RC and RL Time Constants

• transient voltage


  response of capacitors and inductors to sudden changes in DC voltage; unlike resistors (respond instantaneously to applied voltage), capacitors and inductors react over time as they absorb and release energy
• Since capacitors store energy in the form of an electric field, they tend to act like small secondary-cell batteries, being able to store and release electrical energy
• fully discharged


  maintains zero volts across its terminals
• charged


  maintains a steady quantity of voltage across its terminals, just like a battery
• When capacitors are placed in a circuit with other sources of voltage, they will absorb energy from those sources, just as a secondary-cell battery will become charged as a result of being connected to a generator
• A full discharged capacitor, having a terminal voltage of zero, will initially act as a short-circuit when attached to a source of voltage, drawing maximum current as it begins to build a charge

  • the terminal voltage rises to meet applied voltage from the source, and the current through the capacitor decreases
  • once capacitor has reached full voltage of source, it stops drawing current from it, and behave as an open-circuit
• asymptotic


  approach final values, getting closer and closer over time, but never exactly reach their destinations
• While capacitors store energy in an electric field (produced by voltage between two plates), inductors store energy in a magnetic field (produced by current through wire)
• Stored energy in an inductor tries to maintain a constant current through its windings. Inductors oppose changes in current
• fully discharged inductor (no magnetic field)


  zero current through it; initially acts as an open-circuit when attached to a source of voltage (as it tries to maintain zero current), dropping max voltage across its leads

  • over time, inductor's current rises to max value allowed by circuit, and terminal voltage decreases to a minimum (current stays at max level and behaves as short circuit)
• Steps to calculate any values in a reactive DC circuit over time:

  1. identify starting and final values for whatever quantity the capacity or inductor opposes change in (capacitors = voltage, inductors = current)
  2. Calculate the time constant of the circuit (series RC: tau = RC, series L/R: tau = L/R)
• time constant


  the amount of time it takes for voltage or current values to change approximately 63% from their starting values to their final values in a transient situation
• Universal Time Constant Formula


  change = (final - start)(1 - 1/e^[t/tau])

  final = value of calculated variable after infinite time (ultimate value)

  start = initial value of calculated variable

  e = Euler's number

  t = time in seconds

  tau = time constant for circuit in seconds
• discharge


  when energy is being released from the capacitor or inductor to be dissipated in the form of heat by a resistor
• When discharging, heat dissipated by the resistor constitutes energy leaving the circuit, and as a consequence, the reactive component loses its store of energy over time, resulting in either a measurable decrease of voltage (capacitor) or current (inductor)
• The more power dissipated by the resistor, the faster discharging occurs. Thus, a transient circuit's time constant will be dependent upon the resistance of the circuit
• A circuit's time constant will be less (faster discharging rate) if the resistance value is such that it maximizes power dissipation (rate of energy transfer into heat)
• Capacitors and Energy

  capacitors are reservoirs of potential energy
• Inductors and Energy

  inductors are reservoirs of kinetic energy
• Thevenin's Theorem


  we can reduce any linear circuit to an equivalent of one voltage source, one series resistance, and a load component
• Applying Thevenin's Theorem


  1. regard the reactive component as the load and remove it temporarily from the circuit to find Thevenin voltage and resistance
  2. re-connect the capacitor and solve for values of voltage or current over time