Capacitors and Inductors

Unit for Capcitance

Farads (F)

Equation for Capacitor Energy

W = 1/2*C*V²

Capacitors are _____ circuits

Open

Equation for Capacitor Current

i = C * dV/dt

Equation for Capacitor Voltage

V = 1/C * ∫i dt

Unit for Inductance

Henry (H)

Equation for Inductor Energy

W = 1/2*L*i²

Capacitors are _____ circuits

Short

Equation for Inductor Voltage

V = L * di/dt

Equation for Inductor Current

i(t) = 1/L * ∫V(t) dt

Voltage ramped up 10x

Current ramped up 10x

Capacitors in Parallel

Ceq = C₁ + C₂+…

Capacitors in Series

C_{eq} = \frac{1}{\frac{1}{C_1} + \frac{1}{C_2}}

Inductors in Parallel

\frac{1}{L_{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + \dots + \frac{1}{L_n}

Inductors in Series

Leq = L₁+L₂+…

Golden Equation

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

Time Constant RC

\tau = R_{eq}C

Time Constant RL

\tau = \frac{L}{R_{eq}}

Phase 1: The Past (t < 0)

• RC: Find Voltage across open terminals

• RL: Find Current through short wire

Phase 2: The Transition (t = 0)

Capacitor Voltage cannot change instantly: v_{C}(0^{+})=v_{C}(0^{-})

Inductor Current cannot change instantly: i_L(0^+) = i_L(0^-)

Phase 3: The Future (t → infinity)

• Assume the switch is now in its new position forever.

• Draw the circuit in steady state again (Capacitors = Open, Inductors = Short).

Calculate the new final voltage v_C(\infty) or current i_L(\infty).

Finding the Time Constant \tau

• Kill the Sources: Turn off independent sources in your mental model (Voltage sources → Short Circuit; Current sources → Open Circuit).

• Look from the Component: Imagine standing at the terminals of the Capacitor or Inductor. What resistance do you see looking into the rest of the circuit?

• Calculate \tau:

  • RC Circuit: \tau = R_{eq} \cdot C

  • RL Circuit: \tau = \frac{L}{R_{eq}}

  • Req during Phase 1 (t>0) (so before shorted or opened)

Discharge Terminal for Capacitor

Positive Terminal

RLC Natural Parameters

\alpha (Neper Frequency/Damping):

Series RLC: \alpha = \frac{R}{2L}

Parallel RLC: \alpha = \frac{1}{2RC}

\omega_0 (Resonant Frequency):

• \omega_0 = \frac{1}{\sqrt{LC}}

Overdamped

\alpha > \omega_0

Critically Damped

\alpha = \omega_0

Underdamped

\alpha < \omega_0

Overdamped Equation

x(t) = x(\infty) + A_1 e^{s_1 t} + A_2 e^{s_2 t}

(Where s_1, s_2 = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2})

Critically Damped Equation

x(t) = x(\infty) + (A_1 + A_2 t) e^{-\alpha t}

Underdamped Equation

x(t) = x(\infty) + e^{-\alpha t} (B_1 \cos \omega_d t + B_2 \sin \omega_d t)

(Where \omega_d = \sqrt{\omega_0^2 - \alpha^2} is the damped frequency)

RLC Circuits: How to find constants (A_1, A_2 or B_1, B_2)?

Solve a 2x2 system of equations using two initial conditions at t=0^+:

1. The Value: x(0^+) (e.g., v_C(0^+) or i_L(0^+)). Found from t<0 analysis.

2. The Derivative: x'(0^+) (e.g., \frac{dv_C}{dt}(0^+) or \frac{di_L}{dt}(0^+)). Found using the component's main equation:

   • To find \frac{dv_C}{dt}(0^+), first find i_C(0^+) (using KCL) and use: \frac{dv_C}{dt}(0^+) = \frac{i_C(0^+)}{C}

   • To find \frac{di_L}{dt}(0^+), first find v_L(0^+) (using KVL) and use: \frac{di_L}{dt}(0^+) = \frac{v_L(0^+)}{L}