1• Second order circuits: LC undampedECE 201: Lecture 20Borja PeleatoBackground• In the previous lectures we have studied first order circuits, with one capacitor OR one inductor– Occasionally, we had multiple C or L that could be combined into a single one– Applying KVL, KCL, Ohm, and the equations for L or C gave a FIRST order differential equation to describe the circuit– In order to find a general solution to the first order differential equation, we needed to know ONE initial condition (VCor IL)• Now we will study circuits with both capacitors and inductors– They will be described by a SECOND order differential equation– Finding a general solution to the equation will require TWO initial conditions:• initial voltage in C and current in L• Initial voltage in C and its derivative• Initial current in L and its derivative• Etc.2Mechanical oscillator• The general form for a second order differential equation is:• This equation can be understood as describing a damped mechanical oscillator with an external force, where the variable x represents the position of the mass• Despite being called an oscillator, it might not oscillate.– If the damping is strong, it will just slowly go to its equilibrium position and stay there– If the damping is weak, it will oscillate around its equilibrium position, with the oscillations becoming progressively smaller– If there is no damping, it will oscillate forever• In circuits, the equation describes an ELECTRICAL oscillator, where variable x(t) represents voltages or currents3Electrical oscillator• Intuitively, you can think of a series R-L-C with some initial charge on the capacitor– The capacitor discharges creating a current– When it is completely discharged, the inductor has a current– The inductor keeps that current going, charging the capacitor in the opposite direction– This pattern repeats itself. The energy stored in the C, gets transferred to the L, back and forth.– If there is a resistor in the circuit, it can dissipate this energy, eventually stopping the oscillation.• If the resistor dissipates a lot of energy, the oscillation might not happen. All the energy gets dissipated before it can go back and forth.4Solving a 2ndorder diff. eq.• In this course, we just give you the method. You will need to take a math class to understand where it comes from…• In this lecture, we only consider the homogeneous case, ie zero independent term5Case 3.1: Real distinct roots67Case 3.2: Real equal roots89Case 3.3: Complex conjugate roots1011LC oscillator
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