LIMITATIONS OF JUNCTION TRANSISTORS,
IN SWITCHING CIRCUITS
. . .
'
by
Bldhu Bhushan Chaudhuri
Thesis Submitted t o the F a c u lty o f the
DEPARTMENT OF ELECTRICAL ENGINEERING
P a r t i a l F u l f i l l m e n t o f th e Requirements
For th e Degree o f
MASTER OF SCIENCE
In the Graduate Col lege
of
' THE UNIVERSITY OF ARIZONA
.................. 1962
STATEMENT BY AUTHOR
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is in the
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SIGNED
APPROVAL BY THESIS DIRECTOR
This th e s is has been approved on the date shown below:
/ L
W
5 /
DR, ROBERT L. WALKER
Professor o f E l e c t r ic a l , Engineering
,
n
w y
b a te
ACKNOWLEDGEMENT
The author wishes t o take t h i s o p p o rtu n ity t o express a few
words o f h is personal a p p re c ia tio n f o r Dr, Robert L. Walker# Professor
o f E l e c t r ic a l E ngineering.
The author fe e ls a re lu c ta n c e t o fo llo w the
t r a d i t i o n a l way o f making acknowledgements.
Dr. W a lker's c o lla b o r a t io n
r i g h t from the beginning o f f a b r i c a t io n o f the t e s t - c i r c u l t s t o the
p re p a ra tio n o f the manuscript# has been# and w i l l be, a perennial source
o f in s p i r a t i o n t o the a u th o r.
The s t im u la t in g te c h n ic a l discussion the
author had th e o p p o r tu n ity t o have w ith him la id the fou n d atio n s o f th e
present w ork.
His f r i e n d l y and c o o p e ra tive a t t i t u d e made the s t a r t o f
the work a p le a s u re .
The l i t t l e
progress o f the p resent work the author
claim s is la r g e ly due t o h is constant help and l i v e l y encouragement.
The author f e e ls proud t o be a stu d e n t o f a teacher l i k e D r, Walker
who# almost instantaneously# k in d le s a sense o f value in th e hearts
o f h is s tu d e n ts .
ABSTRACT
Using t h e •conventional techniques o f a n a iy s is , tu rn -o n arid
decay-times f o r a sa tu ra te d t r a n s i s t o r under va rio u s loads have been
c a lc u la te d .
C la s s ic a l
( M o l l ' s ) expression f o r sto ra g e -tim e in a
sa tu ra te d t r a n s i s t o r has been in v e s tig a te d .
A few expressions f o r
storage tim e — e s s e n t ia l ly extensions o f M o ll 's r e s u lt s — have been
developed.
I t has been in d ic a te d how, f o r t r a n s i s t o r s in v a l i d a t i n g
th e M oll a p p ro xim atio n s, a g ra p h ica l s o lu t io n o f the governing equation
f o r c u r r e n t flo w in a s a tu ra te d t r a n s i s t o r leads t o accurate r e s u l t s . With
mathematical reasoning i t has been shown t h a t th e d i f f u s i o n equation o f a
s a tu ra te d t r a n s i s t o r cannot be solved under g e neralized d r iv e - c o n d i t i o n s .
To in v e s tig a te th e r e l a t i v e importance o f b a s e -re sista n ce and alpha c u t­
o f f frequency on the tra n s ie n t-re s p o n s e # a simple te tro d e has been s tu d ie d .
.
TABLE OF CONTENTS
Chapter I
In tro d u c tio n
..
Chapter 2
T h e o re tic a l A n a lysis o f T u rn -o n - and Decay-time Under
Varying Load C on d ition s
S w itching C h a r a c te r is tic s
E va lua tio n o f Turn-on and D eiay-tim es
Experimental V e r i f i c a t i o n o f Expressions f o r Turn-onand DeI ay-times
Chapter 3
Storage Time
Physical Basis o f Storage time
Storage Time and i t s Evaluation from T ra n s is to r
Parameters
Measurement o f Storage Time
V e r i f i c a t i o n o f M o ll 's Equation f o r Storage Time
Chapter 4
Ju nction Tetrodes
Chapter 5
Conclusion
Appendix
B ib lio g ra p h y
Chapter I
In tro d u c tio n
Junct io n -tra n s i s to rs ,- among a I I the e le c t r o n ic devices used
f o r la rg e -s ig n a l o r n o n - lin e a r a p p lic a t io n s , are the most amenable to
a n a ly s is .
They lend themselves t o a n a ly s is o f very high accuracy
whether they are used in o f f - s t a t e or o n - s ta te or in t r a n s i t i o n between
o f f and on.
As Mo 11 has shown in h is c la s s i c paper, an ideal t r a n s i s t o r
needs o n ly f i v e parameters f o r i t s complete c h a r a c t e r iz a tio n and these
parameters are w it h in the c o n tr o l o f t r a n s i s t o r d e signers.
A ju n c t io n -
tra n s i s t o r used as a n o n-re g e n e ra tive sw itch can have many megohms
impedance in the o f f - c o n d i t l o n and less than an ohm in the o n - c o n d itio n .
The vo lta g e drop across th e t r a n s i s t o r does not g e n e ra lly exceed a few
100 mVs,
Power s u p p lie d t o the load may be several thousand tim es g re a te r
than th e power necessary t o a c tiv a te the s w it c h ,
In re g e n e ra tiv e sw itche s,
the power may be fa c to r s o f ten hig h e r than f o r n o n -re g e n e ra tive sw itches.
So f a r e v e ry th in g seems t o be p ro m is in g .
But th e t r a n s i t i o n
tim e involved in j u n c t i o n - t r a n s i s t o r s used f o r n o n -1inear a p p lic a tio n s
(n o ta b le among which is s w itc h in g ) poses a g re a t problem, e s p e c ia lly
when we are concerned w ith high-speed s w itc h in g .
Let us now consider a
ju n c t io n -tra n s i s t o r re g e n e ra tiv e s w itc h — a t r a n s i s t o r iz e d m u l t i v i b r a t o r .
In our a n a ly s is o f th e m u l t i v i b r a t o r , we i n i t i a l l y assume t h a t sta te s
are switched in zero tim e , which allow s us t o t r e a t the c i r c u i t behavior
on both sides o f the boundary.
But a c t u a ll y s w itc h in g - tim e is not zero.
In vacuum-tube m u lt iv i b r a t o r s a problem o f t r a n s i t i o n - t i m e is
2
imposed by p a r a s i t i c c a p a c ity and inductance p resent in th e c i r c u i t .
ideal o p e ra tio n , the change o f t u b e - s ta te ,
i . e . , from on t o o f f , should
be accompanied by instantaneous changes in c u r r e n t and v o lta g e .
s tr a y inductance and capacitance prevent t h i s
c u r re n t and v o lta g e .
For
But th e
instantaneous change in
So we r e a l i z e t h a t f o r high frequency o p e ra tio n
load re s is ta n c e should be s m a ll, s tr a y c a p a c ity and inductance should be
minimized and, i f necessary, high frequency compensation should be used.
In t r a n s i s t o r m u l t i v i b r a t o r s , p a r a s i t i c elements p la y a
r e l a t i v e l y unim portant r o le because n o rm aliy-one uses very small values
o f c o l l e c t o r r e s is ta n c e .
In t r a n s i s t o r - m a i t l v i b r a t o r s basic l im i t a t i o n s
on s w itc h in g - tim e are due t o the p ro p e r tie s o f the t r a n s i s t o r i t s e l f .
Three im portant fa c to r s are:
(1 )
The t r a n s i t - t i m e in the t r a n s i s t o r .
(2 )
The c u t - o f f fre q u e n c f o f the t r a n s i s t o r .
(3)
The storage time o f th e m in o r it y c a r r i e r .
T r a n s it - tim e In the t r a n s i s t o r is e q u iv a le n t t o t r a n s i t - t i m e o f e le c tro n s
in vacuum-tube d e v ic e s .
C u r r e n t - c a r r ie r v e l o c i t i e s
in t r a n s i s t o r s (o r In
any o th e r semi-conductor devices) are very slow compared t o th e v e lo c i t y
o f e le c tro n s in the high-vacuum in t e r e le c tr o d e space o f a tu b e !
Any abrupt
change o f the external f o r c in g fu n c tio n re q u ire s some f i n i t e tim e before
i t makes I t s e l f , f e l t a t th e c o l l e c t o r .
c u r re n t w i l l
d iffu s e d o u t.
In tu r n in g o f f th e t r a n s i s t o r ,
continue f l o w i n g . u n t i l the p re v io u s ly in je c te d c a r r ie r s are
The on and o f f times are p ro p o rtio n a l to th e spacing o f
the base-coI le c t o r .and b a s e -e m itte r ju n c t io n s (th ic k n e s s o f the base).
The c u t - o f f frequency o f the t r a n s i s t o r plays an im portant r o le
in high-speed s w itc h in g devices.
In bur a n a ly s is o f r i s e and f a l l tim e
we w i l l see how d i f f e r e n t tim e -c o n s ta n ts f o r d if fe r e n t - c i r c u i t - c o n f i g u r a t i o n s are associated w ith the c u t - o f f fre q u en cie s (alpha and b e ta ) o f the
tra n s is to r.
I f we consider a t r a n s i s t o r m u l t i v i b r a t o r , we r e a l i z e t h a t
a t some s u f f i c i e n t l y high frequency the re d u c tio n in c u r r e n t g a in , p, w i l l
cause the loop-gain t o drop below u n i t . . I f e x ce s s iv e ly narrow pulses
are used f o r s w itc h in g , the lo o p - a m p I if ic a t ion o f t h e i r h ig h -fre q u e n cy
components may be i n s u f f i c i e n t t o insure a change o f c i r c u i t s t a t e .
For
high-speed s w itc h in g we should then s e le c t t r a n s i s t o r s w ith high c u t - o f f
frequency.
The phenomenon, o f s to ra g e -tim e comes in to p lay when t r a n s i s t o r s
are operated in t h e i r s a tu ra te d re g io n .
In t h i s region both th e ju n c tio n s
are forward biased, and both i n j e c t c a r r i e r s .
While we t r y t o tu rn the
t r a n s i s t o r o f f , forward c u r r e n t continues flo w in g u n t i l these c a r r ie r s
are swept o r d iffu s e d o u t.
But t h i s d i f f u s i o n process takes an appreciable
ti.me, sometimes about ten tim es as' long as th e o n - o f f tim e o f th e
t r a n s i s t o r in i t s a c tiv e re g io n .
Moll has shown how, f o r a given t r a n s i s t o r
c o n fig u r a tio n and a given base d r iv e , t h i s storage tim e is re la t e d to
several t r a n s i s t o r parameters.
We s h a l l . a l s o see how, from c o n s id e ra tio n -
o f the d i f f u s i o n e q u a tio n , these parameters can be reduced t o a s in g le
one, the e f f e c t i v e
l i f e - t i m e o f the m in o r it y c a r r i e r s .
Our a n a ly s is
attem pts a t the e v a lu a tio n o f vario u s f a c t o r s which in flu e n c e the r i s e ,
f a l l , . a n d s to ra g e -tim e in ju n c t io n t r a n s i s t o r s used in s w itc h in g c i r c u i t s .
Chapter 2
T h e o re tic a l A n a ly s is o f Turn-on and Decay-time
Under Varying Load C o n d itio n
2.1
:
S w itching C h a r a c t e r is t ic s .
For s w itc h in g a p p lic a t io n s o f a t r a n s i s t o r i t
is common p r a c tic e
t o discuss the behavior in th re e d i s t i n c t regions o f o p e ra tio n , which are
th e OFF, the a c tiv e , and the ON, or s a tu r a tio n , re g io n s.
have analyzed the ju n c t io n t r i o d e
Ebers and Moil
in t h i s manner, and they have shown
t h a t t h i s device may have very d e s ira b le p r o p e rtie s as a s w itc h : a low ON
impedance, a high OFF impedance, and f a s t s w itc h in g tim e .
In the OFF region both e m itte r-b a s e and c o lle c to r - b a s e diodes
are reverse biased, and the OFF c u r r e n t is determined p r i m a r i l y by the
reverse c u rre n ts I
- I ,
COZ EO
On the o th e r hand,
in the ON re g io n , both
diodes are forward biased and the ON .v o lta g e g e n e ra lly is o n ly a f r a c t io n
o f a v o lt,
provided t h a t ' s e r i e s re s is ta n c e in e m itte r and c o l l e c t o r regions
and/or leads is n e g l i g i b l y sm a ll.
In some types o f t r a n s i s t o r s made by
d if f u s i o n techniques, s e r ie s re s is ta n c e , p a r t i c u l a r l y
in th e c o ll e c t o r ,
may l i m i t the ON-impedance.
In the a c tiv e re g io n , which is the t r a n s i t i o n region between
OFF and ON c o n d itio n s , the t r a n s i s t o r
is described, t o a f i r s t approxim ation,
in terms o f the usual small sig n a l parameters ( a t some in te rm e d iate bias
c o n d itio n s ) .
Governing equation f o r the tu rn -o n tim e in v o lv in g the
various t r a n s i s t o r parameters is developed in the next s e c t i o n .
When th e t r a n s i s t o r is' i n i t i a l l y
in the ON, o r s a tu r a tio n
c o n d itio n , the tim e re q u ire d to sw itch t o the OFF c o n d itio n may be
s ig n ific a n tly
la rg e r than tne ON s w itc h in g tim e .
m in o r ity c a r r i e r storage e f f e c t , noted f i r s t
This is due to the
in diodes,
in which the
OFF s w itc h in g time is lim ite d Dy the tim e re q u ire d to sweep c a r r ie r s out
o f the base re g io n .
The OFF s w itc h in g tim e is d ivid e d in to two periods o f
tim e as f o l l o w s :
Storage time = time in te r v a l between tne re d u c tio n o f c o n tro l c u rr e n t
and the entrance o f the o p e ra tin g p o in t in to tne a c tiv e re g io n .
Decay-time = time re q u ire d a f t e r the o p e ra tin g p o in t nas entered tne
a c tiv e region f o r the c o l l e c t o r c u rr e n t to decay to 10# o f i t s c u rre n t
s a tu r a tio n value.
The phenomenon o f storage tim e w i l l
in Chapter 3.
Decay-time,
o f the t r a n s i s t o r ,
2.2
be d e a lt w ith
in d e ta il
in terms o f tne small signal re p re s e n ta tio n
is evaluated in the next s e c tio n .
E valuation o f Turn-on- and d e la y-tim e s
For low fre q u e n c ie rs , the common-base t r a n s i s t o r is u s u a iiy
modelled by tne T - c i r c u i t shown in F ig .
i.
For tne common-emitter, T - c i r c u i t model
Where
a
is as shown in F ig . 2.
= common-base c u rr e n t-g a m under s h o rt c i r c u i t c o n d itio n s .
/
-/3 Ip
c*
te
Vg
n
o-
r>
o
C
O
J
The model repre se n ts low-frequency o r r e s i s t i v e e f f e c t s o n ly ,
but a t h ig h e r frequencies c a p a c itiv e e f f e c t s must be taken in to account.
Across each J u n c tio n , space-charge and d i f f u s i o n capacitances appear.
At th e e m itte r base ju n c t io n the e m it te r d if f u s i o n capacitance has a
much la rg e r value than the space-charge c a p acita n ce .
In th e h ig h -fre q u e n c y
T - c i r c u i t th e e m itte r d if f u s i o n capacitance is placed as shown in F ig . 3,
At the c o l l e c t o r ju n c t io n space-charge capacitance predominates over th e
d if f u s i o n capacita n ce .
For high fre q u e n c ie s , the common base t r a n s i s t o r
is thus u s u a lly modelled by th e T - c i r c u i t shown in F ig . 3,
Where
c^ = e m itte r d i f f u s i o n .
th e e m itte r ju n c t io n
(The space-charge
§
capacitance a t
is n e g le c te d ,)
cc = space-charge capacitance- a t the c o l l e c t o r j u n c t i o n .
(The
d if f u s i o n capacitance a t th e c o l l e c t o r ju n c t io n is n e g le c te d .)
'
|
In many cases r g <§^r~ » Hence ce may be n e g le c te d . Hence
®
the high frequency T -e q u iv a le n t c i r c u i t , f o r common-base c o n f ig u r a t io n ,
•
reduces to t h a t shown, in F ig , 4,
For - the common-em i t t e r , the F ig , 4 is redrawn as in Fig. 5.
We are in te re s te d in f i n d in g o u t t h e o r e t ic a l expressions f o r tu rn -o n and
decay-time under s a tu ra te d c o n d itio n s under vario u s r e s i s t i v e
loads.
The
f o llo w in g c i r c u i t c o n fig u r a tio n ( F ig , 6 ) . was used f o r comparing th e o r e t ic a l
p r e d ic tio n s w ith the experim ental r e s u lt s *
The incremental model f o r common-emitter c o n fig u r a tio n
is shown
d
te
-o
rc
} Yt
o
r/G, 4
F /G
^ 'G
5
G
Cc
-~o
9
Two loop-equations are:
■ ( 1)
Eg = ! b ( r g + r b + r e ) + l cr e
0
= i b ( r 0 - pZd) . + i c ( r
+ Zd + RL ) ----- (2 )
where Zd = r ^
1 + JO Cdr d
From equations ( I ) and ( 2 ) ,
- _ r e ~ Pzd
r e + zd + r L
?b
In Laplace n o ta tio n #
(S)
r
-
;rd
+ SC ,r.
(S)
r e + RL +
rd
I t scdr d
P
r e - Pr d
= _ r ecdr dCs + r ec dr d
c dr d( r e +
[S + r L + .r e + ^
c dr d( r e + RL>
]
'■ Let th e d r iv e - i n p u t i b< t ) be a c u r r e n t step o f magnitude I ^ ,
A c tu a lly
i t was a c u r r e n t pulse# but f o r c a lc u la t io n s on tu rn -o n time# i t may be
■ taken as a c u r r e n t - s te p w ith o u t any loss in g e n e ra fity tV '*
l b CS) =
1 *1
e - 3r d
(S)
and
•b S
I c C+) = - ' b ,
re
r @+RL
s + r ec dr d
S + -Co—i- ^ L
c dr d( r e + RL )
r Q - p rd
( , :
r e + RL + r d
—t / v
+ rc
&
r e + RL
•C3)
10
Where Y = cdr d( r e + rV
r e +■ R|_ + r y
(4)
= Tim e-constant o f the c i r c u i t .
Turn-on tim e To w i l l be c a lc u la te d f i r s t .
U sua lly low-impedance load-
c o n d ltio n s are assumed f o r s i m p l i c i t y , but here we p e rm it
Turn-on tim e w i l l
90# o f i t s
any valu e .
be taken as the tim e re q u ire d f o r th e c u r r e n t to reatih
l i m i t i n g v a lu e .
L et l c be th e l i m i t i n g va lu e ,
i . e . , Vcc/R[_.
For s i m p l i c i t y In m anipulation,"
K4
''
Then from equation ( 3 ) ,
^
- t/q f /
e
f
J
-t/Y
I ( t ) = - Ik SK’ CI - e
) + K
c
11
For tu rn -o n tim e , wew r i t e equation
c
,
I = - ! h S K 'd
'.9
0
I I.
T = Y In
°
*
,9
— -(5)
(5) as
“ To / ^ •
“ T °/t
- e ■. ) -I- K e .
.
.
g iv in g
~ ^
l c + Ij^K*
----- ( 6 )
At th e end o f the storage tim e , th e load c u r r e n t is s t i l l
i t s value in the c u r r e n t s a tu ra tio n r e g io n ,
very ne arly
The slope o f th e load c u r r e n t
t r a n s i e n t is the value corresponding t o a c tiv e region c i r c u i t va lu e s.
decay-time is obtained from th e a c tiv e - r e g io n parameters.
The
So we can assume
t h a t th e value o f the c o l l e c t o r c u r r e n t a t th e s t a r t o f decay-time is 1'
which, a f t e r the t u r n - o f f s ig n a l
l b^ ,
where l ^
is a p p lie d , approaches e x p o n e n tia lly to
is th e b a s e -cu rre n t immediately a f t e r th e beginning o f th e
t u r n - o ff tra n s ie n t.
C o lle c to r c u r re n t is given by
i ( t ) = i + 1, 3 X 1 C
c
2
Let Tg be the decay-tim e.
.1
Ic =
Ic -
( Ic +
I^ p ) (I
-
e
2/^)
2
or .
T2 - T i n
2*3
Then
-
c
y2'-
'
— (7)
! b P + lc /|0
2
Experimental V e r i f i c a t i o n o f Equations ( 6 ) and (7 ) .
in our e v a lu a tio n s o f tu rn -o n anddecay tim e
( 6 ) and( 7 ),
(I)
re cy
(2)
p
from equations
we s h a ll assume t h a t
25
T “ (ohms fo r,v ‘I 'eexpressed
e
in mAs. )
-or
The procedure involves measurement o f r ^ , q-, rg and c_,
(a )
Ib ,
1^^.
Measurement o f or.
The f o llo w in g t e s t - c i r c u i t was f a b r ic a t e d .
(F ig . 8 )
Procedure:
(1 )
C o lle c to r bias was adjusted f o r re q u ire d value o f
as read on
M3 (30V).
(2)
E m itte r bias was adjusted f o r re q u ire d value o f Ie as read on M|
( I mA).
(3 )
Signal generator frequency was a djusted to 1000 CPS.
(4 )
Amplitude o f the sig n a l generator was adjusted such t h a t an ac v o lta g e -
drop ( V |) o f 100 mV was developed across R| as read on M4 .
(5 )
A.C. vo lta g e drop across R? ( V2 ) was measured
- Vo/Ro
a .f L
vi /ri
= .966
m g
/ 1C
,
r
M l/
-O.
!^ C
V
____
\pOTEHTfOMETER
C6
t v m
(3 = 2 8 .6
(b )
Measurement o f r c and c^,
T e s t - c i r c u i t was as shown below ( F ig . 9 ) ,
Procedure:
(1)
I me in p u t was a p p lie d t o th e b rid g e ,
(2)
The brid g e was balanced w ith S| open and $2 c lo se d .
and Cp^ were
noted.
(3)
Proper bias t o e m it te r ( ImA) and c o l l e c t o r (VCg=30V) were a p p lie d .
(4)
The b rid g e was rebalanced w ith sw itch Sj close d .
noted.
and Cp^ were
(S h o rt c i r c u i t by $2 closed is how replaced by o u tp u t impedance
o f the t r a n s i s t o r ) .
C a lc u la tio n s :
When the sw itch $2 is s h o r t - c i r c u i t e d w ith Sj open, the b rid g e balance c o n d itio n s are:
c Ai
Rp = RB c
K
UN
1
= rB
R/y
— (l)
:
1
J^>Cn
(2 )
When th e s h o r t - c i r c u i t is replaced by the unknown impedance, Zx - Rx - Jxx ,
t ions
th e new balance c:ond
o n diitio
n s are:
^2
RP + Rx = RB " 5 ^
M
“ jx x + — L _
J
CP2
=
— da)
%
— !—
( 2a)
r A J1)C"
The unknown re s is ta n c e Rx and reactance xx are th e r e fo r e r e la t e d t o th e
brid g e parameters by th e expressions
A /V W v ___
1 0 0 -Q .
LF
CHOKE
27 K
DC
VTVM
-n
c
* L
'V<ZvXAv.
W VvW —
/ 6)0
rc
o
■o
-o
/O
15
RB
R
n.. (CA
\ v -- CN
a2 - c.a. )
— (lb)
I
“ jx
x
-
j
!$ )(
C
P2
)
( 2b)
A.C. e q u iv a le n t c i r c u i t fo r Fig. 9 is shown in Fig.
Let r b t
10.
RL B R d
c
Output impedance -
+ —----------------
! + j ^ ccrc
Rd + r c + 6a2r c2cc2Rd - j!«;CGr c2
I + W2CG2r ^ 2
Hence
„
^
Rd + r c +
2r e V Rd
—
c
=
I +
I +
Cf:'
~c
c
r
c
Knowing r ^ , one can e a s il y solve equations (3 ) and (4) f o r r c and c .
(c)
Measurement o r r. .
b
r ^ was measured by measuring h j b .
h j b to r @ t r /g .
I t was assumed t h a t
The t e s t c i r c u i t used is id e n tic a l w ith t h a t shown
in Fig. 7.
Procedure:
(1 )
C o lle c to r bias was adjusted t o 30V.
(2)
E m itte r bias was adjusted f o r re q u ire d value o f l^ (lm A ).
(3 )
Signal generator frequency was a djusted to 1000 CPS.
(4 )
Amplitude o f the sig n a l gene ra to r was adjusted such t n a t an a .c .
vo lta g e drop ( V | ) o f 100 mV was developed across R | •
(5 )
Let V. be the a .c .
inp u t vo lta g e .
(6)
hgk was c a lc u la te d
h ib " 1 L
(d)
Measurement o f
lh
and I, ,
_bl
2
Procedure f o r measurement o f Ik, and I,
■
d e t a il
in Chapter 3,
.
1
is e xplained in
2
EXPERIMENTAL RESULTS
S
r
Ohms
Turn-on Time
. " u-S
C alc.
Expt,
Decay Time
>S.
'
Cc '
.Calc,
Expt.
X&lir
rL
s
mA
' b2
mA
1
c
mA
10 K
.71
.05
4.50
560
6.5
4.98
5,46
15 K
.71
.05
3.00
560
6,5
5.31
18 K
.71
.05
2.50
560
6.5
7.67
22 K
,71
.05
2.07
560
6.5
27 K
.71
.05
1,67
560
6,5
Load Resistance
Storage Time
p,S.
1,82
2,10
.84
5.98
1.87
1.26
.84
8.82
1.89
1,24
,84
10.9
1 1,76
2.0 |
1,05
,84
12,97
13.44
2 .3
'
.756
,84
18
The f o llo w in g photographs show the v a r ia t io n s in t u r n - o n - ,
s to r a g e - , and decay-times as fu n c tio n s o f load re s is ta n c e *
Decay time
increases w ith an increase in load re s is ta n c e w h ile storage tim e is
in s e n s it iv e t o lo a d - v a r r a t io n s .
The experim ental r e s u lt s o f decay and
storage tim e agree w ith th e t h e o r e t ic a l p r e d i c t i o n s .
Assuming a c u rre n t
generator f o r the b a se -d riv e and co nstant t r a n s i s t o r parameters, th e o ry
p r e d ic ts an increase in tu rn -o n tim e w ith an increase in load re s is ta n c e ;
but th e experimental r e s u lt s speak to th e c o n t r a r y .
t i o n f o r th e discrepancy is not t a n g ib l e ,
be th e wide v a r ia t io n s
tim e .
A r ig o ro u s explana­
A probable e x p la n a tio n may
in t r a n s i s t o r parameters during th e fu rn -o n
With an Increase in load r e s is ta n c e , the t r a n s i s t o r operates at
a lower range o f c o l l e c t o r v o lta g e and the e f f e c t i v e
th e o th e r hand, s ince i c
increases.
Is s tr o n g ly dependent on vce a t i t s
th e e f f e c t i v e r ^ decreases w ith increase in lo a d -re s is ta n c e .
t o t a l e f f e c t may be t h a t th e tim e -c o n s ta n t o f the c i r c u i t
On
low va lu e s, .
The sumc ^ r ^ l r g + R[_)
V r e + RL
a c t u a ll y decreases ( instead o f in c re a s in g ) "with an increase in load,,
.
N o n - l in e a r it y in t r a n s i s t o r parameters may be a lo g ic a l e xp la n a tio n f o r
a big discrepancy between the c a lc u la te d and experimental values o f
tu rn -o n tim e as re p o rte d by many a u th o rs .
R, = 27 K
Chapter 3
Physical Basis o f S torage-tim e
3.1
Study o f behavior o f a ju n c t io n t r a n s i s t o r in s a tu r a tin g s w itc h ­
ing c i r c u i t s
is o f fundamental
importance f o r the c i r c u i t designer.
Storage
tim e plays a d e c is iv e r o le in l i m i t i n g the speed o f a s w itc h in g device
which d riv e s a ju n c t io n t r a n s i s t o r in t o s a t u r a t io n .
idea about the s to ra g e -tim e and i t s
To get a q u a l i t a t i v e
impact upon the s w itc h in g , one has
t o know the device physics which causes the s to ra g e -tim e .
From a physical
understanding o f the sto ra g e -tim e mechanism, t r a n s i s t o r and c i r c u i t
designers, pressed between c o n f l i c t i n g demands, may gain an idea as to
how to o p tim iz e a t r a n s i s t o r - d e s i g n .
As the term suggests, the phenomenon is associated w ith storage
o f some kind o f charge.
For our case, i t
in the base region o f the t r a n s i s t o r .
is the m in o r it y c a r r i e r storage
To be more s p e c i f i c ,
i f we co n sid e r
a p-n-p t r a n s i s t o r , s to ra g e -tim e is associated w ith th e storage o f holes
( m in o r ity c a r r i e r s )
in the base ( n - r e g io n ).
To understand how the phenomenon occurs,
l e t us in v e s tig a te the
charge d i s t r i b u t i o n
in the base under va rio u s o p e ra tin g c o n d itio n s .
Charge d i s t r i b u t i o n
in the base region is governed by the c o n t i n u i t y
e q uation, which under s te a d y -s ta te c o n d itio n s is :
°p a x i"
= °‘
,
.... . ( i )
A
Yp~ l i f e - t i m e f o r holes
&
Dp- d if f u s i o n c o n sta n t f o r holes
'
20
p= con ce n tra ti o n o f holes as a f u n c t i o n o f k
Nov/ X is taken t o be zero at the e m it te r - e n d and w a t the coI Iector-end
o f the base and
A
pn= e q u i l i b r i u m value o f holes in the base-region.
Since we are concerned here w it h a q u a l i t a t i v e
i n v e s t i g a t i o n o f hole-
d i s t r i b u t i o n , we may n e g l e c t , f o r s i m p l i c i t y , the recombination term in
the above c o n t i n u i t y equation- so t h a t the equation becomes
d^p
DP Tn^ =
(i2 >
A solution
is
P = A + Bk
( i ? § ) where
A & B
are a r b i t r a r y constants t o be evaluated from the boundary c o n d i ti o n s .
Let the e m i t t e r voltage be Ve .
Then
from the Boltzman r e l a t i o n
ship the ho I e -c o n c e n tr a ti o n a t the e m i t t e r - b a r r i e r
p Sve
Pe = Pn
KT
( 3i
Again,
= o) is
j )
i f vc be the c o l l e c t o r v o lta ge , the hoi e - c o nc e n tr a ti o n at the
col I e c t o r - b a r r i e r is
Pc ° Pn e -KT
.
( ik )
For a p-n-p t r a n s i s t o r Vc is -ve and is o f the order -15 v.
temperature KT py 25 mv.
But
So, f o r a l l p r a c t i c a l purposes
q.
Pc « 0
(50
Applying the boundary c o n d i ti o n (13 i ) & (5) t o (12"a ), we have
^ -qVe.
A =.Pn" KT"
and
B = -Pne -^2.
W- •
so t h a t •
Graphical re p re s e n ta tio n o f
p = pn®
( 'O
C1 “ w ]
) Is shown in Fig.
('6i )
I.
at room-
The c u r r e n t f l o w i n g across the, base
j
= -qDp
b
dx
= qDp -L-- e
M
throughout the base-region.
# so t h a t the c u r r e n t is uniform
■
This conclusion
o f the ne gl e ct o f the recombination term.
i s , o f course, the
outcome
Recombination causes the
c o l l e c t o r - c u r r e n t t o be less than the e m i t t e r c u r r e n t .
Hence, a c t u a l l y
- dp. is a small percentage large r at the e m it t e r .than a t the c o l l e c t o r ,
dx
Hence, the e f f e c t o f recombination is t o make the above id e a liz e d curve
s l i g h t l y concave upward, as shown in Fig. 2.
Let us now consider the t r a n s i s t o r re p re se n ta ti o n ( F ig . 3)
p o s i t i v e sense o f c u r re n t s
being represented by the
assume t h a t the t r a n s i s t o r
is o f f ,
biased.
arrows, and l e t us
i . e . both the j u n c t i o n s are reverse
In the o f f - s t a t e
'b « - ' c O .
and hence
.
U +~ 1c = “ 1co
:
■
and hence, because o f the symmetry o f ©up t r a n s i s t o r r e p re s e n t a t i o n ,
- l co/2 w i l l
bution is
flo w through each j u n c t i o n so t h a t the slope o f the d i s t r i ­
l co/2qDp at the
the charge d i s t r i b u t i o n
in
e m i t t e r and - l co/2qDp at t h e c o l l e c t o r .
Hence
the o f f - c p n d i t i o n may be represented as in
Fig . 4.
Now l e t us assume t h a t a tu rn -o n s i g n a l , a -Fve c u r r e n t pulse,
is applied t o the base o f the t r a n s i s t o r .
I t w i l l take some time (known
as tu rn -o n delay time t ^ ) t o br in g the t r a n s i s t o r from t h e o f f - s t a t e to
the edge o f conduction.
J*
Mathematically t h i s may be represented as
l b | ( t , d + = Scrff
23
/ a/
Fts,
F IG
>
2
r t i i
VA
i
-------- >
5
£
x=0
n
rx = N
H
F / G . v3
1
A
'X z h /
F iG ’ S
25.
W h e r e ' i s
the t o t a l
base-charge which must be s u ppl ied t o the
t r a n s i s t o r t o switch i t from the o f f - s t a t e t o the edge o f co n duc ti on .
As the t r a n s i s t o r
in the base.
is switched t o o n - c o n d i t i o n , holes begin t o be stored
In the e a r l y stages o f on-cond i t ion, the slope a t the
e m i t t e r b a r r i e r is much g r e at er than t h a t a t c o l l e c t o r b a r r i e r because
c o l l e c t o r d i f f u s i o n c u r r e n t takes some time t o b u i l d up.
This can be
seen by examining the c o l l e c t o r c u r r e n t b u il d - u p f o r a u n i t step input
at the base.
Assuming t h a t s h o r t - c i r c u i t e d ou tpu t c o n d i t i o n s e x i s t , we
have
#o
' LCS) = — ----- --------------s u - c 0+s / t i n )
=
®o
""
“
u - ffn )s
cv0
I
l "“ °
( l ” c'o ) H i +s
-
so t h a t
a
- ( l - 0fo ) w nti
'fo
i (f) = TZZ~ [ I - 6
]j. i n d i c a t i n g t h a t the c o l l e c t o r
' c ' " " l- a o
c u r r e n t b u il d s up e x p o n e n t i a l l y .
E v e n t u a l l y , o f course, an appreciable
d i f f u s i o n c u r r e n t flows in the c o l l e c t o r and the d i s t r i b u t i o n approxi­
mates a s t e a d y - s t a t e d i s t r i b u t i o n as shown in Fig. 2.
In terms o f charge
storage, the r i s e - t i m e may be represented as
j /
i b l ( t ) d t = l f RE I cs
Where^RE is the t o t a l charge su pplied t o the t r a n s i s t o r per u n i t switched
c o l l e c t o r c u r r e n t I q j during the r i s e - t i m e i n t e r v a l .
Now i f the base
current- continues f l o w i n g , c o l l e c t o r d i f f u s i o n c u r r e n t reaches a p o i n t
lc g beyond which i t cannot Increase f u r t h e r given by IqsRq = -V where
Rq
is the c o l l e c t o r load and V is the c o l l e c t o r bias v o lt a g e .
But no th in g
prevents the increase in the ho I e -s to ra ge in the base so f a r as chargen e u t r a I i t y is maintained.
So i t
is p o s s ib l e t o increase the t o t a l hole
storage in the base up t o a p o i n t where I t i m e s
t o t a l stored charge
26
equals the b a s e- c u r r e n t.
Hole d i s t r i b u t i o n
is now o f the form as shown
in Fig . 5, showing an appreciable amount o f h o l e - d e n s i t y a t X = W.
h o le -s to r a g e a t the c o l l e c t o r - b a r r i e r j u n c t i o n
The
in d ic a te s t h a t the c o l l e c t o r
b a r r i e r vo lt a ge Vc remains clamped a t a vo lta ge equal t o or great er than
zero# s a t i s f y i n g th e Boltzmanslrelationship
' w
Mow, i f a t u r n - o f f signal# say a -ve c u r r e n t pulse a t the base# is appl ied
reverse I
cannot fl o w immediately because o f +ve value o f Vc due t o h o le -
storage a t c o l l e c t o r - b a r r i e r j u n c t i o n .
Forward c u r r e n t continues fl o w in g
u n t i l e x tr a c a r r i e r s are swept or d i f f u s e d out such t h a t Vc goes t o zero.
A f t e r the i n i t i a t i o n o f the t u r n - o f f # holes are depleted by the recombina­
t i o n process and the negative b a s e- cu r r e n t. The t i m e - i n t e r v a l between the
i n i t i a t i o n o f the t u r n - o f f s ig n a l and r e t u r n i n g o f Vc " t o zero is known as
storage ti m e .
M ath ematically, we can represent the s to r a g e - t im e Ts
by
•^0,S' b 2 ^
d"*" "
*BX# where % ' s "*'he excess stored base-charge
per u n i t excess base-current#
1^ .
P h y s i c a l l y we thus r e a l i z e t h a t stor a g e- ti me is c o n t r o l l e d by
the magnitude o f tu rn -o n and t u r n - o f f sign al and by the l i f e - t i m e o f the
m inority c a rrie rs .
We also r e a l i z e t h a t f o r a given b ia s in g and a given
tu r n - o n and t u r n - o f f signal# storage time w i l l decrease f o r a decrease in
c o l l e c t o r Io a d - r e s is ta n ce .
We also see t h a t in very high-speed s w itc hi n g
c i r c u i t s the t r a n s i s t o r should be prevented from s a t u r a t i n g .
27
Increasing D r iv i n g Current
Photographs showing q u a l i t a t i v e l y t u r n - o n , storage and decaytimes ( f o r a common-emitter c o n f i g u r a t i o n ) as f u n c ti o n s o f d r i v i n g
current.
Turn-on time decreases w ith increase in tur n - o n d r i v e ;
s to ra g e- tim e increases w it h an increase in turn -on d r i v e and decreases
w it h an increase in t u r n - o f f d r i v e .
The leading and t r a i l i n g edges
o f the waveforms show the decrease in tur n - o n and decay-times with
increase in d r i v i n g c u r r e n t w hi le the gradual widening o f the waveforms
shows the increase in storage time w it h an increase in d r i v i n g c u r r e n t .
28
3.2
Storage-time and i t s Evaluation from Tr an sis to r-p ar am ete rs
During the l a s t decade many a r t i c l e s have appeared in the l i t e r a ­
t u r e concerning the. behavior o f a j u n c t i o n - t r a n s i s t o r
ing c i r c u i t s .
in s a t u r a t i n g s w itc h ­
Of co nsiderable importance in a saturated switch is an
e f f e c t known as s to ra g e- tim e or t u r n - o f f delay time.
The device-physics
which causes stor a g e- ti m e is q u it e well understood, but the mathematics
involved in r e l a t i n g
i t t o the o r d i n a r y device parameters is very d i f f i c u l t .
Two e n t i r e l y d i f f e r e n t approaches have been made t o f i n d an
expression f o r the storage tim e.
The f i r s t one has been t o solve the d i f f u s i o n
equation under ap pro p ria te boundary c o n d i t i o n s .
The method is beset with
mathematical d i f f i c u l t i e s and moreover, the fu n c ti o n a l r e l a t i o n s are very
d i f f i c u l t t o work w i t h .
The o th e r approach,
i n i t i a t e d by Moll, aims at
breaking the n o n - l i n e a r problem in t o two e s s e n t i a l l y l i n e a r problems which
can be solved more e a s i l y .
A clue t h a t i t might be po ss ib le t o consider the saturated
t r a n s i s t o r as two a c t i v e t r a n s i s t o r s placed back t o back comes from n o ti n g
t h a t the base-charge Qg may be considered due t o two components.
regarded as in Fig.
I t may be
I.
%
= %C + % E
QgC is the charge t h a t would be present i f the c o l l e c t o r - b a s e p o t e n t i a l
were zero.
QBE is the charge t h a t would be present i f th e em itter-base
p o t e n t i a l had i t s real value and the c o l l e c t o r - b a s e p o t e n t i a l were zero.
This d i v i s i o n o f c a r r i e r - d e n s i t y suggests the d i v i s i o n o f e m i t t e r and c o l l e c t o r
c u r re n t s i n t o two components which are associated with emission and c o l l e c t i o n .
Thus f o r ( n - p - n )
!>?
M iN O R /T Y '
CAR
y
30
' e = 'Ef + 'Er
'c = " ' s f + ' c r
where I
1^^. = c u r r e n t due t o emission from the e m i t t e r or c o l l e c t o r
and Igr> l r r = c u r r e n t due t o c o l l e c t i o n a t the e m i t t e r o r c o l l e c t o r .
Since the two processes are independent, the usual c u r r e n t - g a i n r e l a t i o n s
are
""" ( I )
n
Ts/Wn ( l e f (S ) ' where “ n and Wn are +he normal
01
U
s m a ll -s i g n a l
(S)
a c t i v e region c u r r e n t gain
and frequency c u t - o f f
and
Qf,
l»r (S) = ------ 1
---------
Ipf(S)
(2 )
i + s/ t o
where or, and
which r e s u l t
in t e r v a l
are the s m a ll - s i g n a l c u r r e n t gain and frequency c u t - o f f
i f the t r a n s i s t o r
between
is i n v e r t e d .
the i n i t i a t i o n o f the t u r n - o f f signal and the
which Ipf goes t o zero.
( I - Q'nQ' | ) ^
|
Jo
^n
tran sfor m method, namely
»n + W,
s » . g.
■ T, •
*
we a r r i v e at Mol I ' s resu I t s by Lap I ace-
,
,
I t,
l E2 ^2
step.
time at
Making use o f the i n e q u a l i t y
W,
The s u b s c r ip ts I
The s to ra g e- ti m e is the ti m e -
'e,
^|
n
and2 r e f e r t o values before and a f t e r the t u r n - o f f
Or in terms o f ba se- currents,
T,
In
“’n“,| ( l ' V l )
' Bl " !b2
— ( 3 )
'c
Evaluation o f sto ra g e- tim e from (3) involves fou r measurements;
Cfr , or,, 6Un and W, .
i.e .,
But i t can be r e a d i l y shown t h a t W + W|
ULWl ( l-cvncy )
is equal
31
t o the e f f e c t i v e m i n o r i t y c a r r i e r
ly by p la ci n g the t r a n s i s t o r
l i f e - t i m e , which can be measured accurate­
in tandem with a c u r r e n t - s e n s i t i v e device.
Temporary i r r a d i a t i o n o f the t r a n s i s t o r w i l l
and withdrawal o f the r a d i a t i o n source w i l l
exponent i a I Iy w it h the t i m e - c o n s t a n t ? ,
increase i t s c a r r i e r density
r e s u l t in a c u r r e n t f a l l i n g
l i f e - t i m e o f the m i n o r i t y - c a r r i e r
density.
Let I bg denote the minimum ba se-current which s a tu r a te s the
transistor
in a given c i r c u i t .
I - 1 .
D BS
We denote by Qq the
3
I Bg, and by
I
1^ ,
excess base-current, w i l l
represent
base-charge present wit h a base c u r r e n t of
the base-charge in excess o f Qs when a ba se-current
+ I x is f l o w i n g .
BX
Keeping in mind t h a t the r e c o m b in at io n - r a te o f
a p a r t i c u l a r group o f c a r r i e r s ,
p,
is given by
dB. = - _ E e z l = P o ' P
dt
r
r
We note t h a t
T
-
m inority c a rrie r
life-tim e
= Qsx
’ bx
Now Q =
I c , where
is the
common-base s h o r t c i r c u i t c u r r e n t gain c u t -
oO
o f f frequency.
Proceeding as before
9b = J eI +
Just at the s t a r t o f s a t u r a t i o n .
'CF
°'
! EF = —
n
32
Hence Qs
‘C
Excess charge is then
9sx = 9B - 9S = 1 ^
( l Ef " j c )
^ n
I
making use o f the expressions
' bx = ' b " ' bs
E
-
'c + ' b
ER ' " l
Cf
CR " a n Ef
and equation ( I ) we ob tai n
I
'
-<
BX 1 - a-.Q'
and ' c f = i b x (t ^ ^ >
50 t h a t -
+ul
Qsx = 'Bx Jn^u-ofnQf,
. TT n - ^ ; )
so thatcv n +V,
= y
nO | ( I - QfnQf )
=Qsx
1BX
Hence, the storage time T | , may be w r i t t e n as
T, =T In
1R “ 1D
I
2
(4)
C |('-*n ) _ |
—
B2
an
As regards the a p p l i c a t i o n o f the d i f f u s i o n equation t o the e v a lu a ti o n o f
storage time, several attempts have been made.
been successful
Lax and Neustadter have
in e v a lu a ti n g the ho I e- storage deI ay-time in a semiconductor
diode by s o lv i n g the d i f f u s i o n equation wit h a p pro pr iat e boundary c o n di tio n s
33
Kingston has extended the treatment to a j u n c t i o n t r a n s i s t o r ,
assuming
t h a t W 4 Lp and t h a t the c o l l e c t o r j u n c t i o n acts as a sink f o r holes at
a l l times.
But K in gs to n 's assumption t h a t the col I e c t o r - j u n c t i o n
always a si nk f o r holes is not v a l i d as the t r a n s i s t o r
saturation.
As w i l l
is
is run into
be shown below, the d i f f u s i o n equation under general­
ized input and boundary c o n d i ti o n s cannot be solved at the present s ta te
o f mathematical development.
Under the impact o f a t i m e - v a r y i n g input, the c o n t i n u i t y equation
f o r the m i n o r i t y c a r r i e r de ns ity in a p-n-p t r a n s i s t o r takes the form
= - —
+ Dp
f
Tp r
(5)
d
Px
l i f e - t i m e f o r holes
n A
P = Hole d e ns it y in the base-region, p is a fu n c ti o n o f both x (distance
along the base-w i d t h ) and time, t .
Pn = E q u i l i b r i u m
density of
holes
in
the
base r e g i o n .
Since pn is independent o f n and t , equation (5) may be w r i t t e n as
dt ' P - P " ' = " ^ t f + ° P ^ 2
tF
<P-Pn)
[<P-Po> + ( Po " Pn’ l
= -<P-P0 )+(P -Pn >
-----------------------
d2
+Dp^ 2 C<P-P0 ’ + <po ' pn ) ]
Since ( P0- Pn ) i s independent o f t ,
i + ( p- p0 > = ( ^
' " " (6)
the equation ( 6 ) may be w r i t t e n as
+ D p^ 2
'P - P o ’ ’
+ ( - (Pn - pn ) + Dp d2
Tp
L P ^ 2 (P0-Pn ) >
---(7)
Now ( P0- p n ) can be evaluated by s o l v i n g the c o n t i n u i t y equation
34
2
Dp
d (p_pn )
P0 Pn
w ith the boundary c o n d i t i o n s :
T P
dx2
p
= p eqVeo/KT
eo
n
p
co
= p e qVco/KT
n
I f the o r i g i n o f the x -c o o r d i n a t e be chosen as the m i d - p o i n t o f the baseregion (x = -WQ and x = WQ as the coo rdin ate o f e m i t t e r andc o l l e c t o r
j u n c t i o n s r e s p e c t i v e l y ) we have
p
-p
°
=
n
( P Co " Pn )
S in h
[(W o -x )/L p ]
+
(Pc o - P n )
S in h
(wo + x )
-------------------------------------------------------------------------------P=zz( 8 )
Sinh (2W0 /Lp)
(8)
where Lp = ( YpOp)
S u b s t i t u t i o n o f ( 8 ) in (7) leads to
i
(p- p ° ) =, l l f o l + DP
(p_po)
Equation (9) is the d i f f e r e n t i a l
-
(9)
equation f o r the a .c . super-imposed h o ie -
d e n s i ty . Under general ized input c o n d i t i o n , say a square c u r r e n t
the input, the
c l a s s i c a l method o f s o l v i n g (9) f a i l s . Let
wave at
us t r y the
Laplace tra n sf o rm method.
Let p , ( x , s ) = L [ p ( x , t ) - p0 ( x ) ]
Then
p(x,o) -
L [ — ( p ( x , t ) - pQ(x ) Q. = Spj, t a k i n g
p ( x ) = 0, which means t h a t at ze ro-time the a . c .
signal is zero.
The tra n sf o rm o f equation (9) is
SPI = " p - + Dp f l L
T p
dx2
or
dp,
A
~2
dx2
-
I
— 2 ( I + S^Yp) p | = 0
Lp'
A general s o l u t i o n o f equation (10) is
( 10)
( L_ t s Yp,)
_ - ( | +S1fp)
Lp
+ A2e
---- (%i)
p | ( x , s ) - A|e
To evaluate the two a r b i t r a r y co nstants,
it
inverse Laplace tran sfor m o f equation (x i ).
is necessary t o take the
Let us consider the p o s s i b i l i t y
o f t a k i n g inverse tran sfo rm o f p | ( x , S ).
The inverse tran sfor m o f a fu n c ti o n f ( S ) is given by
f(t) =
_L_
f c + jcc
2 7 ij
\
f(S) e
J
The l i n e - i n t e g r a l
dS
(1 2 )
C-joc
f o r f ( t ) is u s u a l l y evaluated by tr a n s fo r m i n g i t in t o
a closed contour and ap plying the c a lc u l u s o f residues.
The contour u s u a l l y
chosen is shown in the Fig. 2.
Let us take as the closed contour P the s t r a i g h t l i n e p a r a l l e l
to the axis o f imaginaries and at a dista nce C t o the r i g h t o f i t and the
i n f i n i t e s e m i - c i r c l e whose center is at (CjO).
fl
d)
SI
e
tO ^ t
f(S)dS = 1
g-(-
e
We then have
St
f(S)dS+-^
e
f (S)dS
J C- ice
where C is chosen g re at enough so t h a t a l l the s i n g u l a r i t i e s o f the
integrand l i e t o the l e f t o f the s t r a i g h t l i n e .
The ev a lu a tio n o f the contour in t e g r a l
is u s u a l l y performed by i n ­
voking Jordan's lemma which in t h i s case may be s tat ed in the f o l l o w i n g
form:
Let f ( S ) be an i n te gr a b Ie f u n c t i o n o f the complex v a r i a b l e S
such t h a t
I im
I S I ->cc
Then
lf(S )l= 0
.
I im
R->o:
f
|J ,
c+
e
f(S)dSI
36
I f f ( S ) s a t i s f i e s the ^conditions f o r Jordan's lemma, the i n t e g r a l around
the i n f i n i t e s e m i - c i r c l e vanishes and we have
G + to=
f(t) =
\
s+
e f(S)dS
c - ice
= 2Aj
W
p
eS+f(S)dS
S"f*
= E Residues o f e f ( S ) in si d e p.
Returning t o equation ( x i ), we note t h a t the fu n c t i o n
x
- ( 1 + s Tp )s
e ——:----------- s a t i s f i e s the c o n d i ti o n s f o r Jordan's lemma, but
■ Lp
• i_
6
—— J l l L . -does n o t . Thus we' conclude t h a t p j ( x , " t ) cannot be found '
■ LP
,
o u t . The d i f f u s i o n equation is, amenable t o s o l u t i o n o n l y when the inpu t
f u n c ti o n s are some elementary fu n c t i o n s (such as s i n e , cosine or e x p o n e n ti a l)
o f time and the ti m e - v a r y i n g s i g n a l s a t the e m i t t e r - j ' u n c t i o n and c o l l e c t o r
j u n c t i o n s are such t h a t they are small compared t o KT/q (%, ,025V).
But these
c o n d i t i o n s are d e f i n i t e l y not v a l i d f o r a saturated t r a n s i s t o r .
Thus we note t h a t a n al y si s o f a saturated t r a n s i s t o r in terms
o f two l i n e a r a c t i v e t r a n s i s t o r s placed back t o back is th e on ly way
(though inadequate and sometimes erroneous) t o p r e d i c t th e storage time
in a given t r a n s i s t o r .
M o i l ' s expression, equation ( i i i ) ,
s to ra g e- ti m e in terms o f parameters arn, afj,
and
,
gives the
As evident from
equation ( ?v), one can p r e d i c t s to ra ge- tim es by a s i n g l e measurement o f
e f f e c t i v e l i f e - t i m e f o r ho les.
has been suggested by Nanabati,
An a l t e r n a t e (and very usefu.l ) approach
Proceeding along l in e s s i m i l a r t o M o ll ,
he w r i t e s ( f o r a p -n -p ) in Laplace tra n s fo r m n o ta ti o n
37
F
f &
t 2.
38
1^(5) = Igp(S) - o’ ! I q^ (S ).
' c (S) = % 'E F < S ' + 'CR(S)
and VCB(S) = Rc I cr (S)
VEB(S) = Re ( S ) I e f (S)
He f u r t h e r assumes t h a t Rq (S) and RP(S) have the form
RC(S) = Rc
I
1 + S / CO|
and w r i t e s f o r
,
n o tin g t h a t a l l
the parameters r e f e r t o t h e i r values
in s a t u r a t i o n :
2
=
21
RP(S) -
(S)
1 e
I - of ( S )of ( S)
I
n
u R
which becomes a f t e r some manipulation
z_ =
■21
7 ^
CS
° l ( r E " Qfny
W^(S +Wg)
" «n'BX
Iq
<
I + W |/u)n
where:
V A = y i +" n
0}
.
= " l a n(l ~
y I +^ n
The ha If-power frequency o f Z2|
is o b v io u s ly :
^Co = %
This is a measurement o f the frequency c u t - o f f o f Z21 and should give
us s to r a g e - t im e , provided the i n e q u a l i t i e s are s a t i s f i e d .
The Z ^
parameter is an open-c i rcu i t measurement which should not be d i f f i c u l t
in s a t u r a t i o n .
I f Ci>n,
U>^,
and a n were known f o r a given
39
typ e, so t h a t the i n e q u a l i t y could be s a t i s f i e d , then w i t h one measurement
the s to ra g e- tim e can be determined.
3.3
Measurement o f Storage-time
Procedures f o r measurement o f stor ag e-t im e are indicated in
Fi g , 3.
The T e s t - C i r c u i t is i d e n t i c a l with t h a t f o r measurement o f tur n -o n
and d e la y- ti m e .
RE.LATH/Z
A M P L IT U D E
40
k~
t
0
—>|
RESPONSE
F /
To
T, =
=
(y- / >3
T U R W -O N
STO R AG E
7^ = D E C A Y
T IM E
'T/ME
41
3 ,4
V e r i f i c a t i o n o f M o l l ' s Equation f o r Storage-time
V e r i f i c a t i o n o f M o l l ' s equation f o r storage time f o r commonem itter configuration
involves measurements o f ^ ^ ,
< ^ |f
Igj and I ( S y m b o l s are explained in sectio n 3 , 2 . )
(Fig,
ant
l Cj ,
The f o l l o w i n g c i r c u i t
I) was used f o r measurements o f of andfe? .
n
n
For measurements o f q/ j and 0 ^
the c i r c u i t was as in Fi g . 1,
Em it te r and c o l l e c t o r were interchanged.
C o l l e c t o r vo lt a g e was adjusted
t o 3V,
F ig , 2 shows the c i r c u i t f o r measuring I 5 * l^ . I
and storage
. 1 . 2 . 1
tim e .
Procedure:
(1)
The d r i v e - i n p u t was adjusted t o a large enough amplitude t o ensure
s a t u r a t i o n o f the t r a n s i s t o r ,
(2)
The vo lt a g e across FT? was examined on a C.R.O,
The wave was o f the
form ( F i g , 3 ) ,
Vj and V? were measured by c a l i b r a t i n g the C.R.O.
I^
and 1^
were obtained
as
•
'b , =
(3)
v!
iq
V2
-
•
By h o r i z o n t a l p o s i t i o n c o n t r o l , the t u r n - o f f i n s t a n t o f the d r i v e -
input (the i n s t a n t corresponding t o p o r t i o n a-b in Fi g , 3) was set a t a
known p o s i t i o n o f the g r a t i c u l e .
Synchronization was su ppli ed d i r e c t l y
42
P R E C iS fO N
/K ,
y 25 X
____
POT,
/ ^
__________A A A A A
M ETEH
M E - 7 'E R
P t G> J
Q U IE S C E N T
V A L U E S ,'E ^ 'T J E R C U R R E N T t mA, COLLECTOR VOLT, 2 S V,
VW W V
56 0 n
r
F/G, 2
43
from the square-wave g e n e r a t o r ,
The sweep c o n tr o l was adjusted such
t h a t I cm o f the C.R*0. scale corresponds t o . Ip,s,
(4)
C o l l e c t o r wave-form was examined and procedure f o r measuring s to r a g e-
time is i l l u s t r a t e d
(5)
in Fi g. 4,
The d e t a i l s involved in measuring l c ^ is i l l u s t r a t e d
in Fig. 5.
— Tf
t
FIG.
t
v3
™
I
' - :T f M E
|
/a //
'
'
C O R G E S P O fV D /fi/G
S / G f\J A L
F /G , F
-> 6
x5
"T O
77/4 r/O A / Q/=' T/yE T U R N - O P E
EXPERIMENTAL RESULTS
Parameter
pn
Qfn
pi
aI
(ra ds/ se c)
rads/sec
' bl
y
1
measured
Storage time (u,s)
Calculated
with i n v a l i d
assumption
Transistor
. type
2NI IS
28,6 ,966 1,3
.56
12.56x10
5
8 . 8x 105
,68
1,6mA 7,5mA
,40
.7-8
no
assumption
.52
4^
Ul
46
Chapter 4
Junct io n -te tro d e s
Our main i n t e r e s t has been the in v e s tig a tio n
in to the problems
o f turn-on# deday-# and storage times in a sa tu ra te d t r a n s i s t o r used f o r
s w itc h in g a p p lic a tio n 's .
In Chapters 2 and 3 we have gained an in s ig h t
in to how these th re e fa c to r s are r e la t e d to some t r a n s i s t o r parameters.
In p a r t i c u l a r , we have n o tic e d the im portant parts' played byY»h, c_ and f
c
<*3
in c o n t r o l l i n g tu rn -o n and decay tim e .
S im ila r ly # these th re e parameters
are re s p o n s ib le f o r d e t e r io r a t i n g h ig h -fre q u e n cy performance in a t r a n s i s t o r .
A designer# s e t to improve the high frequency performance o f a t r a n s i s t o r ,
o fte n faces c o n f l i c t i n g demands on these th re e parameters.
te tro d e
is one innovation t o meet these challe ng e s.
very good high frequency p r o p e r tie s ,
A tra n s is to r
A t e tr o d e , having
improves tu rn -o n and d e la y-tim e to
a g re a t e x te n t.
Storage time also can be minimized in a t e tr o d e .
For a given
input and b ia s in g c o n d itio n , a s a tu ra te d te tro d e w i l l e x h i b i t less storage
tim e (depending on second base b ia sin g ) t h a f i a corresponding t r i o d e .
M o ll 's
expression f o r storage tim e , quoted in Chapter 3, shows t h a t w ith decrease
in of, storage time should f a l l .
In a t r i o d e , a s w itc h in g c i r c u i t designed
has no c o n tro l over a# but in the te tro d e he can vary of according t o h is
p a r t i c u l a r demands and thus can c o n tro l storage tim e .
In the present
chapter we s h a ll see# in the o ry and experiment, how these fa v o ra b le
developments may be brought aigout in a te tr o d e .
47
Tetrodes c o n s t i t u t e an attem pt a t compromise between the
vario u s c o n f l i c t i n g fa c to r s t h a t c o n tr ib u te to the d e t e r io r a t i n g h ig h frequency performance o f a ju n c tio n t r a n s i s t o r .
The h ig h -fre q u e n c y
l i m i t a t i o n s o f a ju n c t io n t r a n s i s t o r r e s u l t mainly from a combination o f
th re e causes:
(1)
The b a r r i e r capacitance o f the c o l l e c t o r ju n c tio n (c c ).
(2 )
The alpha c u t - o f f ( f ^ ) .
(3)
The base re s is ta n c e ( r ^ ) .
When h ig h -fre q u en cy c o n s id e ra tio n s are im po rta n t, i t
base re s is ta n c e as low as p o s s ib le .
w ill
is necessary to keep
This is because th e a m p lif ic a t io n
f a l l o f f more r a p id l y w ith frequency
i f r ^ is increased
w h ile leaving
a l l o th e r c i r c u i t constants unchanged.
The c o l l e c t o r capacitance can be reduced by ( I ) decreasing the
ju n c tio n area; ( 2 ) reducing the c o n d u c t iv it y o f the base and c o ll e c t o r
re g io n s .
F a b ric a tio n d i f f i c u l t i e s and power c o n s id e ra tio n s l i m i t how f a r
one can proceed along the l i n e s ' o f ( I ) .
Reduction o f c o n d u c t iv ity in the
c o l l e c t o r region (w ith o u t a l t e r i n g t h a t o f the e m it t e r )
in g ro w n -ju n c tio n types w h ile re d u ctio n
r e s u lt s
is po ssib le o n ly
in c o n d u c t iv ity o f the base-region
in g re a te r r ^ , which is th e r e fo r e a step in the wrong d ir e c t io n .
The alpha c u t - o f f frequency can be ra is e d by reducing the base
th ic k n e s s .
To be s p e c i f i c , Shockley has shown th a t a lp h a -c u t o f f frequency
should be in v e rs e ly p ro p o rtio n a l to the square o f the base th ic k n e s s .
This r e d u c tio n , however, a lso increases the base re s is ta n c e by decreasing
the c ro s s -s e c tio n a l area through which the base c u r r e n t flo w s .
I t is thus apparent t h a t the obvious steps t h a t w i l l reduce
the c o l l e c t o r capacitance or increase the a lp h a -cu t o f f frequency also
b z
c o llc c t-o r
E M /T -T E -C L
-
F IG .
/
CO LLECTO R
BASE
FIG .
49
increase the u n d e sira ble e f f e c t s due to r b .
Converseiy, steps taken
to reduce r b, such as in cre a s in g the c o n d u c t iv it y o f the b a s e -m a te ria l,
o r incre a sing the b a s e -thickn e ss, have an unfavorable r e s u l t on both c
c
and f .
Of
In the te tr o d e reducing r b does not involve u nfavorable e f f e c t s
on c
c
and f .
a
The geometrical c o n fig u r a tio n and symbolic re p re s e n ta tio n o f a
te tro d e are shown in F ig .
I and F ig . 2.
They are e x a c tly id e n tic a l to
these fo r a ju n c tio n t r a n s i s t o r w ith the exception t h a t a fo u r th e le c tro d e
is added (a second connection) to the base.
same as f o r a normal t r a n s i s t o r .
The b ia s in g arrangement is the
Thus, f o r a p-n-p j u n c t io n - t e t r o d e e m it te r
is tv® w ith respect to base and col le c to r is -ve w ith re sp e ct to base,
W hile a d .c . bias is su p p lie d to the second base te rm in a l w ith such p o l a r i t y
as to fo rc e th e e m it t e r - t o - c o I le c to r c u r r e n t down in to a very small region
in the immediate v i c i n i t y o f base I.
This is made p o s s ib le by making the
e m itte r reverse biased w ith re spect to base 2, since the e m it te r ju n c tio n
near b2, being reverse biased, does not em it e le c tro n s in to the n - la y e r .
This makes a l l the t r a n s i s t o r a c tio n take place very near the base I con­
t a c t , and the whole mechanism is e q u iv a le n t to c o n v e rtin g a t r a n s i s t o r
large enough t o be fa b r ic a t e d in to a t r a n s i s t o r having a very much sm a lIe r
e f f e c t i v e c ro s s -s e c t io n .
The advantage t h a t r e s u lt s is t h a t the base-
re s is ta n c e can be kept low even w ith a very th in
a high fee).
l i g h t l y doped base ( g iv i n g
Thus from the above c o n s id e ra tio n s i t
is apparent th a t one
doped
can use is ju n c t io n - te tr o d e s somewhat th in n e r and more lig h t l y / b a s e layers
than u s u a l. to o b ta in high a lp h a - c u t - o f f and sm aller c o l l e c t o r ju n c tio n
(re du cin g c o l l e c t o r ca p a cita n c e ),
leading t o improvement in h ig h -fre q u e n cy
50
performance w ith o u t, in tro d u c in g any adverse e f f e c t s due t o r. ■
•
b
One can now q u a l i t a t i v e l y expect the fo llo w in g r e s u lt s due to
in tr o d u c tio n o f the a d d itio n a l
(a)
Drop in
base e le c tro d e with proper b ia s in g .
because o f re d u c tio n in e f f e c t i v e
the base-region through which the b a se -c u rre n t flo w s.
the more is the re d u c tio n
length o f
The g re a te r I ^
in r ^ .
( b) . Reduction in alpha because o f some d e t e r io r a t io n
e ffic ie n c y .
a as Ik
is ,
in e m it te r
In t h i s case to o , one can expect more and more re d uctio n in
increases.
This decrease in
band-w i dth a t the expense o f g ain.
a
has the e f f e c t o f increasing
One can fin d an analogous case in an
a m p li f i e r w ith n egative feedback.
(c )
Reduction in ^ .
In an ideal case r c
j
This e f f e c t may be v is u a lis e d as fo llo w s .
should be i n f i n i t e .
But in a l l p r a c tic a l
c const I c
cases I
depends on V
through the dependence o f tr a n s p o r t f a c to r on
through the E a r ly e f f e c t .
The a d d itio n a l e le c tro d e changes the tr a n s p o r t
fa c t o r in such a way as to make the dependence o f l c on Vc more prominent.
This means t h a t r ^ is decreased w ith the in tr o d u c tio n o f I r ^
decreases as Ib ' in c re a se s .
This e f f e c t , to o ,
is in the d ir e c t io n o f
decreasing g a in .
(d)
Increment in r .
e
This is because I -dependence on v
b
e
is
g r e a t ly reduced b y.th e in tr o d u c tio n o f the a d d itio n a l e le c tr o d e .
Among the above fo u r e f f e c t s , the f i r s t two c o n tr ib u te most
towards the improvement in h ig h -fre q u en cy rfesponsi in a ju n c tio n t e t r o d e .
We can now show a n a l y t i c a l l y how the a d d itio n a l e le c tro d e does
reduce the r b and
aQ
and thereby ra is e s the alpha c u t - o f f frequency w ith
consequent improvement in la rg e -s ig n a l t r a n s i e n t response.
As a f i r s t step we develop an e q u iv a le n t c i r c u i t .
With a
constant c u r r e n t bias a p p lie d to b2, the remaining th re e e le c tro d e s
c o n s t i t u t e a t r i o d e w ith p ro p e r tie s which are q u a l i t a t i v e l y the same
as those o f a conventional ju n c tio n t r a n s i s t o r .
The s t a t i c c h a r a c t e r i s t i c s
are o f the same general shape, s i m il a r b ia s in g c o n d itio n s are s u it a b le ,
and the usual t r i o d e e q u iv a le n t c i r c u i t is a p p ro p ria te .
frequency case, we have the e q u iv a le n t c i r c u i t ,
So f o r the low-
(see F ig . 3 ).
The e f f e c t o f the b ia s , a p p lie d to bp, is to m odify the values
o f the re s is ta n c e s in the c i r c u i t as discussed above.
An a p p ro p ria te
e q u iv a le n t c i r c u i t a t high frequencies should take in to account (a) the
e f f e c t o f em1tte r-c a p a c i.ta n c e , ( b) the e f f e c t o f col Ie c to r-c a p a c ita n c e and
(c )
the f a c t t h a t
e v e n tu a lly begins to decrease w ith
a
frequency and to have associated w ith
i t a phase-angle.
incre a sing
Since the
e m itte r ju n c t ion-capac i tance appears ^ e, a very low re s is ta n c e fo r a l l
p r a c t ic a l purposes^
i t s e f f e c t may be neglected.
The e f f e c t o f col le c t o r
capacitance may be taken in to account by rep I acing-iby r / 1 + j 2 X fr- c .
C
c c
Let us now consider the c i r c u i t as shown in F ig . 4.
From the
e q u iv a le n t c i r c u i t f o r the grounded base a m p li f i e r we have, w ith a p p ro p ria te
approxim ations,
V2 _
V" T r +~r + R j f I + R, / r I - a r
S L e
b
9JL
L cJ
ob
a
Now w r i t i n g , cy =
2——
I + jf/f
and r e p la c in g r
a
c
by r / 1 + j o c r , we have
c
c c
^ or L
2 /v g
= [ re
+
D efine r = r
r b + Rg ^ H
e
+ r
b
+ r
52
' + 1rL
r
H
+~jfc) c
c
r ) j - c/ r
c c
° b
I + jf/fcc
g
n = of r / r
o b
X = _L
Xc= (0 c r
c c
“
o
RL
.•.V-2/Vg = --------- H H -----------------r [ I + R|_( I + j x c )J I + jx a
“ oRL / I +
r
[1 + Rj
(1 +
jx c
jx c "
n /l + jx
Of
Qfo RL / r
+ rL( 1 + j x c ) -
(1 +
1 + Jxa
“ oRL
/r
1 + j x + Ri (1 + j x )(1 + j x ) - n
c
O’
L
01
r
c
aoRL
[ I - n + RL/ r c l + j Rl (x tx ) - R, x x + j x
r
c &
r
a c
cv
c
c
To show the d r a s t i c e f f e c t which r b can have on h ig h -fre q u e n cy performance,
l e t us suppose c^ = 0, so th a t
-yvvv vv-
..a
a
/V
%
c
s
<
F iG . v3
szv
6
—6
54
x
c
=0.
Vg
Vg
or
\
2
Now
Q'oRL/ r
(I - n + R|_/rc ) + j x [R|_/r^ + 11
Of
I
Vg
of_RL / r
____________
-----------------------------------------------------------------V ( | - n + R, / r s )2 + x 2 ( I + R. / r
v
Q/
L c
Assuming t h a t R[_/rc
V2
V9 ‘
From t h i s
4
I
( q, R, / r ) ?"
o L
(I + x 2
Of
i t can be seen t h a t th e response w i l l
x
f/f
ci
be down 3db when
= ( l - n ) , t h a t is when
ci r
= 1 ---------- 2J 2----------
r e + r b + Rg
I f r b = 0, then the gain o f the slope is down 3db when f = f _ .
now assume t h a t Rg = 25olTns; w ith
rb «
I ^ = 0,
>2
.99, r_ ( f o r
e
Let us
I o = I mA)
e
25
ik.
Then f / f # = I -
.99 x 1000
25 + 1000 + 25
In t h i s case, the response is down 3db a t f = .057 f Q/.
With a constant
c u r r e n t bias o f Iu
increase.
r ^ and # w i l l decrease and r
2)
lb
w ill
For
e
= 200 p,A, l e t us assume some re p re s e n ta tiv e values f o r # , r^ and r ^ .
Let r Q = 30 ohms, r ^ = 100 ohms and # = .97.
The same c a lc u la t io n as
above shows t h a t the c u t - o f f frequency is now r a is e d , t o f = .31 f c,.
The
e f f e c t o f the te tr o d e bias is , th e r e f o r e , t o ra is e the c u t - o f f frequency
o f the stage by a f a c t o r o f 6 .
This increase is p a r t l y due to the re d u c tio n
or r k and p a r t l y t o I he re d u c tio n o f ofQ«.
The boosting o f a lp h a -c u t o f f
frequency is re s p o n s ib le f o r improvement in h ig h -fre q u e n c y response*
As
have been shown e a r l i e r , r i s e - , s to ra g e - and decay-time are in v e rs e ly
p ro p o r tio n a l t o th e alpha c u t- o ff ^ fr e q u e n c y .
The fo llo w in g o s c illo s c o p e
tra c e s demonstrate a comparison o f the t r a n s i e n t response o f th e te tr o d e
to a pulse in p u t when used as a t r i o d e ( I ^
1^
= ,06 mA., .16 mA, and ,26 mA.
= 0 ) w ith those obtained w ith
The f o llo w in g t e s t - c i r c u i t was used
f o r photographing th e tr a c e s , ( F ig , 5)
/ A
POT.
2 0
K
PO TM E T B M
^
\ j / 25
^-
tV nA/ s/ v
_ n
F /S , 5
K
POT.
<$00
fj
An im portant r e s u l t o f bias on th e added e le c tro d e Is the
re d u c tio n o f b a s e -re s is ta n c e .
o 'S
I b2
IN
The c o l l e c t o r r e s is ta n c e is a p p re c ia b ly reduced by bias on
75
0'8
r z
A
The bias a p plied to th e second-base reduces the c u r re n t
a m p lif ic a t io n f a c t o r , # .
59
Photographs showing th e t r a n s i e n t response o f the te tr o d e
f o r d i f f e r e n t second-base c u r r e n ts .
T u rn -o n -, s to ra g e - and decay-times
decrease w ith increase in second-base c u r r e n t.
I.
= .06 mA
2
S torage-tim e = 1.69 p,S
Decay-time =
.56 p,S
I.
= .16 mA
2
Storage tim e
Decay time
^ 2
~
.36 p,S
.38 p,S
*26 mA
S tora g e -tim e = .95 p,S
Decay tim e
= .25 p,S
Chapter 5
Conclusion
In the present work an attem pt has been made t o present the
h ig h l ig h t s o f th e l i m i t a t i o n s o f sa tu ra te d ju n c tio n t r a n s i s t o r s fo r
s w itc h in g a p p lic a t io n s .
T h e o re tic a l analyses o f tu rn -o n and decay-times
have been made on the assumption t h a t in th e t r a n s i t i o n p e rio d involved
t r a n s i s t o r behavior can be described in terms o f the usual small signal
parameters ( a t some in te rm e d ia te bias c o n d i t i o n ).
The main p o in ts o f
i n t e r e s t are s ta te d below.
From the expressions f o r tu rn -o n and decay-times i t
t h a t the g re a te r the d r iv e on th e t r a n s i s t o r , the f a s t e r i t
is evid e n t
is turned on.
Also, the g re a te r th e t u r n - o f f d r iv e the s h o r te r the decay-tim e.
Both
tu rn -o n and decay-times are p ro p o rtio n a l t o the c i r c u i t c u r r e n t - g a i n .
The expression f o r “jf c l e a r l y in d ic a te s t h a t tu rn -o n and decay-times •
depend on h ig h -fre q u e n cy performance o f th e t r a n s i s t o r .
The hig h -fre q u en cy
performance o f a t r a n s i s t o r is c o n tr o lle d by i t s r b, Cc and
• r^ does
n ot appear in the re le v a n t expressions because o f the assumption ( in th e
e v o lu tio n o f the exp ressio n s) t h a t the d r iv e r was a c u r r e n t source.
in a l l p r a c t i c a l cases one should remember t h a t
But
should be kept as low
as p o s s ib le because o f i t s d e t e r io r a t i n g e f f e c t on high frequency performance
o f a tra n s is to r.
Also we should keep in mind t h a t the i n i t i a l surge o f
c o n t r o l l i n g c u r r e n t passes through the base.so t h a t the d r i v i n g power
re q u ire d f o r a given c u r r e n t increases in p ro p o rtio n t o th e base re s is ta n c e .
The e f f e c t o f c o l l e c t o r capacitance' is t o increase the s w itc h in g tim e .
60
The d e r iv a tio n o f equations f o r tu rn -o n and decay tim e re ve als t h a t i f
Y dCd (a)
« I ( t o be more s p e c i f i c
has no e f f e c t on s w itc h in g tim e .
I)
Then c o l l e c t o r capacitance
As the i n t u i t i v e p ic t u r e o f the
t r a n s i s t o r suggests, both tu rn -o n and decay-times increase w ith increase
in R|_; b u t, as the photographs show, R^-dependence o f decay-time is more
prominent than t h a t o f tu rn -o n tim e .
G e n e ra lly ,
i t may be remarked t h a t
though the tim e constant associated w ith the decay tim e is seen to be
the same as t h a t associated w ith the tu rn -o n tim e , the decay tim e is
u s u a lly somewhat la rg e r than th e r i s e - t im e unless r e l a t i v e l y
o f b a s e -c u rre n t are used d u rin g t u r n - o f f .
F in a lly ,
large values
i t may be remarked
t h a t small discre p a n cie s between th e t h e o r e t ic a l and experimental values
may be a t t r i b u t e d to the f a c t t h a t s w itc h in g times, are influ e n ce d by the ,
bias dependence o f the s m a ll-s ig n a l parameters, which has not-been taken
in to account.
This can be taken in to account by r e s o r t in g t o non-1inear
a n a ly s is method.., as has been developed by Bashkow.
As has been shown a t the end o f Chapter 3, M o l l ’ s expression
does n ot show c o r r e c t values f o r storage tim e f o r a l l t r a n s i s t o r s .
expression is p iv o tte d on the assumption t h a t u),
—
-— - . C l
/_ !_ .
- o ^ |)
His
This
10
40n .
assumption is not v a li d f o r the given t r a n s i s t o r ( 2NI 18). u>,
. z f ; (l - v R
comes o ut to be .322.
The r e s u l t is :
measured value o f sto ra g e -tim e is
. 4 /MS as a g a in s t the c a lc u la te d value o f .78/MS.
But i t may be mentioned
t h a t M o ll's a n a ly s is on a sa tu ra te d t r a n s i s t o r , considered as two a c tiv e
t r a n s i s t o r s placed back to back, gives f a i r l y accurate values i f no
assumption is made on the r e l a t i v e values o f q^ , cYj o)n and cDj.
eq ua tio n ( M o l l ’ s) o f storage tim e is
The basic
where
CO n U ) |
( I
-
o tjX ' )
t o fl = U + CO, A
n
|
COn +£0,
= u n '» l (l ' " n " , ’
(S>n + <V|
In terms o f b a s e -c u rre n t, the above equation may be w r it t e n as
c,
1 '
B2
/I
- a _
^
;
~
-
B
%
•
- WA
In) / CO
V
2 ,
I
Unless i t
,0,7,
D .'
I ~
O'
^
n
I~Q!
n
<9,
_U>aT,
11 =
is assumed t h a t
(I - ofnc£| )•
g raphical s o lu t io n
|
4.
I (e q u iv a le n t to assumption t h a t
^
-™ ) the above equation is not s o lv a b le ; but a
is p o s s ib le .
The procedure is to p l o t the expression
on the r ig h t-h a n d side of. the above equation f o r va rio u s values o f tim e .
Then we note the tim e corresponding to which the value o f rig h t-h a n d side
equals t h a t o f the le ft- h a n d , which is a constant fo r a given t r a n s i s t o r .
The g raphical s o lu t io n r e s u lt s
measured one.
ments to o .
in T| = .52 p,S, a value very close to the
Of course, th e re are some sources o f e r r o r
As f o r example,CO
in our measure­
as measured in the usual small signal a . c .
c i r c u i t accounts o n ly f o r amplitude changes in
w h ile the ph ase-depen den ce
on frequency is also o f importance.
M o l l ’ s l in e a r a n a ly s is o f a s a tu ra te d t r a n s i s t o r suggests some
very useful
in fo rm a tio n .
The f a c to r
tO n
n +
U)
" + ^1
^ n ^1 ( i " a/na'| )
shows th a t the
63
w idth extends almost e n t i r e l y
.
in to the c o l l e c t o r region f o r normal mode
and f o r in v e rte d mode the same extends almost e n t i r e l y in to the base
re g io n .
So perhaps the inner edge o f the b a r r i e r .h its the base-contact
even a t low. values o f c o l l e c t o r v o lta g e .
The nature o f graphs showing the v a r ia t io n s o f y
o f a te tr o d e w ith v a r ia t io n s
,
and a
o
b
in second-base c u rr e n t fo llo w d i r e c t l y from
the i n t u i t i v e p ic t u r e o f the t r a n s i s t o r as shown in Chapter 4.
p o in t o f i n t e r e s t is t h a t y 1^ and f c do not vary r a p id ly w ith
as. t h i s c u r r e n t is g re a te r than about I mA.
I^
The main
so long
This is im portant because i t
means t h a t the good h ig h -fre q u en cy p r o p e r tie s o f . t h e device are not dependent
on a c r i t i c a l s e t t i n g o f t h i s b ia s .
Another p o in t o f i n t e r e s t is the
behavior o f y ^ a t smal l values o f l b .
Time has n ot p e rm itte d a thorough
study o f the behavior o f y ,
made a t I KC.
a t low values o f I b . The measurements were
b
2
is g e n e ra lly frequency dependent and t o come to a
d e f i n i t e conclusion one needs measurements a t various fre q u e n cie s .
I f the
Y^-va I ueSj, as found in the present experiment, remain to be v a l i d a t h ig h frequencies one im portant conclusion is t h a t l b ' should be around .2 mA
( f o r the given t r a n s i s t o r ) f o r the best h ig h -fre q u en cy performance.
APPENDIX
The large discrepancy between t h e o r e t ic a l and experimental
values o f r is e - t im e as evaluated in Chapter 2 has n e c e s s ita te d the present
s e c tio n .
I t is noted t h a t the most im portant discrepancy is in the v a r ia t io n
o f r is e - t im e w ith a change in c o l l e c t o r load re s is ta n c e — th e th e o ry p r e d ic ts
an increase in r is e - t im e w ith incre a sing
speak to the c o n tr a ry .
Also,
w h ile the experimental r e s u lt s
f o r a given load re s is ta n c e th e re e x is t s a
large gap between the t h e o r e t ic a l and experimental values o f r i s e - t im e .
Several fa c to r s are re sp o n s ib le f o r the erroneous th e o r e t ic a l p r e d ic tio n s
made in a d v e r te n tly in Chapter - 2. .
In the e v a lu a tio n o f r is e - t im e ,
i t has been assumed t h a t
c o l l e c t o r - c u r r e n t r is e - t im e is predom inantly c o n tr o lle d by th e tim e con sta n t o f the p a r a l l e l combination o f
Cq ( c o l l e c t o r ca p a c ita n ce ).
( c o l l e c t o r load re s is ta n c e ) and
But i t should be remembered t h a t . c o l le c t o r -
c u r r e n t r is e - t im e a lso depends on the c u t - o f f frequency o f the t r a n s i s t o r
which, under s h o r t - c i r c u i t e d o u tp u t c o n d itio n s ,
and, to a f i r s t o rd er approxim ation,
is not influ e n ce d by Cq
is a fu n c tio n o f e m itte r re s is ta n c e
( r @) and em itter d if f u s i o n capacita n ce -(C e ).
Depending on. the c i r c u i t con­
f i g u r a t io n s one or the o th e r f a c t o r plays a dominant r o le , and f o r a
marginal case both fa c to r s c o n tr ib u te t o the r i s e - t im e o f the c o l l e c t o r
cu rren t s ig n ific a n t ly .
R ise -tim e under the s h o r t - c i r c u i t e d o u tp u t c o n d itio n s has been
th o ro u g h ly stu d ie d by M o ll.
His expression o f r i s e - t im e f o r a common-
64
65
e m it te r c o n fig u r a tio n shows t h a t the r i s e - t im e should decrease w ith an in ­
crease in lo a d -re s is ta n c e .
For t h i s case, the r is e - t im e is predominantly
c o n tr o lle d by the c u t - o f f frequency, having the tim e -c o n s ta n t
The o th e r extreme case, under the assumption t h a t r is e - t im e
by the p a r a t le I combination o f
a u th o r.
and
C q,
.
is c o n tr o lle d
has been in v e s tig a te d by the
For the. l a t t e r case the tim e -c o n s ta n t o f the c i r c u i t
mately R[_ Cq .
I
!n ( | - 0,n >
is a p pro xi­
The th e o ry , under t h i s assumption, p r e d ic ts an increase in
( I -Qf )
r is e - t im e , With an increase
in load r e s is ta n c e . I t remains
what happens f o r a marginal
case.
The o p p o site p u ll
fa c to r s , v a li d f o r the two extreme cases, r e s u lt s
phenomenon f o r the marginal
in
t o be seen
o f the c o n t r o l l i n g
a very in t e r e s t in g
case.
The lin e a r a n a ly s is , ta k in g in to account e f f e c t s o f both CQ and
Cq
on the c o l l e c t o r c u r r e n t r is e - t im e ,
gives the c i r c u i t tim e constant as
approx i mate 1y
* » n + ' /RL Cc
— ——
$n ( I - a0 )
..
where
_ _ l_ _
= r^C^
re sp ect t o
.
- —
RL Cc
- alpha c u t - o f f frequency.
( ! )
.
the c i r c u i t tim e -c o n s ta n t w i l I
I f I/R|_Cq is large w ith
be
I
, which contains
the usual expression f o r grounded e m it te r c u t - o f f frequency.
Let us examine
the expression # i ) f o r R|_ = I OK and Rq = 20K w ith r e p re s e n ta tiv e values o f
6^n,
a0
and C.
.
f n = 5 rncs/sec.
.
■
6
- # n = 2Wfn = 5. x 10- x
0
= 0.966
r
■
6.26 = 51.4 x 10° ra d s /s e c.
66
( I - o-o ) = ( I - 0.966) = 0.034
The measured value o f CU a t V- 35.“V Fit. 6.5 p f.
% ,
o,
:
Assuming t h a t f o r , a grown ju n c t io n u n i t
A
is = 6 p f . This step is to be j u s t i f i e d la t e r on.
- 1/3
% KVq .
, Cq a t Vq = 45V
.
Hence
I
ti
i
=10
_
104 x 6 x I c r ' Z
6
-
RLCC
= I .67 x 10
7
= 16.7 x 10
Thus f o r the present case we note t h a t
tude as
B
n
and both C_ and
e
,
0
I
R, Cc
are to
de te rm in a tio n o f the t r a n s i e n t response.
constants o f th e c i r c u i t f o r
From ( I ) ,
= I OK and
is o f the same o rd er o f magni'
be taken in t o account fo r th e
We can now e v a lu a te the tim e = 20K.
time constant o f the c i r c u i t is given by
$n + I / rlcc .........
For R[_ = I OK
1 +R.Cn
L U
= (31.4 x 10 ) + (16.7 x I0 U)
= (4 6 .0 ) x 106
( 1- 0- )
o.
= 31 .4 x 106 x .034
= 31 .4 x 10^ x 34 '=' 1066 x IO""’
10 0 0 j *
■' : ••••' '' '■ "
'
= I .066 x IO^,
^
(i -
a0 )
(1 .0 6 6 ) x I06 x 16.7 x I0 6
rl cc
= 17.9 x I 0 12,
For RL
= 48.0 x IO6
■ 17.9 x ■t o 12
20K,
j
+ R
LCC
#n ( 1 - O'o )
RLCC.
=
39 . 7 x 106,
= 8.91 x iO12,
6
67
39.7 x 10°
and hence Y = 8.91 x l o ' 2 = 4/45 (j,sec.
Hence we note, as expected, t h a t the tim e -c o n s ta n t o f the c i r c u i t increases
as
increases.
But a t t h i s stage one, tempted by vacuum-tube analogy,,
should not jump t o a conclusion t h a t r i s e - t im e should increase w ith an
increase in tim e -c o n s ta n t,
stopped a t i
re g io n .
v
- i _
ITlaX
= V
Cv
i t should be r e c a lle d t h a t a l l s o lu tio n s are
/R, since they are v a lid o n ly f o r the a c tiv e
L.
Hence f o r the c a lc u la t io n o f tu rn -o n time one should take in to
account both th e tim e -c o n s ta n t and the i max.
The fo llo w in g c a lc u la tio n s
show t h a t r is e - t im e decreases w ith an increase in R^.
Let 'Y be the tim e -c o n s ta n t o f the c i r c u i t .
b a se -d rive ,
I
i ( t ) == - 'b°b
£ -£
( i _ Q;
Let . i
max
[[ I
- . 0.
•
) • . . . •
= Ir ^ Vpp/R,
0 . LL L
and r is e - t im e (0 - 90$) = i
Then 0.9 l c = 1bg0
[ | _ eT° / f . l : .. ..
( i-Q^ )
y ••V-v. v
o
...
,, .
%o'b
o r To =
I" - I "o
0.9 10
a0
■
;|B
For Rj_ = I OK
Y
'C
I
b
=
fj = 2.69 p,sec.
= 45V = 4.5 mA
I OK
= 0.7 mA.
; .. -
■•
o
Then f o r a given
68
T
= 2.69 In
I
I - 0.9 x 4.5
0 .7 x 28,6
= 2.69 In
I
.802
= 2.69 In I .25
= 2.69 x 0.223 = 0.60
For
and
.
'
= 20K
"t
=
I
= 2.25 mA
c
rt%
= 4.45 p,sec.
Ik
= 0.7 mA
-
T
= 4.45 In
I
I - 2.25 x .9
.7 x 28.6
= 4.45 In
I
0.889
= 4.45 In 1.127
= 4.45 x 0 . 1 19
= 0.54 |j,s.
S i m i l a r l y , Tq may be c a lc u la te d f o r o th e r values o f load, and i t can be
shown t h a t T0 decreases w ith an increase in load as demonstrated by the
experimental r e s u l t s - .
To compare a th e o r e t ic a l value o f r is e - t im e w ith one obtained
by experiment, one should- take in to account the f a c t t h a t Cq is a non­
lin e a r fu n c tio n o f Vc .
At any time a f t e r th e base c u r r e n t step is applied
the v o lta ge across the c o l l e c t o r c a p a c ity changes from Vcc t o some value,
V |, and a c e r ta in amount o f charge is d is p la c e d .
An average c a p a c ity , Gao,
can be defined as one which d isplaces the same charge as th e n o n -1inear
c a p a c ity fo r th e same vo lta g e change.
-
-^
= KVC
1/ 3
, or
V = |
KVC
Since ( f o r a grown ju n c t io n u n i t )
c=u
=
For the case o f 0 - 90 per cent r i s e - t im e , V
goes from V^q ^ iq , thus
Vi = V CC/iO
and
2 /3
%
, - I k
Vcc
2/3
- (W
VCC
0)
™ VCC/I0
For the case o f 0 - 90 per ce n t r i s e - t im e , Vc goes from Vqq t o Vqq/ jq,
thus V| =■'VrCC/10
and
=a„ - Ik
voo2/ 3 , - , (vcs/J° )2/3
VCC
=a^
l K
V-
™ VCC/;!0^
/ 3 : ' v = = / to»2/3
vcc ' vcc/io'
|
vc c 2 / 3 . ,
k
f.
'
- J / I O 275)
Vn r ( I - I /1 0 )
,
J /3
'•
^ K Vcc : ( ' - -215)
"
-1 /3
= 1.305 K.VCC. z
= I .305 Cnp
,UU
is the value o f the incremental c a p a c ity measured a t Vqq(4 5 V .).
■
Where
C qq
In passing,
C
;c
...
C'...:'...'.
i t may be noted t h a t f o r an a I l o y - ju n c t i o n t r a n s i s t o r where
- 1/2
= KV
‘
Cgy
= 1.52 Cqq.
"
' " '
Hence the value f o r T0 as obtained above
should be approxim ately m u lt i p l i e d by 1.305 and the t h e o r e t ic a l value o f
T
f o r R, - I OK is % 0.54 x 1.305 = 0.70 jy,s'.
th re e lower than th e experimental one.
This value is a f a c to r o f
No ta n g ib le e x p la n a tio n may be
70
forwarded f o r t h i s b ig discrepancy.
measurement o f base d r iv e ,
Ib.
The author surmises an e r r o r in the
With a maximum generator o u tp u t o f a p p ro xi­
mately 30 v o lt s peak to peak a c u r r e n t o u tp u t (w ith a s e r ie s re s is ta n c e o f
100 K) o f 0 ,7 mA seems t o be in a c c e s s ib le .
A c a lc u la tio n shows th a t a
base c u r r e n t o f 0.26 mA makes the experimental r e s u lt s comparable w ith'
th$se p re d ic te d by the th e o ry .
T his b a se -d rive sounds reasonable, and one
can reasonably tr a c e the large discrepancy between the t h e o r e t ic a l and the
experimental values o f r i s e - t im e to an e r r o r in the measurement o f l b.
BIBLIOGRAPHY
1.
D e w itt, D., and R ossoff, A. L. " T r a n s is to r E le c t r o n ic s , " McGraw-Hill, .
New York, 1957. (Chapter I I , pages 268-281, "Large-S ignal T ra n s ie n t
Response.")
2.
MiddIebrook, R. D., "An In tro d u c tio n t o Junction T ra n s is to r.T h e o ry ."
W iley and Sons, New York, 1957, pages 181-185.
3.
Ebers, J. J ., and M o ll, J . L . , ' "L a rg e -S ig n a l Behavior o f Junction
T r a n s is t o r s , " Proceedings o f I.R .E ., Vol. 42, No. 12, Pages 1761-1772,
December, 1954. '
■ ■ ' •'
' • ■
■
'
4.
M o ll, J. L . , "Large Signal T ra n s ie n t Response o f Junction T ra n s is to r s , "
Proceedings o f I.R .E ., Vo I . 42, No. 12, Pages 1773-1784, December, 1954.
5.
Lebow, I. L ., and Baker, R; :H., "T r a n s ie n t Response o f T r a n s is to r
S w itching C i r c u i t s , " Proceedings o f I.R .E ., Vo I. 42, No. 6, Pages
938-943, June, 1954.
; '
■ '
6.
Lax and N eustadter, " T ra n s ie n t Response o f a P-N J u n c tio n ," Journal
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7.
Kingston, R. P., "S w itc h in g Time in Junction Diodes and Junction
T r a n s is t o r s , " Proceedings o f I.R .E ., V ol. 42, No. 5/ Pages 829-834,
May, 1954.
'
■ : ■ '■ ■■'■
■
■
■'
■ •
'
8.
Nanabati, R. P., " P r e d ic tio n o f Storage Time in Ju nction T r a n s is t o r s ,"
I. R. E. T ransactions o f E le ctro n Devices, Vol. ED-7, No. I, Pages
9-15, January, I960. • ■ ' •
•
’ 9.
Wallace, R. L . , Schimpf, L. G., and D ick te n , E ., "A J u n ction T r a n s is to r
Tetrode f o r High F r e q u e n c y Use," Proceedings o f T* R. E. , Vol. 40,
No, I I , Pages 13 9 5 -1400,- November, 1952.
.
10.
Bashkow, T. R ., " E f f e c t o f N o n -lin e a r C o lle c to r Capacitance on
C o lle c to r C urrent Rise-Tim e," I . R. E. Transactions o f E le ctro n
Devices, V ol. ED-3, Pages 167-171, October, 1956.
-
71