LOG112AIDRE4 Datasheet, PDF (11/20 Pages) Texas Instruments – Precision LOGARITHMIC AND LOG RATIO AMPLIFIERS

INSIDE THE LOG112
Using the base-emitter voltage relationship of matched
bipolar transistors, the LOG112 establishes a logarith-
mic function of input current ratios. Beginning with the
base-emitter voltage defined as:
V_BE
=
V_T
ln
I_C
I_S
where :
V_T
=
kT
q
(1)
k = Boltzmann’s constant = 1.381 • 10^–23
T = Absolute temperature in degrees Kelvin
q = Electron charge = 1.602 • 10^–19 Coulombs
I_C = Collector current
I_S = Reverse saturation current
From the circuit in Figure 12:
V_L = V_BE1 – V_BE2
(2)
Substituting (1) into (2) yields:
V_L
=
V_T1
ln
I₁
I_S1
–
V_T2
ln ^{I 2}
I_S2
(3)
If the transistors are matched and isothermal and
V_TI = V_T2, then (3) becomes:
V_L
=
V_T1

^ln

I₁
I_S
–
ln
I₂
I_S



(4)
V_L
=
V_T
ln ^I1
I₂
and since
(5)
ln x = 2.3log₁₀ x
(6)
V_L
=
n
V_T
log ^I1
I₂
(7)
where n = 2.3
(8)
also
V_OUT
=
V_L
R₁ + R₂
R₁
(9)
V_OUT
=
R₁ + R₂
R₁
n V_T log
I₁
I₂
(10)
or
V_OUT
=
(0.5V)LOG


I₁
I₂


(11)
Q_{1 –}
– ^Q2
I₁
+
+
I₁
V_BE1 V_BE2
A₁
I₂
A₂
V_OUT
V_OUT
=
(0.5V)LOG


I₁
I₂


R₂
I₂
V_L
R₁
FIGURE 13. Simplified Model of a Log Amplifier.
NOTE: R₁ is a metal resistor used to compensate for gain
over temperature.
DEFINITION OF TERMS
TRANSFER FUNCTION
The ideal transfer function is:
V_LOGOUT = (0.5V)LOG (I₁/I₂)
Figure 14 shows the graphical representation of the transfer
over valid operating range for the LOG112 and LOG2112.
ACCURACY
Accuracy considerations for a log ratio amplifier are some-
what more complicated than for other amplifiers. This is
because the transfer function is nonlinear and has two
inputs, each of which can vary over a wide dynamic range.
The accuracy for any combination of inputs is determined
from the total error specification.
TOTAL ERROR
The total error is the deviation (expressed in mV) of the actual
output from the ideal output of V_LOGOUT = (0.5V)LOG (I₁/I₂).
Thus,
V_{LOGOUT(ACTUAL)} = V_{LOGOUT(IDEAL)} ± Total Error
(6)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
–0.5
–1.0
–1.5
–2.0
–2.5
–3.0
–3.5
I2⁼
I2⁼
I2⁼
100pA
1nA
10nA
100pA _{1nA 10nA}
I
I
I
I
I
2
2
2
2
2
=
=
=
=
=
100nA
11100^µ_0µ^A_µAA
1mA
100nA 1^µA 10^µA 100^µA 1mA
10mA
I₁
V_LOGOUT = (0.5V)LOG (I₁/I₂)
FIGURE 14. Transfer Function with Varying I₂ and I₁.
It represents the sum of all the individual components of error
normally associated with the log amp when operated in the
current input mode. The worst-case error for any given ratio
of I₁/I₂ is the largest of the two errors when I₁ and I₂ are
considered separately. Temperature can affect total error.
LOG112, 2112
11
SBOS246D
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