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Op-Amp With Complex Impedance -+-+ Z1Z1 Vin ZLZL ZFZF VoVo A v = - (Z F /Z 1 ) “-” : 180° phase shift Z = a ± j b Z = M <θ (polar form) M = Sqrt(a 2 + b 2 ) θ = tan -1 (b/a) Z = M Cosθ + j M Sinθ Inverting Configuration

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Op-Amp With Complex Impedance -+-+ Z1Z1 Vin ZLZL ZFZF VoVo A v = 1+ (Z F /Z 1 ) A v = (Z 1 +Z F )/Z 1 Noninverting Configuration

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Differentiator Differentiator: circuit whose output is proportional to the derivative of its input Derivative of a function is the instantaneous slope or rate of change of function Output of differentiator is proportional to the rate of change of input signal, with respect to time Output of op amp differentiator will always lag input by 90° (inversion of true derivative) V(t)V’(t) dv/d t

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Differentiator Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

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Differentiator -+-+ C Vin R VoVo RLRL l Av l = l R/(1/jωC) l = l jωRC l = ωRC Av = -ωRC <90 = ωRC <-90

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Differentiator Main problem with op amp differentiator is noise sensitivity Gain of ideal differentiator is zero at dc, and increases with frequency at a rate of 20 dB/decade High frequency noise will tend to be amplified greatly electronics-tutorials.ws

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Practical Differentiator -+-+ C Vin RFRF VoVo RLRL R1R1 To reduce gain to high frequency noise, a resistor is placed in series with the input resistor

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Practical Differentiator Problem: noise at high frequency To reduce noise at high frequency a resistor is placed in series with the input capacitor To reduce noise, R 1 < R F R 1 may be chosen such that 10R 1 < R F to reduce high frequency gain and noise Before adding R 1 : Gain characteristics of unmodified differentiator is superimposed on a typical op-amp open-loop Bode plot; differentiator will act correctly up to f 0 After adding R 1 : differentiator gain levels off at f 1

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Practical Differentiator l A l (dB) AOL f0f0 Log f l A l (dB) AOL f1f1 Log f Before adding R1After adding R1 f1 = 1/(2πR1C)

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Differentiation of Nonsinusoidal Inputs Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey Linear ramp input: V0 = -RCk K: function slope (V/s) Triangular input: V0 = -RCkn Kn: function slope (V/s)

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Integrator Process of integration is complementary to that of differentiation Relationship is analogous to that between multiplication and division Function being integrated is called integrand, and dt is called the differential Integration produces equivalent of the continuous sum of values of function at infinitely many infinitesimally small increments of t Output of integrator will maintain 90° phase lead, regardless of frequency

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Integrator C1C1 -+-+ Vin VoVo RLRL R1R1 V(t)∫ V(t) dt + C ∫ Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

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Integrator A v = - (Z F /Z 1 ) = - 1/(jωCR) l Av l = l 1/(ωRC) l and phase = 90 Av = 1/(ωRC) <90 f 0 (dc), Av ∞ C1C1 -+-+ Vin VoVo RLRL R1R1 Reset switch added to force integrator initial conditions to zero Reset

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Integration of Nonsinusoidal Inputs Constant voltage: V 0 = -V in t / RC V 0 = 0 at starting Ramp input: V 0 = - kt 2 / 2R k is rate of change of V in (V/s) Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

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Integrator (Square Wave Input) V 1 = - (V m t (+) )/RC t (+) = t – t 0 V 2 = V 1 + [-(V m t (-) )/RC] Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

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Integrator Integrator effectively accumulates voltage over time; presence of input offset voltage will cause capacitor to charge up producing error in output Smaller the capacitor, more quickly offset error builds up with time Solutions – Use of larger capacitor – Use of low-offset op amps – Bias compensation resistor R B on noninverting terminal – Use of resistor R C in parallel with feedback capacitor – R C ≥ 10R 1

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Practical Integrator C -+-+ Vin VoVo RLRL R1R1 RCRC RB = R1 ll RC

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