Amplifiers with Variable Gain

In amplifier systems with closed loop stabilization the output signal of the stabilizer must control the gain (or attenuation) of an amplifier stage. Since any control element adversely affects the linearity of the amplifier, the total gain is often controlled by means of the detector voltage variation. Of course, this can be done only when using scin-tillation or proportional counters, but not with semiconductor detectors.

Moreover, changing the photomultiplier voltage also changes the signal delay in the multiplier. The working principle of a closed loop stabili-zation has been described in Chapter 2.57.

'HI-2-<5;'-~ _ _ _ _ _ _ _ _ _ _ _ _ ~V\A"-O+20V 270

In principle the amplifier gain can be controlled by the variation of a resistor in a voltage divider in the signal path or in the feedback loop.

Potentiometers with a servomotor may represent a good solution, though this technique is rather slow and the necessary precision mecha-nics are expensive. Often a Raysistor⁵ is used as a variable resistor according to a proposal by MARLOW [3.060]. The Raysistor consists of

5 Raytheon Company, Newton/Mass., USA.

a photo-sensitive resistor and a light source which are encapsulated together, e.g. in normal transistor housing. The resistance of the photo-sensitive element depends on the light intensity and thus on the voltage applied to the light source (small incandescent bulb). Fig. 3.50 shows an attenuator circuit using the Raysistor type CK 1104 according to PAKKANEN and STENMAN [3.061]' Some of the details of the original circuit are omitted. The nominal Raysistor resistance is 5.6 kO; the fixed resistor 5.6 kO allows the control voltage to be adjusted correctly for nominal attenuation. Precautions must be taken to avoid any de current through the photosensitive Raysistor resistor, which otherwise becomes heated and changes its value. The environmental temperature should also be kept constant.

Instead of Raysistors, the NTC resistors with external heating can be used, e.g. the type THERNEWID F73^{6 .} The slow time response of the Raysistors or NTC resistor of some ten milliseconds is not normally prohibitive.

NTC resistors can also be heated directly by the control current if they are connected to the signal voltage divider by a decoupling capa-citor. Any small incandescent lamp can also be used as a temperature-dependent element. However, the temperature coefficient is positive in this case. ARQUE ALMARAZ [3.062] describes circuits with different lamps, e. g. with the G. E. 344 having a resistance of 300 0 at 3 mA control current. This value rises to about 12000 at 25 mAo The lamp is connected in parallel to the emitter-resistor of a two-stage feedback amplifier (Fig. 3.51) via a 221lF condenser, and the change in its

re-IN

1

.".10k ^151L

12k

39 -12V

r---...--... --<>.12V 300

+----1---. OUT

5.1k CONTROL J-+-'VV\I'--<> VOLTAGE

470

Fig. 3.51. Gain control using the incandescent lamp GE344 [3.062]

6 Siemens & Halske AG, Berlin, Germany.

sistance influences the feedback loop gain. The differential non-linearity of the circuit is better than 0.04

%,

the maximum gain variation is

±20

%

and the linear input voltage range is ± 1 V. The thermal time constant is about 80 msec.

REMIGOLSKY and TEPPER [3.170] pointed out, that the variable element of a linear variable gain amplifier must be outside the feedback loop, if a constant rise time is required.

Field effect transistors offer an alternative as control elements and such attenuators are much faster, having a time constant of about 10llsec [3.062]' PATWARDHAN [3.063] describes a charge-sensitive pre-amplifier, the sensitivity of which could be controlled using a voltage-sensitive Varicap diode capacitor as a part of the feedback capacity.

The control voltage applied to this diode changes the integration ca-pacity.

3.2. Arithmetic Operations on Analog Signals

In many experiments the physicist is interested in results which are given by arithmetic operations on different measured parameters-we need only to instance the identification of particles having different mass and charge by measuring the total energy loss E and dE/dx at the same time. Since these two magnitudes are represented as analog signals it is necessary to perform the arithmetic operations on these.

Of course, it is possible to digitize the pulse height information prior to performing the arithmetic operations and to allow the calculations be made by an on-line or off-line fast digital computer. However, both cases are much more expensive than a simple analog calculator unit. Even if the accuracy of the analog calculation is not high enough and a digital off-line data reduction is accomplished, the additional on-line analog treatment is still carried out because of the instant availability of results.

The techniques used correspond to those used in analog computer technology. In any case, the arithmetic operations on amplitudes of very short pulses cause some additional specific difficulties. Experiments which show the advantages of the analog reduction of measured data prior to the following digital treatment are reviewed by STRAUSS and BRENNER [3.064].

3.2.1. Operational Amplifiers

As already mentioned in Chapter 3.1.1., a parallel feedback amplifier is a suitable element for performing mathematical operations. The work-ing principle can easily be seen if a current is chosen as the input signal

and if the operational amplifier is regarded as a current-to-voltage con-verter (Fig. 3.52). We have

Vout

=

-fin'R--, A 1+A and in the approximation of A -+ 00

Vout= -fin·R.

The impedance at point (X) is

R - - -R

x - 1+A'

(3.41)

(3.42)

(3.43) Rx becomes 0 (i.e. Rx~R) if A-+oo. Because of its extremely low im-pedance, point (X) is often denoted as the "virtual ground". If more than one current generator is connected to (X), the partial currents add according to Kirchhotrs law. Hence the signal addition is performed by simply transforming them into currents flowing to point (X).

Fig. 3.52. Principle of the operational amplifier

The resistor R is a feedback element with an extremely simple re-lationship between the current f and the voltage V, namely V

=

R· f.

If a general component with the voltage-current characteristic described by V

=

b(I) is used instead of R, the output voltage Vout is

resulting in

A

V^out= -b(Iin) 1 +A' (3.44)

(3.45) for A -+ 00. The function b can represent any time-dependent operator (e.g. '" dd, '"

i

^dt' etc.) or any time-independent function

(exponen-t -co

tial, logarithm, square, cube etc.).

Since the input signal exists mostly as a voltage, it must be con-verted into a current. The simplest way to do this consists of using an ohmic resistance (e.g. Rl in Fig. 3.05). However, the conversion can

also be performed by any other circuit element with more general voltage-current characteristic V = a(1). Hence the general circuit con-sists of the two elements, a and b, and of the amplifier - A. Such a circuit shown in Fig. 3.53 will be discussed in more detail in the follow-ing paragraphs.

-

I

Vin

o--I==r-...

---I

V=

am j

^>----oVout Fig. 3.53. Operational amplifier with two general components a and b

Denoting the potential of the point X by Vx we have Vout ^{= -} A· Vx' The currents through a and b are the same (1). Since Vin - Vx = a(1) and Vx - Vout ⁼ b(1), the following equation is valid

b-1(-V _ Vout)=a-1(v

+

Vout)

out A ¹⁰ A ' (3.46)

where I=a-1(V) and I=b-1(V) are the inverse functions of a and b respectively. An explicit solution of (3.46) can be given only for specified functions a and b. Nevertheless, in general it can easily be seen that in the approximation A --> 00

(3.47) By an appropriate choice of the components a and b many functional relationships between Vout and Vin can be realized. Some of them are summarized in Fig. 3.54.

It is hardly necessary to comment on the particular circuits in Fig. 3.54. We need only point out that the given relationships between Vout and Vin are valid only in the approximation A>>> 1, which is gen-erally characteristic for the use of an operational amplifier. The exact relationships between Vout and Vin can be calculated from (3.46) by introducing the actual functions a-¹ and b-1• The equation (3.46), of course, also remains valid for the Laplace transforms Vin(P), Vout(P):

the time-dependent operators a and b become algebraic expressions in p and the equation (3.46) can be solved easily.

~

^R2 Amplifier, multi

plica--A Vo •• = - RI Vin tion by a constant factor

Vin vout

R

.

~

^{-A Vout= - ~n Inverter}

Vin Vout

~

¹

^{f "}

^t Integrator (signal

-A Vo •• = - RC Vin(t )dt duration ~RC(l +A))

vin Vout ^-00

R d DilTerentiator (signal

~~

^{Vo •• = - RC dt Vin(t) duration} ~RC/(l+A))

vin Vout

R

Vii R R Vo •• = - (Vii + ^Vi2 + Vi3)

Vi2 R

&-lv,

Adder

Vi3 Vout

Ria

Vii Rib R Vo •• = - (a' Vii + b· Vi2 + c· Vi3)

Vi2 Ric Linear combination

~o

Vi3 vout

~

_{R Vo ••} = - Vo ^I og RIo ^Vin LOG-converter

Vin -A Vout

~

R ^{Vo •• = -} RIo EV;"'/vo ANTILOG-converter

Vin -A Vout

Fig. 3.54. Various operational amplifier circuits

3.2.2. Arithmetic Operations on Pulse Amplitudes

The true analog infotmation is carried by the pulse amplitude, i.e. by the peak value of the pulse voltage or pulse current. Therefore, two pulses which are to be combined in an analog computer circuit must be exactly simultaneous. However, since the synchronism between two common RC-shaped pulses which frequently pass signal paths with

different propagation times cannot be guaranteed, the pulses are shaped in a pulse stretcher prior to performing the arithmetic operation. The flat top of the stretched pulses must be longer than the maximum pos-sible difference in signal delays. Normally pulse lengths of several microseconds are used.

In Fig. 3.55 a simplified functional block diagram of a stage per-forming the division operation is shown. Both input pulses, A and B, which may be delayed relatively, are first stretched in the pulse stretch-ers PS. The output signal of the division stage (-7-) is proportional to the ratio of the instant values of the input signals. Hence the output signal OUT would have the shape shown by the dotted line if the strech-ed input pulses A' and B' were fstrech-ed directly into the division stage: since B' is delayed with regard to A', the division A': B' at first gives too high a value which decreases to the correct value AlB only after both pulses exhibit the flat top. In order to avoid this overshoot, a sampling pulse T is formed with the aid of a coincidence circuit CC and a monostable multivibrator MMV. A' and B' are fed into the division stage via two linear gates LG which are activated by the sampling pulse T.

A

B

A

~'---

^{_ _ _}

B~

B'~

____

^~nL

______ __

, ^{... _\}

, ,

OUT

! "'---nUl -~

OUT

Fig. 3.55. Synchronizing of the arithmetic operations by means of a sampling pulse T

Sampling pulses limiting the arithmetic operation to the time during which all participating pulses exhibit a flat top are also used in opera-tions other than division. Some of the circuits, such as for example the

LOG converter, are overloaded by negative signals Yin. The sampling pulse prevents the overshoots or pulse parts of false polarity being fed into the following operational stages. Besides the internally generated sampling pulse, external master pulses can be used. Moreover, the selection criterion for the generation of sampling pulses can be made more general as in Fig. 3.55, so that the arithmetic operation is per-formed on preselected pulse pairs or pulse groups only.

3.2.3. Practical Circuits

Normally the multiplication and the division are accomplished by adding or subtracting the logarithms of the original signals, which are formed in LOG converters. If necessary the result, which itself is loga-rithmic, can be linearized in an ANTILOG converter. However, it is somewhat difficult to find a circuit with a strong logarithmic charac-teristic.

Circuits using a logarithmic component. Although some vacuum tubes, such as pentodes and diodes, exhibit an approximately logarithmic characteristic (cf. [3.065J), the use of small semiconductor components which need not be heated offers many experimental advantages. Taking into account only the diffusion current in a semiconductor diode, the theory of SHOCKLEY [3.066J yields the following current-versus-voltage relation

(3.48) with Iso denoting the constant reverse saturation current. With forward bias, i.e. V~kT/e,;::;25 mV at room temperature, the term -1 can be neglected and I depends on V in a purely exponential manner. Various authors ([3.067J to [3.069J) use semiconductor diodes as logarithmic elements. However, the range of validity of the relation (3.48) is limited to one or two decades, since the diffusion current is only one .of the components of I [3.070]. GIANNELI and STANCH! [3.071 J reported the base-emitter pn-junction of a transistor to have logarithmic char-acteristics over much more than two current decades. Although the emitter current of a transistor consists of different components, all ex-cept one flow to the base and only the pure diffusion current flows to the collector, assuming VCB= O. This extremely precise logarithmic relationship between the collector current

Ie

and the emitter-base volt-age VBE up to nine decades [3.064J has been used by PATERSON [3.072J, COOKE-YARBOROUGH [3.073J, LUNSFORD [3.074J, STRAUSS and BRENNER [3.064J and BYRD [3.171J in the design of LOG and ANTILOG converters.

In the LOG converter the "logarithmic" transistor is used in the feedback loop of an operational amplifier, the base being grounded and the collector being connected to the input "virtual ground" in order to fulfil the condition VCB:::::::O [3.072]' A simplified circuit diagram of a LOG converter according to STRAUSS and BRENNER [3.064] is shown in Fig. 3.56. By means of a variable resistor, the quiescent current frer (approximately 25/lA) of the silicon planar transistor 2 N 2219 is ad-justed for operation in the logarithmic region of the flV characteristic.

S1.9 ...

^10V

Fig. 3.56. Operating principle of the LOG-converter according to ^STRAUSS and ^BRENNER [3.064]

The output voltage amplitude VOU! then becomes _ kT

(f ⁺

^free)

The LOG converter in Fig. 3.56 operates satisfactorily with positive input signal Yin only and must be protected against negative voltages

Vln.

PATERSON [3.072J reports a circuit accepting signals of both polarities (Fig. 3.57). The output signal is proportional to log

lVin/VO I

and exhibits a polarity opposite to that of the input voltage.

Fig. 3.57. LOG-converter for both positive and negative input pulses according to PATER-SON [3.072]

The constant Vo=kT/e in (3.49) or (3.50) is a function of T, thus making the logarithmic conversion factor temperature-dependent. Hence the "logarithmic" transistor must be kept at constant temperature, e. g.

with the aid of a small Peltier element [3.074]'

+

Fig. 3.58. The ANTILOG-converter

Fig. 3.58 shows the principle of an ANTILOG converter. The out-put pulse amplitude ^YOU! is given by

( eVin )

- YOU!

=

R . free ^E kT - 1 . (3.51 )

The operating point of the transistor is maintained by Iref • ^YOU! differs from a purely exponential expression by the term R· Iref . Hence a correction current Ie = - I ref must be drawn from the point (X) during the signal pulse duration. Instead of a special additional current gener-ator (, the reference current Iref might also be interrupted during the pulse duration

Since I,.r must flow through the logarithmic transistor, the input voltage Vin = - VBE

must be somewhat negative. The criterion for flowing in the transistor and not into the operational amplifier is of course Vou, = O. Hence the input bias voltage is advantageously controlled by a closed loop control circuit watching Vou,=O (ef. STRAUSS and BRENNER [3.064]). These authors use the "logarithmic" transistor in common emitter configuration with a load resistance in the collector circuit. Obviously the condition VCB=O given by PATERSON [3.072] for logarithmic operation of a transistor is not a strong one.

In both equations (3.49) and (3.51) the scale factor Vo=kT/e has the same temperature dependence. Hence, if by means of an appropriate mounting of all "logarithmic" transistors in the same cooling block local temperature differences are avoided, the scale factors of both LOG and ANTILOG converters become the same. Thus a series circuit consisting of a LOG and an ANTILOG converter is not temperature-dependent and very sophisticated thermostat equipment can be avoided.

10 9

8

7

6 Vou

,=

15 Volll.log IVin II Volt I

5

3 BIAS

a 10 20 ³⁰ 40 50 60 70 80 90 100 110

Fig. 3.59. Approximation of logarithmic characteristics by means of diode function gen-erator according to WAHLIN [3.075]

Approximation of logarithmic characteristics by means of nonlinear voltage dividers. Approximation of a logarithmic or any other function by means of short linear segments can also be used in operational circuits. For instance, Fig. 3.59 shows the approximate realization of the function YOU! = (5 Volt). 10g(Vin/1 Volt) according to WAHLIN [3.075]' When the input voltage Yin is raised, more and more diodes start con-ducting, the input resistance Rl is increasingly loaded and the ^yOU! to

Yin characteristic becomes respectively more flat. With appropriate re-sistor values it becomes approximately logarithmic. A similar circuit, however, using equally valued resistors in each attenuator stage, has been reported by VINCENT and KAINE [3.076]' GOLDSWORTHY [3.077]

describes a pseudo-logarithmic amplifier in which the gradual nonlinear attenuation is performed by means of limiters placed between the par-ticular amplifying stages: the last limiter is overloaded first, then the last but one, etc. A diode power function generator has been described by TUROS et al. [3.172]'

MULTIPLICATION

~--T

DIVISION

Fig. 3.60. Multiplication and division using the amplitude-to-time conversion

Multiplication and division using non-logarithmic techniques. Another possibility of performing the multiplication or division of two pulse amplitudes consists in using the subterfuge of pulse-height-to-time con-version (Fig. 3.60). For the purpose of multiplication, one of the two

pulses (amplitude ~) in an amplitude-to-time converter ATC is converted produced. A difference discriminator compares the ramp voltage with the stretched pulse VA and determines the instant t when both these voltages become equal. Since t is proportional to

VA/~ ~

^{VA/VB' it}

is only necessary to convert the discriminator pulse in a time-to-ampli-tude converter TAC into a pulse of amplitude H ~ t ~ VA/VB'

Multiplication circuits based on this principle are described among others by AITKEN [3.078J and GRIFFITHS et al. [3.079J; division circuits by BAYER [3.080J, TSUKUDA [3.081J and KUHLMANN et al. [3.082]'

KONRAD [3.083J uses the pulse-height-to-time conversion with expo-nential wave forms (due to appropriate RC networks) for forming the logarithms of pulse signals.

Field effect transistors are majority carrier semiconductor devices in which the channel resistances are a reciprocal function of the gate voltage. Hence the channel current is proportional to the product of the channel and gate voltages. This effect can be used for signal multiplication. MILLER and RADEKA [3.084] describe a FET signal multiplier using a bridge configuration for improvement of the circuit linearity. GRUN-BERG et al. [3.085] obtained a linear multiplication by FET transistors by means of a suitable feedback. Another FET multiplier has been described by FISHER and SCOTT [3.173], GERE and MILLER [3.086] used a double emitter transistor 3 N 64 as a multi-plying element (such transistors normally are used in chopper circuits).

According to GRUETER [3.087] a parabolic characteristic can be achieved by feeding the input signal to the two grids g, and g2 of a heptode if these are suitably biased.

An operational amplifier can linearize non-linear data if its characteristic is inverse to the non-linearity, which, of course, must be known [3.089]. This can be used in linearizing the energy-to-pulse-height relationship of detectors with non-linear response.

However, non-linear amplifiers affect the pulse shape as well as the pulse height. These effects are discussed by HORN and KHASANOV [3.090].

3.3. Window Amplifiers

When using high response semiconductor detectors (or in other similar measurements), it may be necessary to spread a part of the pulse height spectrum over the entire range of a multi-channel analyzer. In this case window amplifiers are used. The gain of a window amplifier is zero for signal amplitudes below a well-defined level Vw , and increases (mostly to about 1 to 10) for Yin> Vw ' The amplification characteristic for Vin> V w must be as linear as possible.

The linear range of a window amplifier is limited towards high input amplitudes, too, due to the inherent amplifier nonlinearities 7. However this upper limit is not as pronounced as V_{w '} Fig. 3.61 shows a diode circuit exhibiting the desired characteristic. There is a negative bias - V w applied to the diode. Only that part of pulse which is higher than Vw is passed through the circuit. Due to the exponential form of the diode characteristic (3.48) there is a steady transition from cutoff to conduction. Hence the output voltage YOU! depends upon Yin in a manner shown on the right in Fig. 3.61. The "break point" of the characteristic

The linear range of a window amplifier is limited towards high input amplitudes, too, due to the inherent amplifier nonlinearities 7. However this upper limit is not as pronounced as V_{w '} Fig. 3.61 shows a diode circuit exhibiting the desired characteristic. There is a negative bias - V w applied to the diode. Only that part of pulse which is higher than Vw is passed through the circuit. Due to the exponential form of the diode characteristic (3.48) there is a steady transition from cutoff to conduction. Hence the output voltage YOU! depends upon Yin in a manner shown on the right in Fig. 3.61. The "break point" of the characteristic

In document E. Kowalski. Nuclear Electronics. With 337 Figures. Springer-Verlag New York Heidelberg Berlin 1970 (Page 124-151)