on the sensitivity of oscilloscope tubes - … bound... · one of the most important properties of...

1. Introduction

One of the most important properties of an oscilloscope tube, especially ifsignals with a large bandwidth have to be displayed, is deflection sensitivity,i.e. the deviation of the spot on the screen per unit of deflecting voltage, ex-pressed in millimetres per volt. The following properties are also of considerableimportance: spot diameter, brightness, usefu] scan and length of the tube. Infact, deflection sensitivity and useful scan should be regarded in relation to thespot diameter. Neither the deflecting voltage required to produce a deviation ofone spot diameter nor the number of resolvable spot positions are changed by aproportional reduction of spot diameter, sensitivity and useful scan. However,for practical reasons (convenient viewing distance etc.), a marked reduction ofpicture size is generally undesirable. It will be assumed that spot size, brightnessand useful scan are :fixed by the requirements of the intended application, where-as the operating voltages and the length of the tube can be arbitrarily chosenwithin certain limits.

The problem that will be treated 7an be stated as follows: how can sensitivitybe increased without worsening one or more of the above-mentioned properties,

R643 Philips Res. Repts 22, 515-540, 1967

ON THE SENSITIVITY OF OSCILLOSCOPE TUBES

by E. HJMMELBAUER and J. C. FRANCKEN *)

Abstract

A formula for the spot siz~ of oscilloscope tubes is derived, based upon "-the assumption that the emitted electrons have a Maxwellian velocitydistribution. If the spot size were determined only 'by the thermal-velocity distribution, a higher deflection sensitivity (with equal spotdiameter, brightness and useful scan) could be obtained by reduction ofthe accelerating voltage. This is no longer true, however, if the space-charge repulsion in the field-free region between deflection plates andscreen has to be taken into account. In this case a substantial improve-ment can be achieved only if the tube length is increased. The effect offields acting on the beam after it has been deflected (scan magnification)is investigated for the case of negligible space charge, and it is shownthat an increase of spot size can be avoided by an appropriate change ofbeam width. An accelerating field will yield a higher sensitivity withequal screen voltage, screen current, spot diameter, useful scan and de-flection-plate length. This also holds for a non-accelerating scan-magnification system, provided that it is applied in a tube with flareddeflection plates. In the case of paral1el plates, a gain in sensitivity can-not be obtained unless the length ofthe plates is increased.

2. Calculation of spot size

Let us calculate the spot size by a method' similar to that employed byWeber 1). We choose a Cartesian coordinate system x, y, z so that the cathode

*) University of Groningen, the Netherlands.

516· E. HIM'MELBAUER and J. C. FRANCKEN_ ,.0'

lies in the xy plane, and then we consider the distribution- of the emitted elec-trons 'as a function' of the coordinates Xo, Yo and the velocity components

. ,

di =f(xo, Yo, io, Yo, zo)dxo dyo dio dyo dzo •. (2.1)

Neglecting the effect ofthe spread of the axial velocity component Zo, we assumea single value of Zo, which can be expressed in terms of an equivalent potentialez by means of the relationship

mZ02e =--z 2e' (2.2)

j(xo,yo) = jeo exp ( _ X02~~2Y02).

where R; is the distance from the centre at which the current density has

(2.9)

so that in (2.1) the velocity components io, Yo can be replaced by the anglesxo',Yo':

di = F(xo, Yo, xo', Yo')dxo dyo dxo' dyo'. (2.3)

If we further assume that the angular distribution is independent of the coordi-nates, (2.3) becomes '

di = j(xo, Yo) q(xo', Yo')dxo dyo dxo' dyo', (2.4)

wherej(xo, Yo) denotes the current density and q(xo', Yo') is normalized so that00 00

J J q(xo',Yo')dxo'dyo' = 1.-C() -ex>

(2.5)

In the case of a Maxwellian velocity distribution given by

tp(iij,yo) = .z: exp [- .z: (i02 + Y02)]2nkT 2kT

(2.6)

we find, using (2.2),

. 1 (XO'2 + YO'2)q(xo',yo') = npe2 exp - Pe2 ' (2.7)

in which we have

_ (kT)1/2Pc - .ee,

(2.8)

The current-density distributions of axially symmetrical electron guns of thetype usually employed in cathode-ray tubes have been investigated by Hasker 2).We sháll approximate the actual distribution by a Gaussian distribution,

ON THE SENSITNITY OF OSCILLOSCOPE TUBES 517

dropped to, lie of its value at the centre, jeO' As will be discussed in sec.3,the screen current in oscilloscope tubes mainly originates from the central partof the emitting area and for this part (2.9) is in good agreement with the actualdistribution. The total cathode current I can be found by integration of (2.9),which yields .

I=jeo R/n .. (2.10)

With the help of (2.8) and (2.9), (2.4) can be written as

J. (X 2 + y 2 X'2 + y '2)• ect 0 0 0 0 , ,dl = - exp - - dxodYodxo dyo .

(3/ R/ (3/

An electron beam with such a distribution function will be called a Gaussianbeam, after Dorrestein 3). Itwill be assumed that the electron trajectories satis-fy the paraxial-ray equations (the validity of this assumption has been discussedby Prancken and Dorrestein 4) and by Hasker and Groendijk 5». We shall con-sider systems which are symmetrical with respect to the xz and yz planes sothat the potential in the neighbourhood of the z axis can be expressed as

(2.11)

(2.12)with

(2.13)

The equations for the paraxial electron trajectories x(z) and y(z) are *)

(Vo + 8z)X" + ·~.-vo'x' - a2~x = 0 (2.14a)and

(2.14b)

The electron distribution function for an arbitrary beam cross-section can befound from (2.11) with the help ofthe relationships

x(z) = xogxCz) + xo'hxCz),x'(z) = xogx'(z) + xo'hx'(z) (2.I5a)

. andy(z) = yogy(z) + Yo'h.vCz),y'(z) = yogy'(z) + Yo'hy'(z),

where gx(z), hx(z) and gy(z), h.vCz)are pairs of linearly independent solutions ofthe paraxial-ray equations (2.14a) and (2.14b) with the initial conditions.

(2.I5b)

gx(O) = gy(O) , 1,hxCO)= hy(O) ..:_0,

gx'(O) = gy'(O) = 0,hx'(O) = hy'(O) = 1.

(2.16)

*) Except in the neighbourhood of the cathode, ez is small with respect to Vo and can be ne-'. glected. '. '

518 E. IDMMELBAUER and J. C. FRANCKEN

From the theory of linear second-order differential equations it is known thatthese solutions satisfy the relationship ,

(

8 )1/2g"h,/-g,/hx=g)lhy'-gy'h)l= % •

Vo + 8%

(2.17)

From (2.11) and (2.15) we obtain

j (X2 v2 a.2 a 2)di = _0_ exp - x )1_ dxdydx'dy'fJfJ S2 S2 fJ2 fJ2 ':re" )I ,,)1,,)1

(2.18),

where jo is the current density in the centre of the beam and the quantities s,fJ and a can be expressed in terms of the functions g and h as follows (for thederivation of (2.19), see appendix A):

S 2 = R 2g 2 + fJ 2h 2x ex ex, (2.19)

(2.20)

(2.21)

and similar expressions for S)I' fJ)I and oc)/'Equation (2.20) yields

s"fJ:X(Vo + 8z)1/2 = S)lfJ)I(Vo + 8z)1I2 = RefJeV8% = Q. (2.22)

The quantity Q can be considered as a figure of merit of the beam. Equation(2.18) reveals thatin every cross-section of a Gaussian beam both the current-density distribution and the angular distribution are Gaussian. In each surfaceelement the direction of maximum intensity (a" = a)l = 0) points towards acommon point on the axis if the beam is axially symmetrical. These directionsinterseet in two perpendicular lines of focus if the beam is not rotationally sym-metrical (astigmatic, see fig. I).

Equation (2.22) is a generalization of the Helmholtz-Lagrange law, by whicha similar relationship between radü and beam angles of object and image isestablished, whereas (2.22) is valid for any arbitrary beam cross-section.Using (2.8) and (2.10), we find from (2.22):

_ (kT)1/2 _ (kT)1/2 ( 1)1/2Q-Re - - - -.e ne i.«

(2.23)

For oxide cathodes, k'I'!« is approximately equal to 0·1 volt so that we obtain

Q = 0.18 (/-)1/2.leo

It can be seen from (2.22) that, to produce a circular spot, i.e. a spot with ro-tationally symmetrical current-density distribution (s, = S)I= rs), fJ" must be

(2.24)

ON THE SENSITIVITY OF OSCILLOSCOPE TUBES 519

x

y

-x

y

Fig. 1. Directions of maximum intensity. (a) Rotationally symmetrical beam, (b) astigmaticbeam.

equal to {J.:". This condition can be met by a suitable choice ofthe focusing fields(the magnification factors in the two directions have to be equal). The radius rsof the spot, defined as the distance from the centre at which the current densityhas dropped to lIe of its maximum value, can be found by applying (2.22) tothe image plane (i.e, the screen). Since Sz « Vs,we find

520 E. HIMMELBAUER and i. C. FRANCKEN

GI )1/2 1

's=0'18 -- -.'coVs Ps

(2.25)

In oscilloscope tubes the beam width is limited in both directions by the en-trance spacings of the deflection plates so that the screen current Is is lower thanthe cathode current I, and we define the screen-current efficiency as the ratio

Is'YJ=I' (2.26)

With the position of the diaphragm plane such as encountered in practice, wecan assume that in each surface element of this plane the direction of maximumintensity points to the centre of the focused spot as shown in fig. 2 (the validityof this assumption in the case of a Gaussian beam is discussed in appendix B).

~ Plane of diaphragmFig. 2. Focusing condition at which the current-density distribution in the plane of focuscorresponds to the angular distribution in the diaphragm plane.

Under these conditions, the current-density distribution in the spot correspondsto the angular distribution in the diaphragm plane. Therefore, the Gaussiandistribution and the size of the spot will be preserved;' regardless of the dimen-sions of the diaphragm, and in each surface element of the spot the current den-sity is reduced by the same factor, which is equal to the screen-current efficiencyas defined by (2.26). In order to calculate the value of'YJwe consider the centreöf a circular spot, formed by a beam that is limited in both directions accordingto fig. 3. Only rays for which

and (2.27)

will reach the centre of the spot, all other rays being intercepted. The values ofu'" and u, are proportional to the dimensions of the limiting apertures but theyalso depend on the electron trajectories between the apertures and the screen.

ON THE SENSITIVITY OF OSCILLOSCOp,E TUBES 521

,

y

x

\

Fig. 3. The maximurn beam angles at the screen.

,The value of 'YJ can be found by putting x ~ y = 0 and integrating over theangles. We find

1 u" u" ("2 + /2) ' .. ', Xs Ys ;'YJ= --2 f f exp -. 2 dxs'dys'

nps Ps . '=v« -u"

(2.28)

or

with'YJ= erf (;:) erf(;:} (2.29)

2 "erf x = - f exp (-t2)dt.Jin o

(2.30)

3, The effect of screen-current efficiency on spot size. .Since, for a fixed value of the screen voltage Vs> the brightness is determined

by Is rather than by I, we eliminate I from (2.25) by using (2.26) and obtain

GIs )1/2 1

r=0·18 - --s ur.): PsV'YJ '"

(3.1)

Also, it is convenient to express rs as a function of u" and u" and we can write(3.1) as

(1 )1/2 (u U )1/2

rs = 0.18 s " " •

jcoVsu"u" PsV'YJ(3.2)

.Since, according to (2.29), n depends on u,,/Ps and u,,/Ps> the last factor in (3.2)is, for.a fixed ratio of u" and u., a functionof 'YJdefined by .

522 E. HIMMELBAUER and J.·C. FRANCKEN

where (3.4)

(3.3)

Therefore, (3.1) becomes

G1 )1/2

rs = 0·16 . s fer!).coVsU",uy

(3.5)

The function j'(n) increases monotonically with 'I] so that, in order to obtain asmall spot size, we should choose 'I] as low as possible. For practical purposes,the improvement that can be achieved in this way is limited because of the in-creasing cathode current and by considerations of power supplies, heat dissi-pation at the diaphragms, etc. Apart from this, it can be seen from fig. 4,where f('I]) has been plotted for the case of u'"= u.; that there is little to begained below a certain value oî n. From here onf('I]) will be taken as unity.

0'80.

0·60

0·40

0·20

0-20 0·40 0'60 0·80-1'l,

Fig. 4. The functionf(1) for u'" = uy•

Note: The derivation of (3.4) has been based on the assumption of a Gaussianbeam with a distribution function according to (2.18). Actually, becauseof the finite diameter of the gun, electrons emitted with very high trans-verse velocities will be intercepted before reaching the diaphragm plane.Also, in view of the actual current-density distribution at the cathode,this distribution will not be Gaussian at the diaphragm plane. For thesame reason, the angular distribution near the edge of the beam will differfrom that at the centre unless the diaphragm plane coincides with acathode image, and the screen-current efficiencywill not be exactly given .by (2.29). However, since the exact shape ofthe current-density distribu-tion near the edge of the spot is not essential and since only the central


part of the beam passes through the diaphragm, these effects can beneglected. It should also be noted that, in practice, the spot size will bè

. larger than is given by (3.4), mainly because of space-charge effects andlens aberrations.

4. Calculation of deflection sensitivity and nseful scan

For small angles of deflection the trajectory of the central ray of a beam de-flected in the x direction is given by

. 1 VdX" = - 2V

aE,,(z) = - 2V

aex(~), (4.1)

where E" is the transverse field strength and e" the field strength per unit of thedeflection voltage Vd (e, depends only on the geometrical configuration of thedeflection plates).

Assuming the deflecting field to be effective in the region Zo ~ z ~ Zb wecan find the slope and deviation at the exit of the field by integration of (4.1),which yields

, (4.2a)

and

(4.2b)

The intersection of the back-traced ray with the z axis is called "deflection cen-tre" and its distance from the diaphragm will be denoted by d". If there is áfield-free region between deflection plates and screen, the deviation Xs on thescreen is equal to

(4.3)

where L" is the distance from diaphragm to screen (fig. 5).

Fig. 5. Electrostatic deflection.

524 E. HIMMELBAUER and J. C. FRANCKEN

The interception of the beam takes place gradually so that the definition ofthe useful scan S", has to be based upon a specified decrease of brightness, e.g.to one half of its maximum value. With this definition, the maximum deflectionangle ~'"max is characterized by the fact that the central ray of the beam is justintercepted. From (4.3) we obtain

(4.4)

The deflection voltage corresponding to the maximum deflection angle will bedenoted by Vdmax and the deflection sensitivity N",is found to be

S",N",=---.

2Vdmox(4.5)

For parallel plates with the length I and the spacing 2a we have, neglecting thefringe fields,

1e =-- (4.6)x 2a'

. V IX/(Z1) = _d_ (4.7a)

4Vaaand

Vd[2 (4.7b)X(Z1) =-.8Vaa

Since, in the case of parallel plates, the central ray is just intercepted whenX(Z1) = a, we find

2a~max =- (4.8a)

Iand

(V y/2 2admax (4.8b)-- =-,2Va I

so thata ~mnx (4.8c)Vdmax=4Va--·. I

If, for a given plate length, a larger deflection angle is required, plates with in-creasing spacing can be used. If the entrance and exit spacings are 2a and 2b,respectively, (4.8) can be modified as follows:

(4.9a)

(Vdmnx)1/2 _ a (b)-- --F2-2Va I a

(4.9b)

·ON THE.SENSrr;r:yrry OF. OSCILLOSCOPE TUBES :,.•• 0 ••••• -52~

2F22.F3=--

FtThe. functions Flo 1:2 and, therefore, F3 depend. on the shape of the deflectionplates. In order to obtain the highest sensitivity for specified values of Va, a,c'Jmax and I, the shape should be so chosen that the value of F3 becomes a mini-mum. This optimum shape can be approximately calculated by a method usedby Maloff and Epstein 6), a short outline of which is given in appendix C.Graphs of the functions Flo F2 and F3' with the above-mentioned definition ofuseful scan taken into account, are shown in fig. 6.

and

where

. .' . bV~ max = Va c'Jmax ~'F3 (_.),

.. I a•• 0" •

7

6

5

2 3 .4. : b 5-a

Fig. 6. Characteristics of optimally shaped deflection plates.

(4.9c)

(4.10)

5. Choice of accelerator voltage and 4iaphragm-to-screen distance

One of the basic requirements of an oscilloscope. tube is the brightness B.With the simplified assumption that B is proportional to Ps, the power dissipat-

I ed at the screen, viz.

E. HlMMELBAUER and J. C. FRANCKEN

(5.1)

we can regard this power to be specified instead of the brightness. Eliminatingthe screen current from (3.4) and (5.1), we obtain

0·16 cr )1/2.rs·= -.-.; (UXUy)-1/2.Vs cO

(5.2)

For a tube with a field-free region between the deflection plates and the screen,the angles Ux and Uy are given by .

axu =-=»:x

ayu =-

y L'y

(5.3)

so that .(5.4)

since in this case the screen voltage is equal to the accelerator voltage Va·Equation (5.4) shows that the spot size does not change if a reduction of Va isaccompanied by an appropriate increase in ax and ay and, with the flared platesas usually employed in oscilloscope tubes, such a change yields a higher sensitiv-

ity for equal useful scan.This will be illustrated by a numerical example.Suppose that

1- ~max = F1 = 6·4.a ..

Then" from the graphs in fig. 6, we find the corresponding value

F3 = 7.

If Va is reduced to half its original value, the spot diameter can, according to(5.4), be preserved if both ax and ay are multiplied by two. Therefore, for equal

values of ~max and I,we now have

which corresponds toF3 = 4·87.

The maximum deflection voltage has, therefore, been multiplied by

4·87/7 ~ 0·7,

so that, indeed, the sensitivity has been increased. However, this is only trueif the effect of space charge is negligible. .

For the case of a beam travelling in a field-free region, the combined effectof thermal velocities and space-charge repulsion has been studied by Weber 1).Within the range of parameters as usually encountered in cathode-ray tubes,


his numerical results have been approximated by Barten 7) as follows:

- . 103 IsL2rs = rs + 4·5.

Va3/2rL(5.5)

where is and rs are the spot radii with and without the effect of 'space chargebeing taken into account, and re is the beam radius in a plane with the distanceL from the screen. We may put

(5.6)and

r,Is=-

Va

Inserting (5.4) into (5.5), and replacing rL by a, we find with the help of (5.6)and (5.7):

(5.7)

whence

0·16 GP )1/2 L P.L2r = -_ _: _+ 4,5.103 _s__s V . V 5/2 'a cO a . a a

(5.8)

L [ GP )1/2 . P L ]a = --_- 0·16 __!_ + 4,5.103 _s _.. '.V . V 3/2 .ars c~ . a

(5.9)

From (5.9) we can find, for specified values of Ps> rs andjco, the entrance spac-ing of the deflection plates as a function of Va and L so that, for :fixed values ofuseful scan and deflection-plate length, the sensitivity can be calculated with thehelp of the graphs in fig. 6. It has been found that, for a wide range of param-eters as encountered in practical tube design, the sensitivity is almost inde-pendent of Va, and that it can be improved only by an increase in the distanceL. As to the choice of Va, a higher value of Va implies a smaller value of a (anarrower beam), which is preferable in view of lens and deflection errors. Apractical example is given in fig. 7.

6. The effect of scan magnification on spot size

In the foregoing section it was mentioned that because of space-charge effectsno substantial improvement of sensitivity could be obtained unless the tubelength were increased. Let us now investigate the effect of fields acting on theelectron beam after it has been deflected, for the case where the effect of spacecharge is negligible (the term "scan magnification" will be used even if, in fact,a reduction of the scan is caused).

According to (3.4) and (3.5), for small values of the screen-current efficiency,the spot radius is given by

(6.1)

52S E. HIMMELBAUER and J.,C. FRANC KEN

"

·,·5,..--___..::'---...------------~---,N

(mm/V)

t 1,0 •

... ..

0-5

0L---roLO---~2LOO----3~00---~4~0~0---5~00-L(mmJ

Fig. 7. Deflection sensitivity as a function of diaphragm-to-screen distance; S = 60 mm, rs =0·25 mm, Ps = 0·1 W, l_= 50 mm, jco = 0·01 A/mm2

•

We shall compare two tube designs, respectively with and without scan magni-fication, and all quantities concerning the latter will be denoted by an asterisk.We assume that screen current Is> maximum cathode loading L« and screenvoltage Vs are the same for the two tubes and we postulate that the spot size~hould remain unchanged by the introduetion of scan magnification. obviously,this is the case if .

u" = u,,*and. (6.2)

The values and the derivatives of the electron trajectories, at the screen and theplanes of the diaphragms (fig. 8), are related by the following equations:

x

aA,--1t Ii Lx

Yayt-It I

Ly

Fig. 8. The position of the diaphragms.

ON THE SENSITIVITY OF OSCILLOSCOP.E TUBES

and(~~;) =(Ax .1!~.)(:Ca,) ,x, c, o, x;

(Ys,) = (Ay BY)'(Ya,).Ys c, Dy Ya

(6.3a)

~ (6.3b)

, . .The matrices are determined by the field in the region between the deflectionplates and the screen and their determinants are given by

I~:.~:I= I~: Ä I= :p' (6.4)

withVs

p=-.Va

The focusing condition in the xz plane is

A"xa + Bxxa' = 0,

(6.5)

(6.6)

whence, for an undeflected ray,

Bxxa' = -Xa-· Ax

The value of Ux is equal to Ixs' I corresponding to the initial conditionx, = ax:From (6.3a), (6.4) and (6.7) we find

(6.7)

axUx=--'

BxJ/pFor the tube without scan magnification we have, according to (5.3),

(6.8)

a ** xUx =-., Lx

(6.9)

Let us now consider the deflection of the central ray of the beam. The initialconditions, which can be found by tracing the ray back from the deflectioncentre to the plane of the diaphragm, are given by

(6.10)

so that the deviation Xs on the screen can be found from (6.3a) and (6.10):

Xs = (Bx - Axdx)6x .

Without scan magnification we obtain

Xs* ='(Lx:'_dx)6x

and the factor of scan magnification, M~;,x, is given by

(6.11)

(6.12)

530 E. HIM'MELBAUER and J. C: FRANCKEN

(6.13)

In an analogous manner we find

(6.14)

and(6.15)

The condition (6.2) for constant spot size is met if the following equations hold:

(6.16)and

If, as usually is the case,

(6.17)

(6.13) and (6.15) reduce toB'x

Mscx = - andLx

(6.18)

so that we obtain from (6.16):

(6.19)

The conclusion of the foregoing is: if scan magnification is applied; the spotsize can be kept constant provided that the beam-limiting apertures are changedaccording to (6.16). It should be noted that no assumptions concerning thephysical realization of the scan-magnifier field have been made, so that thevalidity. of the conclusion is not restricted to any particular method of scanmagnification.,.7. Discussion.' For a unipotential scan-magnifier system (p = 1), (6.19) becomes

a :- a*AfSC' (7.1)

Qn the other hand, in order to keep the useful scan unchanged, the maximumdeflection angle can be reduced, resulting approximately in '

1!5max = -- !5max *.

Msc(7.2)

From (7.1) and (7.2) it can be seenthat"

ON THE SENSITIVITY OF pSCILLOSCOPE TUBES 531

(7.3)and

(7.4)

From (7.3) and (4.9c)we see that, with unchangedlength ofthe deflection plates,the maximum deflection voltage will decrease proportionally with F3(b/a). Thevalue of bla, for the tube with scan magnification, can be found from (7.4) and(4.9a). For illustration, let us refer to the numerical example that has beentreated in sec. 5, with

F3* - Vdmax* I = 7

V *.1: * 'a a Umax .

corresponding to

IF * - - (J * - 6·4I - a* max - •

By applying scan magnification with p = 1, Msc = 1,6, this yields, according~a~ -

I 6·4PI = - (Jmax = -- = 2'5,

a 1.62

which corresponds to

so that, with (7.3), we findV *' F * .dmax 3--= -RIj 1·6.Vdmax F3

Therefore, in this case, the gain in sensitivity is about equal to the scan-magni-fication factor. However, if the limiting case of parallel deflection plates hasbeen reached, a further reduction of the maximum deflection angle with con-stant plate length is impossible and a further increase of Msc will cause a gain inuseful scan rather than in sensitivity. This, although generally an advantage,may be quite useless or even objectionable, e.g. in those cases where the maxi-mum deflection signal is limited *). For accelerating scan-magnification systems(p> 1), we have

(7.5)

*) The maximum deflection angle can, of course, be reduced by increasing the plate length,but then the tube with scan magnification has to be compared with a tube without scanmagnification with an equally increased plate length.


anda.5max = Vpa*.5max*. (7.6)

Atypical example of such a system is the post-acceleration field with, generally,Msc< 1. Let us consider the former ·examplewith, as practical values, p = 4and Msc;" 0·75.Then,

a = 1·5a*and

Wefind6·4

F1 = = 5·7,1·5xO·75

which corresponds toF3 = 6·~,

so that, using (6.5) and (7.6);we obtain

Vd max* 7--=-Vp~2·1.Vd rnax 6·6

These examples have shown that an improvement of sensitivity can be obtain-ed by means of scan magnification, and we shall now discuss the limitations ofthis method. As already mentioned, for p = 1, Mç;:» 1, a gain in sensitivity isonly possible if the original plates are flared. The gain is also limited by the factthat the increasing beam width may give rise to aberrations in both the focusingand the scan-magnifier systems. Another limitation is imposed by the require-ment of equal magnification factors in the two directions if systems whichare not rotationally symmetrical are used and a circular spot is required.

Finally-it should be noted that, when the conditions (6.17) do not apply, theaperture dimensions should be calculated from (6.16) rather than from (6.19).As to the effect of space charge, this will largely depend on the specific type

of scan-magnifier system, and a general conclusion cannot be drawn. Whethera similar improvement of sensitivity can be obtained if the spot size is, to anappreciable extent, determined by space charge, will have to be investigated ineach case.

8. Conclusion

Ithas been shown that the deflection sensitivity of an oscilloscope tube with afield-free règion between the deflecting plates and the screen is mainly determin- .ed by the length of this region, if spot size, useful scan and brightness arespecified.. .- . ~~_.-.

For the case ofnegligible space charge, an increase in sensitivity, without de-terioration of the above-mentioned properties, can be obtained by means of


scan magnification. In .order to avoid an enlargement of spot size, a wider beam(and, as a consequence, a larger spacing of the deflecting plates) is required ..Because of the flared shape of the deflecting plates usually employed in prac-tice, a gain in sensitivity will result since the effect of the increased entrancespacing is relatively small.

Appendix A

Solving (2.15) for xo,xo' and Yo, Yo', respectively, we obtain

. (V, .+ 8)1/2Xo = 0 8 . (h,/x - hxx'),

(Vo + 8)1/2 .

xo' = 8 (-gx'x + gxX')

(A. la)

and

(V, + 8)1/2 'Yo' = 0 8" (-gy'y + g)'y').

(A.Ib)

Inserting (A. I) into the exponent of (2.11) yields

x 2 [X'2 V, + 8 [ . ._0_ + _0 _ = 0 (R 2g '2 =t- {J 2h '2)2X2 _ 2(,R 2g g , + {J 2h.h ')xx' +R2 {J2 R2{J2 ex ex \exx exxe e 8 e e

. + (R/g,,? + {Je2hx2)x'2 J=V. + 8 [R 2{J2( h' ,h )2o e e gx x - gx x X2 + (R 2 2 + (J 2h 2) XR 2{J2 R 2 2 {J 2h 2 c gx c x

8 e e e gx + c x

and, in the same manner,

Y02 Yo'2 , y2-+-.-= +R/ {J/ R/g/ + {J/hv2.

+Vo + 8( ,......:.Rc2gviv' + {Je2hvhv'. )2 R/g/ + {J/~/.8 y. R 2g 2 + {J 2h 2 Y R 2{J2 (A.2b)eve vee

(A.2a)

534 E. HlMMELBAUER,and J. C. FRANCKEN

Introducing the quantities defined in (2.19), (2.20)' and (2.21), (A..2) can bewritten as follows:

and (A.3)

The differential in (2.Ü) transforms as follows:

()xo ()xo'

(A.4a)

()x ()x v: + 81 h' 'I (V, + 8)1/2dxdx' = _0__ x -gx dxdx' =' 0 dxdx'. 8 -hx gx 8

öx' öx'and

(v: + 8)1/2

dYodyo' = 0 8 dydy', (A.4b)

so that (2.11) becomes

j (V: + 8) ( x2. y2 ct 2 ct 2)di = cO 0 exp _:_ __ y_ dxdydx'dy'. (A.S)n{J/8 sx2 s/ {Jx2 {J/

The current density jo at the centre of the beam can be found from (1.5) byintegrating over x' and y' :

Vo + 8 {Jx{Jyjo = jco --8-- {J/ ' (A.6)

so that we finally obtain

j (X2 y2 ()(2 ct 2 )di = _o_exp X y_ dxdydx'dy'.n{Jx{Jy sx2. s/ {Jx2 {J/

(A.?)

Appendix B ,

We consider a Gaussian beam that has been made convergent by means of asuitable focusing field. In each cross-section the beam is characterized by thequantities SX, Sy and {Jx, {Jy which are related to each other according to (2.22),and by the directions of maximum intensity. In a cross-section z' = Za, situatedbetween the focusing field and the screen, at a distance L from the screen, thesevalues are Sax, Say and {Jax, {Jaywhereas the directions of maximum intensity aregiven by

d ' Ya .an Ya = -'ljJy,Say . .

(B.I)

ON THE SENSITiVITY OF OSCILLOSCOPE TUBES

where ljJx and ljJy are the directions of maximum intensity corresponding-toXa = sa~ and Ya = Say, respectively. From now on, the subscripts x and Y willbe omitted since all equations will be valid for both the xz and yz planes. Inany arbitrary cross-section, the value of S is given by

where g(z) and h(z) are electron trajectories specified by the initial conditions

g(za) = 1,h(za) = 0,

g'(za) = 'Ijl/Sa,h'(za) = 1.

Equation (B.2) is a property of paraxial Gaussian beams, the proof of which isanalogous to the derivation of (2.19) in appendix A.For the case of a field-free region between the plane z = Za and the screen, the

values of g(z) and h(z) at the screen, ë, and hs> are given by

gs = 1+ Y!_Land

h,=L,

so that the value of s(z) at the screen, Ss>is found to be

Ss = [(sa +LljJ)2 + f}a2L2]1!2

and, for optimumfocusing, the parameters of the focusing fieldhave to be chosenso that Ss becomes a minimum. Now we refer to the focusing conditions asdepicted in fig. 2, i.e. to the case where in each point of the plane z = Za thedirection of maximum intensity points to the intersection of the z axis and thescreen. The values of Sa, Ss> (Ja and 1jJ in this situation will be indicated by thesubscript ° so that, if

then

SaO1jJ = v«=-T'

If the only effect of adjusting the focusing field were a change of 1jJ whereas Saand, therefore, (Ja remained co:nstant, the above-mentioned condition wouldyield minimum spot size. Actually, however, the value of Sawill also depend onthe adjustment of the focusing field and we shall now calculate the minimumvalue of Ss>denoted by Ssm' Putting . ..

535

(B.2)

(B.3)

(B.4)

(B.5)

(B.6)

E. HIMMELBAUER and J. C. FRANCKEN

. Saa'IjJ = 'ljJo + LI'IjJ= ---- + LI'IjJ,. . L .. (B.7)

we may express the dependence of Sa on the focusing field as follows:

Sa = SaO + L:J'IjJ, .. ' (B.8)

where the quantity 1can, with good approximation, be assumed to be equal tothe distance between the plane Z = Za and the position of a thinlensthatisequiv-alent to the focusing field. From (B.7) and (B.8) we fiIid'

Sa + bp = (L + 1)Ll1{J

and from (2.22)and (B.8)

(B.9)

S (1 )-1Pa = Pao ~~ = PaO 1 + -Ll'IjJ •

Sa SaO(B.10)

,Provided that I(1fsao)?J'IjJI «1, we find from (B.10):

.' ( 21. 312 )Pa2 = Pa02 1--Ll'IjJ + --2 LI'ljJ2 .

SaO SaO(B.U)

and insertion of (B.9) and (B.U) into (B.S) gives Ss as a function of LI'IjJ, viz.

_ [ 21 312ss02 + sa02(L + 1)2 2J1/2SS - sso 1--Ll1{J + Ll1{J •

saO sa02ss02(B.12)

This expression has a minimum value Ssm for

Lss02saoLI'IjJ=. 312ss02 + sa02(L + 1)2 '

. (B.13)

which is given by

[

12ss02 J1/2S =s 1- .sm sO 312ss02 + sa02(L + £)2

Since in all practical cases SsO «sa and also 1«L, it can be seen that thefocusing condition yielding minimum spot size does not appreciably differ fromthe situation as shown in fig. 2. Therefore, the conclusion that the insertion ofa diaphragm in the plane Z = Za will reduce the current density in each surfaceelement of the spot by the same factor and that, consequently, the Gaussiancurrent-density distribution of the spot as well as the spot size will be preserved,is justified.

(B.14)

Appendix C

For the calculation of the optimum shape of deflection plates according toMaloff and Epstein it is assumed that the transverse field strength is inversely

,;


proportional to the spacing between the deflection plates, fringe fields beingneglected. The distance between the axis and a plate will be denoted by fez),and if the entrance of the plates is situated at z :::,Zb

(C.I)

where a is one half of the entrance spacing.The field strength Ex is given by .

VdE =---x 2f(z)' (C.2)

so that.jiccording to (4.1), the trajectory of the central ray ofa deflected beamsatisfies the equation

V 1X"=_d --

4Vaf(z)

In order to obtain the maximum sensitivity it is required that for a certain valueof the deflection voltage, Vd' the path of the edge of the beam should be givenby the shape of the deflection plate, so that

(C.3)

fez) = X + a, if Vd = V"' (CA)

From (C.3) and (CA) we obtain the following differential equation forf(z):

l1d 1f"(z) = 4V

af(z)' (C.5)

with the initial conditions

fez)! = a and f'(z!) = O. (C.6)

Equation (C.5) can be solved by means of a series expansion. With the lengthof the plates denoted by I, the deflection angle for Vd = Vd is given by

c5 =f'(Zl+ I). (C.7)

According to the definition of useful scan (interception of one half of the beamcurrent), the maximum deflection angle c5max and the maximum deflection volt:age Vd max are found to be .

bc5max = --f'(z! + I)

b-a(C.S)

and

(C.9)

where we have

b =Jt», + I). (C.IO)


List of symbols *) .

a half width of diaphragni ,a02' a20 coefficients of the second-order terms in the series expansion

of Vex, y,. z) ,half exit spacing of deflection platesdistance between diaphragm and deflection centrecharge of the electrontransversal field strength of deflecting field per unit of de-flecting voltagedistribution functionlinearly independent solutions of the paraxial-ray equationscurrent densitycurrent density at the centre of the beamcurrent density at the centre of the cathodeBoltzmann's constantlength of deflection platesmass of the electron

bdee(z)

f(xo,Yo,xo,yo,zo)g(z), h(z)jjojcokImpq(xo', Yo')r,is

s(z)

x,y,zxo,Yoxo,yo, Zox(z), y(z)x',y'xo',yo'xs',Ys'xa', Ya'·xs, Ysxa, YaA,B,C,DB

ratio of screen voltage to accelerator voltagedistribution functioneffective spot radiuseffective spot radius, with space-charge effects taken intoaccountdistance from the centre of the beam, at which the current-density is lie of its value at the centrevalue of s(z) at the diaphragmvalue of s(z) at the screenmaximum beam angle at the screenCartesian coordinatescoordinates on the cathode surfacecoinponents of electron velocity at the cathodeparaxial electron trajectoriesderivatives with respect to z .derivatives at the cathode'derivatives at the screenderivatives at the diaphragm planevalues of x(z) andy(z) at the screenvalues of x(z) and y(z) at the diaphragm planematrix elements /brightness

*) Quantities which are not identical in the xz and yz planes are denoted by subscriptsx and y, respectively.

ON THE SENSrTIVITY OF OSCILLOSCOPE TUBES 539

N deflection sensitivityE(z) transversal field strength of.deflecting fieldF(xo, Yo, xo', Yo') distribution functionI cathode currentIsLMscPsQRcSTV(x,y, z)Vo(z)Vo', Vo"VaVdVd max

VsX(z)

X'(z), X"(z)Xs

Xa

oe

screen current,distance between diaphragm and screenfactor of scan magnificationpower dissipated at the screenbeam quality factoreffective radius of the emitting area of the cathodeuseful scanabsolute temperatureelectrostatic potentialelectrostatic potentialon the z axisderivatives of Vo(z) with respect to zaccelerator voltagedeflc., ..ing voltagemaximum deflection voltagescreen voltage (with respect to the cathode)trajectory of the central ray of a beam deflected in the xdirectionderivatives of X(z) with respect to zvalue of X(z) at the screenextrapolated value of X(z) at the diaphragm planeangle with respect to the direction of maximum intensityeffective beam angleeffective beam angle at the diaphragmeffective beam angle at the cathodeeffective beam angle at the screenspecific brightness (per unit of power dissipated at the screen)deflection anglemaximum deflection angleequivalent potentialscreen-current efficiencydistribution function

Cathode-ray Tube Development Dept Eindhoven, January 1967


REFERENCES1) C. Weber, Proc. IEEE 52,996,1964.2) J. Hasker, Philips Res. Repts 21, 122, 1966.3) R. Dorrestein, Philips Res. Repts 5, 116, 1950.4) J. C. Francken and R. Dorrestein, Philips Res. Repts 6, 323, 1951.5) J. Hasker and H. Groendijk, Philips Res. Repts 17, 401,1962.6) I.G. Maloffand W. D. Epstein, Electron optics in television, McGraw-Hill, 1938.7) P. G. J. Barten, private communication.

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