Lecture28 Operational Amplifier

Lecture 28 Operational Amplifiers Reading: Jaeger 11.1-11.5 and Notes

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ECE 3040 - Dr. Alan Doolittle

Operational Amplifier •Operational Amplifier or “Op-Amp” is a multistage amplifier that is used for general electrical signal manipulation. •The numbers of applications possible with Op-amps are two numerous to list. •Most everyone agrees: “Op-Amp analysis is significantly easier than transistor analysis.” •Though they are often internally complex, their use in circuits most often simplifies the overall design. •The circuit is modeled by an ideal voltage amplifier. Model

Circuit Symbol

+ Vin -

+ + Vout -

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Vin -

+ Vout -

ECE 3040 - Dr. Alan Doolittle

Ideal Operational Amplifier •Rin = Infinity, •Voltage Gain, Av=Infinity at all frequencies •Rout=0

Model

+ Vin -

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+ Vout -

ECE 3040 - Dr. Alan Doolittle

Ideal Operational Amplifier •Infinite gain means that the device is useless without adding “Feedback” to control the overall gain to a finite value.

i+ v+ iin •Consider the circuit to the right with vin=0 vout = Av (− v − )

vin

R1 v − = vout R1 + R 2 R1 v − = − Av v − R1 + R 2 If A v → ∞, the above equation is only satisfied for v - = 0

iout

vi-

vout i2

•Feedback forces the two input voltages to be equal! This is known as a “virtual ground”. •R1 and R2 form a “Feedback Network” Georgia Tech

ECE 3040 - Dr. Alan Doolittle

Inverting Amplifier •Finite voltage gain results from an infinite voltage gain amplifier with “negative feedback” (feedback that takes a fraction of the output voltage and mixes it back into the negative summation node). i+

1) vin − iin R1 − i2 R2 − vo = 0

v+

2) iin = i− + i2 = i2 due to infinite input resistance iin

v -v i in = in R1 but v - = 0 due to the virtual ground v 3) i in = in R1 Combining 1, 2 and 3, v v vin − in R1 − in R2 − vo = 0 R1 R1 vo R =− 2 v in R1

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iout

vi-

vin

vout i2

Overall circuit gain is finite, negative (for this feedback configuration) and set by the feedback resistor network. ECE 3040 - Dr. Alan Doolittle

Inverting Amplifier •Input Resistance:

vin = R1 Rin = iin

i+ v+ iin

iout

vi-

vin

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vout i2

ECE 3040 - Dr. Alan Doolittle

Inverting Amplifier •Output Resistance:

v t = i1 R 3 + i 2 R 1 but, i1 = 0 since v - = 0 and

v+

it

i1 v-

i2

vt

i1 = i 2 thus, vt = 0 + 0 Rout

vt = =0 it

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ECE 3040 - Dr. Alan Doolittle

Non-Inverting Amplifier iin

Virtual Ground or “short”

v+ vin

vi-

v − = vout

vout

R1 R1 + R2

The virtual ground requires that v + = v - so, v in = v -

i2

so, vout vout R1 + R2 = = vin v− R1 Av =

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vout R = 1+ 2 vin R1

Rin =

vin vin = =∞ iin 0

Rout = 0 (Same circuit as for Non - inverting case)

ECE 3040 - Dr. Alan Doolittle

Unity Gain Buffer or “Voltage Follower”

Same as Non − inverting amplifier except R2 = 0 and R1 = ∞ Av =

vout R = 1 + 2 ⇒ Av = 1 v− R1

Rin =

vin vin = =∞ iin 0

Rout = 0 (Same circuit as for Non - inverting case)

•Can be used to isolate a high impedance circuit from a low impedance circuit Georgia Tech

ECE 3040 - Dr. Alan Doolittle

Summing Amplifier v+ iin,a

v-

i2 = iin ,a + iin ,b + iin ,c vin ,a vin ,b vin ,c v − out = + + R 2 R1a R1b R1c R2 R2 R2 − vin ,b − vin ,c vout = −vin ,a R1a R1b R1c R2  R2 R2  + vin ,b + vin ,c vout = − vin ,a  R c R b 1 1 1 R a  

vin,a

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vout

iin,b vin,b iin,c

•Output is a scaled sum of inputs. •Scaling can be controlled by ratios of resistors

i2

vin,c ECE 3040 - Dr. Alan Doolittle

Difference Amplifier v+

v+

vin,a

vth

v-

v-

vout

vout vin,b

vin,b vth = vin ,a

R2 R1 + R2

and

Rth = R1 || R2

Using Superposition we can combine the results of the Inverting and Non-inverting solutions: vin,b=0 vin,a=0 v out

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 R  = vth 1 + 2  R1  

v out = vin ,a

R2 R1 + R2

v out = vin ,a

R2 R1

vout = − v in,b  R2   1 + R 1  

R2 R1

vout = v in,a

R2 R − v in,b 2 R1 R1

vout = (v in,a − v in,b )

R2 R1

This circuit amplifies the difference of two signals ECE 3040 - Dr. Alan Doolittle

Non-Ideal (Real World) Operational Amplifiers Finite Open-Loop Gain

•Real op-amps do not have “infinite” “open loop (without feedback)” gain. •Voltage gains are typically large but finite: ~104-106 V/V •Finite gain causes a deviation from ideal amplifier behavior v − = v out

R1 = βv out R1 + R 2

where β =

v+

R1 is known as the feedback factor R1 + R 2

v out = A openloop (v + − v − ) = A openloop (v + − βv out )

vin

v-

vout

so, A v, closed loop = If

A openloop v out = , where β A openloop is the loop gain v in 1 + β A openloop

β A openloop >> 1

A v, closed loop =

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1

β

= 1+

R2 R1

⇒ approaches the infinite gain result

ECE 3040 - Dr. Alan Doolittle

Non-Ideal (Real World) Operational Amplifiers Finite Open-Loop Gain

•Finite open-loop gain means the Virtual Ground is not perfect! v + − v − = v + − β vout

 A openloop  v+ v + = = v+ −   1+ β A 1 + β A openloop openloop  

Small but finite offset between + and - terminals

v+ vin

v-

vout

•The Gain Error (GE) that results from the Non-infinite open-loop gain can be quantified as:  1   A openloop  v+ GE =   −  v+ = β (1 + β A openloop )  β   1 + β A openloop  Georgia Tech

ECE 3040 - Dr. Alan Doolittle

Non-Ideal (Real World) Operational Amplifiers Finite Output Impedance •Real Op-Amps have a small but finite output impedance, Ro . •We want to find the Output impedance of the various circuits we have examined . •All the configurations have a common circuit for calculating the output impedance .

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ECE 3040 - Dr. Alan Doolittle

Non-Ideal (Real World) Operational Amplifiers Finite Output Impedance Rout =

vx ix

i x = io + i 2 io =

v x − Av ,openloop (v + − v − ) Ro

v x = i2 (R1 + R2 ) ⇒ i2 = v− =

vx (R1 + R2 )

R1 v = β vx (R1 + R2 ) x

(Rout )

−1

i x 1 + Av ,openloop β 1 = = + (R1 + R2 ) vx Ro

 Ro Rout =  1+ A v ,openloop β 

   

(R1 + R2 )

Ro is very small so this term is EXTREMELY small! Georgia Tech

ECE 3040 - Dr. Alan Doolittle

Non-Ideal (Real World) Operational Amplifiers Finite Input Impedance: Non-Inverting Case

•Real Op-Amps have a large but finite input resistance, RID ix =

v x − v− R ID

Neglecting the current i x compared to i1 and i 2 (due to R ID >> R 1 or R 2 ) v − = i1 R1 ≈ i2 R1 v− = v− =

R1 vout = β vout = Av ,openloop β (v x − v − ) R1 + R2 Av ,openloop β 1 + Av ,openloop β

vx

 Av ,openloop β   vx − vx  1+ A  vx v ,openloop β  = ix = (1 + Av,openloop β )RID R ID Rin = (1 + Av ,openloop β )R ID

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RID is very large so Rin is EXTREMELY large! ECE 3040 - Dr. Alan Doolittle

Non-Ideal (Real World) Operational Amplifiers Finite Input Impedance: Inverting Case

•Real Op-Amps have a large but finite input resistance, RID

Rin R’in

•Rin=R1 +R’in •Find R’in by forming a new test circuit i1 = i− + i2 =

(R'in )−1 =

v1 + Av ,openloop v1 v −v v1 v + 1 out = 1 + R ID R2 R ID R2

i1 1 1 + Av ,openloop = + v1 R ID R2

 R2 R 'in = (RID )   1+ A v , openloop  Thus,

   

R’in

 R2 Rin = R1 + (RID )   1+ A v , openloop 

   

Since RID>>R2/(1+Av,openloop) and Av,openloop is very large,  R2 Rin = R1 + (R ID )   1+ A v ,openloop  Georgia Tech

  R2  ⇒ Rin = R1 +    1+ A v , openloop  

  ≈ R1   ECE 3040 - Dr. Alan Doolittle