Datasheet AD8309 (Analog Devices) - 9

AD8309
As a consequence of this high gain, even very small amounts of
STAGE 1 STAGE 2 STAGE N –1 STAGE N
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
VX A A A A VW
the ideal (Figure 19) at small inputs, toward a fixed baseline. This can either be above or below the intercept, depending on Figure 20. Cascade of Nonlinear Gain Cells the design. Note that the value specified for this intercept is invariably an extrapolated one: the RSSI output voltage will never
Theory of Logarithmic Amplifiers
attain a value of exactly zero in a single supply implementation. To develop the theory, we will first consider a somewhat differ- ent scheme to that employed in the AD8309, but which is sim-
Voltage (dBV) and Power (dBm) Response
pler to explain, and mathematically more straightforward to While Equation 1 is fundamentally correct, a simpler formula is analyze. This approach is based on a nonlinear amplifier unit, appropriate for specifying the RSSI calibration attributes of a which we may call an A/1 cell, having the transfer characteristic log amp like the AD8309, which demodulates an RF input. The shown in Figure 21. We here use lowercase variables to define usual measure is input power: the local inputs and outputs of these cells, reserving uppercase VOUT = VSLOPE (PIN – P0 ) (3) for external signals. VOUT is the demodulated and filtered RSSI output, VSLOPE is the The small signal gain ∆VOUT/∆VIN is A, and is maintained for logarithmic slope, expressed in volts/dB, PIN is the input power, inputs up to the knee voltage EK, above which the incremental expressed in decibels relative to some reference power level and gain drops to unity. The function is symmetrical: the same drop P0 is the logarithmic intercept, expressed in decibels relative to in gain occurs for instantaneous values of VIN less than –EK. the same reference level. The large signal gain has a value of A for inputs in the range The most widely used convention in RF systems is to specify –EK < VIN < +EK, but falls asymptotically toward unity for very power in decibels above 1 mW in 50 Ω, written dBm. (However, large inputs. that the quantity [PIN – P0 ] is simply dB). The logarithmic In logarithmic amplifiers based on this simple function, both the function disappears from this formula because the conversion slope voltage and the intercept voltage must be traceable to the has already been implicitly performed in stating the input in one reference voltage, EK. Therefore, in this fundamental analy- decibels. sis, the calibration accuracy of the log amp is dependent solely on Specification of log amp input level in terms of power is strictly this voltage. In practice, it is possible to separate the basic refer- a concession to popular convention: they do not respond to ences used to determine VY and VX. In the AD8309, VY is trace- power (tacitly “power absorbed at the input”), but to the input able to an on-chip band-gap reference, while VX is derived from voltage. In this connection, note that the input impedance of the the thermal voltage kT/q and later temperature-corrected by a AD8309 is much higher that 50 Ω, allowing the use of an im- precise means. pedance transformer at the input to raise the sensitivity, by up Let the input of an N-cell cascade be VIN, and the final output to 13 dB. VOUT. For small signals, the overall gain is simply AN. A six- The use of dBV, defined as decibels with respect to a 1 V rms sine stage system in which A = 5 (14 dB) has an overall gain of amplitude, is more precise, although this is still not unambiguous 15,625 (84 dB). The importance of a very high small-signal ac complete as a general metric, because waveform is also involved gain in implementing the logarithmic function has already been in the response of a log amp, which, for a complex input (such noted. However, this is a parameter of only incidental interest in as a CDMA signal) will not follow the rms value exactly. Since the design of log amps; greater emphasis needs to be placed on most users specify RF signals in terms of power—more specifi- the nonlinear behavior. 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
High speed, high dynamic range log amps use a cascade of
AEK SLOPE = 1
nonlinear amplifier cells (Figure 20) to generate the logarithmic
A/1
function from a series of contiguous segments, a type of piece-
OUTPUT
wise-linear technique. This basic topology offers enormous gain-
SLOPE = A
bandwidth products. For example, the AD8309 employs in its main signal path six cells each having a small-signal gain of
0 EK INPUT
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 Figure 21. The A/1 Amplifier Function gain is thus 100,000 (100 dB) and the bandwidth to –10 dB Thus, rather than considering gain, we will analyze the overall point at the limiter output is 525 MHz. This very high gain- nonlinear behavior of the cascade in response to a simple dc bandwidth product (52,500 GHz) is an essential prerequisite to input, corresponding to the VIN of Equation (1). For very small accurate operation under small signal conditions and at high inputs, the output from the first cell is V1 = AVIN; from the frequencies: Equation (2) reminds us that the incremental gain second, V2 = A2 VIN, and so on, up to VN = AN VIN. At a certain decreases rapidly as VIN increases. The AD8309 exhibits a loga- value of VIN, the input to the Nth cell, VN–1, is exactly equal to rithmic response over most of the range from the noise floor of the knee voltage EK. Thus, VOUT = AEK and since there are N–1 –91 dBV, or 28 µV rms, (or –78 dBm/50 Ω) to a breakdown- cells of gain A ahead of this node, we can calculate that VIN = limited peak input of 4 V (requiring a balanced drive at the EK /AN–1. This unique point corresponds to the lin-log transition, differential inputs INHI and INLO). –8– REV. B