21.6 DC Circuits Containing Resistors and Capacitors

RC circuits blend resistance and capacitance, creating fascinating electrical behavior. The time constant, τ = RC, is key, determining how quickly capacitors charge or discharge. This concept is crucial for understanding voltage changes and applications in technology.

RC circuits find use in everyday devices like camera flashes and touchscreens. They exhibit transient and steady-state behaviors, with the electric field playing a vital role in energy storage and release. These principles are essential for grasping DC circuit dynamics.

DC Circuits with Resistors and Capacitors

Time constant calculation for RC circuits

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• Measure of how quickly a capacitor charges or discharges in an RC circuit
• Calculated by multiplying the resistance $R$ and the capacitance $C$: $\tau = RC$
  • Measured in seconds (s)
• Represents the time for the capacitor voltage to reach approximately 63.2% of its final value when charging or discharging
  • After one time constant, the capacitor voltage is $V_C = V_f(1 - e^{-1}) \approx 0.632V_f$, where $V_f$ is the final voltage
  • After five time constants, the capacitor is considered fully charged or discharged (99.3% of final value)
• Examples:
  • In an RC circuit with a 100 kΩ resistor and a 10 μF capacitor, the time constant is $\tau = (100 \times 10^3 \Omega)(10 \times 10^{-6} F) = 1$ s
  • If a 220 kΩ resistor is used with a 47 nF capacitor, the time constant is $\tau = (220 \times 10^3 \Omega)(47 \times 10^{-9} F) \approx 10.3$ ms

Voltage changes during capacitor charge/discharge

• Charging in an RC circuit:
  1. Capacitor voltage increases exponentially from zero to the final voltage $V_f$
  2. Charging equation: $V_C = V_f(1 - e^{-t/\tau})$, where $V_C$ is the capacitor voltage at time $t$
• Discharging in an RC circuit:
  1. Capacitor voltage decreases exponentially from the initial voltage $V_0$ to zero
  2. Discharging equation: $V_C = V_0e^{-t/\tau}$, where $V_C$ is the capacitor voltage at time $t$
• Rate of voltage change depends on the time constant $\tau$
  • Smaller $\tau$ results in faster charging or discharging
  • Larger $\tau$ results in slower charging or discharging
• Examples:
  • If $\tau = 1$ s and $V_f = 5$ V, after 1 s of charging, $V_C \approx 3.16$ V (63.2% of $V_f$)
  • For $\tau = 10$ ms and $V_0 = 3.3$ V, after 20 ms of discharging, $V_C \approx 0.41$ V (12.4% of $V_0$)
• The charging and discharging processes follow an exponential decay pattern

RC circuit applications in technology

• Timing circuits in various applications:
  • Camera flashes: RC circuit controls flash duration by determining capacitor discharge time through the flash tube
  • Touchscreens: RC circuits detect and locate touch position by measuring capacitance change caused by a finger
  • Pacemakers: RC circuits generate timing pulses to stimulate heart muscle contraction at a steady rate
    • Time constant determines the pacing rate
  • Defibrillators: RC circuits control charging and discharging of high-voltage capacitors for delivering therapeutic electrical shocks to the heart during cardiac arrest
• Other applications:
  • Debouncing switches: RC circuits filter out rapid voltage fluctuations caused by mechanical switch contacts
  • Smoothing power supply ripples: RC low-pass filters reduce voltage fluctuations in DC power supplies
  • Audio equalizers: RC high-pass and low-pass filters adjust the balance between high and low frequencies in audio signals

Transient and Steady-State Behavior in RC Circuits

• Transient response: The initial period when the circuit is adjusting to a change in voltage or current
  • Characterized by rapid changes in voltage and current
  • Duration depends on the time constant of the circuit
• Steady state: The condition reached after the transient response has died out
  • Voltage and current remain constant or follow a repeating pattern
  • In DC circuits, capacitors act as open circuits in steady state
• Electric field: Stores energy in the capacitor during charging and releases it during discharging
  • Strength of the electric field is proportional to the charge stored on the capacitor plates