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 301 Discussion #18 – Operational Amplifiers 1 Lecture 18 – Operational Amplifiers Continue with Different OpAmp configurations ECEN 301 Discussion #18 – Operational Amplifiers 2 Op-Amps – Open-Loop Model 1. How can v– ≈ v+ when vo is amplifying (v+ - v-) ? 2. How can an opAmp form a closed circuit when (i1 = i2 = 0) ? i1 – + vin + + v– v+ i1 – + i2 v– io – Rin + vin v+ – + Rout + vo – – – + AOLvin– + vo – i2 NB: op-amps have near-infinite input resistance (Rin) and very small output resistance (Rout) vo AOL vin AOL (v v ) ECEN 301 AOL – open-loop voltage gain Discussion #17 – Operational Amplifiers 3 Op-Amps – Open-Loop Model 1. How can v– ≈ v+ when vo is amplifying (v+ - v-) ? 2. How can an opAmp form a closed circuit when (i1 = i2 = 0) ? i1 v– vo AOL vin AOL (v v ) vo v v AOL – – Ideally i1 = i2 = 0 (since Rin → ∞) Rin vin v+ + Rout + + AOLvin– + i2 vo – What happens as AOL → ∞ ? → v– ≈ v+ ECEN 301 Discussion #17 – Operational Amplifiers 4 Op-Amps – Closed-Loop Mode 1. How can v– ≈ v+ when vo is amplifying (v+ - v-) ? 2. How can an opAmp form a closed circuit when (i1 = i2 = 0) ? RF RS iS vS(t) + – v– i S i F vS v vo v RS RF iF – i1 v+ + + vo – ECEN 301 vS vo / AOL vo vo / AOL RS RS RF RF vS v v v S o o RS RS AOL RF AOL RS 1 1 1 vS vo RF / RS AOL RF / RS AOL Discussion #17 – Operational Amplifiers 5 Op-Amps – Closed-Loop Mode 1. How can v– ≈ v+ when vo is amplifying (v+ - v-) ? 2. How can an opAmp form a closed circuit when (i1 = i2 = 0) ? RF RS iS vS(t) + – v– 1 1 1 vS vo RF / RS AOL RF / RS AOL iF – NB: if AOL is very large these terms → 0 i1 v+ + + A CL vo – ECEN 301 Closed - LoopGain : NB: if AOL is NOT the same thing as ACL Discussion #17 – Operational Amplifiers vo vS RF RS 6 Op-Amps – Closed-Loop Mode 1. How can v– ≈ v+ when vo is amplifying (v+ - v-) ? 2. How can an opAmp form a closed circuit when (i1 = i2 = 0) ? va R R – + R1 R – + vo R2 vb R ECEN 301 R – + NB: Current flows through R1 and R2 Discussion #17 – Operational Amplifiers 7 Op-Amps – Closed-Loop Mode 1. How can v– ≈ v+ when vo is amplifying (v+ - v-) ? 2. How can an opAmp form a closed circuit when (i1 = i2 = 0) ? va R R – + R R1 – + vo R2 vb R R – + NB: Inverting amplifiers and (RS = RF) → vo = -vi ECEN 301 Discussion #17 – Operational Amplifiers 8 Op-Amps – Closed-Loop Mode 1. How can v– ≈ v+ when vo is amplifying (v+ - v-) ? 2. How can an opAmp form a closed circuit when (i1 = i2 = 0) ? iF R1 -va i1 R2 -vb i2 i1 i2 iF R – + vo va v vb v vo v R1 R2 R va 0 vb 0 v 0 o R1 R2 R va vb vo R1 R2 R va vb vo R R1 R2 ECEN 301 Discussion #17 – Operational Amplifiers 9 More OpAmp Configurations ECEN 301 Discussion #18 – Operational Amplifiers 10 Op-Amps – Closed-Loop Mode The Differential Amplifier: the signal to be amplified is the difference of two signals RF iF RS iS vS1(t) + – vS2(t) + – ECEN 301 RS v– – i1 v+ + + i1 RF vo – Discussion #18 – Operational Amplifiers 11 Op-Amps – Closed-Loop Mode The Differential Amplifier: the signal to be amplified is the difference of two signals RF NB: an ideal op-amp with negative feedback has the properties iF RS + – iS vS1(t) vS2(t) + – RS v– – i1 v+ + + i2 RF vo v v i1 i2 0 – iF iS ECEN 301 Discussion #18 – Operational Amplifiers 12 Op-Amps – Closed-Loop Mode The Differential Amplifier: the signal to be amplified is the difference of two signals RF Voltage Divider : RF v vs 2 RF RS iF RS + – iS vS1(t) vS2(t) + – ECEN 301 RS v– – i1 v+ + + i2 RF vo – v v vS 1 iS RS vS 1 v iS RS Discussion #18 – Operational Amplifiers vo v iF RF 13 Op-Amps – Closed-Loop Mode The Differential Amplifier: the signal to be amplified is the difference of two signals RF vo iF RF v iF RS + – iS vS1(t) vS2(t) + – ECEN 301 RS v– iS RF v – i1 v+ + + i2 RF vo – vS 1 vS 2 vS 2 RF vo RF R R R R R R S F S S F S RF vS 2 vS1 RS Discussion #18 – Operational Amplifiers 14 Op-Amps – Level Shifter Level Shifter: can add or subtract a DC offset from a signal based on the values of RS and/or Vref RF Vsensor RS AC voltage with DC offset v– – v+ + Vref + – DC voltage ECEN 301 Discussion #18 – Operational Amplifiers + vo – 15 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find Vref) RS = 10kΩ, RF = 220kΩ, vs(t) = 1.8+0.1cos(ωt) RF Vsensor RS v– – v+ + Vref ECEN 301 + – + vo – Discussion #18 – Operational Amplifiers 16 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find Vref) RS = 10kΩ, RF = 220kΩ, vs(t) = 1.8+0.1cos(ωt) RF Vsensor RS – v+ + Vref ECEN 301 + – Find the Closed-Loop voltage gain by using the principle of superposition on each of the DC voltages v– + vo – Discussion #18 – Operational Amplifiers 17 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find Vref) RS = 10kΩ, RF = 220kΩ, vs(t) = 1.8+0.1cos(ωt) RF RS Vsensor + – DC from sensor: Inverting amplifier v– – v+ + + vo – ECEN 301 vosen ACL vsenDC RF vsenDC RS Discussion #18 – Operational Amplifiers 18 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find Vref) RS = 10kΩ, RF = 220kΩ, vs(t) = 1.8+0.1cos(ωt) RF DC from reference: Noninverting amplifier v– RS – v+ + Vref + – + vo – ECEN 301 voref ACL vref RF 1 RS Discussion #18 – Operational Amplifiers vref 19 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find Vref) RS = 10kΩ, RF = 220kΩ, vs(t) = 1.8+0.1cos(ωt) RF Vsensor RS v– v+ + Vref ECEN 301 + – voDC vosen voref – + vo RF RF vref vsenDC 1 RS RS – Discussion #18 – Operational Amplifiers 20 Op-Amps – Level Shifter Example1: design a level shifter such that it can remove a 1.8V DC offset from the sensor signal (Find Vref) RS = 10kΩ, RF = 220kΩ, vs(t) = 1.8+0.1cos(ωt) RF Vsensor RS v– – v+ + Vref ECEN 301 + – Since the desire is to remove all DC from the output we require: R RF vsDC 1 F RS RS RF / RS vsDC 1 RF / RS 220/ 10 (1.8) 1 220/ 10 1.714V 0 + vo – vref Discussion #18 – Operational Amplifiers vref 21 Op-Amps – Ideal Integrator The Ideal Integrator: the output signal is the integral of the input signal (over a period of time) CF The input signal is AC, but not necessarily sinusoidal RS iS(t) vS(t) + – iF(t) v– – i1 v+ + + vo(t) – ECEN 301 Discussion #18 – Operational Amplifiers NB: Inverting amplifier setup with RF replaced with a capacitor 22 Op-Amps – Ideal Integrator The Ideal Integrator: the output signal is the integral of the input signal (over a period of time) CF iS (t ) iF (t ) RS iS(t) vS(t) + iF(t) v– – i1 v+ vS (t ) d vo (t ) v (t ) C F RS dt + + – vo(t) – dvo (t ) v (t ) S dt RS C F 1 vo (t ) RS C F t vS ( )d dv (t ) Recall i(t ) C dt ECEN 301 Discussion #18 – Operational Amplifiers 23 Op-Amps – Ideal Integrator Example2: find the output voltage if the input is a square wave of amplitude +/–A with period T T = 10ms, CF = 1uF, RS = 10kΩ CF iS(t) vS(t) + – v– – vs(t) RS A iF(t) i1 v+ + T/2 + vo(t) -A – ECEN 301 T Discussion #18 – Operational Amplifiers time 24 Op-Amps – Ideal Integrator Example2: find the output voltage if the input is a square wave of amplitude +/–A with period T T = 10ms, CF = 1uF, RS = 10kΩ CF RS iS(t) vS(t) + – iF(t) v– 1 vo (t ) RS CF – i1 v+ + + vo(t) – ECEN 301 t vS ( )d t 1 0 v ( ) d v ( ) d S S 0 RS CF 1 vo (0) RS CF Discussion #18 – Operational Amplifiers 0 t vS ( )d 25 Op-Amps – Ideal Integrator Example2: find the output voltage if the input is a square wave of amplitude +/–A with period T T = 10ms, CF = 1uF, RS = 10kΩ NB: since the vs(t) is periodic, we can find vo(t) over a single period – and repeat CF RS iS(t) vS(t) + – 0t T /2 iF(t) v– – i1 v+ + + 1 vo (t ) vo (0) RS C F 0 t vS ( )d t vo(t) – 0 100 Ad 0 100A t 0 100At ECEN 301 Discussion #18 – Operational Amplifiers 26 Op-Amps – Ideal Integrator Example2: find the output voltage if the input is a square wave of amplitude +/–A with period T T = 10ms, CF = 1uF, RS = 10kΩ NB: since the vs(t) is periodic, we can find vo(t) over a single period – and repeat CF RS iS(t) vS(t) + – T /2t T iF(t) v– – i1 v+ + + vo(t) – ECEN 301 1 t T vo (t ) vo v ( ) d S 2 RS C F T / 2 t T 100A 100 ( A)d T /2 2 T t 100A 100A T / 2 2 100A(T t ) Discussion #18 – Operational Amplifiers 27 Op-Amps – Ideal Integrator Example2: find the output voltage if the input is a square wave of amplitude +/–A with period T T = 10ms, CF = 1uF, RS = 10kΩ CF T/2 iS(t) vS(t) + – iF(t) v– – i1 vo(t) RS T v+ + + vo(t) -50AT – time ECEN 301 Discussion #18 – Operational Amplifiers 28 Op-Amps – Ideal Differentiator The Ideal Differentiator: the output signal is the derivative of the input signal (over a period of time) RF The input signal is AC, but not necessarily sinusoidal iF(t) v– CS – iS(t) vS(t) + – i1 v+ + + vo(t) – ECEN 301 Discussion #18 – Operational Amplifiers NB: Inverting amplifier setup with RS replaced with a capacitor 29 Op-Amps – Ideal Differentiator The Ideal Differentiator: the output signal is the derivative of the input signal (over a period of time) RF iF(t) – iS(t) vS(t) + i1 v+ d vS (t ) v (t ) vo (t ) CS dt RF v– CS iS (t ) iF (t ) + + – vo(t) – dv (t ) dvS (t ) vo (t ) RF CS dt NB: this type of differentiator is rarely used in practice since it amplifies noise Recall i(t ) C dt ECEN 301 Discussion #18 – Operational Amplifiers 30 Op-Amps – Closed-Loop Mode Circuit Diagram RS Inverting Amplifier + – vS RS1 Summing Amplifier + – vS1 RS2 + – vS2 vo ACL vS RF – + – + ACL RF vS RS + vo – N RF + vo – vo AOLn vSn RSn + – ECEN 301 vSn Discussion #18 – Operational Amplifiers n 1 N RF vSn n 1 Rn 31 Op-Amps – Closed-Loop Mode Circuit Diagram Noninverting Amplifier Voltage Follower ECEN 301 – + R vS + – – + + – vo ACL vS RF RS ACL RF vS 1 RS + vo – + vo – vo ACL vs Discussion #18 – Operational Amplifiers vs 32 Op-Amps – Closed-Loop Mode Circuit Diagram RF RS Differential Amplifier ECEN 301 + – – + RS v1 + – v2 RF + vo – ACL RF vS 2 vS1 vo RS Discussion #18 – Operational Amplifiers 33 Op-Amps – Closed-Loop Mode Circuit Diagram RS Ideal Integrator + – vS CF – + + vo(t) – ACL 1 vo RF CS t vS ( )d RF CS – Ideal Differentiator ECEN 301 + – vS + + vo(t) – dv S (t ) vo RF CS dt Discussion #18 – Operational Amplifiers 34 Op-Amps Example 3: find an expression for the gain if vs(t) is sinusoidal CF = 1/6 F, R1 = 3Ω, R2 = 2Ω, CS = 1/6 F CF vs(t) R1 R2 i1(t) i2(t) iS(t) ECEN 301 iF(t) v+ + vo(t) i1 CS v– – Discussion #18 – Operational Amplifiers 35 Op-Amps Example 3: find an expression for the gain CF = 1/6 F, R1 = 3Ω, R2 = 2Ω, CS = 1/6 F ZF=1/jωCF 1. 2. Node b Vs(jω) Z1 I1(jω) Node a IF(jω) Z2 I2(jω) v+ Iin ZS v– IS(jω) NB: v+ = v– = vo and Iin = 0 ECEN 301 + – Transfer to frequency domain Apply KCL at nodes a and b KCL at a : I1 I F I 2 0 Vo(jω) VS Va Vo Va Va V 0 Z1 ZF Z2 1 1 1 1 1 1 VS Va Vo Z1 Z F Z 2 Z2 ZF Z1 1 j 1 1 j 1 Va Vo VS 3 6 2 2 6 3 Discussion #18 – Operational Amplifiers 36 Op-Amps Example 3: find an expression for the gain CF = 1/6 F, R1 = 3Ω, R2 = 2Ω, CS = 1/6 F ZF=1/jωCF 1. 2. Node b Vs(jω) Z1 I1(jω) Node a ECEN 301 IF(jω) Z2 I2(jω) v+ + Iin ZS IS(jω) v– – Transfer to frequency domain Apply KCL at nodes a and b KCL at b : I 2 I S I in 0 Vo(jω) Va Vo Vo 0 0 Z2 ZS 1 1 1 Va Vo 0 Z2 ZS Z2 1 j 1 Va Vo 0 2 6 2 Discussion #18 – Operational Amplifiers 37 Op-Amps Example 3: find an expression for the gain CF = 1/6 F, R1 = 3Ω, R2 = 2Ω, CS = 1/6 F ZF=1/jωCF Vs(jω) Z1 I1(jω) IF(jω) Z2 I2(jω) v+ + Iin ZS IS(jω) v– – 1. 2. 3. Vo(jω) Transfer to frequency domain Apply KCL at nodes a and b Express Vo in terms of Vs Va 5 j Vo 3 j 2VS 3Va Vo 3 j 0 6VS Vo 2 j5 6 ECEN 301 Discussion #18 – Operational Amplifiers 38 Op-Amps Example 3: find an expression for the gain CF = 1/6 F, R1 = 3Ω, R2 = 2Ω, CS = 1/6 F ZF=1/jωCF Vs(jω) Z1 I1(jω) IF(jω) Z2 I2(jω) + Vo(jω) Transfer to frequency domain Apply KCL at nodes a and b Express Vo in terms of VS Find the gain (Vo/VS) Iin ZS IS(jω) ECEN 301 v+ 1. 2. 3. 4. v– – Vo 6 2 VS j5 6 Discussion #18 – Operational Amplifiers 39 Op-Amps Example 3: find an expression for the gain CF = 1/6 F, R1 = 3Ω, R2 = 2Ω, CS = 1/6 F Frequency Response ZF=1/jωCF Vs(jω) Z1 I1(jω) IF(jω) Z2 I2(jω) v+ + Vo(jω) Iin ZS IS(jω) v– – 2nd order Lowpass filter ECEN 301 Discussion #18 – Operational Amplifiers 40 Instrumentation Op-Amp • Special configuration used for common-mode voltage rejection 2R (V2 V1 ) Vout 1 Rgain Advantages: - Common-mode voltage rejection - Very high input impedance for V1 and V2 - Set gain with one resistor Connect sensor with twist pair in differential configuration to minimize external noise pickup. Using grounded shield also helps.