logarithmic amplifier

7/23/2019 Logarithmic Amplifier

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Alexandru Alice431G

Logarithmic amplifier

1. Introduction

The logarithmic amplifier (logarithmic convertor) creates a logarithmic dependence

between the input and output. Usually, the output is in voltage; the input can be current or

voltage.

The symbol used for convertors and their transfer functions are:

Fig .

The logarithmic convertor has a large scale of applications. !t is used in analogic

computations (multiple"ing, dividing, e"ponential with higher#lower than one e"ponent,effective value) and in dynamic compression circuits (given the property of the logarithmic

function to increase slower than its argument)

!n this paper we study a logarithmic convertor in which the logarithmic conversion element is

a bipolar transistor connected in a negative feedbac$ of an operation amplifier.

Used in a basic circuit the connection topology of the transistor is called %trans&diode'. This

circuit is found in the maority of logarithmic convertors found on sale.

Fig.


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The collector current (ic) depends on emitter&base voltage (*+) and on the collector&base

voltage (*-+) and is given by the bers /oll e0uation:

!s 1 saturation current

23 1 4.5 6 4.7 is the ratio between ie and !c when the transistor is wor$ing in the inverse

active region

8 operates in active normal region at *-+ 1 4 (ideal 9). ubstituting *-+14 in the first

e0uation we obtain for the output voltage *o :

For the case of current control:

nd for the case of voltage control:


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For a small power transistor, at

T1>-, the saturation current !s has

a typical value of 1014

6

1015

.

The logarithmic convertor shown in

Fig. has a few maor disadvantages

which ma$es it useless in practice.

These disadvantages will be shown

below.

(a) !t is observed that at high

currents the 9 load output

becomes e"cessive, whichleads to the decrease of the

gain in open loop. This fact

determines a big deviation

from the ideal model

described earlier, the transfer

characteristic being non&logarithmical.

(b) n important effect of the trans&diode connection is that the feedbac$ loop is not

passive. The transfer function of the circuit is given by the e0uality:

having a value according to the level of input signal. 9bserving the very large

bandwidth in which the logarithmic transistor amplifies and the fact that the value of

the gm3i gain can get higher than one we notice that the circuit might oscillate.

/odifying the circuit in fig. according to fig. ? offers the solution for the

problems. The group -c, 3c does the fre0uency compensation of the converter,

providing the reduction for the loop gain at high fre0uencies. The load resistance at

output 9 is 3c@re.

Fig. ? Fre0uency compensation

(c) 0uality (?) which gives the transfer characteristic shows that the influence of

temperature is present due to !s and T.

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0uality (>) shows the maor gain which is obtained through this altered circuit; the

saturation current of the logarithmic transistor is replaced with a reference current

!31*3#33. The imperfect bonding of the two transistors is reflected by the ratio !si#!3,

defined by unity but independent of temperature.

The error introduced by the non&unitary ratio !si#!3 is compensated by the adustment of

reference current; for this reason, e0uation (=) is usually written as:

The compensation of the temperature effect can be done by using a thermostat for the

transistor pair which e"ecutes the conversion, or by choosing 3 dependent on

temperature.

The transfer characteristic of a logarithmic converter is defined by the conversion slope:

nd the value of the input voltage *io, for which the output voltage is 4

s e"ample, a convertor with a conversion slope of *#decade and a voltage *io14 m*

has the transfer characteristic

4


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(d) For small values of the input signal, the 9 bottlenec$s become significant. These

bottlenec$s are determined by the offset voltage, the input polariAation currents (offset

current) and by the thermic derivatives of these parameters.

2. Laboratory Work

The circuit for the logarithmic converter is given below:

-ompensating the temperature variation effect is done by using a pair of thermostatresistors (+B=C). For a value of >4DE of 35 D (74 H-) and *314 * we will

determine the ideal (theoretical) relations for the transfer characteristic *o(*i) and

*o(!i). Ge will detail the values of the conversion slopes and of the input

voltage#current, *io#!io, for which the output voltage is Aero. $T#0 has the value

>.I m* at T1544D.

Note: The values obtained for the estimation of the theoretical characteristic willbe used also for the setup of the module in which we will graphically represent the

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e"perimental characteristics. The graphic representation will be done on an ?

sheet of paper. 9n the abscissa for *i and !i we will have a scale of >d+#decade

and on the ordinate a scale of .>cm#* for *o and .>cm#44 m* for *+.

slope=

R2+R

1

R1

kT

q ln10=1.17V/decade

Vio=VRRi

RR=100V , for Ri=10M

Vo=1.17 lgVi

100

Iio=Vio

Ri=10

5A Vo=1.17 lg

Ii

105

.5 The switch D is set on the /+!JT position.

.? The circuit is connected to a >* source.

Observations:

() ll voltages will be measured with 54 digital multimeter.

() The value of the input current is set by fi"ing the potentiometer K, by

choosing the right position for D5 and by choosing the right value for the input

resistance. etting the right values will be done using the e0uality:

.> Ge compensate the offset voltage of 9. Ge must switch D on position 9 and

pin is short&circuited to ground. K is used such that the output voltage *o is set

to Aero.

.= The switch D is set to L

.B Ge fi" !i1!i4, computed at point ., *314* and we read *o. From e0uality (>)

we determine the value of the ratio !i#!31.

We fix Ii=Iio=0.01mA, VR=10V and we read Vo:

Vo=508,425,9( ln (0.0110

31M10 )ln

ISI

ISR

)

105mV=13.6lnISI

ISR ln

ISI

ISR=7.7210

3

ISI

ISR1.002

.7 Ge fi" !i1!io computed at point . and we adust the voltage *3 from K5 until*o14.

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Vo=0 V for VR=12.58 V

.I Ge determine and plot the characteristic *o(*i) for 3i1$M and *i1 > *, *,

*,... m*, m*, 4,> m* in the following situations:

() The offset voltage of 9 is compensated

() The cursor of K is at the higher end (the value of the input voltage on the non&inverting input 9 is also measured)

(5) The cursor of K is at the lower end (the value of the input voltage on the non&

inverting input 9 is also measured)

i!" # 2 1 $%# $%2 $%1 $%$# $%$2

.*o(offset 4) &5,> &,I= &,? &,I5 &,5 &,4 &4,B &4,

.*o(offset ) &5,= &,I7 &,?? &,7B &,5B &,4B &4,B7 &4,?7

5.*o(offset ) &5,5? &,B7 &,5= &,7I &,? &4,I5 &4,=B &4,5=

i!" $%$1 $%$$# $%$$2 $%$$1

.*o(offset 4) &4,5 &4, &4,7 &4,=

.*o(offset ) &4,55 &4,I &4,> &4,4?

5.*o(offset ) &4,B &4,4> 4,? 4,4>

0 0.01 0.1 1 10

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Vo(ofset 0)Vo(ofset )

Vo(ofset )

.4 Ge repeat the voltage compensation procedure of the 9 (points .> and .=).

. Ge determine and plot the characteristics *o(!i) and !-(*+) of transistor

8 for the values of the collector current indicated in table T..

Note: For simplicity:

() Ge set the value for the input current and measure *o, *+;

() Ge pass to the ne"t value of the input current and measure *o, *+;

&urrent

&onnecte

d

'oom

temperature Warm

value pins ' i!" o!" be!m" o!" be!m"

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> m & . $ > &5,>? &4,=I &5,5= &4,B

m & . $ &5,4= &4,= &,75 &4,==

m & . $ &,? &4,>> &,5B &4,=>

>44 A & . $ 4,> &,75 &4,>= &,BB &4,>I

44 A & . $ 4, &,5 &4,?= &,> &4,>?44 A & . $ 4, &4,I &4,? &4,75 &4,?>

>4 A & 5. 44$ > &4,= &4,?B &4,5? &4,?

4 A & 5. 44$ &4, &4,5 4, &4,5?

4 A & 5. 44$ 4,?> &4,7 4,> &4,7

> A & 5. 44$ 4,> 4,B= &4,? 4,7> &4,I

A & 5. 44$ 4, , &4, , &4,>

A & 5. 44$ 4, ,> &4,5 ,> &4,B

>44 nA & ?. 4/ > &4,4I ,BB &4,7

44 nA & ?. 4/ ,5B &4,47 , &4,=

44 nA & ?. 4/ ,B> &4,4> ,>5 &4,?>4 nA & ?. 4/ 4,> 5, &4,4? ,I &4,

4 nA & ?. 4/ 4, 5,57 &4, 5, &4,5

4 nA & ?. 4/ 4, 5,=? &4,4> 5,> &4,7

> nA & ?. 4/ 4,4> 5,75 &4,4 5,= &4,

nA & ?. 4/ 4,4 5,77 4,4 5,7 &4,4=. From the value e"perimental value of the conversion plot we compute the

value of the environment temperature.

.5 Ge set D to N9T (the electroluminescent diode lights)

.? tep . is repeated, the characteristics are plotted on the same graphic.

.> Ge fi" *o14. The switch D is placed on /+!JT; this way the chip

(including 8i and 83 ) start to cool down. Ge verify that *o is Aero, such that

!i#!31 is temperature independent.

0 0 0 0 0.01 0.1 1 10

-4

-3

-2

-1

0

1

2

3

4

5

Vo[V]-room temperature

Vo!V"

#


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0 0 0 0 0.01 0.1 1 10

-4

-3

-2

-1

0

1

2

3

4

5

Vo[V]-warm

Vo!V"

IcI

S(e

vBE

vT 1)

room warm

*beOm*P !c#!s *beOm*P !c#!s

&4,=I &4,4=I &4,B &4,4=I

&4,= &4,45>= &4,== &4,4>&4,>> &4,44I5 &4,=> &4,4?B

&4,>= &4,45 &4,>I &4,4?

&4,?= &4,4B>? &4,>? &4,44=

&4,? &4,4>B &4,?> &4,4B

&4,?B &4,4BI &4,? &4,4>5

&4,5 &4,47> &4,5? &4,45

&4,7 &4,44B &4,7 &4,44B

&4,? &4,44II &4,I &4,4

&4, &4,44B== &4,> &4,44>7

&4,5 &4,4477 &4,B &4,445

$


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Alexandru Alice431G

&4,4I &4,445?= &4,7 &4,44B

&4,47 &4,4454B &4,= &4,4

&4,4> &4,44I &4,? &4,44I

&4,4? &4,44>? &4, &4,447?

&4, &4,4457? &4,5 &4,4477&4,4> &4,44I &4,7 &4,44=I

&4,4 &4,44457 &4, &4,44?=

4,4 4,444BB &4,4= &4,445

-0.# -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

-0.03

-0.03

-0.02

-0.02

-0.01

0

0

0.01

Ic/Is-room temperature

%c&%s

-0.# -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

-0.03

-0.03

-0.02

-0.02

-0.01

0

0

Ic/Is-warm

%c&%s

(. )uestions

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5.. Ghich are the factors that determine de deviation of the transfer characteristic of the

logarithmic convertor at high input currents (voltages) Q

The deviation is given by the voltage drop on the series resistance of the emittor, by

the offset currents and the offset voltage. Therefore, the characteristic wonRt be logarithmic

anymore.

5.?. For the studied logarithmic convertor, the *o(!i) characteristic has more decades thanthe characteristic *o(*i). GhyQ !s this a general situation or is it specific for this convertorQ

The *o(!i) characteristic is larger the the *o(*i) characteristic because, in the case of

voltage %attac$', the effects of the offset currents and the offset voltage is greater than in the

case of current %attac$'.

5.>. Ghich are the criteria for determining 3gQ

3gand -5 form a supplementary compensation in fre0uency of 95 amplifier. 3gis

computed from the following relations:

RCg III +=

RgBE UVU +=

5.B. "plain why the fre0uency compensations of 9 and 95 are different.The compensations are different because the feedbac$ loops are different. Therefore,

9 has a smaller feedbac$ loop and a better stability in time. 9 has an active feedbac$

loop, high fre0uency compensation, but the ris$ of oscillations appears.

5.7. !f the transistor 8!and 83are not $ept at constant temperature, the compensation of

the temperature variation can be made by ma$ing the 3, 3 divider temperature dependent.

Ghich resistor would be chosen as temperature dependentQ

!f 3SS3 is temperature dependent, the temperature effect $T#0 would be

compensated.

5.I. Ghat is the role of 9Q -an the potentiometer K be connected directly at the

convertorRs inputQ

9 has the role to maintain constant the output impedance, role which cannot befulfilled if we connect K directly to 9Rs input.

5.. The described convertor is an inverter. Now can a non&inverting behavior be obtainedQ

To obtain a non&inverting behavior, the two inputs must be switched.

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logarithmic amplifier

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