RC Circuits and RL Circuits Study Notes

1. Classification of Differential Equations

• Differential Equation Types:
  • Second Order Ordinary Differential Equations:
  • Highest number of derivatives: two time derivatives.
  • Example form: For LC circuit, $\frac{d^2q}{dt^2} = -\frac{1}{LC}q$.
  • Key concepts involved:
    • Newton’s second law relates acceleration to position: $a_x = \frac{d^2x}{dt^2}$.
    • Hooke’s law relates position to force: $F = -kx$.
  • First Order Ordinary Differential Equations:
  • Differential equations characterized by first derivatives.
  • Used for RC and RL circuits.
  • Examples include how current varies with charge in capacitors and inductors.

2. RC Circuits

• Definition: Circuits formed by resistors and capacitors.
• Applications: Used in time-keeping mechanisms.
• Basic Configuration: Involves one resistor and one capacitor.

2.1 RC Circuit - Discharging a Capacitor

• Initial Setup: Fully charged capacitor in a circuit with an open switch. When the switch is closed, the capacitor discharges.
• Key Variables:
  • $Q$: Charge on the capacitor
  • $Q_0$: Initial charge
• Loop Law Application: Kirchoff's loop law states:
  • $\Delta V<em>C + \Delta V</em>R = 0$,
  • where $\Delta V<em>C = \frac{Q}{C}$ and $\Delta V</em>R = iR$.
  • For clockwise current flow: $\frac{Q}{C} - iR = 0$ (Equation 6).
• Current and Charge Relationship: During discharge:
  • $i = -\frac{dQ}{dt}$ (Equation 9).
• Differential Equation Formation: Substituting into loop law gives:
  • $\frac{Q}{C} + \frac{dQ}{dt} R = 0$.
  • Rearranging yields: $\frac{dQ}{dt} + \frac{Q}{RC} = 0$ (Equation 11).

Solving the Differential Equation for Discharging Capacitor

1. Separate Variables:
   • $\frac{dQ}{Q} = -\frac{1}{RC}dt$ (Equation 12).
2. Integrate Both Sides:
   • From time $t=0$ (initial charge $Q_0$) to time $t$ (charge $Q$).
   • $\int<em>{Q</em>0}^{Q} \frac{1}{Q}dQ = -\frac{1}{RC}\int_{0}^{t}dt$ (Equations 13-14).
3. Results of Integration:
   • $\ln(Q) - \ln(Q_0) = -\frac{t}{RC}$ (Equation 16).
   • Simplifying gives: $\ln\left(\frac{Q}{Q_0}\right) = -\frac{t}{RC}$ (Equation 17).
4. After Exponentiation:
   • $\frac{Q}{Q_0} = e^{-\frac{t}{RC}}$ (Equation 18).
   • Thus, $Q = Q_0 e^{-\frac{t}{RC}}$ (Equation 19).

• Time Constant Definition:
  • The time constant $\tau = RC$ (Equation 20).
• Voltage Across Capacitor:
  • $\Delta V<em>C = \Delta V</em>{C,0} e^{-\frac{t}{\tau}}$ (Equation 23).
• Current through Resistor:
  • $i = - \frac{dQ}{dt}$ (Equation 24).
  • After calculation: $i = I_0 e^{-\frac{t}{\tau}}$ (Equation 29).

2.2 RC Circuit - Charging a Capacitor

• Charging Process: When charging, a battery is added; capacitor charges until potential difference equals battery voltage (emf).
• Loop Law Application:
  • $\epsilon + \Delta V<em>R + \Delta V</em>C = 0$ (Equation 30).
  • For charging: $\epsilon - iR - \frac{Q}{C} = 0$ (Equation 31).
• Defining Current Relative to Charge:
  • $i = \frac{dQ}{dt}$ (Equation 32).
• Differential Equation Formation for Charging:
  • Rearranging provides: $\frac{dQ}{dt} = \frac{1}{R}(\epsilon - \frac{Q}{C})$ (Equation 34).

Solving the Differential Equation for Charging Capacitor

1. Separate Variables:
   • $\frac{dQ}{\epsilon - \frac{Q}{C}} = \frac{1}{R}dt$ (Equation 35).
2. Integration Boundaries:
   • Integrate from $0$ to $Q$ on the left, and from $0$ to $t$ on the right (Equation 38).
3. Results of Integration:
   • Steps yield: $Q = Q<em>f(1 - e^{-\frac{t}{RC}})$ (Equation 44), where $Q</em>f = C\epsilon$ is the final charge on the capacitor.

• Voltage Over Time During Charging:
  • $\Delta V_C = \epsilon(1 - e^{-\frac{t}{RC}})$ (Equation 46).

3. RL Circuits

• Definition: Circuits formed by resistors and inductors.

3.1 Current Increase in RL Circuit

• Setup: Circuit includes resistor, inductor, and battery. On closing the switch, current increases.
• Kirchoff's Loop Law Application:
  • $\epsilon - iR - L\frac{di}{dt} = 0$ (Equation 47).
• Rearranging for Differential Equation:
  • $\frac{di}{dt} = \frac{\epsilon - iR}{L}$ (Equation 48).
  • It is a first order differential equation.

Solving the Differential Equation for Current Increase

1. Separating Variables:
   • $\int \frac{di}{\epsilon - \frac{iR}{L}} = \int dt$ (Equation 49-50).
2. Integrate Both Sides:
   • Find current function $i = \frac{\epsilon}{R}(1 - e^{-\frac{t}{\tau}})$ where $\tau = \frac{L}{R}$ (Equation 62).

• Final Current Value:
  • Approaches: $I = \frac{\epsilon}{R}$ (Equation 63).

3.2 Current Decay in RL Circuit

• Scenario: Removing the battery leads to current decreasing over time.
• Kirchoff's law for Decay Situation:
  • The governing equation is similar to the charging case, leading to an expression of the form:
  • $i = I<em>0 e^{-\frac{t}{\tau}}$, where $I</em>0$ is the initial current (Equation 73).