6. RLC CIRCUITS CIRCUITS by Ulaby & Maharbiz

Overview

Second Order Circuits A second order circuit is characterized by a second order differential equation  Resistors and two energy storage elements  Determine voltage/current as a function of time  Initial/final values of voltage/current, and their derivatives are needed

Initial/Final Conditions  v C, i L do not change instantaneously  Get derivatives dv C /dt and di L /dt from i C, v L  Capacitor open, Inductor short at dc Guidelines

Example 6-2: Determine Initial/Final Conditions Circuit t = 0 ‒

Example 6-2: Initial/Final Conditions (cont.) t = 0 + Given:

Example 6-2: Initial/Final Conditions (cont.) t  

Series RLC Circuit : General Solution Solution Outline Transient solution Steady State solution

Series RLC Circuit: Natural Response Find Natural Response Of RLC Circuit 0 Natural response occurs when no active sources are present, which is the case at t > 0.

Series RLC Circuit: Natural Response Find Natural Response Of RLC Circuit 0 Solution of Diff. Equation Assume: It follows that:

Solution of Diff. Equation (cont.) 0 Invoke Initial Conditions to determine A 1 and A 2

Circuit Response: Damping Conditions Damping coefficient Resonant frequency s 1 and s 2 are real s 1 = s 2 s 1 and s 2 are complex

Overdamped Response Overdamped,  >  0

Underdamped Response Underdamped  <  0 Damping: loss of stored energy Damped natural frequency

Critically Damped Response Critically damped  =  0

Total Response of Series RLC Circuit Need to add Forced/Steady State Solution Natural solution represents transient response, decays to 0 as t  . v(  ) represents forced/steady state solution. Overdamped (  >  0 ) Critically Damped (  =  0 ) Underdamped (  <  0 ) Now find unknown constants from initial conditions v(0 + ) and dv/dt at t = 0 +

Example 6-7: Overdamped RLC Circuit Cont.

Example 6-7: Overdamped RLC Circuit

Example 6-8: Pulse Excitation

Example 6-9: Determine Capacitor Response Circuit t = 0 ‒ At t = 0 ‒ :

Example 6-9: Capacitor Response (cont.) t = 0 + Initial values of the capacitor voltage and its derivative will be needed to evaluate constants D 1 and D 2

Example 6-9: Capacitor Response (cont.) t > 0 This is just a series RLC circuit!

Example 6-9: Capacitor Response (cont.)

Parallel RLC Circuit Overdamped (  >  0 ) Critically Damped (  =  0 ) Underdamped (  <  0 ) Same form of diff. equation as series RLC

Oscillators If R=0 in a series or parallel RLC circuit, the circuit becomes an oscillator

General Second Order Circuits  Setup differential equation  Determine   Natural solution  Forced solution (steady state)  Unknowns from initial conditions

Example 6-13: Op-Amp Circuit Substitute v out into KCL expression, rearrange for diff. equation in terms of i L

Example 6-13: Op-Amp Circuit (cont.) Cont.

Example 6-13: Op-Amp Circuit (cont.) Cont.

Example 6-13: Op-Amp Circuit (cont.)

Multisim Example of RLC Circuit

RFID Circuit

Tech Brief 12: Micromechanical Sensors and Actuators

Tech Brief 13: Touchscreens and Active Digitizers

Summary