National Bureau cf qt««H«^

Library, n. w. Bldg

MAY 6 1964

Reference book not to be

Unical ylote 240

AVERAGE POWER DISSIPATED IN A DIODE

SWEPT ALONG ITS REVERSE CHARACTERISTIC

HARRY A. SCHAFFT

i

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

Technical Note 240

ISSUED APRIL 30# 1964

AVERAGE POWER DISSIPATED IN A DIODE

SWEPT ALONG ITS REVERSE CHARACTERISTIC

Harry A. Schafft

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

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CONTENTS

PAGE

LIST OF 5Y!1BOL5 IN ORDER OF APPEARANCE iii

INTRODUCTION 1

DISCRETION AND DISCUSSION 2

SUMMARY 7

APPENDIX A 8

APPENDIX B 9

APPENDIX C 10

APPENDIX D 13

APPENDIX E 14

ACKNOWLEDGEMENT 15

FIGURE 1 16

FIGURE 2 17

FIGURE 3 18

FIGURE 4 19

li
List of Svmbols in Order of Appearance

AVT

P (A) av

p L

- maximum voltage across the device.

- current through the device at the voltage Vp

.

- positive constant less than or equal to unity and serves

to characterize a hypothetical device.

- voltage across the hypothetical device above which the

current is non-zero.

- average power dissipated in a hypothetical device,

characterized by A, swept along its reverse characteristic using a full-wave rectified sinusoidally varying

voltage. When A=l the symbol P av d) i- 3 used.

- current limiting resistance.

n - defined at Rj^Ip/Vp, the ratio of the maximum voltage

across the current limiting resistor to that across

the device.

<$> - defined as sin (A/1 + n).

v - frequency of the sinusoidally varying source voltage.

E - energy dissipated in the device during one sweep cycle

of duration l/2v.

P. - instantaneous power dissipation in the device.

1

t. - time during the sweep cycle at which the voltage across

the hypothetical device is first equal to AV

t - time at which P.(l) = P. (A) for the first time in the

o sweep cycle.11

A+ , A - areas as indicated in Fig. 2b with dimensions of power.

in

^^^^^HHH^l^^M
List of Symbols in Order of Appearance (Cont'd)

X ,X' - constants such that X < X* < 1 and where X refers to

the hypothetical characteristic that is required to

enclose the real characteristic over the region of significant power dissipation.

? [R] - average oower dissipated in a real device. av to *

S - area as indicated in Fig. 4a with dimension of power

and described in Appendix D.

T - defined as j P (1) - P (X')| av av 'maximum

V" - maximum voltage of the voltage suoply.

A,B,C,D,F,L - areas as indicated in Fig. Ua with dimensions of power.

? „(X') - average nower dissipated in a hypothetical device with

av? I . a perturbed characteristic.

IV
AVERAGE POWER DISSIPATED IN A DIODE

SWEPT ALONG ITS REVERSE CHARACTERISTICHarry A. Schafft

The commonly used method of sweeping to a fixed power in order to compare the reverse characteristics of a group of similar diodes is found

lacking under conditions for which an average temperature is meaningful. It is shown how a determination may be made of the average power dissipation

in a diode (or in any device with a similarly shaped

characteristic) when it is swept along its reverse characteristic by a full-wave rectified sinusoidally

varying voltage in series with a resistive load. The

uncertainty in the determination due to the variability

of the characteristic is for most cases less than +5%

if the ratio of the maximum voltage drop across the

current limiting resistor to the maximum voltage

across the device is larger tnan or equal to two.

Errors introduced by small uncertainties in the

pertinent parameters are also presented. Finally

it is shown how the reverse characteristics of

similar diodes can be examined under relatively

equivalent heating conditions.

INTRODUCTION

The reverse characteristic of a diode, as well as the open base

characteristic of a transistor, is often examined on an oscilloscope

screen while the device is swept by a full-wave rectified sinusoidally

varying voltage in series with a current limiting resistor. One is

usually concerned about the power being dissipated in the device, not

only because of the existence of a power rating, but also because these

characteristics are temperature-dependent.

It will be shown how a determination may be made of the average

power dissipation in any device which has a characteristic that can be

enclosed over regions of significant power dissipation by a right triangle in the way drawn in Fig. 1 to enclose the curve R. The approach

to the problem involves the calculation of the average power dissipated

in a hypothetical device having a characteristic with an adjustable

parameter. The calculation is used to develop an upper and lower bound

on the value of the average power dissipated in a real device.

When dealing with devices with thermal time constants that are large enough compared with the oeriod of the power supplv to make an average temperature meaningful a method is presented which will make

it possible for the characteristics of a group) of similar devices to be

examined under approximately the same heating conditions. This method

consists of extending the swept characteristics to a calculable curv^> in

the volt age -current, plane which may be drawn on the cnthode ray tube.

Klnis v:orv; was partially suoportod by Rone Air Dev> io:> ! -:ent Center \:r\'l^r

Contract #A"30(602) - 3253.

^^H^^m^H^^^H^^^H^^^H^^HBH
It is common practice to use a curve determined by some fixed value for

the product of the peak voltage and peak current. It will be shown that

the use of such a curve can result in unnecessarily large variations in

the heating of the devices being examined. When measurements are made

under adverse operating conditions, which are defined in the paoer, then

the use of any curve can result in very large differences in heating.

DESCRIPTION AND DISCUSSION

Consider a hypothetical device which has a characteristic represented by H in Figure 1. Here no current is drawn until a voltage AV

is reached. V is the voltage to which the device is ultimately swept

and A is a constant where < A < 1, As the voltage increases beyond

AV the current increases linearly with the voltage across the device

until it reaches a current, I , corresponding to the voltage V_.

p P Hypothetical devices with different A's can, in the extreme cases,

take on the characteristic of a resistor for which A = or the characteristic of a diode with an "ideal" reverse characteristic for which

A = 1. The average power, P (A), dissioated in such a device while it

is being swept along its breakdown characteristic using a full-wave

rectified sinusoidally varying voltage in series with a current limiting

resistor R , is shown in ApDendix A to be given by:

Li

P (x) = Li— {(l+n) cos $ -A(i -$)} av ird+n-A)

2V I p (l-A) n .x2

t -LI {^~ (tt-24 + sin 2*)

7i(l+n-A)

2

O TT -2A(l+n) co", $ + A (--*)} , (1)

where n

= R (I /V ), and <J> = sin" (A/1 + n). More generally, the energy,

E, dissioated during one sweep cycle of duration l/2v, where v is the

frequency of the sine wave, is related to P (A) bv the expression

2uE = P (A), av Consider two hypothetical devices, H and H , characterized by A<1

and A=l resoectively. It will be shown that the average oower dissipation in a real device can be bounded and so be known to a soecifiable

degree of accuracv if its characteristic, R, can be bounded by H and H

in the way shown in Fig. 1, where the point (I ,V

p

) is common to all

three curves. It is assumed that the power dissioation in the real
device while its operating point is outside the right triangle defined

by the points (I D ,V ), (0,V ), (O.AVp

) constitutes a negligible fraction

of the average power dissipation. this contribution to the uncertainty

in the calculated values is not included in the uncertainties to be

quoted later.

A brief examination how of the instantaneous power dissipation, P.,

during one-half of the sweep cycle will aid in obtaining a qualitative

understanding of the relative dependence of P_„. (A) on RT and V_. This

7

" . " L P deoendence will be presented graphically later. The examination will

also make clearer the way in which the average power dissipation of a real device will be bounded. If a hypothetical device is sweot out to

a voltage V and a current I p and the instantaneous power is plotted as

a function of time, then a curve similar to that shown in Figure 2a will

be obtained. The area under this curve is a measure of the energy dissipated during one-half of the sweep cycle or one-quarter of the sine

wave cycle. This curve will depend on the A associated with the hypothetical device and on R

T

. For all other variables held constant the

dissipation of power in the hypothetical device will begin later in the

sweep cycle, i.e. t as shown in Figure 2a will increase, as devices

with larger A's are examined. On the other hand, t.. will decrease as

R is made larger for a constant V .2

A olot of the product of the instantaneous oower dissipation and

twice the sine wave frequencv as a function of time is shown in Figure

2b for the characteristics of two hypothetical devices. The scale of

the vertical axis is chosen to make the area under each curve equal to

one-half the average power dissipation, i.e. Pav (A)/2. Curve 1 represents the dissipation in a device with A = 1 while curve 2 in one with

A<1. In general the two curves will intersect not only at a time

t = l/4v but also at some time earlier in the cycle. The time at which

this other intersection occurs is designated as t . The relative

magnitude of the areas A and A , shown in Figure 2b, determines whether

P (A) will be smaller or larger than P (1). An outline of the derivation of t in Apnendix B shows that t is a function of A and n. As

n is decreased, t increases toward l/4v and A- decreases to zero while

A + remains finite; thus for small n, P (A) > Pav (l). On the other hand,

if n is increased to very large values, t Q will decrease toward zero and

A decreases to zero v/hile A_ remains finite; thus for large n,

P av (A) < P av (l). At some intermediate value of n, A+

= A_ and so

P av (A) = P av (l). This behavior can be seen in Figure 3 where the norm- alized average power dissipation, P ay(^)/IpVp, is plotted as a function

of n for four different values of A. An important point to observe in

1. The product of the voltage and current at the point of enclosure

should be much less than v

p

I p

. 2. The deDendence of t, on A and RT is given in Appendix A.

1 L

^M^^I^^^^^^H
the plot is that Pav (A) is a monotonic function of X except in the

region 2.0 < n < 2.6, where the dependence on A is small. Thus for any

hypothetical device characterized by A', P av (*') can be bounded in the

following way:

P (A)<P (A» ) * P (1) for n > 2.6 av o av av = P (A ) > P (A')>P (1) for n < 2.0 av o av av = and P (1) + C z P (X') i P (1) - C for 2.0 < n < 2.6 , av av av = = ' where C is some appropriately chosen constant, and A Q < A' < 1. However, as can be seen in Figure 3, P av (X) decreases as r, decreases to zero and the dependence on X increases sharply. Thus while Pav (A') can be indeed be bounded, the uncertainty in Pav (A') can soon approach the

magnitude of P av (A') as n decreases much below unity.

The reverse characteristics of real devices are, at least, only

similar to those of the hypothetical devices at the higher currents.

Real devices have characteristics that appear more like that pictured

by R in Figure 1. It is shown in Anoendix C that the average Dower dissipated in a real device, Pav[R], with an arbitrarily shaped characteristic, can be bounded if the characteristic itself can be bounded over

the region of significant power dissipation. Here, as in the case of

P av (A'), the usefulness of the bound decreases markedly at low values

of n. For ease in presentation and ultimate usefulness, Appendix C con- siders only conditions for which n > 1. It will be seen later in

Appendix D that if the uncertainties for the cases examined are to re- main tolerably small, then onlv conditions for which n 2. 2 should be

considered; therefore the data that follows will be presented for such

values of n.

If the characteristic of the real device is bounded by the characteristics of two hypothetical devices, one with A = 1 and the other

with A < 1, then from Aopendix C:

P (A ) - 2A+

< P [R] < P (1) + 2A+ + 2S for n > 2.6 av o

T av av = and either

P (A ) - 2A.< P [R] < P (1) + 2A. + 2(S + T) av o

T av av T or

P (1) - 2A+ < P [K] < P (A ) + 2A+ + 2S for 1 < n < 2.6 av T av av o r = = where T = I P (1) - P (A f )| . for A < X' < 1 1 av av ' maximum o and S is a complicated function of n and A described in AnDendix D. The constant, A Q

, will refer to that hypothetical characteristic which

is required to enclose the real characteristic over the region of significant Dower dissipation. A discussion of the magnitudes of the terms
involved in the bounds for P av[R] as well as how they were determined

is presented in Appendix D for XQ

- .6 and A Q

= .8. Two values of A Q were used, with the stated magnitudes, in order to make the calculations

adaptable to as wide a variety of devices as seemed feasible. Because

both S and T were found to be negligibly small over most of the range

of n, the value for P av[R] was chosen to be a value mid-way between

the values of P av (*o) anc* ^av^^ at anPspecific value of n. This was

done so that the uncertainties could be made symmetric about Pav[R],

The results of Appendix D show that for real devices which have characteristics which can be bounded by a hypothetical characteristic with

A = .6 the uncertainty in P av[R] is within +_5%. If a hypothetical

characteristic with A = . 8 can be used, the uncertainty in P av[R] is

reduced to within +3%. For a given A the dependence on n of the

normalized average power dissipation in a real device, P av[R3/IpVp, can

be sufficiently well represented by two simple expressions to make no

change in the above specified uncertainties necessary. These expressions are shown below for A Q

= .6 and A = . 8 and will give the value

of Pav[R] once the values of Ip, Vp and Rl are specified.

For A q

= .6,

P av

CR] 18W

Vp

. uuiu —

n

and P av^

Vp

= .6090 - .2530

n

For A o= .8,

P av

CR]

Vp

= .6140 - .2120

n

and

Vp

r .6230 - .2980

n

, for 2 < n < 10 (2a)

, for n > 10. (2b)

, for 2 s n <, 10 (3a)

for n > 10. (3b)

The stated uncertainties do not include, as mentioned earlier, the

contribution due to the dissipation while the real device's characteristic is not bounded. It is assumed that A Q

is chosen to make this

contribution negligible. Additional sources of error which can become

appreciable are the uncertainties in Vp , Ip, and R^. It is shown in

Appendix E that the fractional change in the average power dissipation,

AP av[R]/P av[R] introduced by small changes in Vp, Ip, and R^ is given

approximately by the relation

AP

av

[R] AI

P

AV

P 1

iR

L

"T^TrT <

i p

+

v p

+ 2n R

L
Thus it is obvious that unless considerable care is used in the

measurement procedures the errors introduced in P av[R] can easily be as large as the already specified uncertainties used to take into account

the variability of the shape of the characteristic.

Using equations (2) or (3), depending on which A is chosen, a curve in the voltage-current plane can be obtained to which the characteristics of similar devices under test can be swept in order to as- sure uniform average power dissipation and thus approximately uniform

heating conditions. For a specific value of the average power dissipation desired, R^ must be chosen so that n will be larger than or equal

to two for all test conditions. The value of n will be a minimum at

the maximum peak voltage, Vp, expected for the devices under test.

Since n is the ratio of the voltage drop across the current limiting

resistor to the voltage droo across the device it is advantageous to adjust conditions so that n is small at the maximum expected peak voltage.

Therefore n should be chosen to be equal to two at this voltage and the

aooropriate expression is then solved for Ip. Knowing Ip, Vp and n the

current limiting resistor R^ to be used in the test can be determined.

Finally by substituting IpRr/Vp for n in the appropriate equations,

i.e. (2) or (3), and solving for Ip the other points on the voltagecurrent plane may be obtained over the desired voltage range and a curve may then be drawn "joining the Doints obtained. The uncertainty in

P av[R] when the characteristic of the device is swept to this curve is

within either +3% or +5%, deoending on the value of X Q required. To

this uncertainty the errors introduced by the uncertainty R^ and the

maladjustment of the peak current should be added. These errors, as shown in Aopendix E, contribute a fractional change in the average

oower dissipation of

flP av

[R]

, „ AI

P 1 iRL

It is often assumed that approximately uniform heating conditions

can be obtained by extending the characteristics of the test devices to

the curve IpVp = constant <* Pav[R]. If this assumption is correct then

P av[R]/IpVp should be a linear function of n. That this is not the case can be seen by the variation of P av (^)/IpVp with n shown in Figure 3

and from equation (2) and (3). The departure from linearity is greatest

at the lower values of n. The use of such a curve can result in

variations in P av[R] of almost 25 percent as similar devices with different characteristics are examined. Of course if measurement con- ditions are such that n<2 then the variations of P [R] can be very much

larger.
SUMMARY

It has been shown how the average power dissipation in a diode can be calculated when it is sweat along its reverse characteristic by a full-wave rectified sinusoidallv varying voltage. These calculations

will also apply to any similarly shaped characteristic, for example,

the open-base characteristic of some transistors. The average power

dissipation may be obtained from a simple expression once the maximum

voltage across the diode, the corresponding current and the current

limiting resistance is specified. The uncertainty arising from the

variability of the shape of the characteristic can in most cases be within +5% if measurement conditions are such that the ratio of the maximum

voltage across the current limiting resistor to the maximum voltage

across the device is greater than or equal to two. When this ratio is

less than two, the average power dissipation in the device becomes

very sensitive to the value of this ratio as well as the shape of the

characteristic of the device.

Estimates of the fractional change in the average power dissioation

introduced by small changes in the pertinent parameters have shown that

the uncertainty of the average oower dissipation in the device due to

poor measurement techniques can easily be larger than the already quoted

uncertainty resulting from the variability of the shape of the characteristic.

The ability to determine the average power dissipation in a device

permits the construction of a curve to which the characterisitic of each

similar diode can -be swept and achieve relatively equivalent heating

conditions. The use of this curve represents a distinct imorovement

over the use of a fixed oeak power curve.
APPENDIX A

Let the supply voltage be given a V^ sin 2 Tr ^t for < t < l/4v.

Let V represent the voltage across and I represent the current through

the device under consideration. From the description of the characteristic of the hypothetical device, it follows that for < t < tj_, 1 = and for t]_ < t < 1/Uv,

V = j- (1-A) I + XV

Di

1 -1 Vl where t n = -^— sin (A/1 + ——) . 1 Z7TV V„ (Al)

The explicit form of I for t-, < t < l/4v is obtained from the relation

V

P

V„ sin 2Tivt = IRT + rp- (1-A) I + AV n . M L 1„ P The solution for I is,

-I = V„sin 2nvt-AVn H P

R

L+V

p

(l-A)/ I p

(A2)

If E is the energy dissipated during one sweep cycle of the full wave rectified sinusoidally varying supply voltage then P av (*) = 2vE.

l/4v

But

Therefore

E = 2 VIdt.

tl

P (A) = Uv av

i/Uv

f VIdt. (A3)

If (Al) and (A2) are substituted into (A3) and the integration performed

then the following result is obtained,

2V

p

I A p <*) = i\j x\ t(l+ n) cos <J> -A(l -$)} av Tr(l+n-A) l 2

'

2V I (1-A) , .2

-2A(l+n) cos <$> + A 2 (j -4>)},

where n = R (I /V ), and <J> - sin_1 (A/l +n) = 2irvt .
APPENDIX B

The instantaneous power dissipation P^(t,X) in a hypothetical device

is given by P^CtjX) = IV where V and I are given by equations Al and A2

in Appendix A. When Pi(t,X) is equated to Pi(t]_,l), a quadratic equation in terms of sin 2irvt is obtained. One root of the equation is

unity since the two characteristics have a common point at t = l/4v.

The other root is found to be

sin 2-rrvt = o

(l - x + n)

(1 - x)(i + n)

:

X 2

n (l - x + n

)'
APPENDIX C

In Figure 4a the product of twice the sweep frequency and the

instantaneous power dissipation for three different hypothetical curves

are superimposed and plotted as a function of time. One of these curves

is for X = 1 while the other two curves are for X and X' where XQ

< X

? , The size of the areas shown, which have dimensions of power, are exaggerated for ease of identification. These same curves are also plotted

in Figure 4b on the V-I plane.

Note that it can be shown that for n > 1, 3t /9X|

n > 0; so that in

Figure 4a the way in which the curves are drawn intersecting the X = 1 curve is correct.

From Figure 4a it can be seen that

P (1) -£L_ A B + F.

P (X«) av

and P (X ) av o

= B + C+L + F

= C + D + F .

Consider 'now that the X' curve is perturbed in such a manner that on the V-I plane the slope, dl/dV, decreases in some uniform way as is

indicated by the dotted line in Figure 4b. The perturbation is assumed

to be such that the X = X and X = 1 curves on the V-I Diane still enclose the perturbed A' curve.

As the device with the perturbed characteristic is swept in the

V-I plane, the instantaneous power dissipation at anv time during the

sweep cycle is equal to that of some hvpothetical device with an un- perturbed characteristic with X such that XQ

< X > X'. The average

power dissipated, P avp(*')> by the device with the perturbed characteristic can be given by the relation,

7

P avP

(A ' } = F + C + a

l B + a2 L + a3 D + S

where < aj_ < 1, < ap < 1, <, a i 1. The nature of S and its magnitude is discussed in Aopendix D. Convenient maximum and minimum

limits on Pavp(*' ) are

i- P (X» ) max. .sF + C + B + D + S,

2 avP

10
and

7T P „(** ) min. 2 F + C.

2 avP

There are four ways that P avp(*')» Pav^o) and p av(l) TMay be found

to be ordered with respect to their relative magnitudes over the full

range of n. This ordering will now be examined to see how bounds on

Pavp(*') may be constructed for all n.

1.) P (1) > P _(AM > P (A ). av avP av o This implies that A > C + L or that A = C + L + c, where Z, > 0.

Now

i P _(A f ) <B+C+D+F+L+S

2 avP or i-P D U')<~P (1) - A + C + D + L + S

2 avP 2 av

= i P (1)-C+D+S<ip (1)+(C+D)+S. 2 av 2 av ip _(AM i C + F = ^P U)-D>Jp (X ) - (C + D).

2 avP 2 av o 2 av o ip (X)-(C+D)<ip JAM <ip (1) + (C + D) + S.

2 av o 2 avP 2 av

2.) P (X > > P _UM> P (1). av o avP av This implies that C + D > A + B and that C+D-A-B>C + L-A

or that D > B + L. Now

i P „(x») <b+c+d+f+l+s = 4p (a ) + B + L + S

2 avP 2 av o

And,

Thus

And

Thus

< i P (A ) + (C + D) + S.

2 av o ~P _(A' ) > C + F = i P (1)-A-B>ip (1) - (C + D).

2 avP -

2 av 2 av ip (1)-(C + D)<~P (A')<i-P (A ) + (C + D) + S.

2 av 2 avP 2 av o

3.) P (AM > P (1) > P (A ). avP av av o 4.) P (AM > P (A ) > P (1). avP av o av For both cases, P n (AM > P (A ) - (C + D), avP av o

'

11
and

T p TD (X')<3+C+D+F+L+S = -i-P (D-A+C + D + L+S 2 avP 2 av

< i- P (1) + T + S + (C + D)

2 av where

Thus

T = P (A» ) - P (1) . * L - A. 1 av av ' maximum

i P av U

Q

) - (C D) < i P

avp

U' ) < i PavU) + T S + (C D).

For n £ 2.6 the first case will hold and depending on X and n, one of the other cases will hold for 1 < n £ 2.6.

The treatment in this section will also apply to any real device

as long as the slope dl/dV on the V-I plane remains constant or de- creases in some uniform manner. Therefore, the bounds that apply to

Pavp(A') will also apply to the P av[R]. Accordingly,

P (X ) - 2A+ < P [R] < P (1) + 2A+ + 2S for n > 2.6 av o

T av av T = and either

P (1) - 2A+ < P [R] < P (X ) + 2A+ + 2S av T av av o

T or

P (A ) - 2A+ < P [R] < P (1) + 2A. + 2S + 2T av o

T av av + for

1 < n < 2.6 where A+

= C + D.

12
APPENDIX D

The bounds for P av [R] involve T, S, A+ and |P av (l) - P av (x)| which

depend on A and n. Estimates and upper bounds for these values were obtained by a number of numerical calculations and graphical constructions,

The value for T was obtained by plotting P av (A)/IpVp over a limited

range of A for different values of A. From the graphic representation

it was determined that the contribution of T to the uncertainty of

PavtR] was less than 0.3% for AQ = . 8 and less than 1.6% for AQ

= .6.

Consider Figure 4a. If curves for all possible A between 1 and AQ are also drawn, then the area enclosed by these curves that does not

include the area A + B + C + D + F will be equal to or larger than any

S that can arise. If Pj_(A) is the instantaneous power dissipation in a device with an A characteristic and if t x

(A) is the time at which

Pi(A+A) = P^(A) where A + A <1 and finally if t (A) is the time at which

Pi(A) = Pj.(l) then

t (A +iA)

(1-A )/A r lim I |e(a

A-»o i = l

t x(;^o+iA)

(E(AQ

S < +iA) - E(A +[i-l]A)}dt.

To estimate the magnitude of S, A was set equal to 0.5 and several terms

were evaluated. It appears that these terms, over the range of n > 2, are largest for n = 2 and decrease rapidly as n is increased, and as

A increases.

At n = 2 for A = .6, S is estimated to contribute an uncertainty

to P av[RJ of less than 0.1% of P av[Rj.

The values for A+ /IpVp and P av (l) - P av (X ) |/Vplp were calculated for several values of n for A Q

= . 6 and A Q= .8. The results are indicated below:

n i P (1) - P (.6)| + 4A. |P (1) - ? (.8)| + 4A 1 + av av t ' av av ^

p TT61 p CT)

av av

2.0 .065 .041

2.6 .042 .028

4.0 .051 .025

6.0 .060 .030

.095 .045

As can be seen, allowing for T and S, + 5% of P av[R]' for ^ = '^ anc* + ^

?° of P pv[R] for A Q

= .6 are conservative estimates for the uncertainty of

?avCRJ:

13
APPENDIX E

If P av[R] iVIpVp is plotted against n it is found that to a good

approximation the two quantities are linearly related for n Z 2. Thus

PavCRJl/IpVp = P av[R]RL/Vp

- Fn + G where F and G are constants. If

both sides of the equation are multiplied by Vp/R^ then

P av

CR3 FI

p

V

p

t GV

p

/R

L (El)

If small changes in Vp, Ip and Rl are allowed and denoted by AVp, Alp

and AR^ respectively, then the fractional change in P av is given by

AP TR]

av

P [Rj av

V„I„ P?

P [R] av

AI. + (1 + G Vp i. ^p jgraV v

p

Vp AR

1 L

P [R] n av

[low

Thus

F = .61, G - -.25 and ? av[R]/I

p

V

p

> .51.

AP [R] AI. AVT

av

1

flKL

2n RT (E2)

At currents close to Ip the slope of the real breakdown characteristic should not change greatly, so it can be described by some A characteristic. . This voltage is related to the current by the equation

V = — (l-A)I + AV

p

For small changes in Vp and Ip,

AV

P

AI

P -T-= (l-A) -/.

P P

and equation (E2) for A > .6 becomes

&Pav

[RJ

, , "p

2n RT

14
ACKNOWLEDGEMENT

The author wishes to express his thanks to L. J. Swartzeridruber and

R. L. Raybold for programming equation (J,), and J. C. French for his en- couragement.

15
I_lI

figure 1 - Volt ape- current characteristic

H»(A = 1) an; H(X < 1), and real device R. hypothetical devices

16
TIME

Figure 2a - Qualitative features of the instantaneous power,

P^ vs. time characteristic of a hypothetical device.

Ql

CM

TIME

.gure 2b - Product of the instantaneous power and twice the sine wave frequency as a function of time for fewc hypothetical

devices, one with A = 1 and the other with A < 1.

17
r>0

CXJ

CD LO ro CXJ

o

a

o>

>

X)

d A d

'A

d I/^)

AD

d

a P c

<-*

-j > c , a. 3

<y

I

X

1 3

18
Q_~

CVJ

1/4 v

TIME

Figure 4a - Qualitative features of the product of the instantaneous

oower and twice the sine wave frequency as a function of

tine for three hypothetical devices and one real device.

REAL

DEVICE

\,v

p H v

p

V

Figure 4b - Voltage-current characteristics of three hypothetical

devices and one real device.

19

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