6 log anti log amplifiers

8/13/2019 6 Log Anti Log Amplifiers

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Log and AntiLogAmplifiers

Recommended Text: Pallas-Areny, R. & Webster, J.G.,

Analog Signal Processing, Wiley (1999) pp. 293-321


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Introduction

Log and Antilog Amplifiers are non-linear circuits inwhich the output voltage is proportional to the

logarithm (or exponent) of the input. It is well known that some processes such as

multiplication and division, can be performed by

addition and subtraction of logs. They have numerous applications in electronics, such

as:

Multiplication and division, powers and roots Compression and Decompression

True RMS detection

Process control


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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, Ic, andthe base-emitter voltage, v

be, in silicon bipolar transistors.


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Transdiode Log Amplifier

The input voltage is converted by R1 into a current, which then flows throughthe transistor's collector modulating the base-emitter voltage according to theinput voltage.

The opamp forces the collector voltage to that at the noninverting input, 0 V From Ebers-Moll model the collector current is

where Is is saturation current, q is the charge of the electron 1.6x10-9Coulombs, k is the Boltsmans constant 1.38x10-23 Joules, T is absolutetemperature, VT is thermal voltage.

For room temperature 300oK

The output voltage is therefore

TT VVbe

s

VVbe

s

kTqVbe

sc eIeIeII/// )1()1( ==

Vbes

Vbesc eIeII 6.386.38 )1( =

=

=

== IsRi

Vin

IR

vV

I

iVVbeVout

S

iT

S

CT ln0259.0lg3.2ln 1



Thermal and Frequencystability

This equation yields the desired logarithmic relationship over awide range of currents, but is temperature-sensitive because of

VT and IS resulting in scale-factor and offset temperature-dependent errors.

The system bandwidth is narrower for small signals becauseemitter resistance increases for small currents.

The source impedance of voltage signals applied to the circuitmust be small compared to R1. Omitting R1 yields a current-input log amp.

Using a p-n-p transistor changes the polarity of input signalsacceptable but limits the logarithmic range because of thedegraded performance of p-n-p transistors compared to n-p-ntransistors


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IC Log Amps.

These basic circuits needs additional components to improovethe overall performance, i.e:

to provide base-emitter junction protection, to reduce temperature effects,

bulk resistance error and op amp offset errors,

to accept bipolar input voltages or currents,

and to ensure frequency stability.

Such circuit techniques are used in integrated log amps:AD640, AD641, ICL8048, LOG100, 4127.

IC log amps may cost about ten times the components needed tobuild a discrete-component log amp.

Nevertheless, achieving a 1% logarithmic conformity over

almost six decades for input currents requires careful design.



TemperatureCompensation

=

S

iTo

IR

vVv

1

ln

The equation for output voltage shows that the scale factor ofthe basic transdiode log amp depends on temperature because of

VTand that there is also a temperature-dependent offset because ofIS.

Temperature compensation must correct both error sources.

Figure (next slide) shows the use of a second, matched,transistor for offset compensation and a temperature-dependentgain for gain compensation.




Temperature compensation in a transdiode log amp:

a second transistor (Q2) compensates the offset voltage and

a temperature-sensitive resistor (R4) compensates the scalefactor

T t




For transistors Q1 & Q2 we have

whereIr is a reference, temperature-independent, current.

The output voltage will be

Matched transistors (IS1 =IS2) will cancel offset.

In order to compensate the gain dependence on temperature, R4 must be

much smaller thanR3 and such that d(VT/R4)/dT= 0. This requires dR4/R4 == dVT/VT(= l/T).

At T= 298 K, the temperature coefficient ofR4 must be 3390 x 10-6K.

D1protects the base-emitter junction from excessive reverse voltages.

=

11

1 lnS

iTBE

IR

vVv

=

2

2 lnS

rTBE

I

IVv

( )

+=

+=

+=

1

21

4

3

4

3

124

3

1ln111

S

S

i

r

TBEBEo I

I

v

IR

R

RV

R

Rvv

R

Rvv


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Stability Considerations

Transdiode circuits have a notorious tendency to oscillate due to the presenceof an active element in the feedback that can provide gain rather than loss.

Consider the voltage-input transdiode. Ignoring op amp input errors, we have

and

The feedback factor for a given value of Vi, is determined as

Differentiating and using the fact thatIc = Vi/R, we obtain

indicating that b can be greater than unity.

For instance, with Vi = 10 V we have = 10/0.026 = 400 = 52 dB, indicatingthat in the Bode diagram the |1/b| curve lies 52 dB below the 0 dB axis.

Thus, the |1/b| curve intersects the\a\curve atfc >> ft, where the phase shiftdue to higher-order poles is likely to render the circuit unstable; an additional

source of instability is the input stray capacitance Cn

cin IRVV = BEo VV =

BEcon dVdIRdVdV // ==

TiTc VVVIR // ==


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Range Considerations

The transdiode circuit is compensated by means of an emitterresistor RE to decrease the value of and a feedback capacitorCp to combat Cn, as shown.

To investigate its stability, refer to the incremental model,where the BJT has been replaced by its common-base small-signal model.


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Transistor parameters re and ro depend on the operating currentIc,

where VA is called the Early voltage (typically ~ 100 V). C is the base-

collector junction capacitance. Both Cand Cn are typically ~10 pF range.

CTCTe IVIVr // = CAo IVr /=

Eedo RrRRrrRR +=+= 2and)(||||1

KCL at the summing junction yields

Eliminatingie and rearranging yields

whereie=-vo/R2 and C1=Cn+C+CF

)()(1/1 onFenn vvCjiCCjRv +++++

221

1111

RCRjv

RCRjv Fon +=+

11121

121

CRjCRj

RR

vv F

n

o

++=


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The | 1/b| curve has a low-frequency asymptote at R2/R1, a high-frequencyasymptote at C1/CF, and two breakpoints atf=fz andf= fp.

While C1/CF and fz are constant, R2/R1 and fp depend on the operatingcurrentIC. As such, they can vary over a wide range of values.

F

zz

p

z

n

o

CRf

CRf

ffj

ffj

R

R

v

v

12

1and

112

1where

)/(1

)/(1

1

21

==

+

+==

The hardest condition is when Ic =Ic(max), since this minimizes the valueofR2/R1 while maximizing that offp,.

As a rule of thumb,RE is chosen to makeR2(min)/R1 ~ 0.5 for a reasonably lowvalue of ||max,

CF is chosen to makefp(max) ~ 0.5fc forreasonable phase margin.


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Frequency stability

As the input level is decreased, we witness an increasing dominance of fp ,which slows down the dynamics of the circuit.

Since at sufficiently low current levels re>>RE, we havefp=1/(2reCF)

The corresponding time constant is = reCF=(VT/IC)CF =(VT/ Vi)RCFindicating that is inversely proportional to the input level, as expected.

For instance, withIc = 1 nA and Cp = 100 pF, we have = (26 x 10-3/10-9) x

100 x 10

-12

= 2.6 ms.

It takes 4.6 for an exponentialtransition to come within 1 percent of itsfinal value, therefore our circuit willtake about 12 ms to stabilize to within 1percent.

his limitation must be kept in mind

when operating near the low end of thedynamic range.


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Diode-connected Log Amp

In the second circuit a BJT connected as a diode to achieve thelogarithmic characteristic.

The analysis is the same as above for the transdiode connection,but the logarithmic range is limited to four or five decadesbecause the base current adds to the collector current.

On the pro side, the circuit polarity can be easily changed by reversing the transistor, the stability improves, and

the response is faster.


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Input Current Inversion

The basic log amp in only accepts positive input voltages or currents.

Negative voltages or currents can be first rectified and then applied to the logamp, but this adds the errors from the rectifier.

Alternatively, the log amp can be preceded by a precision current inverter.

The current inverter in Figure below uses two matched n-p-n transistors and aprecision op amp to achieve accurate current inversion.

The collector-base voltage in both Ql and Q2 is 0 V, so that the Ebers-Mollmodel for BJT transistors leads to

=

=

)1(

)1(/

22

/11

2

1

TBE

TBE

Vv

ESe

Vv

ESe

eIi

eIi

whereIES1 andIES2 are the respective

emitter saturation currents ofQlandQ2.


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Input Current Inversion

From circuit inspection, assuming an op amp with infinite open-loop gain butfinite input currents and offset voltage,

Solving for the output current in termsof the input current yields

+=

+=

+=

ioBEBE

boe

bie

Vvv

Iii

Iii

12

22

11

22

/

1

1

2

/

1

2

1 bESVV

ES

b

ES

VV

ES

ES

io IIeI

I

IeI

I

ii

TioTio

++=

which shows that, in order to havesmall gain and offset errors, the

offset voltage must be smallcompared to VT,

the op amp offset current must besmall compared to the input current,

and Ql and Q2 must be matched.

Exponential (Antilog)


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Exponential (Antilog)Amplifiers

An exponential or antilogarithmic amplifier (antilog amp), performs thefunction inverse to that of log amps:

its output voltage is proportional to a base (10, e) elevated to the ratio

between two voltages. 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


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Antilog Amplifier

Interchanging the position of resistor and transistor in a log amp yields abasic antilog amp.

The base-collector voltage is kept at 0 V, so that collector current is given by

and for negative input voltages we have:

There is again a double temperature dependence because ofIS and VT.

Temperature compensation can be achieved by the same technique shown for

log amps.

( )TBEsc VvIi /exp )/exp(11 TiSCo VvRIRiv ==

Temperature


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The input voltage is applied to a voltage divider that includes a temperaturesensor. IfR3 R4, vBC1 ~ 0V and applying to Ql yields

where Vris a reference voltage and we have assumed VBE1>>VT(25 mV).

In Q2 VBC2 = 0V and hence : Also:

Substituting vBE1 and vBE2, and solving for vo, if Ql and Q2 are matched yields

)1)/(exp( = TBEsc VvII

5/)/exp(11 RVVvIi rTBEsc =

5/)/exp( 222 RVVvIi oTBEsc =

21

34

4BEBEi vv

RR

Rv =

+

)43

4exp(

5

1

RR

R

V

v

R

RVv

T

iro

+

Therefore, if the temperaturecoefficient of R4 is such that

dR4/R4 == dVT/VT = l/T thevoltage divider will compensatefor the temperature dependence

of VT. At T = 298 K, thetemperature coefficient of R4must be 3390 x 10-6K.


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Log-Antilog

Log and antilog amp circuits include the same elements butarranged in different feedback configurations.

Some integrated log amps have uncommitted elements allowingus to implement antilog amps.

Some IC (like ICL8049) are a committed only antilog amp.

Some so-called multifunction converters (AD538, LH0094,4302) include op amps and transistors to simultaneouslyimplement log and antilog functions, or functions derivedthereof, such as multiplication,

division,

raising to a power,

or taking a root


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Basic Multiplier

Multipliers are based on the fundamental logarithmicrelationship 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) = lna + lnb This formula shows that two signal voltages are

effectively multiplied if the logarithms of the signal

voltages are added.


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Multiplication Stages

The multiplication procedure take three steps:

1. 1. To get the logarithm of a signal voltage use a Log amplifier.

2. 2. By summing the outputs of two log amplifiers, you get the

logarithm of the product of the two original input voltages.

3. 3. Then, by taking the antilogarithm, you get the product of thetwo input voltages as indicated in the following equations:

[ ] 2121*

)ln(exp)exp( VVVVVV OO ===

)ln(and)ln( 2*21*1 VVVV ==

)ln()ln()ln( 2121*

2*

2* VVVVVVVO =+=+=

block diagram of an


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block diagram of ananalog multiplier

The block diagram shows how the functions are connected tomultiply two input voltages.

Constant terms are omitted for simplicity.

B i M lti li Ci it


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Basic Multiplier Circuitry

The outputs of the log amplifierare stated as follows:

where K1 = 0.025 V, K2 = RIebo

andR = R1 = R2= R6.The two output voltages from thelog amplifiers are added andinverted by the unity-gain

summing amplifier to produce thefollowing result:

=

2

11)1(log ln

K

VKV inout

=

2

21)2(log ln

K

VKV inout

=

=

+

=

22

211

2

2

2

11)(

ln

lnlnln

K

VVK

K

V

K

VKV

inin

ininsumout


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F Q d t M lti li


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Four-Quadrant Multipliers

Four-Quadrant Multiplier is a device with two inputs and one output.

Typically k = 0.1 to reduce the possibility of output overload.

It is called four-quadrant since inputs and output can be positive or negative.

An example device is Motorola MC1494, powered by 15 V power supply

VoutV2

V1

21 VVkVout =

M lti li A li ti


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Multiplier Applications

Alongside the multiplication Multipliers havemany uses such as:

Squaring

Dividing

Modulation / demodulation Frequency and amplitude modulation

Automatic gain control

Amplitude Modulation


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p& Squaring

Amplitude Modulation

Squaring circuit

VRF

VoutVLF

VoutVin

Divider


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Divider

Divider VoutVxKVm =

22 R

Vm

i =11 R

Vin

i =

VoutVxKVmVin ==

VxK

Vin

VxK

VmVout ==

Square root: If VoutVx=

VoutK

VinVout

=

KVinVout =


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6 log anti log amplifiers

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