Chapter 3-Webster Amplifiers and Signal Processing

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24 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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Chapter 3
-
Webster

Amplifiers and Signal Processing

The three major operations done on biological signals using Op
-
Amp:



1)
Amplifications and Attenuations

2)
DC offsetting: add or subtract a DC

3)
Filtering:

Shape signal’s frequency content

Applications of Operational Amplifier

In Biological Signals and Systems

Ideal Op
-
Amp

Figure 3.1 Op
-
amp equivalent circuit.


The two inputs are

1

and


2
. A differential voltage between them
causes current flow through the differential resistance
R
d
. The
differential voltage is multiplied by A, the gain of the op amp, to
generate the output
-
voltage source. Any current flowing to the output
terminal
v
o

must pass through the output resistance
R
o
.

Most bioelectric signals are small and require amplifications

20 transistors

11 resistors

1 capacitor

Inside the Op
-
Amp (IC
-
chip)

Ideal Characteristics

1
-

A

=


(gain is infinity)

2
-

V
o

= 0, when
v
1

=
v
2

(no offset voltage)

3
-

R
d

=


(input impedance is infinity)

4
-

R
o

= 0 (output impedance is zero)

5
-

Bandwidth =


(no frequency response limitations) and no phase shift

Two Basic Rules

Rule 1

When the op
-
amp output is in its linear range, the two input terminals
are at the same voltage.


Rule 2

No current flows into or out of either input terminal of the op amp.

Inverting Amplifier

Figure 3.3 (a) An inverting amplified. Current flowing through the
input resistor
R
i

also flows through the feedback resistor
R
f
. (b) The
input
-
output plot shows a slope of
-
R
f
/
R
i
in the central portion, but the
output saturates at about
±
13 V.

R
i


i


o

i

R
f

i

+

-

(a)

10 V

10 V

(b)


i


o

Slope =
-
R
f
/
R
i

-
10 V

-
10 V

i
f
i
o
i
i
f
o
R
R
v
v
G
v
R
R
v
-


-

Summing Amplifier










-

2
2
1
1
R
v
R
v
R
v
f
o

1


o

-

+

R
2

R
1

R
f


2

Example 3.1


i

v

b


i


o


o

-

+

+15V

+10

0

Time


i
+

b
/2



-
10

(a)

(b)

5 k
W

-
15 V

R
b

20 k
W

R
i

10 k
W

R
f

100 k
W

Voltage, V

The output of a biopotential preamplifier that measures the electro
-
oculogram is an undesired dc voltage of
±
5 V due to electrode half
-
cell potentials, with a desired signal of
±
1 V

superimposed. Design a
circuit that will balance the dc voltage to zero and provide a gain of
-
10 for the desired signal without saturating the op amp.

Follower ( buffer)

Used as a buffer, to prevent a high source resistance from being
loaded down by a low
-
resistance load. In another word it prevents
drawing current from the source.


o


i

+

-

1


G
v
v
i
o
Noninverting Amplifier


o

10 V

10 V


i

Slope = (
R
f
+
R
i
)/
R
i

-
10 V

-
10 V

R
f


o


i

i

+

-

i

R
i















i
f
i
i
f
i
i
i
f
o
R
R
R
R
R
G
v
R
R
R
v
1
Differential Amplifiers

)
(
3
4
3
4
v
v
R
R
v
o
-

3
4
3
4
R
R
v
v
v
G
o
d

-

R
4

R
4

R
3

R
3

v
3

v
4

v
o

Differential Gain
G
d


Common
-
mode rejection ratio
CMMR

c
d
G
G
CMRR

Common Mode Gain
G
c

For ideal op amp if the inputs are equal then the
output = 0, and the
G
c


=0. No differential
amplifier perfectly rejects the common
-
mode
voltage.

Typical values range from 100 to 10,000

Disadvantage of one
-
op
-
amp differential amplifier is its low input resistance

Instrumentation Amplifiers

Advantages: High input impedance, High CMRR, Variable gain

Differential Mode Gain



1
2
3
4
1
1
2
2
v
v
R
R
R
R
R
v
o
-










1
1
2
2
1
4
3
1
2
1
2
1
2
4
3
2
)
(
R
R
R
v
v
v
v
G
iR
v
v
R
R
R
i
v
v
d


-
-


-



-
Comparator
–

No Hysteresis


o


i


ref

10 V

-
10 V

-
10 V

v
2

+15

-
15


i


o

-

+

R
1

R
1

R
2


ref

If (
v
i
+
v
ref
) > 0 then
v
o

=
-
13 V


else

v
o

= +13 V

R
1

will prevent overdriving the op
-
amp

v
1

>
v
2
,
v
o

=
-
13 V

v
1

<
v
2
,
v
o

= +13 V

Comparator
–

With Hysteresis


i


o

-

+

R
1

R
1

R
2

R
3


ref


o


i

-


ref

10 V

-
10 V

With hysteresis

-
10 V

10 V

Width of the Hysteresis = 4V
R3

Reduces multiple transitions due to mV noise levels by moving the
threshold value after each transition.

Rectifier

10 V

(b)

-
10 V


o


i

-
10 V

10 V

-

+

(a)

D
3

R

R


o
=


i

-

+

D
2

D
1

D
4

xR

(1
-
x
)
R


i

x

Full
-
wave precision rectifier:


a)

For

i

> 0,

D
2

and D
3

conduct, whereas D
1

and D
4

are

reverse
-
biased.

Noninverting amplifier at the top is active


(a)

D
2

v

o


i

-

+

xR

(1
-
x)R

Rectifier

10 V

(b)

-
10 V


o


i

-
10 V

10 V

-

+

(a)

D
3

R

R


o
=


i

-

+

D
2

D
1

D
4

xR

(1
-
x
)
R


i

x

Full
-
wave precision rectifier:


b)

For

i

< 0,

D
1

and D
4

conduct, whereas D
2

and D
3

are reverse
-
biased.

Inverting amplifier at the bottom is active

(b)

D
4

v

o


i

-

+

xR
i

R

One
-
Op
-
Amp Full Wave Rectifier

For

i

< 0, the circuit behaves like the inverting amplifier rectifier with
a gain of +0.5. For

i

> 0, the op amp disconnects and the passive
resistor chain yields a gain of +0.5.

(c)

D

v

o


i

-

+

R
i

= 2 k
W

R
f

= 1 k
W

R
L

= 3 k
W

Figure 3.8

(a) A logarithmic amplifier makes use of the fact that a
transistor's
V
BE

is related to the logarithm of its collector current.

For range of I
c

equal 10
-
7

to 10
-
2

and the range of
v
o

is
-
.36 to
-
0.66 V.

(a)

R
f

I
c

R
f

/9


o

R
i


i

-

+

Logarithmic Amplifiers











-
13
10
log
06
.
0
i
i
o
R
v
v









S
C
BE
I
I
V
log
06
.
0
V
BE

Uses of Log Amplifier

1.
Multiply and divide variables

2.
Raise variable to a power

3.
Compress large dynamic range into small ones

4.
Linearize the output of devices

Figure 3.8

(a) With the switch thrown in the alternate position, the
circuit gain is increased by 10. (b) Input
-
output characteristics show
that the logarithmic relation is obtained for only one polarity;

1 and

10 gains are indicated.

(a)

R
f

I
c

R
f

/9


o

R
i


i

-

+

(b)

10 V

-
10 V

v

o


i

-
10 V


1


10

10 V

Logarithmic Amplifiers

V
BE

V
BE

9V
BE

Integrators









C
R
j
R
R
j
V
j
V
C
R
R
j
R
R
j
V
j
V
i
f
i
i
o
i
f
i
f
i
o







-


-

1
f
f
c
i
f
i
o
C
R
f
R
R
v
v

2
1

-

for
f

<
f
c

i
f
i
o
t
ic
i
f
i
o
Z
Z
j
V
j
V
v
dt
v
C
R
v
-


-


)
(
)
(
1
1
0


A large resistor R
f

is used to prevent saturation

Integrators

Figure 3.9 A three
-
mode integrator

With S
1

open and S
2

closed, the
dc circuit behaves as an inverting amplifier. Thus

o

=

ic

and

o

can
be set to any desired initial conduction. With S
1

closed and S
2

open,
the circuit integrates. With both switches open, the circuit holds

o

constant, making possible a leisurely readout.

R

+

FET

Piezo
-
electric

sensor

-


o

C

i
s

i
s
R

i
s
C

dq
s
/
dt
=
i
s
=
K dx/dt

Long cables may be used without changing sensor sensitivity or time
constant.

Example 3.2

The output of the piezoelectric sensor may be fed directly into the
negative input of the integrator as shown below. Analyze the circuit
of this charge amplifier and discuss its advantages.

i
sC

=
i
sR

= 0

v
o

=
-
v
c

C
Kx
dt
dt
Kdx
C
v
t
o
-

-


1
0
1
Differentiators

Figure 3.11 A differentiator

The dashed lines indicate that a small
capacitor must usually be added across the feedback resistor to
prevent oscillation.

RC
j
Z
Z
j
V
j
V
dt
dv
RC
v
i
f
i
o
i
o



-

-

-

)
(
)
(
Active Filters
-

Low
-
Pass Filter

+

-

R
i

R
f

(a)

C
f


i


o

Active filters


(a) A low
-
pass filter attenuates high frequencies





f
f
i
f
i
o
C
R
j
R
R
j
V
j
V




-

1
1
Gain = G =

|G|

freq

f
c

= 1/2

R
i
C
f

R
f
/R
i

0.707 R
f
/R
i

Active Filters (High
-
Pass Filter)

C
i

+

-

R
i


i


o

(b)

R
f

Active filters


(b) A high
-
pass filter attenuates low frequencies and blocks dc.





i
i
i
i
i
f
i
o
C
R
j
C
R
j
R
R
j
V
j
V





-

1
Gain = G =

|G|

freq

f
c

= 1/2

R
i
C
f

R
f
/R
i

0.707 R
f
/R
i

Active Filters (Band
-
Pass Filter)

+

-


i


o

(c)

R
f

C
i

R
i

Active filters


(c) A bandpass filter attenuates both low and high frequencies.









i
i
f
f
i
f
i
o
C
R
j
C
R
j
C
R
j
j
V
j
V







-

1
1
|G|

freq

f
cL

= 1/2

R
i
C
i

R
f
/R
i

0.707 R
f
/R
i

f
cH

= 1/2

R
f
C
f

C
f

Frequency Response of op
-
amp and Amplifier

Open
-
Loop Gain

Compensation

Closed
-
Loop Gain

Loop Gain

Gain Bandwidth Product

Slew Rate

Offset Voltage and Bias Current

Read section 3.12


Nulling, Drift, Noise


Read section and 3.13


Differential bias current, Drift, Noise

Input and Output Resistance

d
i
i
ai
R
A
i
v
R
)
1
(





+

-

R
d

i
i

R
o

R
L

C
L

i
o

A

d


d


o


i

+

-

1





A
R
i
v
R
o
o
o
ao
Typical value of R
d

= 2 to 20 M
W

Typical value of R
o

= 40
W

Phase Modulator for Linear variable
differential transformer LVDT

+

-

+

-

Phase Modulator for Linear variable
differential transformer LVDT

+

-

+

-

Phase
-
Sensitive Demodulator

Used in many medical
instruments for signal detection,
averaging, and Noise rejection

The Ring Demodulator

v
c



2
v
i

If
v
c

is positive then D
1

and D
2

are forward
-
biased and
v
A

=
v
B
. So
v
o
=
v
DB

If
v
c

is negative then D
3

and D
4

are forward
-
biased and
v
A

=
v
c
. So
v
o
=
v
DC