Log and AntiLog Amplifiers. Recommended Text: Pallas-Areny, R. & Webster, J.G., Analog Signal Processing, Wiley (1999) pp

Transcription

1 Log and AntiLog Amplifiers Recommended Text: Pallas-Areny, R. & Webster, J.G., Analog Signal Processing, Wiley (999) pp. 93-3

2 ntroduction Log and Antilog Amplifiers are non-linear circuits in which the output oltage is proportional to the logarithm (or exponent) of the input. t is well known that some processes such as multiplication and diision, can be performed by addition and subtraction of logs. They hae numerous applications in electronics, such as: Multiplication and diision, powers and roots Compression and Decompression True RMS detection Process control

3 Two basic circuits There are two basic circuits for logarithmic amplifiers (a) transdiode and (b) diode connected transistor Most logarithmic amplifiers are based on the inherent logarithmic relationship between the collector current, c, and the base-emitter oltage, be, in silicon bipolar transistors.

4 Transdiode Log Amplifier The input oltage is conerted by R into a current, which then flows through the transistor's collector modulating the base-emitter oltage according to the input oltage. The opamp forces the collector oltage to that at the noninerting input, 0 V From Ebers-Moll model the collector current is where s is saturation current, q is the charge of the electron.6x0-9 Coulombs, k is the Boltsman s constant.38x0-3 Joules, T is absolute temperature, V T is thermal oltage. c For room temperature 300 o K qvbe / kt Vbe / V s ( e ) s ( e ) s The output oltage is therefore c 38.6Vbe s ( e ) s e Vbe / T V T e 38.6Vbe Vout Vbe V T ln i C S V T lg.3 i R S ln Vin Ri s

5 Thermal and Frequency stability This equation yields the desired logarithmic relationship oer a wide range of currents, but is temperature-sensitie because of V T and S resulting in scale-factor and offset temperaturedependent errors. The system bandwidth is narrower for small signals because emitter resistance increases for small currents. The source impedance of oltage signals applied to the circuit must be small compared to R. Omitting R yields a currentinput log amp. Using a p-n-p transistor changes the polarity of input signals acceptable but limits the logarithmic range because of the degraded performance of p-n-p transistors compared to n-p-n transistors

6 Transfer Function

7 C Log Amps. These basic circuits needs additional components to improoe the oerall performance, i.e: to proide base-emitter junction protection, to reduce temperature effects, bulk resistance error and op amp offset errors, to accept bipolar input oltages or currents, and to ensure frequency stability. Such circuit techniques are used in integrated log amps: AD640, AD64, CL8048, LOG00, 47. C log amps may cost about ten times the components needed to build a discrete-component log amp. Neertheless, achieing a % logarithmic conformity oer almost six decades for input currents requires careful design.

8 Temperature Compensation o V T ln i R S The equation for output oltage shows that the scale factor of the basic transdiode log amp depends on temperature because of V T and that there is also a temperature-dependent offset because of S. Temperature compensation must correct both error sources. Figure (next slide) shows the use of a second, matched, transistor for offset compensation and a temperature-dependent gain for gain compensation.

9 Temperature Compensation Temperature compensation in a transdiode log amp: a second transistor (Q) compensates the offset oltage and a temperature-sensitie resistor (R4) compensates the scale factor

10 Temperature Compensation For transistors Q & Q we hae BE VT ln i R S BE VT ln r S where r is a reference, temperature-independent, current. n order to compensate the gain dependence on temperature, R4 must be much smaller than R3 and such that d(v T /R4)/dT 0. This requires dr4/r4 dv T /V T ( l/t). At T 98 K, the temperature coefficient of R4 must be 3390 x 0-6 K. The output oltage will be R + 3 R + 3 R + 3 R o BE BE VT ln R4 R4 R4 Matched transistors ( S S ) will cancel offset. r S ( ) i S D protects the base-emitter junction from excessie reerse oltages.

11 Stability Considerations Transdiode circuits hae a notorious tendency to oscillate due to the presence of an actie element in the feedback that can proide gain rather than loss. Consider the oltage-input transdiode. gnoring op amp input errors, we hae V and n Vi R V c o VBE The feedback factor β for a gien alue of V i, is determined as β dv / dv R d / dv n o Differentiating and using the fact that c V i /R, we obtain β R c / VT Vi / VT indicating that b can be greater than unity. c BE For instance, with V i 0 V we hae β 0/ db, indicating that in the Bode diagram the /b cure lies 5 db below the 0 db axis. Thus, the /b cure intersects the \a\ cure at fc >> ft, where the phase shift due to higher-order poles is likely to render the circuit unstable; an additional source of instability is the input stray capacitance C n

12 Range Considerations The transdiode circuit is compensated by means of an emitter resistor R E to decrease the alue of β and a feedback capacitor C p to combat C n, as shown. To inestigate its stability, refer to the incremental model, where the BJT has been replaced by its common-base smallsignal model.

13 Range Considerations Transistor parameters r e and r o depend on the operating current c, where V A is called the Early oltage (typically ~ 00 V). C µ is the basecollector junction capacitance. Both C µ and Cn are typically ~0 pf range. C T C T e V V r / / α C A o V r / E e d o R r R R r r R R + + and ) ( KCL at the summing junction yields Eliminating i e and rearranging yields where i e - o /R and CC n +C µ +C F ( ) ) ( ) ( / o n F e n n C j i C C j R ω α ω µ R C R j R C R j F o n ω ω + + C R j C R j R R F n o ω ω β + +

14 Range Considerations β o n R + R + j( f j( f / / f f z p ) ) where The /b cure has a low-frequency asymptote at R/R, a high-frequency asymptote at C/C F, and two breakpoints at ffz and f fp. While C/C F and fz are constant, R/R and fp depend on the operating current C. As such, they can ary oer a wide range of alues. The hardest condition is when c c(max), since this minimizes the alue of R/R while maximizing that of fp,. As a rule of thumb, R E is chosen to make R(min)/R ~ 0.5 for a reasonably low alue of β max, C F is chosen to make fp(max) ~ 0.5 fc for reasonable phase margin. f z and πr C f z πr C F

15 Frequency stability As the input leel is decreased, we witness an increasing dominance of fp, which slows down the dynamics of the circuit. Since at sufficiently low current leels r e >>R E, we hae fp/(πr e C F ) The corresponding time constant is τ r e C F (V T / C )C F (V T / V i )RC F indicating that τ is inersely proportional to the input leel, as expected. For instance, with c na and Cp 00 pf, we hae τ (6 x 0-3 /0-9 ) x 00 x ms. t takes 4.6 τ for an exponential transition to come within percent of its final alue, therefore our circuit will take about ms to stabilize to within percent. his limitation must be kept in mind when operating near the low end of the dynamic range.

16 Diode-connected Log Amp n the second circuit a BJT connected as a diode to achiee the logarithmic characteristic. The analysis is the same as aboe for the transdiode connection, but the logarithmic range is limited to four or fie decades because the base current adds to the collector current. On the pro side, the circuit polarity can be easily changed by reersing the transistor, the stability improes, and the response is faster.

17 nput Current nersion The basic log amp in only accepts positie input oltages or currents. Negatie oltages or currents can be first rectified and then applied to the log amp, but this adds the errors from the rectifier. Alternatiely, the log amp can be preceded by a precision current inerter. The current inerter in Figure below uses two matched n-p-n transistors and a precision op amp to achiee accurate current inersion. The collector-base oltage in both Ql and Q is 0 V, so that the Ebers-Moll model for BJT transistors leads to i i e e ES ES ( e ( e BE BE / V T / V T ) ) where ES and ES are the respectie emitter saturation currents of Ql and Q.

18 nput Current nersion From circuit inspection, assuming an op amp with infinite open-loop gain but finite input currents and offset oltage, Soling for the output current in terms of the input current yields i i e e BE i i i o + + b BE b + V io i o i i e + + e ES Vio / VT b Vio / VT ES ES b ES ES which shows that, in order to hae small gain and offset errors, the offset oltage must be small compared to V T, the op amp offset current must be small compared to the input current, and Ql and Q must be matched.

19 Exponential (Antilog) Amplifiers An exponential or antilogarithmic amplifier (antilog amp), performs the function inerse to that of log amps: its output oltage is proportional to a base (0, e) eleated to the ratio between two oltages. Antilog amps are used together with log amps to perform analog computation. Similar to Log Apms there are two basic circuits for logarithmic amplifiers (a) transdiode and (b) diode connected transistor

20 Antilog Amplifier nterchanging the position of resistor and transistor in a log amp yields a basic antilog amp. The base-collector oltage is kept at 0 V, so that collector current is gien by i c s exp( BE / VT ) and for negatie input oltages we hae: o ic R S R exp( i / VT ) There is again a double temperature dependence because of S and V T. Temperature compensation can be achieed by the same technique shown for log amps.

21 Temperature Compensation The input oltage is applied to a oltage diider that includes a temperature sensor. f R3» R4, BC ~ 0V and applying c s (exp( BE / VT ) ) to Ql yields where Vr is a reference oltage and we hae assumed V BE >>V T (5 mv). n Q V BC 0V and hence : Also: Substituting BE and BE, and soling for o, ic s exp( BE / VT ) Vr / R5 if Ql and Q are matched yields ic s exp( BE / VT ) Vo / R5 o V r i R4 R4 + R3 R exp( R5 V i T BE R4 ) R3 + R4 BE Therefore, if the temperature coefficient of R4 is such that dr4/r4 dv T /V T l/t the oltage diider will compensate for the temperature dependence of V T. At T 98 K, the temperature coefficient of R4 must be 3390 x 0-6 K.

22 Log-Antilog Log and antilog amp circuits include the same elements but arranged in different feedback configurations. Some integrated log amps hae uncommitted elements allowing us to implement antilog amps. Some C (like CL8049) are a committed only antilog amp. Some so-called multifunction conerters (AD538, LH0094, 430) include op amps and transistors to simultaneously implement log and antilog functions, or functions deried thereof, such as multiplication, diision, raising to a power, or taking a root

23 Basic Multiplier Multipliers are based on the fundamental logarithmic relationship that states that the product of two terms equals the sum of the logarithms of each term. This relationship is shown in the following formula: ln(a x b) ln a + ln b This formula shows that two signal oltages are effectiely multiplied if the logarithms of the signal oltages are added.

24 Multiplication Stages The multiplication procedure take three steps:.. To get the logarithm of a signal oltage use a Log amplifier. * V ln( V ) and V ln( V).. By summing the outputs of two log amplifiers, you get the logarithm of the product of the two original input oltages. * * V O V * + V * ln( V ) + ln( V ) ln( V V ) Then, by taking the antilogarithm, you get the product of the two input oltages as indicated in the following equations: V O exp( V * O ) exp [ ln( V V ) ] V V

25 block diagram of an analog multiplier The block diagram shows how the functions are connected to multiply two input oltages. Constant terms are omitted for simplicity.

26 Basic Multiplier Circuitry The outputs of the log amplifier are stated as follows: V V out(log) out(log) K K V ln K V ln K in in where K 0.05 V, K R ebo and R R R R6. The two output oltages from the log amplifiers are added and inerted by the unity-gain summing amplifier to produce the following result: V in + Vin V out ( sum) K ln ln ln K K V in Vin K ln K

27 This expression is then applied to the antilog amplifier; the expression for the multiplier output oltage is as follows: V out(exp) K K V in V K V exp in V out( sum) K in V K K in exp K The output of the antilog (exp) amplifier is a constant (/K) times the product of the input oltages. The final output is deeloped by an inerting amplifier with a oltage gain of K. Vout K V V V V in in in in K K V ln in V K in

28 Four-Quadrant Multipliers Four-Quadrant Multiplier is a deice with two inputs and one output. V V out V out k V V V Typically k 0. to reduce the possibility of output oerload. t is called four-quadrant since inputs and output can be positie or negatie. An example deice is Motorola MC494, powered by ± 5 V power supply

29 Multiplier Applications Alongside the multiplication Multipliers hae many uses such as: Squaring Diiding Modulation / demodulation Frequency and amplitude modulation Automatic gain control

30 Amplitude Modulation & Squaring Amplitude Modulation V LF V out V RF Squaring circuit V in V out

31 Diider Diider Vm K Vx Vout i Vin R i Vm R Vin Vm K Vx Vout Vm Vin Vout K Vx K Vx Square root: f Vx Vout Vin Vout K Vout Vin Vout K

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