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