Differential Equations
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General
 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