Datasheet

MAX4207
Precision Transimpedance Logarithmic
Amplifier with Over 5 Decades of Dynamic Range
_______________________________________________________________________________________ 9
Detailed Description
Theory
Figure 2 shows a simplified model of a logarithmic
amplifier. Two transistors convert the currents applied
at LOGIIN and REFIIN to logarithmic voltages accord-
ing to the following equation:
where:
V
BE
= base-emitter voltage of a bipolar transistor
k = 1.381 x 10
-23
J/K
T = absolute temperature (K)
q = 1.602 x 10
–19
C
I
C
= collector current
I
S
= reverse saturation current
The logarithmic amplifier compares V
BE1
to the refer-
ence voltage V
BE2
, which is a logarithmic voltage for a
known reference current, I
REF
. The temperature depen-
dencies of a logarithmic amplifier relate to the thermal
voltage, (kT/q), and I
S
. Matched transistors eliminate
the I
S
temperature dependence of the amplifier in the
following manner:
where:
k = scale factor (V/decade)
I
LOG
= the input current at LOGIIN
I
REF
= the reference current at REFIIN
The MAX4207 uses internal temperature compensation
to virtually eliminate the effects of the thermal voltage,
(kT/q), on the amplifier’s scale factor, maintaining a
constant slope over temperature.
Definitions
Transfer Function
The ideal logarithmic amplifier transfer function is:
Adjust K (see the Scale Factor section) to increase the
transfer-function slope as illustrated in Figure 3. Adjust
I
REF
using REFISET (see the Adjusting the Logarithmic
Intercept section) to shift the logarithmic intercept to
the left or right as illustrated in Figure 4.
Log Conformity
Log conformity is the maximum deviation of the
MAX4207’s output from the best-fit straight line of the
V
LOGV1
versus log (I
LOG
/I
REF
) curve. It is expressed as
a percent of the full-scale output or an output voltage.
Referred-to-Input and Referred-to-Output Errors
The log nature of the MAX4207 insures that any addi-
tive error at LOGV1 corresponds to multiplicative error
at the input, regardless of input level.
VK
I
I
IDEAL
LOG
REF
=×






log
10
VVV
kT
q
I
I
kT
q
I
I
kT
q
I
I
I
I
kT
q
I
I
OUT BE BE
LOG
S
REF
S
LOG
S
REF
S
LOG
REF
= −
=












−




















=












−














=










12
ln ln
ln ln
ln










=






()














=×






kT
q
I
I
K
I
I
LOG
REF
LOG
REF
ln( ) log
log
10
10
10
V
kT
q
I
I
BE
C
S
=












ln
LOGIIN
CMVIN
V
CC
REFIIN
V
CC
V
BE1
V
BE2
V
EE
V
EE
I
LOG
I
REF
Figure 2. Simplified Model of a Logarithmic Amplifier
(see Figure 3)