ECEN 301Discussion #18 – Operational Amplifiers1 Give to Receive Alma 34:28 28 And now behold, my beloved brethren, I say unto you, do not suppose that this is all; for after ye have done all these things, if ye turn away the needy, and the naked, and visit not the sick and afflicted, and impart of your substance, if ye have, to those who stand in need—I say unto you, if ye do not any of these things, behold, your prayer is vain, and availeth you nothing, and ye are as hypocrites who do deny the faith.

ECEN 301Discussion #18 – Operational Amplifiers2 Lecture 18 – Operational Amplifiers Continue with Different OpAmp configurations

ECEN 301Discussion #17 – Operational Amplifiers3 Op-Amps – Open-Loop Model 1.How can v – ≈ v + when v o is amplifying (v + - v - ) ? 2.How can an opAmp form a closed circuit when (i 1 = i 2 = 0) ? – + +v+–+v+– +v––+v–– +vo–+vo– ioio i2i2 i1i1 – v in + – + +–+– R out R in i1i1 A OL v in +vo–+vo– – v in + v–v– v+v+ NB: op-amps have near-infinite input resistance (R in ) and very small output resistance (R out ) A OL – open-loop voltage gain i2i2

ECEN 301Discussion #17 – Operational Amplifiers4 Op-Amps – Open-Loop Model 1.How can v – ≈ v + when v o is amplifying (v + - v - ) ? 2.How can an opAmp form a closed circuit when (i 1 = i 2 = 0) ? – + +–+– R out R in i1i1 A OL v in +vo–+vo– – v in + v–v– v+v+ i2i2 Ideally i 1 = i 2 = 0 (since R in → ∞) What happens as A OL → ∞ ? → v – ≈ v +

ECEN 301Discussion #17 – Operational Amplifiers5 Op-Amps – Closed-Loop Mode RFRF – + +vo–+vo– i1i1 RSRS v S (t) +–+– v+v+ v–v– iFiF iSiS 1.How can v – ≈ v + when v o is amplifying (v + - v - ) ? 2.How can an opAmp form a closed circuit when (i 1 = i 2 = 0) ?

ECEN 301Discussion #17 – Operational Amplifiers6 Op-Amps – Closed-Loop Mode RFRF – + +vo–+vo– i1i1 RSRS v S (t) +–+– v+v+ v–v– iFiF iSiS NB: if A OL is very large these terms → 0 NB: if A OL is NOT the same thing as A CL 1.How can v – ≈ v + when v o is amplifying (v + - v - ) ? 2.How can an opAmp form a closed circuit when (i 1 = i 2 = 0) ?

ECEN 301Discussion #17 – Operational Amplifiers7 Op-Amps – Closed-Loop Mode – + vava R R – + vbvb R R – + R1R1 R R2R2 vovo 1.How can v – ≈ v + when v o is amplifying (v + - v - ) ? 2.How can an opAmp form a closed circuit when (i 1 = i 2 = 0) ? NB: Current flows through R 1 and R 2

ECEN 301Discussion #17 – Operational Amplifiers8 Op-Amps – Closed-Loop Mode – + vava R R – + vbvb R R – + R1R1 R R2R2 vovo 1.How can v – ≈ v + when v o is amplifying (v + - v - ) ? 2.How can an opAmp form a closed circuit when (i 1 = i 2 = 0) ? NB: Inverting amplifiers and (R S = R F ) → v o = -v i

ECEN 301Discussion #17 – Operational Amplifiers9 Op-Amps – Closed-Loop Mode 1.How can v – ≈ v + when v o is amplifying (v + - v - ) ? 2.How can an opAmp form a closed circuit when (i 1 = i 2 = 0) ? – + R1R1 R R2R2 vovo -v a -v b iFiF i1i1 i2i2

ECEN 301Discussion #18 – Operational Amplifiers10 More OpAmp Configurations

ECEN 301Discussion #18 – Operational Amplifiers11 Op-Amps – Closed-Loop Mode The Differential Amplifier: the signal to be amplified is the difference of two signals – + +vo–+vo– i1i1 RFRF v S2 (t) +–+– v+v+ v–v– iFiF RSRS iSiS i1i1 v S1 (t) +–+– RFRF RSRS

ECEN 301Discussion #18 – Operational Amplifiers12 Op-Amps – Closed-Loop Mode The Differential Amplifier: the signal to be amplified is the difference of two signals RFRF – + +vo–+vo– i1i1 v S2 (t) +–+– v+v+ v–v– iFiF RSRS iSiS i2i2 v S1 (t) +–+– RFRF RSRS NB: an ideal op-amp with negative feedback has the properties

ECEN 301Discussion #18 – Operational Amplifiers13 Op-Amps – Closed-Loop Mode The Differential Amplifier: the signal to be amplified is the difference of two signals RFRF – + +vo–+vo– i1i1 v S2 (t) +–+– v+v+ v–v– iFiF RSRS iSiS i2i2 v S1 (t) +–+– RFRF RSRS

ECEN 301Discussion #18 – Operational Amplifiers14 Op-Amps – Closed-Loop Mode The Differential Amplifier: the signal to be amplified is the difference of two signals RFRF – + +vo–+vo– i1i1 v S2 (t) +–+– v+v+ v–v– iFiF RSRS iSiS i2i2 v S1 (t) +–+– RFRF RSRS

ECEN 301Discussion #18 – Operational Amplifiers15 Op-Amps – Level Shifter Level Shifter: can add or subtract a DC offset from a signal based on the values of R S and/or V ref – + +vo–+vo– RFRF V ref +–+– v+v+ v–v– RSRS V sensor AC voltage with DC offset DC voltage

ECEN 301Discussion #18 – Operational Amplifiers16 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find V ref ) R S = 10kΩ, R F = 220kΩ, v s (t) = cos(ωt) – + +vo–+vo– RFRF V ref +–+– v+v+ v–v– RSRS V sensor

ECEN 301Discussion #18 – Operational Amplifiers17 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find V ref ) R S = 10kΩ, R F = 220kΩ, v s (t) = cos(ωt) – + +vo–+vo– RFRF V ref +–+– v+v+ v–v– RSRS V sensor Find the Closed-Loop voltage gain by using the principle of superposition on each of the DC voltages

ECEN 301Discussion #18 – Operational Amplifiers18 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find V ref ) R S = 10kΩ, R F = 220kΩ, v s (t) = cos(ωt) – + +vo–+vo– RFRF v+v+ v–v– RSRS V sensor +–+– DC from sensor: Inverting amplifier

ECEN 301Discussion #18 – Operational Amplifiers19 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find V ref ) R S = 10kΩ, R F = 220kΩ, v s (t) = cos(ωt) – + +vo–+vo– RFRF v+v+ v–v– RSRS V ref +–+– DC from reference: Noninverting amplifier

ECEN 301Discussion #18 – Operational Amplifiers20 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find V ref ) R S = 10kΩ, R F = 220kΩ, v s (t) = cos(ωt) – + +vo–+vo– RFRF V ref +–+– v+v+ v–v– RSRS V sensor

ECEN 301Discussion #18 – Operational Amplifiers21 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find V ref ) R S = 10kΩ, R F = 220kΩ, v s (t) = cos(ωt) – + +vo–+vo– RFRF V ref +–+– v+v+ v–v– RSRS V sensor Since the desire is to remove all DC from the output we require:

ECEN 301Discussion #18 – Operational Amplifiers22 Op-Amps – Ideal Integrator The Ideal Integrator: the output signal is the integral of the input signal (over a period of time) – + + v o (t) – i1i1 CFCF RSRS v S (t) +–+– v+v+ v–v– i F (t) i S (t) The input signal is AC, but not necessarily sinusoidal NB: Inverting amplifier setup with R F replaced with a capacitor

ECEN 301Discussion #18 – Operational Amplifiers23 Op-Amps – Ideal Integrator The Ideal Integrator: the output signal is the integral of the input signal (over a period of time) – + + v o (t) – i1i1 CFCF RSRS v S (t) +–+– v+v+ v–v– i F (t) i S (t) Recall

ECEN 301Discussion #18 – Operational Amplifiers24 Op-Amps – Ideal Integrator Example2: find the output voltage if the input is a square wave of amplitude +/–A with period T T = 10ms, C F = 1uF, R S = 10kΩ – + + v o (t) – i1i1 CFCF RSRS v S (t) +–+– v+v+ v–v– i F (t) i S (t) A -A T/2T

ECEN 301Discussion #18 – Operational Amplifiers25 Op-Amps – Ideal Integrator Example2: find the output voltage if the input is a square wave of amplitude +/–A with period T T = 10ms, C F = 1uF, R S = 10kΩ – + + v o (t) – i1i1 CFCF RSRS v S (t) +–+– v+v+ v–v– i F (t) i S (t)

ECEN 301Discussion #18 – Operational Amplifiers26 Op-Amps – Ideal Integrator Example2: find the output voltage if the input is a square wave of amplitude +/–A with period T T = 10ms, C F = 1uF, R S = 10kΩ – + + v o (t) – i1i1 CFCF RSRS v S (t) +–+– v+v+ v–v– i F (t) i S (t) NB: since the v s (t) is periodic, we can find v o (t) over a single period – and repeat

ECEN 301Discussion #18 – Operational Amplifiers27 Op-Amps – Ideal Integrator Example2: find the output voltage if the input is a square wave of amplitude +/–A with period T T = 10ms, C F = 1uF, R S = 10kΩ – + + v o (t) – i1i1 CFCF RSRS v S (t) +–+– v+v+ v–v– i F (t) i S (t) NB: since the v s (t) is periodic, we can find v o (t) over a single period – and repeat

ECEN 301Discussion #18 – Operational Amplifiers28 Op-Amps – Ideal Integrator Example2: find the output voltage if the input is a square wave of amplitude +/–A with period T T = 10ms, C F = 1uF, R S = 10kΩ – + + v o (t) – i1i1 CFCF RSRS v S (t) +–+– v+v+ v–v– i F (t) i S (t) T/2T -50AT

ECEN 301Discussion #18 – Operational Amplifiers29 Op-Amps – Ideal Differentiator The Ideal Differentiator: the output signal is the derivative of the input signal (over a period of time) – + + v o (t) – i1i1 CSCS RFRF v S (t) +–+– v+v+ v–v– i F (t) i S (t) The input signal is AC, but not necessarily sinusoidal NB: Inverting amplifier setup with R S replaced with a capacitor

ECEN 301Discussion #18 – Operational Amplifiers30 Op-Amps – Ideal Differentiator NB: this type of differentiator is rarely used in practice since it amplifies noise The Ideal Differentiator: the output signal is the derivative of the input signal (over a period of time) – + + v o (t) – i1i1 CSCS RFRF v S (t) +–+– v+v+ v–v– i F (t) i S (t) Recall

ECEN 301Discussion #18 – Operational Amplifiers31 Op-Amps – Closed-Loop Mode Circuit DiagramA CL Inverting Amplifier Summing Amplifier – + +vo–+vo– +–+– vSvS RSRS RFRF – + +vo–+vo– +–+– +–+– +–+– R Sn R S2 R S1 v Sn v S2 v S1 RFRF

ECEN 301Discussion #18 – Operational Amplifiers32 Op-Amps – Closed-Loop Mode Circuit DiagramA CL Noninverting Amplifier Voltage Follower – + +vo–+vo– +–+– R RSRS RFRF vSvS – + +vo–+vo– +–+–

ECEN 301Discussion #18 – Operational Amplifiers33 Op-Amps – Closed-Loop Mode Circuit DiagramA CL Differential Amplifier – + +vo–+vo– +–+– +–+– RSRS RSRS RFRF RFRF v1v1 v2v2

ECEN 301Discussion #18 – Operational Amplifiers34 Op-Amps – Closed-Loop Mode Circuit DiagramA CL Ideal Integrator Ideal Differentiator – + + v o (t) – +–+– vSvS CSCS RFRF – + + v o (t) – +–+– vSvS RSRS CFCF

ECEN 301Discussion #18 – Operational Amplifiers35 Op-Amps Example 3: find an expression for the gain if v s (t) is sinusoidal C F = 1/6 F, R 1 = 3Ω, R 2 = 2Ω, C S = 1/6 F + – v o (t) i1i1 CFCF R2R2 v+v+ v–v– i F (t) i 2 (t) R1R1 i 1 (t) CSCS i S (t) v s (t)

ECEN 301Discussion #18 – Operational Amplifiers36 Op-Amps + – Vo(jω)Vo(jω) I in Z F =1/jωC F Z2Z2 v+v+ v–v– IF(jω)IF(jω) I2(jω)I2(jω) Z1Z1 I 1 (jω) ZSZS IS(jω)IS(jω) Vs(jω)Vs(jω) Node a Node b 1.Transfer to frequency domain 2.Apply KCL at nodes a and b NB: v + = v – = v o and I in = 0 Example 3: find an expression for the gain C F = 1/6 F, R 1 = 3Ω, R 2 = 2Ω, C S = 1/6 F

ECEN 301Discussion #18 – Operational Amplifiers37 Op-Amps + – Vo(jω)Vo(jω) I in Z F =1/jωC F Z2Z2 v+v+ v–v– IF(jω)IF(jω) I2(jω)I2(jω) Z1Z1 I 1 (jω) ZSZS IS(jω)IS(jω) Vs(jω)Vs(jω) Node a Node b 1.Transfer to frequency domain 2.Apply KCL at nodes a and b Example 3: find an expression for the gain C F = 1/6 F, R 1 = 3Ω, R 2 = 2Ω, C S = 1/6 F

ECEN 301Discussion #18 – Operational Amplifiers38 Op-Amps + – Vo(jω)Vo(jω) I in Z F =1/jωC F Z2Z2 v+v+ v–v– IF(jω)IF(jω) I2(jω)I2(jω) Z1Z1 I 1 (jω) ZSZS IS(jω)IS(jω) Vs(jω)Vs(jω) 1.Transfer to frequency domain 2.Apply KCL at nodes a and b 3.Express V o in terms of V s Example 3: find an expression for the gain C F = 1/6 F, R 1 = 3Ω, R 2 = 2Ω, C S = 1/6 F

ECEN 301Discussion #18 – Operational Amplifiers39 Op-Amps + – Vo(jω)Vo(jω) I in Z F =1/jωC F Z2Z2 v+v+ v–v– IF(jω)IF(jω) I2(jω)I2(jω) Z1Z1 I 1 (jω) ZSZS IS(jω)IS(jω) Vs(jω)Vs(jω) 1.Transfer to frequency domain 2.Apply KCL at nodes a and b 3.Express V o in terms of V S 4.Find the gain (V o /V S ) Example 3: find an expression for the gain C F = 1/6 F, R 1 = 3Ω, R 2 = 2Ω, C S = 1/6 F

ECEN 301Discussion #18 – Operational Amplifiers40 Op-Amps + – Vo(jω)Vo(jω) I in Z F =1/jωC F Z2Z2 v+v+ v–v– IF(jω)IF(jω) I2(jω)I2(jω) Z1Z1 I 1 (jω) ZSZS IS(jω)IS(jω) Vs(jω)Vs(jω) Example 3: find an expression for the gain C F = 1/6 F, R 1 = 3Ω, R 2 = 2Ω, C S = 1/6 F Frequency Response 2 nd order Lowpass filter

Instrumentation Op-Amp Special configuration used for common-mode voltage rejection Advantages: - Common-mode voltage rejection - Very high input impedance for V 1 and V 2 - Set gain with one resistor Connect sensor with twist pair in differential configuration to minimize external noise pickup. Using grounded shield also helps.