operational amplifiers - physicscvelissaris/fall13/phy3722/... · 2013. 7. 25. · 1) the voltage...
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Operational Amplifiers
• The three rules of the op amp
• The inverting and the non-inverting configuration.
• The (weighted) summer.
• The difference amplifier. The CMRR (common mode rejection ratio).
• An instrumentation amplifier.
• Frequency effects on open and closed loop amplifiers. The gain-bandwidth theorem.
• The real op amp. Large signal limitations and DC imperfections.
• Integrators and differentiators.
Operational Amplifier is a device that can amplify the difference
between two input signals. Among its pins, there are two signal
inputs (the inverting and the non inverting input), a ground, an output
pin and two dc power supply pins that provide the necessary power
for the op amp IC chip to work.
-
+
V+
1υ
2υ( )12 υυυ −= Ao
V−
Inverting input
Non inverting input The amplification factor A
(or open loop gain) depends
highly on temperature, varies a
lot from op amp to op amp and
depends a lot on frequency.
-
+
V+
1υ
2υ ( )12 υυυ −= Ao
V−
The open loop op amp acts as a voltage
(differential) amplifier. It has huge input and
tiny output resistances.
It can be modeled as:
inR
+
-1υ
2υ
+
-
( )12 υυυ −= Ao
When op amps are being employed in
circuits they are being used almost
exclusively with negative feedback
(closed loops). The gain of the
configuration is reduced but the syste
is much more predictable, stable and
with improved bandwidth.
The closed loop op amp.
When op amps are being employed in circuits they are being used
almost exclusively with negative feedback (closed loops). The gain
of the configuration is reduced but the system is much more
predictable, stable and with improved bandwidth.
Negative feed back means that a portion of the output is being fed
back at the inverting input of the op amp.
When we analyze circuits employing closed loop op amps we are
using the three (approximate) rules of the ideal op amp.
The rules of an ideal op-amp
connected in a closed loop.
1) The voltage gain A of the ideal open loop op-amp is infinitely large.
2) The current through the ideal op-amp is zero. That is, the ideal
op-amp has infinite input resistance.
3) Both terminals of the ideal op-amp are at the same voltage.
(Consequence of rule #2)
The inverting amplifier.
oυiυ
i
2R
1R i
virtual ground
1
2
iR
iR
i
o
=
=−
υ
υA
R
R
i
o =−=1
2
υ
υ
-
+
Input resistance:
1RRiR inini =⇒=υ
may be too small
output resistance 0≈outR
An improved inverting amplifier.
oυiυ
i
2R
1R i
-
+
Input resistance:
1RRin =
can be chosen to be large
4R
3R
++−=
3
4
2
4
1
2 1R
R
R
R
R
R
i
o
υ
υ
amplifier gain:
Choose R1=R
2=R
4=R as large as possible to maximize the input
resistance and minimize its effects on the closed loop voltage gain.
Choose R3
as small as possible to obtain the desired voltage gain.
An improved inverting amplifier.
oυiυ
i
2R
1R i
-
+
4R3R
3ii −
3i
++−=
⇒
++−=⇒−+=−
−==
3
4
2
4
1
2
3
4242422
221
1
)(
,
R
R
R
R
R
R
R
RRRRiRiiiR
iRRiiR
io
oo
i
υυ
υυ
υ
The weighted summer.
oυ
1υfR
1R
-
+
2υ
2R
NυNR
N
N
fff
oR
R
R
R
R
Rυυυυ −−−−= ....2
2
1
1
Effects of finite open loop gain A
on the inverting amplifier.
oυiυ
i
2R
1R i
-
+A
0υ−
A
R
R
R
R
Gi
o
1
2
1
2
1
1
+
+
−==υ
υ
Since1
21R
RA +>>
1
2
R
RG
i
o −≈=υ
υ
The non-inverting amplifier.
oυiυ
2R
1R i
same
voltage ( )21
2
RRi
iR
o
io
+=
=−
υ
υυ
AR
R
i
o =+=1
21υ
υ
-
+
The input resistance
is infinite and the
output resistance is 0.
Effects of finite open loop gain A
on the non-inverting amplifier.oυ
iυ
2R
1R i
-
+
Ai
0υυ −
A
R
R
R
R
Gi
o
1
2
1
2
1
1
1
+
+
+
==υ
υ
Since1
21R
RA +>>
1
21R
RG
i
o +≈=υ
υ
The voltage follower.
iυ
-
+1021 =⇒== GRR
iυ
The voltage follower has infinite input resistance 0 output resistance
and voltage gain unity (the output voltage is equal, follows the input
voltage). It is used as a “buffer” a “matching” element between
different stages of electronic instrumentation.
The difference amplifier.An amplifier that responds to the difference υ
d=υ
2-υ
1 between two
input signals υ1
and υ2. The output voltage υ
o(response) of the
amplifier can be written as:
cmcmddo AA υυυ +=
2
12 υυυ
+=cm
12 υυυ −=dwith the difference input signal
and the common mode input signal
Acm
is the common mode gain and ideally must be 0
Ad
is the difference gain and ideally must be very large.
The common mode rejection ratio CMRR of a difference amplifier
(ideally infinite) is defined as:
( )dBinA
ACMRR
cm
dlog20=
The difference amplifier.
oυ1υ
2R
1R
-
+2υ
4R
3R
43
42
RR
R
+υ
( )( ) 1
21
431
2142
R
R
RRR
RRRo υυυ −
+
+=
straightforward analysis gives:
12 υυυ −=d
2
12 υυυ
+=cm
define
and therefore:2
,2
12d
cmd
cm
υυυ
υυυ −=+=
The output voltage now can be expressed as
a function of the common mode and
difference voltages
The difference amplifier.
oυ1υ
2R
1R
-
+2υ
4R
3R
( )( )
( )( )
−
+
++
+
++=
1
2
431
214
431
214
1
2
2 R
R
RRR
RRR
RRR
RRR
R
Rcm
d
o υυ
υ
By choosing R1
= R3
and R2
= R4
the common mode gain becomes 0.
The difference gain becomes R2/R
1
The difference amplifier.2R
1R
-
+dυ
1
2
R
Rdo υυ =
2R
1R
ii
11 22 RRRi ind =⇒=υ
The input resistance of this difference amplifier
is relatively low (since we want high difference
gain). The instrumentation amplifier is a superior
difference amplifier with infinite input resistance.
Instrumentation amplifier.
oυ
1υ
-
+
2υ
4R
3R
-
+
2R
2R12R-
+
3R
doR
R
R
Rυυ
+=
1
2
3
4 1
A major advantage of this configuration is that the first stage op amps
do not amplify the common mode signal therefore are not prone to
saturation. It is a superior difference amplifier.
4R
1i1i
1i
2i
3i
2i
3i
1
21
2R
Rdυυ −
1
22
2R
Rdυυ +
Frequency dependence of open
loop op amps|)(|log20 fA
f
dB3
Hzfb 10=
The open loop gain of an op amp has a very strong
dependence on frequency. It reaches its 3dB
attenuation very quickly at 10 Hz very low value.
bf
fj
AfA
+
=
1
)( 0
We can model this behavior with the function:
A0
is the dc open loop gain and fb
is the 3 dB frequency. The product
ft=A
0 fb
is called the “unity-gain bandwidth” and it is specified in the
data sheet by the manufacturer of the op amp.
Frequency dependence of closed
loop op amps
Let’s examine the following question. How does the external feedback
affect the frequency response of the gain of an inverting amplifier?
+
++
+
−=
+
+
−=
1
2
0
1
2
1
2
1
2
1
2
1
11
)(
1
1
)(
RR
f
fj
A
RR
RR
fA
R
R
R
R
fG
t
where A0
is very large
The gain bandwidth theorem
Since is A0
very large we can write
+
+
−≈
1
2
1
2
1
1
)(
RR
f
fj
RR
fG
t
We see that the new 3dB frequency of the closed loop is:
( ) .1
13
1
2
3 constfKf
R
R
ff tdB
t
dB ==+⇒
+
=This is the gain bandwidth
theorem. It can also be
proven for a non inverting
amplifier.
Frequency response of closed loop
op amps
|)(|log20 fG
f
dB3
dBf3
Amplifier bandwidth
io fG υυ |)(|=
Op Amp large signal limitations.
Output voltage saturation. The output voltage of an op amp cannot be larger than a specified
value, the rated output voltage of the op amp.
Maximum output current. The output current supplied by an op amp is limited to a specified
maximum value. (about 20 mA for an741C op amp).
Slew Rate (SR).There is a maximum rate of change of the output voltage that an op
amp can handle. It is called the slew rate (SR) of the op amp.
max
=
dt
dSR oυ
Op Amp full power bandwidth
For a sinusoidial input signal to an op amp. )sin()( tVt ii ωυ =
)cos()( tVtdt
di
i ωωυ
=
The full power bandwidth fM
of an op amp is defined by the relationship:
SR=ωM
Vomax
=2π fM
Vomax
where Vomax
is the rated output voltage of the op amp (the maximum
output voltage the op amp can deliver without clipping).
The physical meaning of the op amp’s full power bandwidth fM
is the
maximum frequency a harmonic input to the op amp can have when the
output amplitude is equal to the op amps rated voltage. If the frequency
is increased the output signal is distorted due to slew rate limitation of
the op amp. The full power bandwidth fM
of an op amp is specified by
the manufacturer in the date sheet of the device.
DC imperfections of an op amp
Offset voltageIf the op amp is supplied with two equal voltages at its inputs the
output voltage is not zero. The non ideal op amp is equivalent to
an ideal op amp with a small dc offset voltage connected to one
of its terminals.
Input bias and offset currents.In order for a non ideal op amp to operate its two input terminals
are supplied with two small input bias currents IB1
and IB2
Op Amp Offset Voltage
In order to account for the offset voltage effect a non ideal op amp
can be modeled as shown
-
++-
VOS
The small dc voltage at the positive
terminal of the op amp is called the
“offset voltage” of the op amp and
it is specified by the manufacturer.
Op Amp Offset Voltage
The offset voltage of the op amp appears amplified at its output.
Consider the inverting amplifier below where both its terminals
are connected to ground. What is its output voltage?
-
++-
VOS
2R
1R
+=
1
21R
RVOSoυ
The offset dc voltage of the op amp
appears amplified at the output.
Op Amp Offset Voltage
By connecting a capacitor at the input of the inverting amplifier
the offset voltage appears with unity gain at the output.
Useful, but only if we want to amplify ac signals.
-
++-
VOS
2R
1RC
OSOSo VR
V =
∞+= 21υ
Op Amp Offset VoltageSome op amps have two additional terminals to which a specified
circuit should be connected to trim the output voltage due to VOS
1V+
2V−
Op Amp input bias and offset currents
In order for a non ideal op amp to operate its two input terminals
are supplied with two small input bias currents IB1
and IB2
Input offset current 21 BBOS III −=
Input bias current
2
21 BB
B
III
+=
typical value 10 nA
typical value 100 nA
A non ideal op amp with input bias and offset currents is modeled as
2BI
1BI
3
1
2
Effect of op amp input offset and bias
currents to a closed loop amplifier
2BI
1BI
-
2R
1R
+
0 V
1BI
221 RIRI BBo ≈=υ
By introducing the appropriate resistance R3
to the non inverting input
of the op amp we can fix the dc output voltage due to the bias current
Effect of op amp input offset and bias
currents to a closed loop amplifier
2BI
1BI
-
2R
1R
+
We can get a more meaningful
expression for the output voltage
if we do the substitution:
2BI
3R
32 RI B−
1
32
R
RI B
1
3
21R
RII BB −
+−=
1
23221 1
R
RRIRI BBoυ
21
OS
BB
III +=
22
OSBB
III −=
Effect of op amp input offset and bias
currents to a closed loop amplifier
2BI
1BI
-
2R
1R
+2BI
3R
32 RI B−
1
32
R
RI B
1
3
21R
RII BB −
( )
( )
+++
+
+−=
21
2132
21
2132
12
1
RR
RRRRI
RR
RRRRI
OS
Boυ
By choosing21
21
3RR
RRR
+= we get 2RIOSo ≈υ
Op Amp differentiators
-
++-
VOS
R
C
oυ
iυ( )
( ) ( )tdt
dRCt
dt
dqRtiR
C
q
i
O
oo
i
υυ
υυ
υ
−=
⇒−=⇒=−
=
Op Amp integrators
-
+
R
C ( ) ( )
( ) ( ) tdtRC
tC
q
tdtR
tqdt
dq
RI
t
ioo
t
ii
′′−=⇒=−
′′=⇒==
∫
∫
0
0
1
1
υυυ
υυ
oυiυ
Effect of op-amp offset voltage
on integrators
-
++-
VOS
R
CV
OS
oυI
I
C
qV
dt
dqI
IRV
OSo
OS
=−
=
=
υ
+=
=⇒=
RC
tV
tR
Vq
R
V
dt
dq
OSO
OSOS
1υ
Because of the offset voltage the integrator
will saturate soon after its power on.
A remedy for the offset voltage
effect on integrators
-
++-
VOS
R
CV
OS
oυI
FR
1I
FIThe presence of a small R
Fwill
limit the output voltage at an
asymptotic value a little larger
than VOS
.
Effect of op-amp offset current on
integrators
-
+I
B2
R
C0
oυI
B1
IB1
tC
I
C
q
tIqIdt
dq
BO
BB
1
11
==
=⇒=
υ
Remedy of op-amp offset current
effect on integrators
-
+
IB2
R
C
-IB2
R
oυI
B1
IB2
-IB1
R
IB2