As a consequence of this high gain, even very small amounts of
thermal noise at the input of a log amp will cause a finite output
for zero input, resulting in the response line curving away from
the ideal (Figure 19) at small inputs, toward a fixed baseline.
This can either be above or below the intercept, depending on
the design. Note that the value specified for this intercept is
invariably an
extrapolated
one: the RSSI output voltage will never
attain a value of exactly zero in a single supply implementation.
Voltage (dBV) and Power (dBm) Response
STAGE 1
STAGE 2
STAGE N –1
STAGE N
V
X
A
A
A
A
V
W
Figure 20. Cascade of Nonlinear Gain Cells
Theory of Logarithmic Amplifiers
While Equation 1 is fundamentally correct, a simpler formula is
appropriate for specifying the RSSI calibration attributes of a
log amp like the AD8309, which
demodulates
an RF input. The
usual measure is input power:
V
OUT
=
V
SLOPE
(P
IN
–
P
0
)
(3)
V
OUT
is the demodulated and filtered RSSI output,
V
SLOPE
is the
logarithmic slope, expressed in volts/dB,
P
IN
is the input power,
expressed in decibels relative to some reference power level and
P
0
is the logarithmic intercept, expressed in decibels relative to
the same reference level.
The most widely used convention in RF systems is to specify
power in decibels above 1 mW in 50
Ω,
written dBm. (However,
that the quantity [P
IN
– P
0
] is simply dB). The logarithmic
function disappears from this formula because the conversion
has already been implicitly performed in stating the input in
decibels.
Specification of log amp input level in terms of power is strictly
a concession to popular convention: they do
not
respond to
power
(tacitly “power absorbed at the input”), but to the input
voltage.
In this connection, note that the input impedance of the
Ω,
allowing the use of an im-
pedance transformer at the input to raise the sensitivity, by up
to 13 dB.
The use of dBV, defined as
decibels with respect to a 1 V rms sine
amplitude,
is more precise, although this is still not unambiguous
complete as a general metric, because waveform is also involved
in the response of a log amp, which, for a complex input (such
as a CDMA signal) will not follow the rms value exactly. Since
most users specify RF signals in terms of power—more specifi-
cally, in dBm/50
Ω—we
use both dBV and dBm in specifying
the performance of the AD8309, showing equivalent dBm levels
for the special case of a 50
Ω
environment.
Progressive Compression
To develop the theory, we will first consider a somewhat differ-
ent scheme to that employed in the AD8309, but which is sim-
pler to explain, and mathematically more straightforward to
analyze. This approach is based on a nonlinear amplifier unit,
which we may call an A/1 cell, having the transfer characteristic
shown in Figure 21. We here use lowercase variables to define
the local inputs and outputs of these cells, reserving uppercase
for
external
signals.
The small signal gain
∆V
OUT
/∆V
IN
is A, and is maintained for
inputs up to the knee voltage E
K
, above which the
incremental
gain drops to unity. The function is symmetrical: the same drop
in gain occurs for instantaneous values of V
IN
less than –E
K
.
The
large signal
gain has a value of A for inputs in the range
–E
K
< V
IN
< +E
K
, but falls asymptotically toward unity for very
large inputs.
In logarithmic amplifiers based on this simple function, both the
slope voltage and the intercept voltage must be traceable to the
one reference voltage, E
K
. Therefore, in this fundamental analy-
sis, the
calibration accuracy
of the log amp is dependent solely on
this voltage. In practice, it is possible to separate the basic refer-
ences used to determine V
Y
and V
X
. In the AD8309, V
Y
is trace-
able to an on-chip band-gap reference, while V
X
is derived from
the thermal voltage kT/q and later temperature-corrected by a
precise means.
Let the input of an N-cell cascade be V
IN
, and the final output
V
OUT
. For small signals, the overall gain is simply A
N
. A six-
stage system in which A = 5 (14 dB) has an overall gain of
15,625 (84 dB). The importance of a very high small-signal ac
gain in implementing the logarithmic function has already been
noted. However, this is a parameter of only incidental interest in
the design of log amps; greater emphasis needs to be placed on
the
nonlinear
behavior.
High speed, high dynamic range log amps use a cascade of
nonlinear amplifier cells (Figure 20) to generate the logarithmic
function from a series of contiguous segments, a type of piece-
wise-linear technique. This basic topology offers enormous gain-
bandwidth products. For example, the AD8309 employs in its
main signal path six cells each having a small-signal gain of
12.04 dB (×4) and a –3 dB bandwidth of 850 MHz, followed by
a final limiter stage whose gain is typically 18 dB. The overall
gain is thus 100,000 (100 dB) and the bandwidth to –10 dB
point at the limiter output is 525 MHz. This very high gain-
bandwidth product (52,500 GHz) is an essential prerequisite to
accurate operation under small signal conditions and at high
frequencies: Equation (2) reminds us that the incremental gain
decreases rapidly as V
IN
increases. The AD8309 exhibits a loga-
rithmic response over most of the range from the noise floor of
–91 dBV, or 28
µV
rms, (or –78 dBm/50
Ω)
to a breakdown-
limited peak input of 4 V (requiring a balanced drive at the
differential inputs INHI and INLO).
OUTPUT
AE
K
A/1
SLOPE = 1
SLOPE = A
0
E
K
INPUT
Figure 21. The A/1 Amplifier Function
Thus, rather than considering
gain,
we will analyze the overall
nonlinear behavior of the cascade in response to a simple dc
input, corresponding to the V
IN
of Equation (1). For very small
inputs, the output from the first cell is V
1
= AV
IN
; from the
second, V
2
= A
2
V
IN
, and so on, up to V
N
= A
N
V
IN
. At a certain
value of V
IN
, the input to the Nth cell, V
N–1
, is exactly equal to
the knee voltage E
K
. Thus, V
OUT
= AE
K
and since there are N–1
cells of gain A ahead of this node, we can calculate that V
IN
=
E
K
/A
N–1
. This unique point corresponds to the
lin-log transition,
–8–
REV. B
AD [ ANALOG DEVICES ]
