AD8307

Data Sheet

Rev. F | Page 10 of 24

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, because waveform is involved as well.

Because 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.

PROGRESSIVE COMPRESSION

Most high speed, high dynamic range log amps use a cascade of

nonlinear amplifier cells (see 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 (86 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.

VX

VW

STAGE 1

STAGE 2

STAGE N–1

STAGE N

A

A

A

A

Figure 22. Cascade of Nonlinear Gain Cells

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.

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 ≤

VIN ≤ +EK, but falls asymptotically toward unity for very large

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, in the case of

the AD8307, VY is traceable to an on-chip band gap reference,

whereas VX is derived from the thermal voltage kT/q and is later

temperature corrected.

SLOPE = A

SLOPE = 1

AEK

EK

0

INPUT

A/1

Figure 23. A/1 Amplifier Function

Let the input of an N-cell cascade be VIN, and the final output

be VOUT. For small signals, the overall gain is simply AN. 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

gain in implementing the logarithmic function has been noted;

however, this parameter is only of incidental interest in the design

of log amps.

From this point forward, 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 because 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, whereas

for higher values of VIN, it enters into a series of segments that

lie on a logarithmic approximation (dotted line).

RATIO

OF A

2

1

3

3

2

EK/AN–1 EK/AN–2 EK/AN–3 EK/AN–4

LOG VIN

(4A–3) EK

VOUT

(3A–2) EK

(2A–1) EK

AEK

0

(A–1) EK

Figure 24. First Three Transitions

Continuing this analysis, the next transition occurs when the

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 of A.

At the next critical point (labeled 3 in Figure 24), the input is

again A times larger and VOUT has increased to (3A − 2)EK, that

is, by another linear increment of (A − 1)EK.