Op Amp Basics
op amp = operational amplifier
Prepared by Scott Robertson
Fall 2007
Diefenderfer and Holton (D&H), Ch. 9
Horowitz and Hill (H&H), Ch. 4,5,6,7
1
What are they good for?
•
•
•
•
•
Amplifiers: sum and difference of voltages
Differentiators
Integrators
Buffers or follower (cable drivers)
Filters: high pass, low pass,
bandpass, band reject, notch
• Comparators
• Limitations of op amps (nonideal behavior)
• Oscillators (another lecture)
2
1
Theorists view:
3 pins:
Experimentalists view:
7 pins + ground:
V+
Inverting input
VA
VB
VA
−
Vout
+
VB
Non-inverting input
Positive power supply
Offset adjust
(optional)
−
Vout
+
V−
Negative power supply
Ground is sometimes not
connected to the op amp.
Vout = Gain × (VB – VA)
Vout from Golden Rules
3
Manufacturer’s views
Notch
The socket also has a notch
at pin 1 so you don’t put the op
amp in backwards.
Dual-in-line (DIP) Package
4
2
What’s inside?
National’s LF356
5
“Bypass” the impedance of the power
supply wires using small capacitors
V+
0.01 μF, >15 V
To power
supply
VA
VB
To power
supply
Failure to do this can cause
oscillation and “cross talk”
between circuits because the
power supply voltage will
otherwise vary when the load
varies.
A screwdriver-adjust potentiometer
is used for “one time” offset adjustment
−
Vout
+
Small disc ceramic capacitors are used
for “bypass.” They MUST be located
near (a few cm) to the chip and
attached to a “good” ground.
V−
6
3
Characteristics of the ideal op amp
1. Input impedance is infinite
inputs draw no current
2. Gain G is infinite
in the math, terms with 1/G → 0
3. Negative feedback determines the
function performed
Negative feedback is a connection from the
output to the inverting input
4. Output impedance is zero
7
From D & H ch 9-1
Inverting x1 amplifier (G not ∞)
VB = 0
Negative
feedback
thus
Vout = G × (0 − VA )
R
(1)
I1 = I 2
Vin
R
I2
VA
−
I1
+
VB
Vout
Vin − VA VA − Vout
=
R
R
Vout = − Vin + 2 VA
(2)
2 equations, 2 unknowns
Vout = − Vin − 2 VGout ⎯G⎯
⎯→ − Vin
→∞
VB − VA =
Vout
G
; VA ⎯G⎯
⎯→ VB
→∞
8
4
Golden rules of ideal op amps
1. The output does what is necessary to
make VB = VA.
2. The inputs draw no current.
With these rules, knowledge of G is not
needed as long as G>>1.
Reference: H&H 4.03
9
Inverting amplifier from Golden Rules
Golden rules :
R2
Vin
R1
I1 = I2
I2
VA
−
I1
+
VB
Vout
and
VB = VA = 0
Vin − 0 0 − Vout
=
R1
R2
1 equation, 1 unknown
Vout = −
R2
Vin
R1
X10 inverting amplifier if R2 = 100KΩ and R1 = 10 kΩ.
10
5
Summing amplifier
R
R,R,R
V1
V2
V3
VA
−
Vout
+
VB
VA = 0
Vout = -(V1 + V2 + V3)
“intuitively obvious”
11
Current to voltage converter
or electrometer
R
I1
-I1
VA
→ Ileak
Vout = −I1 R + Voffset
-
+
Vout
VB
Used to measure small currents.
Offset error: Voffset = -Ileak R,
where Ileak is the nonideal input offset current
(also called Ibias).
12
6
Non-inverting amplifier with gain >1
Vin = VA
R2
R1
Feedback is simple voltage divider :
⎛ R1 ⎞
⎟⎟
VA = Vout ⎜⎜
⎝ R1 + R2 ⎠
I2
VA
−
I1
Vin
+
(Golden rule)
Vout
VB
⎛ R + R2 ⎞
⎟⎟
Vout = Vin ⎜⎜ 1
⎝ R1 ⎠
Vout
R
= 1+ 2
Vin
R1
Answer is NOT R2/R1.
Can’t make a x1 or x0.5 amplifier this way!
13
Non-inverting amplifier with gain = 1 or <1
VA
VA
−
Vin
VB
+
−
Vin
Vout
+
Vout
VB
x1 amplifier is a buffer or follower
It is used to drive long lengths of
cable, otherwise signal is decreased.
Volume control, gain <1
High current buffer
amplifiers with no “-B”
input are available.
BUF634 can put out 250 mA !
14
7
Difference or “B-A” amplifier
R2
R1
VA
−
V1
V2
+
Vout = ( V2 − V1 )
Vout
VB
R1
R2
R1
R2
Gain can be 1, >1 or <1.
The mathematical analysis is in D&H,
ch. 9-5, (3 eqns., 2 unknowns)
15
Differentiator from Golden Rules
R
Vin
C
Golden rules :
I1 = I2
I2
VA
+
− Vout
d
or
Vin =
dt
R
1 equation, 1 unknown
Vout
VB
Here is what it
will do to a
square pulse:
VB = VA = 0
C
−
I1
and
Vout = −RC
− Vout
d
Vin =
dt
RC
d
Vin
dt
Vin
The derivative of a step fcn.
is a delta fcn.
Vout
What determines the width of
the delta fcn. in this case?
What determines the height?
16
oscilloscope
8
Simple integrator is “ruined” by nonideal Ibias
Golden rules :
I1 = I2 and VB = VA = 0
Vin
d
− Ileak = −C Vout
R
dt
1 equation, 1 unknown
C
Vin
R
I2
VA
t
→ Ileak
I1
Vout
+
Vout = −
I
V
d
Vout = − in + leak
RC C
dt
or
t
1
1
Vin dt ′ − ∫ Ileak dt ′ + Vout (t = 0)
14243
RC ∫0
C0
constant of
14243
integration
→∞
VB
Eventually, the capacitor is charged to the supply
voltage by the leakage current and integration stops.
Ileak is the tiny input current
to terminal VA, often ignored.
Ileak is also called Ibias.
17
Practical op amp integrator with extra resistor
R2 tends to
discharge C
Vin
R1
VA
I1
R2
Golden rules : I1 = I2
C
V
Vin
d
= −C Vout − out
R1
dt
R2
VB = VA = 0
and
or
V
V
d
Vout = − in − out
dt
R 1C R 2 C
I2
→ Ileak
+
t
Vout = −
Vout
VB
t
1
1
Vin dt ′ −
Vout dt ′
∫
R 1C 0
R C ∫0
124
4244
3
error term
error term is small if
t
Ileak is the tiny input current
to terminal VA, often ignored.
1
Vout dt ′
R 2 C ∫0
t
<< 1 or
<< 1
Vout
R 2C
so make R 2 big. The cost of this is : Voffset = −Ileak R 2
18
9
Error of practical integrator is “droop”
Vin
Input signal
Vout
True integral
Output of practical integrator
“droops” with time constant R2C
Not significant if R2C >> t
t’ →
t
oscilloscope
19
The comparator
a special kind of op amp
20
10
Comparator
output is either high (V+) or low (V-)
V+
VA
Vref
−
Vout
+
VB
Vtest
Vtest > Vref
Vout is V+
Vtest < Vref
Vout is V-
Op amps designed to be comparators are not
damaged if VA ≠ VB.
21
Comparator as temperature controller
V+
“Thermostat”
Colder
RT
V+
R
Vref
−
Hotter
Buffer
+
Vout
Heater
Vtest
R
R
Note that V- can be ground.
Thermistor RT is heated and gets more resistive,
driving Vtest down below Vref, and the heater turns off
until the comparator changes state again.
22
11
Some comparators only “sink” current
the output is an “open collector”
V+
LF311
V+
Vref
VA
Rload
−
Vout
+
Vtest
VB
Wait a bit and we will
talk about transistors
and what this means.
23
Schmitt trigger
Definition:
A comparator that goes high at a higher reference voltage than the
reference voltage for going low.
If off, and then
Vtest > Vref - ΔV
then output goes high (furnace turns on at 68 F)
If on, and then
Vtest < Vref + ΔV
then output goes low (furnace turns off at 70 F)
Why do this?
So that your furnace (for example) will stay on until the house heats up a
few degrees, then turn off. The furnace should run 5 minutes per hour
rather than 5 seconds per minute. ΔV determines how much the
parameter being controlled is allowed to vary.
24
12
Comparator made into Schmitt trigger
V+
V+
100 RT
RT ≅ R
R
RT
Vref
−
“thermostat”
Buffer
+
R
R
Vout
Heater
Vtest
Positive feedback (100RT) raises Vtest relative to Vref when the
heater is on, which has the same effect as lowering Vref (by about
1% in this case). This circuit provides control to order 1%.
25
Schmitt trigger for digital logic
prevents noise from changing the output
Noisy signal
Vref+ΔV
Vref – ΔV
Vin
Special symbol
For Schmitt trigger
Clean output
Vout
Ordinary
comparator
Vout
time →
Noisy output
26
13
Comparator made into time delay
V+
Vref
VA
−
+
Vin
Vout
VB
R
C
Input pulse (solid)
Vref
C voltage (dotted)
Vin
Delayed output pulse
Delay is of order RC
Vout
27
time →
Filters
28
14
Filters are for
1. removal of noise
2. selecting a feature (radio station?)
Low pass
High pass
Vout
Vin
Vout
Vin
f
Vout
Vin
Bandpass,
Band reject
f
f
Q ≅ f / Δf
Quality factor
29
Unsophisticated low pass filter (3db per octave)
Z2 = [(R2)-1+jC2ω]−1
C2
R2
VA
Vin
−
Z1= R1
Vout
+
VB
Replace R by Z in op amp circuit analysis
Vout = −
R2
Z
Vin becomes Vout = − 2 Vin
R1
Z1
⎛ 1
⎞
+ jC 2ω ⎟⎟
Let : Z 2 = ⎜⎜
⎝ R2
⎠
⎛R
Vout
= −⎜⎜ 2
Vin
⎝ R1
−1
and Z1 = R1
⎞
1
⎟⎟
1
+
jR
2 Cω
⎠
Vout/Vin
Compare to simple RC :
Vin
Vout
Vout
1
=
Vin 1 + jRCω
f
30
15
Unsophisticated high pass filter
R2
Vin
R1
C
Vout = −
Let : Z 2 = R1 + 1 / jCω and Z 2 = R2
I2
VA
⎛
⎞ − jR2Cω
⎞
Vout
R2
R ⎛
1
⎟⎟ =
⎟⎟
= −⎜⎜
= − 2 ⎜⎜
+
+
+
Vin
R
jC
jR
C
R
jR
C
1
/
ω
1
ω
1
1
/
ω
⎝ 1
⎠
⎠
1
1 ⎝
1
−
I1
Z2
Vin
Z1
+
VB
Compare to simple RC :
Vout
R
jRCω
1
=
=
=
Vin R + 1 / jCω 1 + jRCω 1 + 1 / jRCω
Vin
Vout/Vin
31
f
Sophisticated filters
• You can easily get 6 db per octave with 1 op amp, 2 Rs
and 2 Cs, so don’t settle for 3 db per octave (previous
slides).
• It is standard practice to copy the filter design from a
book.
• There are “too many” filter designs
Butterworth, Bessel, Chebyshev, etc.
Some designs are flatter near the cutoff.
• See textbooks for examples
f
32
16
Typical 6 db per octave filter
4 Rs, 2 Cs
C
R
R
−
C
+
R
See H&H, Fig. 5-16.
R
33
Twin – T notch reject filter
Vin
Vout/Vin
90 degree
phase delay
at f0
C,C
R,R
2C
90 degree
phase advance
at f0
Vout
R/2
f0
f
At frequency f0, currents from the two sides of the circuit are
180 degrees out of phase and cancel at location Vout .
34
17
Non ideal op amps
• There is a nonzero input current (20 pA?)
• Op amp can’t put out more volts than V+ or V• The output current is limited (usually it can’t
drive a long 50 ohm cable)
5 V / 50 Ω = 100 mA = too much
• The frequency response is limited
(pay more $ above 1 MHz)
35
Input current is not zero
LF156
family
And it increases with temperature strongly!
36
18
Output impedance
LF156
family
Output impedance is
higher at higher
frequencies and
higher gains Av.
Don’t ask for more than
about 15 mA from this
chip. This chip cannot
put 1 V onto 50 ohms!
37
Op amp frequency response
An expression for the gain G valid
at all frequencies is:
G (ω ) =
G (ω ) =
G0
G0
=
1 + j (ω ω0 ) 1 + j ( f f 0 )
G0
1 + j (ω / ω0 )
Open loop gain G(ω) is less at high
frequencies.
f0 and ω0 are 3 db points of op amp.
Open loop gain-bandwidth product
(~5 MHz) is a constant at the higher
frequencies.
G (ω ) × ω ≅ G0ω0
ω >> ω0
38
19
Using gain bandwidth product
Suppose the gain-bandwidth product G(ω)ω = 5 MHz.
Then I can have a gain of 5 at 1 MHz.
Or a gain of 1 at 5 MHz.
Or a gain of 50 at 100 kHz, etc.
This helps you SELECT the proper op amp for the job.
39
Op amp frequency response with feedback
Open loop response
Closed (feedback)
loop response for
30 db gain
If the feedback sets the gain at 30 db, then the frequency
response of the circuit is flat to 100 kHz for this chip.
40
20
The op amp has internal noise
1/f (one-over-f) noise
below ~100 Hz (varies)
Ordinary noise
(independent of f)
Noise is covered later in
PHYS3330. Please wait a
while to find out what √Hz is
all about.
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21