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AD8307AR Datasheet(PDF) 7 Page - Analog Devices |
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AD8307AR Datasheet(HTML) 7 Page - Analog Devices |
7 / 20 page AD8307 –7– REV. A LOG AMP THEORY Logarithmic amplifiers perform a more complex operation than that of classical linear amplifiers, and their circuitry is signifi- cantly different. A good grasp of what log amps do, and how they do it, will avoid many pitfalls in their application. The essential purpose of a log amp is not to amplify, though amplifi- cation is utilized to achieve the function. Rather, it is to com- press a signal of wide dynamic range to its decibel equivalent. It is thus a measurement device. A better term might be logarith- mic converter, since its basic function is the conversion of a signal from one domain of representation to another, via a precise nonlinear transformation. Logarithmic compression leads to situations that may be con- fusing or paradoxical. For example, a voltage offset added to the output of a log amp is equivalent to a gain increase ahead of its input. In the usual case where all the variables are voltages, and regardless of the particular structure, the relationship between the variables can be expressed as: VOUT = VY log (VIN /VX) (1) where: VOUT is the output voltage, VY is called the slope voltage; the logarithm is usually taken to base-ten (in which case VY is also the volts-per-decade), VIN is the input voltage, and VX is called the intercept voltage. All log amps implicitly require two references, here VX and VY, which determine the scaling of the circuit. The absolute accu- racy of a log amp cannot be any better than the accuracy of its scaling references. Equation 1 is mathematically incomplete in representing the behavior of a demodulating log amp such as the AD8307, where VIN has an alternating sign. However, the basic principles are unaffected, and we can safely use this as our starting point in the analyses of log amp scaling which follow. VOUT 5VY 4VY 3VY 2VY VY –2VY VOUT = 0 LOG VIN VSHIFT LOWER INTERCEPT VIN = 10–2VX –40dBc VIN = 102VX +40dBc VIN = 104VX +80dBc VIN = VX 0dBc Figure 19. Ideal Log Amp Function Figure 19 shows the input/output relationship of an ideal log amp, conforming to Equation 1. The horizontal scale is loga- rithmic and spans a wide dynamic range, shown here as over 120 dB, or six decades. The output passes through zero (the log-intercept) at the unique value VIN = VX and would ideally become negative for inputs below the intercept. In the ideal case, the straight line describing VOUT for all values of VIN would continue indefinitely in both directions. The dotted line shows that the effect of adding an offset voltage VSHIFT to the output is to lower the effective intercept voltage VX. Exactly the same alteration could be achieved raising the gain (or signal level) ahead of the log amp by the factor VSHIFT/VY. For example, if VY is 500 mV per decade (that is, 25 mV/dB, as for the AD8307), an offset of +150 mV added to the output will appear to lower the intercept by two tenths of a decade, or 6 dB. Adding an offset to the output is thus indistinguishable from applying an input level that is 6 dB higher. The log amp function described by Equation 1 differs from that of a linear amplifier in that the incremental gain ∂V OUT/∂VIN is a very strong function of the instantaneous value of VIN, as is apparent by calculating the derivative. For the case where the logarithmic base is e, we have: ∂V OUT ∂V IN = VY VIN (2) That is, the incremental gain is inversely proportional to the instantaneous value of the input voltage. This remains true for any logarithmic base, which is chosen as 10 for all decibel- related purposes. It follows that a perfect log amp would be required to have infinite gain under classical small-signal (zero- amplitude) conditions. Less ideally, this result indicates that, whatever means are used to implement a log amp, accurate response under small-signal conditions (that is, at the lower end of the dynamic range) demands the provision of a very high gain-bandwidth product. A further consequence of this high gain is that, in the absence of an input signal, 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 shown in Figure 19 toward a finite baseline, which can be either above or below the intercept. Note that the value given for this intercept may be an extrapolated value, in which case the output may not cross zero, or even reach it, as is the case for the AD8307. While Equation 1 is fundamentally correct, a simpler formula is appropriate for specifying the calibration attributes of a log amp like the AD8307, which demodulates a sine wave input: VOUT = VSLOPE (PIN – P0) (3) where: VOUT is the demodulated and filtered baseband (video or RSSI) output, VSLOPE is the logarithmic slope, now expressed in volts/dB (typically between 15 and 30 mV/dB), PIN is the input power, expressed in decibels relative to some reference power level, and P0 is the logarithmic intercept, expressed in decibels relative to the same reference level. The most widely used reference in RF systems is decibels above 1 mW in 50 Ω, written dBm. Note that the quantity (P IN – 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 popu- lar convention: log amps manifestly do not respond to power (tacitly, power absorbed at the input), but, rather, to input |
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