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Op-Amp Circuits Alan Murray

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Agenda Op-Amp circuit Analysis –non-inverting amplifier circuit –inverting amplifier circuit... from first principles –(i.e. only Ohm's Law!) Op-Amp "GOLDEN RULES" –simplified Op-Amp circuit analysis –non-inverting amplifier circuit –inverting amplifier circuit Positive and Negative Feedback –By analogy and in the Op-Amp context

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Reminder … ideally … Differential Amplifier i.e. V out = A × ( V non - V inv ) –ideally, V inv = V non V out = 0 Input impedance Z in = –i.e. no current to/from the input terminals Output impedance Z out = 0 –i.e. a large current can flow to/from the output Gain A = + - V out =GV in Z out Z in V in

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Clickers out 10V + I V 10Ω 1) I=+1A, V=+10V 2) I=-1A, V=+10V 3) I=+1A, V=-10V 4) I=-1A, V=-10V Solution

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Reminder : Potential Dividers VaVa VbVb VcVc I R1R1 R2R2 V =(V c -V a ) (V b -V a ) DC Potential divider simulation AC Potential divider simulation

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+ - V out V in i=0 I into Op-Amp or to load NB Inverting Circuit Animation

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Op-Amp Circuit Analysis From First Principles The process … –V out =A x (V non -V inv ) –A –Current into V inv and V non terminals= 0 –So I through R 1 = I through R 2 and use Ohm’s Law … + - V out V in i=0 R2R2 R1R1 I I

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Op-Amp Circuit Analysis From First Principles V out =A x (V non -V inv ) ÷A V out /A=V non -V inv A = V out /A =0 V inv = V non = 0V –V inv is a “virtual ground” Ohm’s law, R 1 : I = (V in - 0)/R 1, I = V in /R 1 Ohm’s law, R 2 : I = (0 - V out )/R 2, I = - V out /R 2 V in /R 1 = - V out /R 2, –rearrange to get … V out = -R 2 V in R V out V in i=0 R2R2 R1R1 Have you seen this before? Simulate V tail V tip I I V tail V tip

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- + V out V in i=0 R2R2 R1R1 I I Non-Inverting Circuit The process … –V out =A x (V non -V inv ) –A –I into V inv and V non = 0 –So I in R 1 = I in R 2 and use Ohm’s Law … NB Animation

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- + V out V in i=0 R2R2 R1R1 I I Non-Inverting Circuit V out =A x (V non -V inv ) ÷A V out /A=V non -V inv A = V out /A =0 V non = V inv = V in R 1 and R 2 = a potential divider R2R2 R1R1 V out V inv = V in Here it is again Simulate

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The Golden Rules ANALYSIS OF IDEAL OP-AMP CIRCUITS CAN BE REDUCED TO TWO "GOLDEN RULES"..... 1) No Current enters the "inv" and "non" terminals of the Op-Amp, I inv = I non = 0 2) With negative feedback present, V out will change such that V inv = V non Engrave these on your heart... they are very useful, as long as you remember that they are idealisations Idealisations? More later – all we mean is that, in reality, I inv ≈ I non ≈ 0 V inv ≈ V non For an initial analysis of an Op-Amp circuit with negative feedback, use I inv = I non = 0 V inv = V non

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Alan Murray – University of Edinburgh Non-Inverting Circuit Revisited Golden Rule#1 Golden Rule#1 I(R 1 ) I(R 2 ) =II(R 1 ) I(R 2 ) =I so R 1 & R 2 form a potential dividerso R 1 & R 2 form a potential divider V non = V out × R 1 /(R 1 +R 2 ) V non = V out × R 1 /(R 1 +R 2 ) Golden Rule#2 Golden Rule#2 V in = V inv = V nonV in = V inv = V non V in = V out × R 1 /(R 1 +R 2 ) V in = V out × R 1 /(R 1 +R 2 ) V out = V in × (R 1 +R 2 )/R 1 V out = V in × (R 1 +R 2 )/R V out V in i=0 R2R2 R1R1 I I R2R2 R1R1 V out V inv = V in

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Alan Murray – University of Edinburgh Inverting Circuit, by nodal analysis DO NOT use V out as a node DO NOT use V out as a node (b) is a boring node (b) is a boring node Sum currents at (a) Sum currents at (a) I R + I Rf + I inv = 0 I R + I Rf + I inv = 0 I inv = 0 I inv = 0 Golden RuleGolden Rule (V in -V inv )/R + (V out -V inv )/R f + 0 = 0 (V in -V inv )/R + (V out -V inv )/R f + 0 = 0 (V in -V inv )/R = -(V out -V inv )/R f (V in -V inv )/R = -(V out -V inv )/R f V inv = V non = 0 V inv = V non = 0 Golden RuleGolden Rule V in /R = -V out /R f V in /R = -V out /R f V out = -V in R f R V out = -V in R f R R RfRf (a) (b) + - V in V out V inv V non

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Alan Murray – University of Edinburgh Procedure Check for negative feedback Check for negative feedback Apply Golden Rules Apply Golden Rules Using Nodal Analysis? Using Nodal Analysis? No Current to input terminals of the Op-AmpNo Current to input terminals of the Op-Amp V inv = V nonV inv = V non Rearrange to get V out = function(V in ) Rearrange to get V out = function(V in )

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Alan Murray – University of Edinburgh Try This … + - V out V in = 3V 1kΩ V out a) 3V b) 1.5V c) 6V d) 15V Solution

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Negative Feedback V out = A [V non - V inv ] V non - V inv = V out /A –A = , V out /A = 0 unless V out = V non = V inv –and V out = V inv V out =V non + - V non V inv V out

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Positive Feedback V out = A [V non - V inv ] V non - V inv = V out /A –A = , V out /A = 0 unless V out = V non = V inv –and V out = V non V out =V inv Same result?!?! Positive feedback = negative feedback?!?! NO! + - V non V inv V out

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Analogy - Central Heating Negative Feedback + THERMOSTAT TEMPERATURE TOO HIGH TURN DOWN RADIATOR -

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Analogy - Central Heating Negative Feedback + THERMOSTAT TEMPERATURE TOO LOW TURN UP RADIATOR -

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Analogy - Central Heating Positive Feedback + THERMOSTAT TEMPERATURE TOO LOW TURN DOWN RADIATOR +

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Analogy - Central Heating Positive Feedback + THERMOSTAT TEMPERATURE TOO HIGH TURN UP RADIATOR +

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Analogy - Central Heating Positive or Negative Feedback + DO NOTHING THERMOSTAT TEMPERATURE EXACTLY CORRECT

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Negative Feedback V out = A[ V a - V b ] so a +ve change in V b a -ve change in V out V b = V out ×R 1 /(R 1 +R 2 ) so a +ve change in V out a +ve change in V b "LOOP GAIN" < V in V out (a) (b) R2R2 R1R1 "Forward gain" < 0 "Backward gain" > 0 Run Simulation

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Positive Feedback V out = A[ V b - V a ] so a +ve change in V b a +ve change in V out V b = V out ×R 1 /(R 1 +R 2 ) so a +ve change in V out a +ve change in V b "LOOP GAIN" > V in V out (a) (b) R2R2 R1R1 "Forward gain" > 0 "Backward gain" > 0 Run Simulation

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So what happens in an op-Amp circuit with positive feedback? At temperature = absolute zero, with a perfect Op-Amp and perfect initial conditions, all is well. Otherwise …the smallest disturbance will be amplified and fed back positively The Op-Amp's output will head for and then crash into the power supplies (or close to them) The output may then stick there or oscillate

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Summary You should now know... – How to analyse any simple Op-Amp circuit –(a) From first principles –(b) Using the "Golden Rules" –What is meant by feedback positive and negative

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Small Reminders Power supplies, V + and V -, are NOT the same as V inv and V non although some books use confusing notation No power supplies, no Op-Amp function Golden rules apply strictly to Ideal Op-Amps only Real Op-Amps are not ideal Golden rules are almost true –near enough for most purposes

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