ECE 201 Lecture 20 Borja Peleato Second order circuits LC undamped 1 Background 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 VC or 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 2 Mechanical 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 currents 3 Electrical 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 4 Solving a nd 2 order 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 term 5 Case 3 1 Real distinct roots 6 7 Case 3 2 Real equal roots 8 9 Case 3 3 Complex conjugate roots 10 11 LC oscillator undamped 12 13
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