Post on 12-May-2019
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
Georg Wirth, Institut für Laser-Physik 3
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
Georg Wirth, Institut für Laser-Physik 4
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)
Georg Wirth, Institut für Laser-Physik 5
• 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'
Georg Wirth, Institut für Laser-Physik 6
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
Georg Wirth, Institut für Laser-Physik 7
→ 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
Georg Wirth, Institut für Laser-Physik 8
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
Georg Wirth, Institut für Laser-Physik 9
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
Georg Wirth, Institut für Laser-Physik 10
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).
Georg Wirth, Institut für Laser-Physik 11
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
Georg Wirth, Institut für Laser-Physik 12
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
Georg Wirth, Institut für Laser-Physik 13
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
rule junction skirchhoff'
Georg Wirth, Institut für Laser-Physik 14
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
rule junction skirchhoff'
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
Georg Wirth, Institut für Laser-Physik 15
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
Georg Wirth, Institut für Laser-Physik 16
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)
Georg Wirth, Institut für Laser-Physik 17
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
Georg Wirth, Institut für Laser-Physik 18
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
Georg Wirth, Institut für Laser-Physik 19
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
Georg Wirth, Institut für Laser-Physik 20
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
Georg Wirth, Institut für Laser-Physik 21
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
Georg Wirth, Institut für Laser-Physik 22
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
Georg Wirth, Institut für Laser-Physik 23
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
Georg Wirth, Institut für Laser-Physik 24
Comparison OP27 – OP37
OP27 (unity-gain stable)
OP37 (not unity-gain stable)
Georg Wirth, Institut für Laser-Physik 25
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) )
Georg Wirth, Institut für Laser-Physik 26
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
Georg Wirth, Institut für Laser-Physik 27
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
Georg Wirth, Institut für Laser-Physik 28
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
Georg Wirth, Institut für Laser-Physik 29
Avoiding ground loops
photo- diode
con- troller
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
Georg Wirth, Institut für Laser-Physik 30
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)