LOG104_14 Datasheet, PDF (8/15 Pages) Texas Instruments – LOGARITHMIC AND LOG RATIO AMPLIFIER

INSIDE THE LOG104
Using the base-emitter voltage relationship of matched
bipolar transistors, the LOG104 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 11, we see that:
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 = 2.3 log₁₀ x
(6)
V_L
=
n V_T
log ^I1
I₂
(7)
where n = 2.3
(8)
also
or
V_OUT
=
V_L
R₁ + R₂
R₁
V_OUT
=
R₁ + R₂
R₁
n
V_T
log
I₁
I₂
V_OUT
=
0.5V • log ^I1
I₂
(9)
(10)
(11)
I₁
I₁
Q_{1 –}
– ^Q2
+
+
V_BE1 V_BE2
I₂
A₂
V_OUT
A₁
V_OUT = (0.5V) LOG
I₁
I₂
R₂
I₂
V_L
R₁
FIGURE 11. Simplified Model of a Log Amplifier.
DEFINITION OF TERMS
TRANSFER FUNCTION
The ideal transfer function is:
V_OUT = 0.5V • logI₁/I₂
(5)
See Figure 12 for the graphical representation of the transfer
over valid operating range for the LOG104.
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.
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
–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
V_OUT = (0.5V) • LOG (I₁/I₂)
10mA
I₁
FIGURE 13. Transfer Function with Varying I₂ and I₁.
8
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LOG104
SBOS243C