AD8307
The most widely used reference in RF systems is decibels above
1 mW in 50 Ω, written dBm. Note that the quantity (PIN – P0) is
just dB. The logarithmic function disappears from the formula
because the conversion has already been implicitly performed
in stating the input in decibels. This is strictly a concession to
popular convention; log amps manifestly do not respond to
power (tacitly, power absorbed at the input), but rather to input
voltage. The use of dBV (decibels with respect to 1 V rms) is
more precise, though still incomplete, since waveform is involved,
too. Since most users think about and specify RF signals in terms of
power, more specifically, in dBm re: 50 Ω, this convention is used in
specifying the performance of the AD8307.
in the case of the AD8307, VY is traceable to an on-chip band
gap reference, while VX is derived from the thermal voltage
kT/q and is later temperature corrected.
AE
K
SLOPE = 1
A/1
SLOPE = A
0
E
INPUT
K
PROGRESSIVE COMPRESSION
Figure 23. A/1 Amplifier Function
Most high speed, high dynamic range log amps use a cascade of
nonlinear amplifier cells (Figure 22) to generate the logarithmic
function from a series of contiguous segments, a type of
piecewise linear technique. This basic topology immediately
opens up the possibility of enormous gain bandwidth products.
For example, the AD8307 employs six cells in its main signal
path, each having a small signal gain of 14.3 dB (×5.2) and a
−3 dB bandwidth of about 900 MHz. The overall gain is about
20,000 (8ꢀ dB) and the overall bandwidth of the chain is some
500 MHz, resulting in the incredible gain bandwidth product
(GBW) of 10,000 GHz, about a million times that of a typical op
amp. This very high GBW is an essential prerequisite for
accurate operation under small signal conditions and at high
frequencies. In Equation 2, however, the incremental gain
decreases rapidly as VIN increases. The AD8307 continues to
exhibit an essentially logarithmic response down to inputs as
small as 50 μV at 500 MHz.
Let the input of an N-cell cascade be VIN, and the final output
OUT. For small signals, the overall gain is simply AN. A six stage
V
system in which A = 5 (14 dB) has an overall gain of 15,ꢀ25
(84 dB). The importance of a very high small signal gain in
implementing the logarithmic function has been noted;
however, this parameter is only of incidental interest in the
design of log amps.
From here onward, rather than considering gain, analyze the
overall nonlinear behavior of the cascade in response to a
simple dc input, corresponding to the VIN of Equation 1. For
very small inputs, the output from the first cell is V1 = AVIN.
The output from the second cell is V2 = A2 VIN, and so on, up to
VN = AN VIN. At a certain value of VIN, the input to the Nth cell,
VN–1, is exactly equal to the knee voltage EK. Thus, VOUT = AEK
and since there are N–1 cells of gain A ahead of this node,
calculate VIN = EK /AN–1. This unique situation corresponds to
the lin-log transition, (labeled 1 in Figure 24). Below this input,
the cascade of gain cells acts as a simple linear amplifier, while
for higher values of VIN, it enters into a series of segments that
lie on a logarithmic approximation (dotted line).
STAGE 1
STAGE 2
STAGE N–1
STAGE N
V
V
X
W
A
A
A
A
V
OUT
Figure 22. Cascade of Nonlinear Gain Cells
(4A–3) E
(3A–2) E
(2A–1) E
AE
K
K
K
To develop the theory, first consider a scheme slightly different
from that employed in the AD8307, but simpler to explain and
mathematically more straightforward to analyze. This approach
is based on a nonlinear amplifier unit, called an A/1 cell, with
the transfer characteristic shown in Figure 23.
2
3
3
2
(A–1) E
K
1
RATIO
OF A
The local small signal gain δVOUT/δVIN is A, maintained for all
inputs up to the knee voltage EK, above which the incremental
gain drops to unity. The function is symmetrical: the same drop
in gain occurs for instantaneous values of VIN less than –EK. The
large signal gain has a value of A for inputs in the range −EK ≤
K
0
LOG V
IN
N–1
/A
N–2
N–3
N–4
E
E
/A
E
/A
E
/A
K
K
K
K
Figure 24. First Three Transitions
VIN ≤ +EK, but falls asymptotically toward unity for very large
Continuing this analysis, the next transition occurs when the
inputs. In logarithmic amplifiers based on this amplifier
function, both the slope voltage and the intercept voltage must
be traceable to the one reference voltage, EK. Therefore, in this
fundamental analysis, the calibration accuracy of the log amp is
dependent solely on this voltage. In practice, it is possible to
separate the basic references used to determine VY and VX and
input to the (N–1) stage just reaches EK; that is, when VIN
=
EK /AN–2. The output of this stage is then exactly AEK, and it is
easily demonstrated (from the function shown in Figure 23)
that the output of the final stage is (2A–1) EK (labeled 2 in
Figure 24). Thus, the output has changed by an amount (A–1)EK
for a change in VIN from EK /AN–1 to EK/AN–2, that is, a ratio change
Rev. C | Page ±0 of 24
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