The Ideal Op-amp (Operational amplifier) v+ VOUT v– +15V VIN VOUT –15V saturation VOUT VOUT [V] VOUT=A(v+–v–) A~105 VIN [μV]

Op-amp Feedback v+ VOUT v– v+ VIN VOUT v– R1 R2 Non-Inverting Amplifier Circuit

Op-amp Feedback v+ v– Assumptions: Gain is very large (A) Inputs draw no current (ZIN=) Output attempts to make input voltage difference zero (v+=v–) VOUT + – v+ v– VIN R1 R2 Non-Inverting Amplifier Circuit

R2 R1 v– VIN i VOUT v+ R2 i R1 v– V1 VOUT v+ V2 V3 Inverting Amplifier Circuit R2 – v– v+ i Summing amplifier R2 – + v– v+ VOUT i V1 R1 V2 V3

RF i V1 R1 V2 R2 V3 R3 v– – VOUT v+ +

Difference amplifier R2 R1 v– V1 – VOUT v+ R1 + V2 R2

Integrator VIN VOUT Capacitor as integrator R Vi C Vint If RC>>t VC<<Vi

Op-amp Integrator C – v– v+ i R VIN VOUT +

Differentiation C Vi Vdiff R Small RC 

Op-amp Differentiator VIN VOUT R – v– v+ i C VIN VOUT +

Complex analysis R2 i C R1 v– – i VOUT v+ V0ejωt + High pass filter

Exploiting op-amp saturation + – v+ v– VOUT Saturation voltage VOUT = A(v+–v–) A VOUT = +VSat v+>v– VOUT = –VSat v+<v–

Analogue – digital conversion (ADC) + – R + – R + – VIN R + – R

Oscillator VOUT  + R1 C R

Op-amp applications Building block of analogue electronics Signal amplifiers Audio amplifiers Integrators / differentiators Voltage / current sources Active filters Oscillators Digital-analogue and analogue-digital convertors

Bridge circuits Z1 Z3 VA VB V0 VOUT Z4 Z2 Bridge balanced when VAVB=0 + – VB V0 VOUT Z2 Z4 Bridge balanced when VAVB=0

End of Op-Amps!