Question

show that the relaxation time for a heavily damped rlc circuit is RC

Answer 1

Relaxation oscillators can be divided into two classes

• Sawtooth oscillator: In this type the energy storage capacitor is charged slowly but discharged rapidly, essentially instantly, by a short circuit through the switching device. The charging period thus takes up virtually the entire period of the waveform. The voltage across the capacitor is a sawtooth wave, while the current through the switching device is a sequence of short pulses.
• Astable multivibrator: In this type the capacitor is both charged and discharged slowly through a resistor, so both the charge and discharge periods contribute to the period and the output waveforms consist of two parts. The voltage generated by the capacitor is a triangle waveform, while the voltage from the switching device is a square wave.

• is set by across a resistive voltage divider:

  

  is obtained using Ohm's law and the capacitor differential equation:

  

  Rearranging the differential equation into standard form results in the following:

  

  Notice there are two solutions to the differential equation, the driven or particular solution and the homogeneous solution. Solving for the driven solution, observe that for this particular form, the solution is a constant. In other words, where A is a constant and .

  

  

  Using the Laplace transform to solve the homogeneous equation results in

  

  is the sum of the particular and homogeneous solution.

  

  

  Solving for B requires evaluation of the initial conditions. At time 0, and . Substituting into our previous equation,