PROCEEDINGS OF THE I.R.E.

Transient Response of Cathode Followersin Video Circuits*

B. Y. MILLSt

Summary-The behavior of cathode followers handling the ir-regular video signals of radar or television is discussed. It is shownthat the signal-handling ability of a cathode follower may be reducedif it has a capacitive load. A general expression for the magnitude ofthis reduction is given, and its effect on design procedures is dis-cussed.

I. INTRODUCTION- HE USE OF cathode followers for providing low-

I impedance outputs with a small current drain andlow distortion is well known, but in general their

behavior when operating into a capacitive load is thesubject of some misconception. The object of the presentpaper is to show the limitations this behavior imposeson the design of cathode followers required for handlingthe irregular video signals of radar or television.The exact shape of these signals depends on the na-

ture of the circuits preceding the cathode follower. Thesignals may be represented approximately, however, asa series of positive and negative linear transitions ofconstant transition time T, as illustrated in Fig. 1. Thetransition time of the actual signals is conventionally

v93LT t

Fig. 1-A typical signal.

taken as the time required for the voltage to changefrom 10 to 90 per cent of its final value.' When appliedto a linear transition, this value will be conservative,provided that there is little or no overswing in the origi-nal.The requirement usually imposed on a cathode fol-

lower is that such transitions be reproduced with littledistortion. It will be shown that the presence of capaci-tance in parallel with the cathode load resistor may thenresult in a reduction of the maximum signal amplitude-which may be handled. Although this effect is quali-tatively familiar to most circuit designers, no quantita-tive analysis immediately applicable to video signals has

* Decimal classification: R139.21. Original manuscript receivedby the Institute, December 8, 1947; revised manuscript received,November 17, 1948.

t Council for Scientific and Industrial Research, Chippendale,N. S. W., Australia.

I Transition times for common circuits are given by a number ofauthors. See, in particular, H. E. Kallmann, R. E. Spencer, and C. P.Singer, "Transient response," PROC. I.R.E., vol. 33, pp. 169-196;March, 1945.

so far been published. It is common to see uneconomicaldesigns based on an exaggerated respect for the limita-tion.

II. THE EQUIVALENT CIRCUITIn the analysis it has been found convenient to use

the equivalent circuit of Fig. 2(b). The cathode followerof Fig. 2(a) can be represented there by putting its in-ternal resistance R equal to 1/gm Mu/g, + 1.

R

t,9 Pk XV9

(a) (b)Fig. 2-The cathode follower and equivalent circuit.

Nonlinearity of tube characteristics is neglected in themathematical analysis, although its effect is demon-strated qualitatively in the final equations. As explainedlater, its presence does not greatly affect the usefulness-of the results.

III. RESISTIVE CATHODE LOAD

Before commencing a discussion of transient behaviorin the equivalent circuit, it is desirable to write downsome of the relations needed which apply to the simpleresistive-loaded case. These relations hold when thetransition time of the applied signal is long comparedwith the time constant of the cathode load Rk Ck, sothat the capacitance Ck may be ignored.The gain of a cathode follower with cathode load Rk

is given by the ratio of the cathode voltage Vk to the gridvoltage v,:

Vk Rk /1A = - =

v g R+Rk IA+ 1 (1)

A more useful concept turns out to be the gain of theequivalent circuit A', given by the ratio of the cathodevoltage Vk to the input voltage of the equivalent circuit

(2)f Vk RkA = _

vg' R+Rk

It is also necessary to define the maximum outputvoltage swing Vo as the maximum amount by which theoutput voltage can change in a given direction from anyparticular reference level. In a negative direction thelimit is set by cathode current cutoff, or sometimes sim-

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PROCEEDINGS OF THE I.R.E.

ply an increase in distortion. In a positive direction theusual limitation is the drawing of grid current. If Io isthe corresponding cathode-current change,

VO= RkIo (3)

IV. CAPACITIVE CATHODE LOADTurning now to the case of an applied signal with a

short transition time, we may divide the analysis intotwo parts. The first part deals only with the small-signalresponse. The second deals with the limitation on maxi-mum output voltage caused by the short transition timeof the applied signal.As far as the small-signal response is concerned, it is

sufficient to note that the transient response of a cath-ode follower is the same as that of a resistance-coupledpentode amplifier with plate load A' R (consisting of Rand Rk connected in parallel) shunted by a capacitanceof Ck. The behavior of such a circuit to various signalforms is well known.' In particular its response to a lin-ear transition is given by equation (9) of the Appendix.It is seen that, provided exp (- 1.25T/A'RCk) is small,the signal is reproduced with little distortion and de-layed by a time A'RCk.A cathode follower which has an adequate small-sig-

nal transient response may, however, display severetransient distortion on a signal which would appear tohave an amplitude well within the capabilities of thetube. The reason is that in reproducing accurately alinear transition, a cathode follower must supply asteady current of dVk/dt' Ck to the capacitance Ck duringthe rise (or fall) time of the signal. The available currentswing through the load resistance Rk is therefore re-duced, and consequently the signal-handling capabili-ties of the tube may be severely restricted. The followingrelation between the maximum output-voltage swingpossible with a signal transition time T and the maxi-mum available voltage swing (i.e., for T== oo) is derivedin the Appendix:

VVO

T =

1 + A' RkCk1.25T

where VT is the maximum output voltage swing withtransition time of T, and Vo is the maximum availableoutput voltage swing.

It is assumed that exp (-1.25 T/A' RCk) is small or,in other words, that the transient distortion with a smallinput signal is not large. The equation applies to bothpositive and negative transitions.

In a cathode follower handling a positive signal andbiased near anode-current cutoff, the limitation occursusually on the negative-going trailing edge, and its ef-fect is often apparent as a considerable lengthening ofthis portion of the signal. In such a case the distortion isthe result of insufficient available voltage swing in anegative direction, which causec the cathode current to

be cut off. The effective time constant of the cathodecircuit is thereby changed from A'RC, to RkCk, withconsequent distortion.

Equation (4) can be shown to apply also when Ck isreplaced by an open-circuited transmission line withpropagation time considerably shorter than T, and witha total capacitance of Ck. This is analogous to the simi-lar and more familiar representation of an open-cir-cuited transmission line carrying a sine-wave signal of afrequency considerably less than the lowest resonantfrequency of the line.

V. CATHODE FOLLOWER DESIGNFor design purposes it is better to write (4) in a dif-

ferent form. In Fig. 3 are represented the cathode volt-ages in a cathode follower handling a typical positivesignal. Consider the negative transition, as it is the onein which trouble is usually expreienced. It will be as-sumed that the signal amplitude is the maximum whichcan be handled without cathode-current cutoff.

I~~~~~~~~~~~~~~~~~~~I

VJ T T 13 Rk

Fig. 3-Cathode voltages with a typical positive signal.

The cathode current necessary to maintain a steadyvoltage across Rk equal to the minimum cathode voltageis taken as IL. The available voltage swing Vo is thenequal to VT+I8Rk, whence (4) takes the form

VT + I,RkVT = S

RkCk;1 + A'

1.25T

which, on rearranging, becomes

A'Ck2I., = VT,.

1.25 T (5)

In the design of a cathode follower in which the nega-tive transition is the determining factor, it is thereforenecessary to ensure that the minimum steady cathodecurrent I8 is greater than that obtained from (5). Thisequation is based on the ideal linear circuit of Fig. 2(b).Actually, of course, the internal resistance of a cathodefollower increases steadily as the anode current is de-creased. This will result in a higher possible value of VT

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Mills. Transient Response of Cathode Followers in Video Circuits

before cutoff occurs because of the decrease in A', but alower value before transient distortion increases per-ceptibly. In practice, the equation as given will oftenrepresent a suitable compromise. If it is necessary to en-sure that the internal resistance R should never exceeda certain value, then Is must be increased by an amountequal to the cathode current corresponding to the maxi-mum allowable internal resistance.

It will be observed that the signal-handling capacityapparently does not depend to any great extent on Rkprovided that the gain A' is near unity. This is becausethe negative transition only has been considered. In acathode follower designed to have the minimum value ofIT, the peak cathode current will be given by

VTI(max) = - + 218,

Rk

tion time T. It is required to find the relationship be-tween this applied signal and the maximum change ofcathode current lo at time 1.25T.

vtf +Ck Ilk

Fig. 4-Simplified equivalent circuit for calculation. ofthe instantaneous value of the cathode voltage Vk.

The slope of the applied signal is equal to VT/1.25T;that is,

(6)

The current IS is doubled because an extra current ofthis magnitude is required to reproduce the positivetransition.The peak current must, of course, be within the capa-

bilities of the tube, and usually it is desirable that itshould be as small as possible in order to keep the tubedissipation to a minimum.

Usually, therefore, the cathode resistor should bemade as large as possible. In the most common practicalcircuit the signal is coupled to the cathode follower gridvia a capacitor and grid-leak combination in which thecold end of the grid leak is at earth potential. The mini-mum grid voltage is then zero, or even negative, if theduty cycle of the signal is appreciable.The maximum value of Rk is then limited to that

which produces a cathode current I, given by (5) whenthe grid has this minimum voltage. Often a more eco-nomical design is, therefore, one in which the cathoderesistor is larger than this, and the grid leak is returnedto a positive voltage sufficient to keep I, above the mini-mum value.

VI. ACKNOWLEDGMENT

This work was carried out as part of the research pro-gram of the Division of Radiophysics, Council for Scien-tific and Industrial Research, Commonwealth of Aus-tralia.

VTv = ~~t.

1.25T (7)

It may be shown that, if a signal consisting of a lin-early changing voltage with the above slope is appliedto the circuit of Fig. 4, the magnitude of the current i,flowing into the capacitor during the transition time ofthe signal is given by

VT { ARC/ic = *5 Ck I exp1.25T A'RCoJ (8)

The voltage Vk across the capacitor is then given by

Vk = V-A'Ric

= VT [ {1 - exp(7)]-1.25T 1.25T A'RCk/} (9)

The total cathode current is the sum of current flow-ing into Ck and Rk. The former is given by (8) above;the latter is equal to vk/Rk. Thus, on adding these andrearranging, we have, for the cathode current,

VT r_t A'RkCk ( / t 1

i=- ~~+ 41 -exp (.T )]Rk 1.25T 1.25T A1' RCk -

(10)

The maximum value of i, equal to Io, occurs at a timeof 1.25T. Inserting these values for i and t in (10) and re-arranging, we have

APPENDIXMAXIMUM VOLTAGE OUTPUT

A cathode follower with capacitive load may be rep-resented by the simplified circuit of Fig. 4, on applyingThevenin's theorem to the euqivalent circuit of Fig.2(b). The input signal to this circuit is then the outputof the cathode follower in the absence of the capacitanceCk, while the voltage across the capacitance Ck is theactual cathode voltage Vk. The input signal to be consid-ered is a linear transition of magnitude VT and transi-

VT =IoRk

1 +A' -{1 - exp Al1.25T A'RC, J

(t1)

Since the requirement of small transient distortiondictates a small value for exp(-1.25T/A'RCk), thisequation may normally be simplified to

VOVT =

1 RkCk1.25T

(4)

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