Operational Amplifiers
Electronics:
Experimental techniques in the photon sciences
Georg Wirth
Institut für Laser Physik
October 2012
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
Georg Wirth, Institut für Laser-Physik
2
Complex impedance
u(t), i(t)
Sinusoidal signals

u (t )  uˆ  exp( jt )
i (t )  iˆ  exp( jt   )
voltage
current
t
impedance
•
Z
u (t )
 Z  exp(i )
i (t )
Capacities have to be
charged first to have a
certain voltage
du
dt
1
Z ( ) 
j C
iC
Im
•
Voltage drop over ohmic
resistor is proportional to
current
uL
Im
Re

Re
2
Georg Wirth, Institut für Laser-Physik
Current change through
coil induces voltage drop
di
dt
Z ( )  jL
u  R i
Z ( )  R
Im

•
3


2
Re
The decibel scale
•
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
P 
G  10  log10  E 
 PA 
•
In electronics, the power of a signal is usually proportional to the squared field amplitude
(e.g. P = u2 / R for an ohmic resistor)
 u E2 
u 
G  10  log10  2   20  log10  E 
 uA 
 uA 
dB
power
amplitude
•
-20
-3
0
3
10
20
30
1/100
1/2
1
2
10
100
1000
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
4
The low-pass filter
•
•
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
u : t  0
input signal step u E   0
 0 : else
uR
R
uE
i
du A u A

0
dt
R
 u A (t )  u0  exp(t /  ) with   RC

•
•
•
for t  0 is iC  iR  0
1
j C
uE(t), uA(t)
uA
 C
t 0
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)
ohm' s law
u E  i  Z R  Z C 
u A  i  ZC
transfer function
H ( j ) 
uA
1 / j C
1


u E R  1 / jC 1  jRC
1
gain
G ( )  H ( j ) 
phase
 ( )  arctanH ( j )  arctan 
Georg Wirth, Institut für Laser-Physik
1  ( / 0 ) 2
with 0 
 

 0 
5
1
RC
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”
DC gain
-3dB at cut-off
-20 dB / decade
1
gain
G ( ) 
phase
 ( )  arctan 
1  ( / 0 )
with 0 
1
RC
 

 0 
cut - off
Z R  ZC
output
uA 
phase
2
 
1
 0
RC
1
uE
2
  arctan(1)   / 4
Georg Wirth, Institut für Laser-Physik
- π/4
6
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.
→
–
circuit symbol
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
TO-99 (transistorsingle-outline)
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
Georg Wirth, Institut für Laser-Physik
SOIC-8 (small-outlineintegrated-circuit)
CERDIP / PDIP (dual-inline-package)
7
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)
uA
10 V
2
A0 =
105
uD
uD
-100 µV
u
100 µV
linear
Georg Wirth, Institut für Laser-Physik
3
saturation
8
7
+
V-
=
8
-
1
6
–
4
-
u
-10 V
saturation
+
V+
=
+
uA
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
i
+
z
uD
u
i
~
zD
–
zA
z
uA
Georg Wirth, Institut für Laser-Physik
u
virtual GND
real GND
9
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
Solution
•
•
op-amp
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
input
error
output
plant
±
feedback
controller
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
10
The noninverting amplifier
•
realize controller by voltage divider R2-R1 , rising output
voltage hinders input difference u+-u-
R1
uA
R1  R2
negative feedback
u 
output signal
R1
u A  A0 (u  u )  A0 (u E 
uA )
R1  R2
u
closed- loop- gain
 1
u
R1 

A  A   
u E  A0 R1  R2 
ideal OP - Amp
A
•
•
+
–
uE
R2
uA
1
R1
R1  R2
R
 1 2
R1
R1
R1  R2
 u
R1
from feedback
u A  A0 (u  u ) 
virtual short- circuit
 1 R  R2 
  u  u  u
u    1
R1 
0
A


u
u
uE
0
•
u
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)
Georg Wirth, Institut für Laser-Physik
11
+
–
u A  uE
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
kirchhoff' s junction rule
open loop gain
u D  u A A0
high op - amp impedance
0  i
 0
uE  uD u A  uD

z1 ( )
z 2 ( )
 u A  z1 
•
•
z2 ( )
0  i1  i2  i
closed loop gain
A( ) 
input impedance
z E ( ) 
uE
z1 ( )
u
i1
u
–
+
i2
uA
u A  z 2 u A  z1

 u E  z 2
A0
A0
uA
z2
z ( )

 2
uE
z1  z 2 A0  z1 A0
z1 ( )
du E
 z1 ( )
diE
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
12
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
kirchhoff' s junction rule
Rn
0  i1  i2    in  iF
in
u
u u
u
 1  2  n  A
R1 R2
Rn RF
R2
R1
i1
closed - loop - gain
Georg Wirth, Institut für Laser-Physik
R  R1  R2    Rn
uA  

RF
u1  u2    un 
R
13
iF
i2
u u
u 
u A   RF  1  2    n 
Rn 
 R1 R2
assume equal inputs
RF
u2
u1
u
u
–
+
uA
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
virtual short - circuit
u  u  u2 
kirchhoff' s junction rule
0  i1  iN 
R
uA   N
R1
insert u
if R1  R2 and RP  RN
uA  
RP
R2  RP
RN
u1  u u A  u

R1
RN

R R 1 
 u1  1 N
 u2 
R2 RP  1 

R1
u
u
u1
u2
R2
–
+
RP
RN
 u1  u2 
R1
Caution
•
assume RN = α · R1 and thereby the closed-loop-gain A = - α, resistors have tolerance Δα
CMRR  A
•
•

du A
 (1   )
d (u  u )


error, if resistor values differ
different input impedances require source with low output impedance
better results with instrumentation amplifier
Georg Wirth, Institut für Laser-Physik
14
uA
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
R3
use
z1 ( )  R1 and z 2 ( )  (C2  R2 ) || R3 
closed loop gain A( )  
•
•
R3  (1  jR2C2 )
1  j ( R2  R3 )C2
R
z 2 ( )
1  jR2C2
 3
z1 ( )
R1 1  j ( R2  R3 )C2
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
lower limit
A(  0)  ADC  
1
1


 ADC  1 
 A(1 ) 
2
2
2


( R3  2 R3 R2  R2 )  C2
upper cut - off
 A( ) 
upper limit
2  A
A(  )  A  
Georg Wirth, Institut für Laser-Physik

R1
u
i1
u
uE
log  A( ) 
R3
R1
lower cut - off
2
R2
ADC
i3
C2
–
i2
+
uA
integrating
range
R3
( R3  2 R3 R2  R2 )
2 
R2 ( R2  R3 )  C2
2
2
R2  R3
R1  ( R2  R3 )
 20 dB/dec
A
0
15
R2 log 
1
2
The integrator – operating method
closed loop gain A( )  
kirchhoff' s
R
z 2 ( )
1  jR2C2
 3
z1 ( )
R1 1  j ( R2  R3 )C2
R3
0  i1  i2  i3
u
d (u A  u R 2 ) u A
 E  C2

R1
dt
R3
junction rule
R2
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
u
du
0  E  C2 A
R1
dt
•
•
uE
R1
u
i1
u
i3
–
i2
+
t

1
u A (t )  
 u E (t ) dt   u A (0)
R1C2 0
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
du A
1  u0


   i 
dt
C2  R

•
C2
i- = 1 µA results with C = 1 µF in a change of 1 V per second (offset compensation)
Georg Wirth, Institut für Laser-Physik
16
uA
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
use
z1 ( )  R1 
closed loop gain A( )  
•
•
•
1
1  jR1C1

and z 2 ( )  R2
jC1
jC1
R2
z 2 ( )
jR2C1

z1 ( )
1  jR1C1
C1
R1
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
u
i1
uE
lower cut - off
 A( )  1
1 
upper cut - off
1


 A 
 A(2 ) 
2


2 
1
upper limit
Georg Wirth, Institut für Laser-Physik
i2
–
+
uA
log  A( ) 
differentiating
range
A(  0)  ADC  0
lower limit
u
1
R2  R1  C1
2
2
 20 dB/dec
1
R1 R2  R1  C1
A(  )  A  
R1
A
2
R2
R1
17
log 
0
1
2
The differentiator – operating method
closed loop gain A( )  
kirchhoff' s
z 2 ( )
jR2C1

z1 ( )
1  jR1C1
0  i1  i2
junction rule
 C1
R2
d (u E  u R1 ) u A

dt
R2
R1
i1
Approximation
•
•
u
u
i2
–
+
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
0  C1
•
uE
C1
du E u A

dt
R2

u A (t )   R2C1 
du E
dt
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
Georg Wirth, Institut für Laser-Physik
18
uA
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
iC  iCS (T , uCE )  exp uBE uT 
iCS
uT = kT/e
•
•
•
•

C
iC
R
uBE  uT  ln iC iCS 
uE  0
–
+
B
E
uA
reverse leakage current
thermal voltage
since
u A  u BE
,
iC  u E R

 u 
u A  uT  ln  E 
 iCS R 
(for u E  0)
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
19
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-
u
–
uA
u
uA
for u  u
 u
u A   A max
 u A max for u  u
•
•
+
uD
u A max
useful for regeneration of digital signals or as trigger
special devices for fast applications
uD
uA
uD
 u A max
t
saturation
Georg Wirth, Institut für Laser-Physik
20
saturation
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
input
•
•
u 
R4
 uref
R3  R4
u 
R2
R1
u
u
uE
R3
uref
+
–
uA
R4
R2
 u E  u A 
R1  R2
assume high positive input uE, so that output is uAmax
with falling input voltage uE, output uA remains unchanged until u+ = u- = 0
upper switching point
u E ,on 
lower switching point
u E ,off 
R4  R1  R2 
 uref  u A max 
R2  R3  R4 
R4  R1  R2 
 uref  u A min 
R2  R3  R4 
uA
uD
uA
u A max
u E ,on
u E ,off
Georg Wirth, Institut für Laser-Physik
uE
u A min
t
u E ,off
21
u E ,on
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, opamp 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
Georg Wirth, Institut für Laser-Physik
22
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
log  A0 ( ) 
(subsequent stages usually have increasing
bandwidths)
if A0(ω) is open-loop-gain, and |kF| ≤ 1 is feedback
factor, requirement for oscillation is (negative
A0(ωC)
input adds phase shift φ = π)
k F ( )  A0 ( )  1
•

 k F  A0  1

  arg k F A0   0,2 ,...
-40 dB/dec
-60 dB/dec
log 
1
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)

0
 2

If an op-amp is not internally frequencycompensated (and by that not unity-gain
stable), external frequency compensation or
minimal closed-loop-gain is necessary.
Georg Wirth, Institut für Laser-Physik
-20 dB/dec
3 2
23
 2 ωC  3
Comparison OP27 – OP37
OP27
(unity-gain stable)
OP37 (not
unity-gain stable)
Georg Wirth, Institut für Laser-Physik
24
Bandwidth
•
frequency-compensated op amps can be
regarded as 1st order low-pass
ADC
A0 ( ) 
1  i  ( 1 )
•

+
1  Ri Ci
Ri
–
for frequencies ω >> ω1 above cut-off, openloop gain is approximately
Ci
uE
R2
A0 ( )  i (1  ADC  )
1  ADC
 gain- bandwidth- product
R1
(unity - gain bandwidth)
•
•
uA
usually ADC is in the range of 120 dB, but cutoff 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
1
 1
A0 ( )
u
R1 
 
A( )  A  

u E  A0 ( ) R1  R2 
1  k F A0 ( )
insert A0 ( )
for ω  1
A
A

1  i  ( 1)  i (1  )  A for ω  1
1
R1
A

(DC) closed- loop gain
k F R1  R2
1  k F 1 ADC
closed- loop cut - off frequency
ADC
log( A0(ω) )
A( ) 
Georg Wirth, Institut für Laser-Physik
25
log( A(ω) )
A = 1 / kF
log 
1
ω1´
Slew Rate
amplitude-dependant distortion
•
•
•
•
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
u E  u E max  sin   t 
•
uA

t
frequency-dependant distortion
 du 
max  A   A( )    u E max
 dt 
uA
slew rate limits amplitude of undistorted
sine-wave output swing above some
critical frequency (power bandwidth)
u E max 
t
SR

Georg Wirth, Institut für Laser-Physik
26
Input offset voltage and bias current
Offset voltage
•
•
•
•
+
output signal uA ≠ 0 even for no input signal
u+ = u- = 0, due to unsymmetrical differentialamplifier 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
V+
2
3
-
=
10k
7
+
8
1
6
–
4
-
=
V-
+
Bias current
•
input transistors need constant base- or gatecurrent for operation, which is delivered by
power supply
Bipolar
Darlington
FET
R2
R1
100 nA
1 nA
1 pA
–
+
uE
•
can be compensated by additional
bias resistor RB
RB 
RB
R1 R2
R1  R2
Georg Wirth, Institut für Laser-Physik
27
uA
Grounding problems
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
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
Georg Wirth, Institut für Laser-Physik
casing 1
~
28
casing 2
Avoiding ground loops
powersupply
photodiode
controller
oscilloscope
optical
table
1.
2.
3.
4.
5.
6.
7.
AOMdriver
AOM
ADwin
switchboard
Don’t use shielding as conductor, i.e. signal path
Isolate all casings from metal surfaces
Use separate power supplies
Power supplies don’t need a ground reference
Use digital / analog isolators (e.g. ADUM524x and AD210)
Measure signals differentially
No oscilloscopes at unbuffered / source signals
Georg Wirth, Institut für Laser-Physik
RFamp
29
optical
table
References
Control theory
•
G. v. Wichert, Grundlagen der Regelungstechnik (http://tams-www.informatik.unihamburg.de/lehre/2004ws/vorlesung/regelungstechnik/folien/Grundlagen%20der%20Regelungstec
hnik_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)
Georg Wirth, Institut für Laser-Physik
30