Op-Amp Circuits

Alan Murray

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

+

- VoutVini=0

i=0

I

into Op-Amp or to load

NB

Inverting CircuitInverting Circuit

Animation

Op-Amp Circuit Analysis From First Principles

• The process …– Vout=A x (Vnon-Vinv)

– A – Current into Vinv and Vnon terminals= 0

– So I through R1 = I through R2 • and use Ohm’s Law …

+

-VoutVin

i=0

i=0

R2

R1

I

I

Op-Amp Circuit Analysis From First Principles

• Vout=A x (Vnon-Vinv)÷A

• Vout/A=Vnon-Vinv

• A = Vout/A =0• Vinv = Vnon = 0V

– Vinv is a “virtual ground”• Ohm’s law, R1 : I = (Vin - 0)/R1, I = Vin/R1

• Ohm’s law, R2 : I = (0 - Vout)/R2, I = - Vout/R2

• Vin/R1 = - Vout/R2,

– rearrange to get …• Vout = -R2 Vin

R1

+

-VoutVin

i=0

i=0

R2

R1

Have you seen this before? Simulate

Vtail Vtip

I

I

Vtail

Vtail

Vtip

Vtip

-

+Vout

Vin

i=0

i=0

R2

R1

II

Non-Inverting Circuit

• The process …– Vout=A x (Vnon-Vinv)

– A – I into Vinv and Vnon = 0

– So I in R1 = I in R2 • and use Ohm’s Law …

NB

Animation

-

+Vout

Vin

i=0

i=0

R2

R1

II

Non-Inverting Circuit

• Vout=A x (Vnon-Vinv)

÷A

• Vout/A=Vnon-Vinv

• A = Vout/A =0

• Vnon = Vinv = Vin

• R1 and R2 = a potential divider

R2

R1

Vout

Vinv= Vin

1in out

1 2

RV = V

R +R

1 2out in

1

R +RV = V

R

Here it is again

Simulate

The Golden Rules ANALYSIS OF IDEAL OP-AMP CIRCUITS CAN BE REDUCED TO TWO "GOLDEN RULES" .....

1) No Current enters the "inv" and "non" terminalsof the Op-Amp, Iinv = Inon = 0

2) With negative feedback present,Vout will change such that Vinv = Vnon

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,

Iinv ≈ Inon ≈ 0

Vinv ≈ Vnon

For an initial analysis of an Op-Amp circuit with negative feedback, use

Iinv = Inon = 0

Vinv = Vnon

Alan Murray – University of EdinburghAlan Murray – University of Edinburgh

Non-Inverting CircuitNon-Inverting CircuitRevisitedRevisited

Golden Rule#1Golden Rule#1• I(RI(R11) I(R) I(R22) =I) =I

• so Rso R11 & R & R22 form forma potential dividera potential divider

VVnonnon = V = Voutout × R × R11/(R/(R11+R+R22))

Golden Rule#2Golden Rule#2• VVinin = V = Vinvinv = V = Vnonnon

VVinin = V = Voutout × R × R11/(R/(R11+R+R22)) VVoutout = V = Vinin × (R × (R11+R+R22)/R)/R11

-

+Vout

Vin

i=0

i=0

R2

R1

II

R2

R1

Vout

Vinv= Vin

Alan Murray – University of EdinburghAlan Murray – University of Edinburgh

Inverting Circuit, by nodal analysisInverting Circuit, by nodal analysis

DO NOT use VDO NOT use Voutout as a node as a node (b) is a boring node(b) is a boring node Sum currents at (a)Sum currents at (a) IIRR + + IIRfRf + + IIinvinv = 0 = 0 IIinvinv = 0 = 0

• Golden RuleGolden Rule (V(Vinin-V-Vinvinv)/R)/R + + (V(Voutout-V-Vinvinv)/R)/Rff + 0 = 0 + 0 = 0 (V(Vinin-V-Vinvinv)/R)/R = = -(V-(Voutout-V-Vinvinv)/R)/Rff VVinvinv = V = Vnonnon = 0 = 0

• Golden RuleGolden Rule VVinin/R = -V/R = -Voutout/R/Rff VVoutout = -V = -VininRRff

RR

R Rf

(a)

(b) +

-

Vin

Vout

Vinv

Vnon

Alan Murray – University of EdinburghAlan Murray – University of Edinburgh

ProcedureProcedure

Check for negative feedbackCheck for negative feedback Apply Golden RulesApply Golden Rules

Using Nodal Analysis?Using Nodal Analysis?

• No Current to input terminalsNo Current to input terminalsof the Op-Ampof the Op-Amp

• VVinvinv = V = Vnonnon

Rearrange to get VRearrange to get Voutout = function(V = function(Vinin))

Alan Murray – University of EdinburghAlan Murray – University of Edinburgh

Try This …Try This …

+

-

VoutVin = 3V

1kΩ

1kΩ

1kΩ

VVoutout

a) 3Va) 3V

b) 1.5Vb) 1.5V

c) 6Vc) 6V

d) 15Vd) 15V

Solution

Negative Feedback

• Vout = A [Vnon - Vinv]

• Vnon - Vinv = Vout/A

– A = , Vout /A = 0

• unless Vout =

• Vnon = Vinv

– and Vout = Vinv

• Vout =Vnon

+-

Vnon

Vinv

Vout

Positive Feedback

• Vout = A [Vnon - Vinv]

• Vnon - Vinv = Vout/A

– A = , Vout /A = 0

• unless Vout =

• Vnon = Vinv

– and Vout = Vnon

• Vout =Vinv

• Same result?!?!• Positive feedback = negative feedback?!?!• NO!

+-

Vnon

VinvVout

Analogy - Central HeatingNegative Feedback

+ THERMOSTAT

TEMPERATURE TOO HIGH

TURN DOWN RADIATOR

-

Analogy - Central HeatingNegative Feedback

+ THERMOSTAT

TEMPERATURE TOO LOW

TURN UP RADIATOR

-

Analogy - Central HeatingPositive Feedback

+ THERMOSTAT

TEMPERATURE TOO LOW

TURN DOWN RADIATOR

+

Analogy - Central HeatingPositive Feedback

+ THERMOSTAT

TEMPERATURE TOO HIGH

TURN UP RADIATOR

+

Analogy - Central HeatingPositive or Negative Feedback

+

DO NOTHING

THERMOSTAT

TEMPERATURE EXACTLY CORRECT

Negative Feedback

• Vout = A[ Va - Vb ]

• so a +ve change in Vb

a -ve change in Vout

• Vb = Vout×R1/(R1+R2)

• so a +ve change in Vout

a +ve change in Vb

• "LOOP GAIN" < 0

+

-

Vin

Vout

(a)

(b)

R2R1

"Forward gain" < 0

"Backward gain" > 0

Run Simulation

Positive Feedback

• Vout = A[ Vb - Va ]

• so a +ve change in Vb

a +ve change in Vout

• Vb = Vout×R1/(R1+R2)

• so a +ve change in Vout

a +ve change in Vb

• "LOOP GAIN" > 0

+

-Vin

Vout

(a)

(b)

R2R1

"Forward gain" > 0

"Backward gain" > 0

Run Simulation

So what happens in an op-Amp circuit with positive feedback?

• At temperature = absolute zero, with a perfectOp-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

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

Small Reminders

• Power supplies, V+ and V-, are NOT the same as Vinv and Vnon 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