EE100Su08 Lecture #9 (July 16 th 2008) Outline HW #1s and Midterm #1 returned today Midterm #1 notes HW #1 and Midterm #1 regrade deadline: Wednesday, July 23 rd 2008, 5:00 pm PST. Procedure: HW #1: Bart s office hours Midterm #1: Attach a note to the FRONT of your test with your complaint and drop it in HW box Questions? This week: Operational Amplifiers (Op-Amps) Op-Amp Model Negative Feedback for Stability Components around Op-Amp define the Circuit Function Nonlinear circuits Op-Amp from 2-Port Blocks Slide 1
The Operational Amplifier The operational amplifier ( op amp ) is a basic building block used in analog circuits. Its behavior is modeled using a dependent source. When combined with resistors, capacitors, and inductors, it can perform various useful functions: amplification/scaling of an input signal sign changing (inversion) of an input signal addition of multiple input signals subtraction of one input signal from another integration (over time) of an input signal differentiation (with respect to time) of an input signal analog filtering nonlinear functions like exponential, log, sqrt, etc Isolate input from output; allow cascading Slide 2
Op Amp Terminals 3 signal terminals: 2 inputs and 1 output IC op amps have 2 additional terminals for DC power supplies Common-mode signal= (v 1 v 2 )/2 Differential signal = v 1 -v 2 V positive power supply Inverting input v 2 Non-inverting input v 1 - v 0 output V negative power supply Slide 3
Op Amp Notation and Model Slide 4
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Summing-Point Constraint Check if under negative feedback Small v i result in large v o Output v o is connected to the inverting input to reduce v i Resulting in v i =0 Summing-point constraint v 1 = v 2 i 1 = i 2 =0 Virtual short circuit Not only voltage drop is 0 (which is short circuit), input current is 0 This is different from short circuit, hence called virtual short circuit. Slide 16
Ideal Op-Amp Analysis Technique Assumption 1: The potential between the op-amp input terminals, v () v (-), equals zero. Assumption 2: The currents flowing into the op-amp s two input terminals both equal zero. No Currents R 1 No Potential Difference R 2 VIN EXAMPLE V 0 Slide 17
Ideal Op-Analysis: Non-Inverting Amplifier Assumption 1: The potential between the op-amp input terminals, v () v (-), equals zero. Assumption 2: The currents flowing into the op-amp s two input terminals both equal zero. R 1 VIN EXAMPLE KCL with currents in only two branches R 2 V 0 V IN appears here v R v in v in v R out = 1 2 R1 R2 out = v in R1 0 Non-inverting Amplifier Slide 18
Ideal voltage amplifier v in - _ R 2 R 1 Non-Inverting Amplifier v 0 v1 = v2 = vin, i1 = i2 = 0 v 2 Use KCL At Node 2. 2 R L Closed loop gain ( v0 v2) ( v2 0) i = = R R 2 1 vo ( R1 R2) A = = v R in Input impedance 1 = A = v i v v v o in in = Slide 19
Ideal Op-Amp Analysis: Inverting Amplifier R 1 R 2 I 2 V IN -V R R L V OUT Voltage is V R Only two currents for KCL V V R V R OUT 1 IN = V R V R R R V R 2 OUT = 0 ( V V ) Inverting Amplifier with reference voltage 2 1 in R Slide 20
Negative feedback checked Use summing-point constraint i v in - R 1 2 v 2 v 1 _ R 2 Inverting Amplifier v 0 R L v Closed loop gain = Av = v v1 = v2 = 0, i1 = i2 = 0 Use KCL At Node 2. ( vin v2) ( vout v2) i = = R1 R2 Rv 2 o vo = R1 vin Input impedance = = R1 i o in Ideal voltage source independent of load resistor Slide 21
Voltage Follower v in - v 2 v 0 _ RL R R i 2 1 = 0 ( v0 v2) ( v2 0) = = R R 2 1 vo ( R1 R2) R2 A = = = 1 = 1 v R R in 1 1 Slide 22
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Summing Amplifier v 1 R 1 - R 0 - v 2 - R 2 _ v 0 v 3 R 3 Slide 29
Difference Amplifier R 2 v 1 - - R 1 _ v 0 v 2 R 3 R 4 Slide 30
Want v o Integrator = K vindt What is the difference between: v in - R C V 0 - Slide 31
Differentiator Want R v in - C _ v 0 Slide 32
Nonlinear Opamp Circuits Start reading through online notes: Introduction to nonlinear circuit analysis. Outline: Differences between positive and negative feedback. Oscillator circuit. Slide 33
High Quality Dependent Source In an Amplifier V AMPLIFIER SYMBOL Differential Amplifier V V A 0 V 0 = A(V V ) AMPLIFIER MODEL Circuit Model in linear region R i V 1 AV 1 V 0 V 0 depends only on input (V V - ) See the utility of this: this Model when used correctly mimics the behavior of an amplifier but omits the complication of the many many transistors and other components. Slide 34
Model for Internal Operation A is differential gain or open loop gain Ideal op amp A Ri R = 0 o v 1 Circuit Model i 1 R o i o Common mode gain = 0 ( v v ) = = 2 v = A v A v 1 2 vcm, vd v1 v2 o cm cm d d Since v = A( v v ), A = 0 o 1 2 cm v 2 i 2 _ R i A(v 1 v 2 ) v o Slide 35
Model and Feedback Negative feedback connecting the output port to the negative input (port 2) Positive feedback connecting the output port to the positive input (port 1) Input impedance: R looking into the input terminals Output impedance: Impedance in series with the output terminals v 1 v 2 Circuit Model i 1 i 2 R o i o R i v o _ A(v 1 v 2 ) Slide 36
Op-Amp and Use of Feedback A very high-gain differential amplifier can function in an extremely linear fashion as an operational amplifier by using negative feedback. R 1 R 2 R 1 R 2 VIN Summing Point Negative feedback Stabilizes the output V 0 V IN R i - V 1 Circuit Model Hambley Example pp. 644 for Power Steering AV 1 V 0 We can show that that for A and R i, V 0 V IN R 1 R 1 R 2 Stable, finite, and independent of the properties of the OP AMP! Slide 37
Application: Digital-to-Analog Conversion A DAC can be used to convert the digital representation of an audio signal into an analog voltage that is then used to drive speakers -- so that you can hear it! Weighted-adder D/A converter S4 10K 8V - S3 S2 S1 4-Bit D/A 20K 40K 80K (Transistors are used as electronic switches) 5K V 0 S1 closed if LSB =1 S2 " if next bit = 1 S3 " if " " = 1 S4 " if MSB = 1 Slide 38 Binary number (volts) 0 0 0 0 0 0 0 0 1.5 0 0 1 0 1 0 0 1 1 1.5 0 1 0 0 2 0 1 0 1 2.5 0 1 1 0 3 0 1 1 1 3.5 1 0 0 0 4 MSB 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 LSB Analog output 4.5 5 5.5 6 6.5 7 7.5
Analog Output (V) 8 7 6 5 4 3 2 1 0 0000 0 2 4 6 8 10 12 14 16 0001 Characteristic of 4-Bit DAC 0100 Digital Input Slide 39 1000 1111