three values for the formula are ready

   Narration transcript

   Let us apply this formula to an R C charging circuit. We have a voltage source V S connected through a switch to a resistor R and capacitor C in series. The switch closes at t equals zero. Before switching, the capacitor is uncharged, so V C of zero equals zero. After a long time, the capacitor fully charges to the source voltage, so V C of infinity equals V S. The time constant is tau equals R C. These are our three values: initial is zero, final is V S, and tau is R C.

4. 4. RC charging voltage and current

   

   Choose x=v_C in the general formula

   v_C(t)=V_S+[0−V_S]e⁽−t/RC)

   v_C(t)=V_S(1−e⁽−t/RC))

   Voltage rises from zero to V_S

   i(t)=(V_S/R)e⁽−t/RC)

   Current falls from V_S/R to zero

   At t=τ, v_C is 63.2 percent of final

   At t=5τ, v_C reaches 99.3 percent

   Voltage rises while current falls

   Narration transcript

   Plugging into the general formula: V C of t equals V S plus the quantity zero minus V S times e to the negative t over R C. Simplifying, V C of t equals V S times the quantity one minus e to the negative t over R C. This is the R C charging equation. The voltage starts at zero and rises exponentially toward V S. At t equals tau, V C reaches sixty three point two percent of V S. At five tau, it reaches ninety nine point three percent, which is essentially fully charged. The current follows: i of t equals V S over R times e to the negative t over R C. The current starts at a maximum of V S over R and decays exponentially to zero. Notice: the voltage rises as the current falls. Energy flows from the source through the resistor into the capacitor.

5. 5. Numerical 12 V RC charging example

   

   $R=10 k\Omega, C=100 \muF, V_{\mathrm{S}}=12 V$

   R=10 000 Ω and C=100×10⁻⁶ F

   τ=RC=1 s

   $v_{\mathrm{C}}(t)=12(1-e^{(}-t)) V$

   $i(t)=12/10 000\cdot e^{(}-t) A$

   i(t)=1.2e⁽−t) mA

   $v_{\mathrm{C}}(1 s)=7.59 V$

   $v_{\mathrm{C}}(3 s)=11.40 V$

   $v_{\mathrm{C}}(5 s)=11.92 V$

   Narration transcript

   Let us work a numerical example. Given R equals ten kilohms, C equals one hundred microfarads, and V S equals twelve volts. Step one: the initial condition is V C of zero equals zero volts, the capacitor starts uncharged. Step two: the final value is V C of infinity equals twelve volts. Step three: the time constant is tau equals R C equals ten thousand times one hundred times ten to the negative six, which equals one second. Now apply the formula: V C of t equals twelve times one minus e to the negative t. At t equals one second, V C equals twelve times one minus e to the minus one, which is twelve times zero point six three two, giving seven point five nine volts. At t equals three seconds, V C equals twelve times one minus e to the minus three, which equals eleven point four volts. At t equals five seconds, V C equals eleven point nine two volts, essentially twelve. The current: i of t equals twelve over ten thousand times e to the negative t, which is one point two milliamps times e to the negative t.

6. 6. Step-response solution checks

   

   Draw the initial and final circuits first

   The energy-storage variable is continuous

   $v_{\mathrm{C}}(0⁺)=v_{\mathrm{C}}(0⁻)$

   A capacitor is open at steady state

   Verify the x(0⁺), x(∞), and τ trio

   Test the result at t=0 and t→∞

   Check that the exponential argument is dimensionless

   One tau is 63.2 percent; five tau is 99.3 percent

   Next: apply the same method to an RL step response

   Narration transcript

   Let us summarize. The general first order step response formula is: x of t equals x of infinity plus x of zero minus x of infinity times e to the negative t over tau. The three step method: find x of zero, find x of infinity, find tau, then substitute. For R C charging: V C of t equals V S times one minus e to the negative t over R C. Voltage rises from zero to V S. Current decays from V S over R to zero. At one tau, sixty three percent complete. At five tau, ninety nine percent complete. Next, we will apply the same formula to the R L step response.