Operational Amplifiers - uni-hamburg.dephoton.physnet.uni-hamburg.de/fileadmin/user_upload/GRK/Internal/... · Operational Amplifiers Electronics: Experimental techniques in the photon

October 2012

Operational Amplifiers

Electronics:

Experimental techniques in the photon sciences

Georg Wirth

Institut für Laser Physik

Georg Wirth, Institut für Laser-Physik 2

Outline

Basics

• Complex impedance & the low pass-filter

• The decibel scale and Bode plots

Introduction

General characteristics

• Basic operation

• The ideal op-amp

The Concept of Feedback

• Basic idea of feedback

• The noninverting and inverting amplifier

Circuit Examples

• Summing and differential amplifier

• Integrator and Differentiator

• Nonlinear applications

Real op-amps

• Frequency Response and Slew Rate

• Input Offset Voltage and Bias Current

Grounding problems

Sinusoidal signals


CjZ

dt

duCi

1)(

RZ

iRu

)(LjZ

dt

diLu

)(

)exp()(

)(

)exp(ˆ)(

)exp(ˆ)(

iZti

tuZ

tjiti

tjutu

impedance

current

voltage

t

u(t), i(t)

2

Re

Im

• Capacities have to be charged first to have a certain voltage

• Voltage drop over ohmic resistor is proportional to current

2

Re

Im

• Current change through coil induces voltage drop

Re

Im

Complex impedance

• decibels (dB) measure the power (intensity) ratio on a logarithmic scale → transfer functions of consecutive elements can be “added”, e.g. gain of two amplifiers


A

E

A

E

u

u

u

uG 102

2

10 log20log10

The decibel scale

• In electronics, the power of a signal is usually proportional to the squared field amplitude (e.g. P = u2 / R for an ohmic resistor)

A

E

P

PG 10log10

dB -20 -3 0 3 10 20 30

power 1/100 1/2 1 2 10 100 1000

amplitude 1/10 1/√2 1 √2 √10 10 √1000

• Suffix indicates fixed reference → absolute quantity

– dBV → relative to 1 V (0 dBV = 1 V, 20 dBV = 10 V)

– dBm → relative to 1 mW (0 dBm = 1 mW, 30 dBm = 1W)

– dBc → power ratio to carrier (e.g. for sideband modulation)


• Many physical systems show low-pass characteristic, i.e. they linearly transmit a signal with a certain time delay (PT1 element in controller theory)

• Exponential step response characteristic

EuAu

Ru

R

Cj

1i

RCtutu

R

u

dt

duCiit

tuu

A

AARC

E

with

is for

else step signal input

)/exp()(

000

:0

0:

0

0

0t

uE(t), uA(t)

The low-pass filter

• The complex transfer function H(jω) relates the output uA(t) to the input uE(t) signal via a

(magnitude) and a phase shift

• translate linear DE into frequency space via Laplace- / Fourier-transformation, transfer function is a fraction of polynomials

• poles and zero characterize transfer function (→ control theory)

0

02

0

arctan)(arctan)(

1

)/(1

1)()(

1

1

/1

/1)(

jH

RCjHG

RCjCjR

Cj

u

ujH

Ziu

ZZiu

E

A

CA

CRE

phase

with gain

functiontransfer

law sohm'


Bode plots

• Characterize transfer function by plotting magnitude (in dB) and phase on a log. frequency scale

• Polynomials appear as straight lines, poles and zeros can be seen as “kinks”

-3dB at cut-off

- π/4

DC gain

-20 dB / decade

4/)1arctan(

2

1

1

arctan)(

1

)/(1

1)(

0

0

02

0

phase

output

off-cut

phase

with gain

EA

CR

uu

RCZZ

RCG


→ Why not use a transistor?

Advantages of op-amps

• Versatility - operation only determined by external surrounding circuit

• No bias current needed to determine operation point

• High input impedance, low output impedance

• High intrinsic gain

• Very linear and precise amplification over broad voltage an frequency range

Disadvantages

• higher noise due to multiple amplifier stages

• lower cut-off-frequency than single-stage transistor amplifiers

• Strong feedback over multiple amplifier stages causes non-linear distortions

Pros and Cons of op-amps

+

–

Basic idea:

Use integrated circuit (black box – internal realization unknown) as a modular device to manipulate signals, so that behavior of the circuit is purely characterized by external elements.

CERDIP / PDIP (dual-inline-package)

TO-99 (transistor- single-outline)

SOIC-8 (small-outline-integrated-circuit)

circuit symbol


Basic operation and wiring

Typical pin assignment

• inverting (-) and noninverting (+) input (2,3)

• voltage difference uD = u+ - u- between inputs is amplified linearly by factor A0 (open-loop-gain) to output uA = A0∙( u+ - u- ) (6)

• every potential is measured relatively to ground potential (usually GND or 0V)

• two connections (4,7) for power supply V ± (usually

±15V)

• no separate ground connection, output ground reference by uA = 0 for u+ - u- = 0 (for ideal op-amp)

• usually two extra pins for external offset compensation (8,1)

+

–

V+

V- =

= + -

- +

2

3

7 8

1

6

4

u

u

Du

AuDu

Au

10 V

-10 V

-100 µV 100 µV

A0 = 105

linear saturation saturation


The ideal op-amp

Parameter Symbol Ideal Real (OP27)

Input Impedance Differential-Mode

zD ∞ 4 MΩ || ≤ 1 pF

Input Impedance Common-Mode

z+ , z- ∞ 2 GΩ || ≤ 1 pF

Input Bias Current

i+ , i- 0 ±15 nA

Input Offset Voltage

u0 0 30 µV

Output Impedance

zA 0 70 Ω

Unity Gain Bandwidth

f (A0 = -3 dB) ∞ 8 MHz

Open-Loop-Gain A0 = duA/duD ∞ 106 = 120 dB

Common-Mode Rejection Ratio

A0 /

duA/d(u+ + u-) ∞ 106 = 120 dB

Slew rate max(duA/dt) ∞ 2.8 V/µs

+

–

~

u

u

Au

Dz

z

z

Az

i

virtual GND

real GND

i

Du


Basic idea of feedback

Problem

• open-loop-gain A0 too high, input voltage range of ≤100 µV to small

• amplification A not adjustable

• gain A0 determined by device (and hence differs between various devices), instabilities

• feed factor kF of output signal back into input

• signal experiences certain amplification / attenuation and phase shift while passing the loop, effect determined by phase difference at inputs

plant ±

controller

input error

feedback

output

Solution op-amp

Consequences

• reduced gain, but improved linearity, frequency response, bandwidth and stability

• more negative feedback results in less dependency on device parameters

• in general, feedback can be frequency-dependant (equalizers and filters) or amplitude-dependant (logarithmic amplifiers or multipliers)

The “golden rules” concerning op-amps

1. The output attempts to do whatever is necessary to make the voltage difference between the inputs zero (infinite open-loop-gain).

2. The inputs draws no current (infinite input impedance).


The noninverting amplifier

• realize controller by voltage divider R2-R1 , rising output voltage hinders input difference u+-u-

uuuR

RR

Au

uR

RRuuAu

R

R

R

RRA

RR

R

Au

uA

uRR

RuAuuAu

uRR

Ru

A

E

A

AEA

A

0

1

21

0

1

210

1

2

1

21

1

21

1

0

21

100

21

1

1

)(

1

1

)()(

circuit-short virtual

feedback from

Amp-OP ideal

gain-loop-closed

signal output

feedback negative

• because generally A0 >> A, the input voltage difference tends

to zero (virtual short-circuit) (complies with 1st golden rule)

• high input impedance (1012Ω), low output impedance (101Ω)

• for R2 → 0 and R1 → ∞ the closed-loop-gain A → 1, op-amp works as buffer (voltage follower)

+

– Eu

Au

u

2R

1R

u

+

– Eu

EA uu

u

u


The inverting amplifier

• connect input signal uE to lower end of feedback voltage divider R2-R1

• assume input voltage uE = 1V , but no output uA = 0V

– since noninverting input is connected to GND u+ = 0V, op-amp sees high input unbalance

– this forces the output uA to go negative, until both input are on GND u+ = u- = 0V

–

+ Eu

Au

u

u

)(2 z

)(1 z

1i

2i

)()(

)(

)()(

)()(0

0

0

1

1

2

01021

2

2

0

1

0

21

21

0

21

zdi

duz

z

z

AzAzz

z

u

uA

zuA

zu

A

zuzu

z

uu

z

uu

i

Auu

iii

E

EE

E

A

EAA

A

DADE

AD

impedance input

gain loop closed

impedance amp-op high

gain loop open

rule junction skirchhoff'

• frequency-dependant (but linear) feedback provides active filters, integrators, differentiators

• non-linear feedback allows to build amplitude-dependant devices, i.e. exponential or logarithmic amplifiers


The summing amplifier

Problem

• summing currents is easy, but since all potentials are grounded, how to add potentials?

Solution

• take inverting amplifier and add currents in feedback loop

• assume that high input impedance i- = 0 and virtual GND u+ ≈ u- = 0

–

+ 1u

Au

u

u

1R

1i

Fi

2R

nR

FR

2u

2i

ni

nF

A

n

n

nFA

F

A

n

n

Fn

uuuR

Ru

RRRR

R

u

R

u

R

uRu

R

u

R

u

R

u

R

u

iiii

21

21

2

2

1

1

2

2

1

1

210

gain-loop-closed

inputs equal assume



The differential amplifier

Problem

• measure voltage drop over some impedance, what if none of both connections has GND contact?

Solution

• use features from both inverting and noninverting amplifier

–

+

2uAu

u

u

1R

NR

1u2R

PR

2

2

11

1

1

11

2

2

1

1

0

uRR

RRu

R

Ruu

R

uu

R

uuii

RR

Ruuu

P

NNA

N

AN

P

P

insert


circuit-short virtual

21

1

21 uuR

RuRRRR N

ANP and if

Caution

• assume RN = α · R1 and thereby the closed-loop-gain A = - α, resistors have tolerance Δα

• different input impedances require source with low output impedance

• better results with instrumentation amplifier

differ valuesresistor if error, CMRR

)1(

)( uud

duA A


The integrator – frequency response

• use inverting amplifier with RC-high-pass C2-R1 filter in feedback

• strong negative feedback for high frequencies (respectively weak feedback for low frequencies) results in low-pass-characteristic

232

22

1

3

1

2

232

223322211

)(1

1

)(

)()(

)(1

)1(||)()()(

CRRj

CRj

R

R

z

zA

CRRj

CRjRRRCzRz

gain loop closed

and use

–

+ Eu

Au

u

u

2C

1R

1i

2i

R2

R3

i3

)()(

)(

)2(2)(

)2(

1

2

1)(

)0(

321

32

2322

2

223

2

3

22

2

2

223

2

3

11

1

3

RRR

RRAA

CRRR

RRRRAA

CRRRRAA

R

RAA

DC

DC

limitupper

off-cutupper

off-cutlower

limitlower

• missing feedback at ω = 0 makes circuit instable, insertion of bypass R3 limits feedback at low frequencies

• attenuation (|A| < 1) at high frequencies can be avoided by insertion of R2

)(log A

log

DCA

A

1 20

dB/dec 20

R3

R2

integrating range


The integrator – operating method

–

+ Eu

Au

u

u

2C

1R

1i

2i

R2

R3

i3

3

22

1

321

232

22

1

3

1

2

)(

0

)(1

1

)(

)()(

R

u

dt

uudC

R

u

iii

CRRj

CRj

R

R

z

zA

ARAE

rule junction

skirchhoff'

gain loop closed

Approximations

• for frequencies far above lower limit ω >> ω1 , current i3 = uA / R3 can be neglected (shortened by C2)

• for frequencies far beneath upper limit ω << ω2, impedance of C2-R2 is dominated by condensator, so the voltage change duR2/dt over R2 can be neglected

)0()(1

)(0021

2

1

A

t

EAAE utdtu

CRtu

dt

duC

R

u

• integration constant uA(0) = Q0 / C is determined by initial condensator charge and capacity

• input bias current i- and input offset voltage u0 result in additional condensator current u0 / R + i-,

which changes output voltage by

i

R

u

Cdt

duA 0

2

1

• i- = 1 µA results with C = 1 µF in a change of 1 V per second (offset compensation)


The differentiator – frequency response

• swap condensator and resistor in integrator

• RC-low-pass filter R2-C1 delivers in high negative feedback for low frequencies, resulting in a

high-pass characteristic

–

+ Eu

Au

u

u

1C

2R

2iR1

i1

11

12

1

2

22

1

11

1

11

1)(

)()(

)(11

)(

CRj

CRj

z

zA

RzCj

CRj

CjRz

gain loop closed

and use

)(log A

log

A

1 20

dB/dec 20

R1

differentiating range

• at high frequencies ω, missing feedback makes

circuit unstable

• HF noise is strongly amplified, circuit tends to oscillate due to phase delay of op-amp and feedback

• gain limiting at high frequencies can be achieved by insertion of R1

1

2

1

2

121

22

1

2

1

2

2

11

)(

1

2

1)(

11)(

0)0(

R

RAA

CRRRAA

CRRA

AA DC

limitupper

off-cutupper

off-cutlower

limitlower


The differentiator – operating method

–

+ Eu

Au

u

u

1C

2R

2iR1

i1

2

11

21

11

12

1

2

)(

0

1)(

)()(

R

u

dt

uudC

ii

CRj

CRj

z

zA

ARE

rule junction

skirchhoff'

gain loop closed

Approximation

• for frequencies far beneath upper limit ω << ω2, impedance of R1-C1 is dominated by condensator, so the voltage change duR1/dt over R1 can be neglected

• differentiator is bias-stable, optional roll-off-condensator in feedback can reduce bandwidth (bandpass)

• capacitive input impedance draws current from source, problems possible at high frequencies

dt

duCRtu

R

u

dt

duC E

AAE 12

2

1 )(0


The logarithmic amplifier

Problem

• measured signal has large dynamic range

Idea

• instead of linear characteristic curve iR = uR / R

of a resistor, use exponential I-U-dependency of semiconductors

• use collector current iC of a bipolar transistor

CSCTBETBECECSC iiuuuuuTii lnexp),(

iCS reverse leakage current

uT = kT/e thermal voltage

–

+ 0EuAu

R

Ci

C

B

E

0)(for

, since

E

CS

ETA

ECBEA

uRi

uuu

Ruiuu

ln

• no error due to collector-base-current iCB, since uCB = 0, but temperature drift

• with appropriate transistor and op-amp with low bias-current, usually nine decades available

• swap resistor and transistor: exponential amplifier

• multiply / divide signals by taking logarithm, add / subtract, take exponential value


The comparator

Problem

• device for comparison of two signals, which decides if voltage is below / above some threshold

Solution

• use op-amp without feedback to compare voltages (comparator)

• because of high open-loop-gain, circuit is sensitive to small voltage differences uD and switches between saturated values -uAmax and uAmax

• in saturation no virtual short-circuit between inputs, i.e. u+ ≠ u-

Du

Au

saturation saturation

maxAu

maxAu

+

–

uu

Du

Au

uuu

uuuu

A

A

Afor

for

max

max

• useful for regeneration of digital signals or as trigger

• special devices for fast applications

Du

Au

t


The noninverting Schmitt trigger

Problem

• comparator has no well-defined output for small input signals uD ≈ 0

• different switching values for high and low state desired

Solution

• use positive feedback to create hysteresis for switching

min

432

214,

max

432

214on,

ArefoffE

ArefE

uuRRR

RRRu

uuRRR

RRRu

point switchinglower

point switchingupper

–

+

refuAu

u

u

1R

2R

Eu3R

4R

AEref uuRR

Ruu

RR

Ru

21

2

43

4 input

• assume high positive input uE, so that output is uAmax

• with falling input voltage uE, output uA remains unchanged until u+ = u- = 0

Eu

Au

maxAu

minAu

off,Eu on,Eu

Du

Au

t

on,Eu

off,Eu


The real op-amp

• differential open-loop-gain A0 = duA/duD is usually not infinite -> error in approx. A0 = ∞

• even with shortened inputs u+ - u- = 0, op-amp amplifies common-mode voltage u+ + u-,

usually expressed by ratio of differential open-loop-gain A0 to duA/d(u+ + u-) -> common-mode rejection ratio

• finite input impedance draws current from source, common-mode input impedance (input to GND) usually negligible, effect compensated if adjusted

• nonzero output impedance negligible, since decrease of output uA by output load is

compensated by feedback

• transistors as well as resistors produce noise, which is amplified towards the output

Parameter Symbol Ideal Real (OP27)

Input Impedance Differential-Mode

zD ∞ 4 MΩ || ≤ 1 pF

Input Impedance Common-Mode

z+ , z- ∞ 2 GΩ || ≤ 1 pF

Input Bias Current

i+ , i- 0 ±15 nA

Input Offset Voltage

u0 0 30 µV

Output Impedance

zA 0 70 Ω

Unity Gain Bandwidth

f (A0=-3 dB) ∞ 8 MHz

Open-Loop-Gain A0 = duA/duD ∞ 106 = 120 dB

Common-Mode Rejection Ratio

A0 /

duA/d(u+ + u-) ∞ 106 = 120 dB

Slew rate max(duA/dt) ∞ 2.8 V/µs


Real op-amps - frequency response

• real op-amps are multi-stage transistor-amplifiers

• every single stage represents a low-pass filter with a particular cut-off-frequency ω, that

reduces the gain by -20 db per decade and adds a certain phase shift of maximal φ = π/2

(subsequent stages usually have increasing bandwidths)

• if A0(ω) is open-loop-gain, and |kF| ≤ 1 is feedback

factor, requirement for oscillation is (negative input adds phase shift φ = π)

• to prevent oscillation at ωC, feedback factor has to be maximal kF ≤ 1 / A0(ωC) (signal gain in one loop

passage must be smaller than one)

,...2,0arg

11)()(

0

0

0

Ak

AkAk

F

F

F

If an op-amp is not internally frequency-compensated (and by that not unity-gain stable), external frequency compensation or minimal closed-loop-gain is necessary.

)(log 0 A

log

1 2 3

-20 dB/dec

-60 dB/dec

0

2

23

ωC

A0(ωC) -40 dB/dec


Comparison OP27 – OP37

OP27 (unity-gain stable)

OP37 (not unity-gain stable)


Bandwidth

• frequency-compensated op amps can be regarded as 1st order low-pass

iiDC CR

i

AA

1

1

0)(1

)(

Eu

Au2R

1R

+

– iR

iC

frequency off-cut loop-closed

gain loop-closed (DC)

for

for

insert

DCF

F

FE

A

Ak

RR

R

kA

ωAi

ωA

i

AA

AAk

A

RR

R

Au

uA

11

21

1

11

1

1

0

0

0

1

21

1

0

1

)()(1)(

)()(1

)(

)(

1)(

• for frequencies ω >> ω1 above cut-off, open-

loop gain is approximately

• usually ADC is in the range of 120 dB, but cut-off frequency very low ω1 / 2 π = 10 Hz

• use closed-loop gain from noninverting amplifier with feedback factor kF = R1 / (R1 +R2)

to calculate new frequency response

bandwidth) gain-(unity

product-bandwidth-gain DC

DC

A

AiA

1

10 )()(

log

1

ADC

ω1´

A = 1 / kF

log( A0(ω) )

log( A(ω) )


Slew Rate

• internal capacities and frequency compensation act as a low-pass, whose output for a unit-step input is a (more or less fast) exponential function

• output transistors and capacities need driving current from (previous) driving stage, whose output current is limited

• output change rate is limited to the slew rate SR = max(duA/dt)

• image sine-wave input signal

maxmax )(maxsin EA

EE uAdt

dutuu

• slew rate limits amplitude of undistorted sine-wave output swing above some critical frequency (power bandwidth)

Au

t

t

Au

amplitude-dependant distortion

frequency-dependant distortion

SRuE max


Input offset voltage and bias current

Offset voltage

• output signal uA ≠ 0 even for no input signal u+ = u- = 0, due to unsymmetrical differential-

amplifier at input (usually several µV)

• amplified towards output

• can be compensated by potentiometer at extra pins

• problem is temperature drift of usually 1 µV/°C

+

–

10k V+

V- =

= + -

- +

2

3

7 8

1

6

4

Bias current

• input transistors need constant base- or gate-current for operation, which is delivered by power supply Bipolar 100 nA Darlington 1 nA FET 1 pA

• can be compensated by additional bias resistor RB

21

21

RR

RRRB

–

+ Eu

Au

2R

1R

BR


Grounding problems

icon

icon

Ground loops

• RF-signal need shielding to avoid crosstalk → use coaxial cables as waveguides

• Problem: DC-signals need two wires, i.e. a clear voltage reference (usually GND)

→ shielding provides conductor loop between two grounded casings

• Time-varying stray fields from transformers induce voltages / currents in loop

→ a 10 µT stray field of a 50 Hz transformer induces ~100 mV in a 10 m2 conductor loop

~

casing 1 casing 2

Ground connections

• Conductive tracks have nonzero resistance, so flowing currents produce a voltage drop (ground shift) -> separate signal ground from consumer ground

• If possible, connect every device/component starlike to one common ground reference


Avoiding ground loops

photo- diode

controller

AOM- driver

RF- amp

AOM

power- supply

optical table

optical table

switch- board

oscillo- scope

ADwin

1. Don’t use shielding as conductor, i.e. signal path

2. Isolate all casings from metal surfaces

3. Use separate power supplies

4. Power supplies don’t need a ground reference

5. Use digital / analog isolators (e.g. ADUM524x and AD210)

6. Measure signals differentially

7. No oscilloscopes at unbuffered / source signals


References

Control theory

• G. v. Wichert, Grundlagen der Regelungstechnik (http://tams-www.informatik.uni-hamburg.de/lehre/2004ws/vorlesung/regelungstechnik/folien/Grundlagen%20der%20Regelungstechnik_02.pdf)

Electronics

• U. Tietze, C. Schenk, E. Gamm: Halbleiter – Schaltungstechnik, Springer, Berlin 2002

• P. Horowitz, W. Hill: The Art of Electronics, Cambridge University Press 1989

• T. Matsuyama: Vorlesungsskript Elektronik 1 und 2, Hamburg 2005

Op-amp circuits

• Ron Mancini: Op Amps for Everyone. Design Reference. 2 Auflage. Elsevier, Oxford 2003 (http://focus.ti.com/lit/an/slod006b/slod006b.pdf)

• Walter G. Jung: OP AMP Applications. Firmenschrift Analog Devices, Norwood 2002 (http://www.analog.com/library/analogDialogue/archives/39-05/Op_Amp_Applications.zip)

• Wikipedia (http://en.wikipedia.org/wiki/Operational_amplifier_applications)

Datasheets / Manufacturers

• http://www.datasheetcatalog.com

• ANALOG DEVICES (http://www.analog.com)

• Texas Instruments / Burr Brown / National Semiconductor (http://www.ti.com)

http://tams-www.informatik.uni-hamburg.de/lehre/2004ws/vorlesung/regelungstechnik/folien/Grundlagen der Regelungstechnik_02.pdf






http://focus.ti.com/lit/an/slod006b/slod006b.pdf

http://www.analog.com/library/analogDialogue/archives/39-05/Op_Amp_Applications.zip



http://en.wikipedia.org/wiki/Operational_amplifier_applications

http://www.datasheetcatalog.com/

http://www.datasheetcatalog.com/

http://www.analog.com/

http://www.ti.com/

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Operational Amplifiers - uni-hamburg.dephoton.physnet.uni-hamburg.de/fileadmin/user_upload/GRK/Internal/... · Operational Amplifiers Electronics: Experimental techniques in the photon