Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7EE201 Lecture 29 P. 1First Order Op-amp Circuits (RC)Capacitors can be added to op-amp circuits for increased control of circuit outputs. However, there are subtleties concerning this approach:1. All resistors do not determine time constant2. Op-amp circuits may deviate from ideal response3. Stray capacitances (a capacitance between a conductor and ground) may influence results.EE201 Lecture 29 P. 2Example: Find vo(t) for the op amp circuit shown assuming ideal op amp behavior.+_+--+vo(t)icRCiRvsInput characteristics for op-amp v- = v+ = 0Initial condition for capacitor voltagevc(0- ) = vc(0+ ) = vo(0) vo(t) = vc(t ) = vc(0+ ) + ic()d (1)1Ct0ic(t) = - iR(t) (Property of ideal op amp i-=0 )EE201 Lecture 29 P. 3xample: Find vout(t) assuming vc(0-) = 0Step 1: Find voltage across capacitor vout(t) = v+Solution for voltage across capacitorvc(t) = vc() + [vc(0+) - vc()] exp{- t/(RTHC)}iC(t) = - vs(t)/ R put this into (1) vo(t) = vc(0+ ) - vs()d0 1 RCt+_+_-+vout(t)v+(t)1k1k0.1F5u(t)V_1kEE201 Lecture 29 P. 4Since i+= 0, resistors and capacitor are in series RTH = 2 kRTHC = 0.2 x 10 -3secvC(0+) = vC(0-) = 0vC() = 5VvC(t) = 5 - 5e-5000t VStep 2: Find current through 1k resistor between capacitor and ground.Differentiate voltage across capacitoriC = C = (0.1F)(-5000)(-5e-5000t)iC(t) = 2.5 x 10-3 e-5000t Ad vC(t) dtEE201 Lecture 29 P. 5Step 3: Calculate vout(t) from iC(t) vout(t) = (1k)(iC(t))vout(t) = (103)(2.5x10-3) e-5000tvout(t) = 2.5 e-5000tBecause properties of op amp (v+ = v-) , vout(t) does not depend on load resistor.EE201 Lecture 29 P. 6Example: Find vout(t) for the op amp circuit shown assuming ideal op amp behavior.+_+--+vout(t)if0.01Fvin(t)i2Step 1: Realize v+= v-Therefore i1= i2= -if i1= i2= vin(t)/2RAlso, v+ =v-=vin/2100k100k100ki1EE201 Lecture 29 P. 7Step 2 : Find feedback current from capacitor equation if = C [vout - vin/2 ] = [vout(t) – vin(t)/2 ]Integrating,vout(t) = vin(t) 2ddt-vin2RCddtStep 3: Solve for vout(t) d vout(t) d vin(t) vin dt 2dt 2RC =_ 12RC0tvin() d_ vin(t) 20t_vin() dvout(t)
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