Differential Equations

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  • Get rate equation and initial condition
  • Compare analytic solution to numerical solution given by scipy.integrate.odeint

RLC Analytic Solution

  • Kirchoff's current law "KCL"
  • current through a capacitor
  • current through a inductance
  • current through a resistor
  • Ohm's law for voltage drop across resistor
  • Kirchoff's voltage law "KVL" - voltages in the circuit = 0

Natural response

  • no voltage, maybe charge on capacitor, will decay to steady state
  • KCL current sums to 0
  • Take derivative of everything
  • Get second derivative by itself
  • Turn differential equation into Characteristic equation - the s equations
  • Use quadratic formula to find the roots of s, s1 and s2
  • Assume form of the solution is
  • Response can fall into three categories
    • Overdamped - roots s1 and s2 are both real numbers and distinct from each other.
    • Underdamped - both are complex and distinct (conjugates)
    • Critically damped - both are real and equal
  • To find out which response we have, compare two frequencies, whether or not one is bigger than the other, or are they equal.
    • Resonant radian frequency [radians/s]
    • Neper frequency [radians/s]
  • Plug in the values of the coefficients
  • Overdamped:
  • Overdamped:
      • Damping frequency
  • Critically damped:
  • Initial conditions, plug in t=0 to find coefficients A12 / B12 / D12, take derivitive, plug in t=0

Step Response

  • Same as above, just set equation to constants I0 and V0 respectively