for
circuit
equivalent
high-frequency
Small-signal,
transistor
field-effect
themetal-oxide-semiconductor
J . W . H a s l e t t , B . E . , M . S c . , a n d F . N . T r o f i m e n k o f f ,B ' E . , M . S c . ' P h . D .
Abstract
'intrinsic' m.o.s.f.e.t.
The differentialequationsdescribingthe small-signalsinusoidaloperation of the
structureare solved using modified Besselfunctions of the first kind. Expressionsfor the small-signal
short-circuitadmittanceparametersare obtained in seriesform. By retaining appropriateterms in the
series,the elementsof a convenientequivalentcircuit are computedfor both the nonpinchofland the
by other authors,to show that previouscalculapinchoffcases.
Resultsare comparedwith thosepresented
tions for the nonpinchoffcaseare incorrect.
I
r0 : resistance between drain and source of the
intrinsic f.e.t.
1o : inductance between drain and source for the
intrinsic Portion of the f.e.t.
Introduction
The m.o.s.f.e.t. is rapidly gaining popularity as a
small-signal, high-frequency amplifier because of its high
input resistance, low input capacitance and large transconductance. The design of high-frequency circuits using the
m.o.s.f.e.t. requires an appropriate equivalent circuit for the
device. In this work, the differential equations describing the
small-signal sinusoidal operation of the device are solved,
yielding an equivalent circuit for the f.e.t. in both the
nonpinchoff mode and the pinchoff mode of operation.
'intrinsic' or active
The transistor can be divided into an
'extrinsic'
portion and an
or passive portion, as shown in
Fig. t. The analysis presentedhere yields an equivalent circuit
for the intrinsic portion of the device only. Extrinsic components may be added in a manner indicated by Reddy and
Trofimenkoff,e in order to obtain a complete equivalent
circuit for the actual transistor.
A number of equivalent circuits have been proposed for
the m.o.s.f.e.t.r-6 In general, short-circuit admittance parameters are used to obtain the form shown in Fig. 2. Hofstein
and Heimanr proposed the most simple model for the device
in 1963. Later Sah2 and Das3 carried out charge-control
analyses, in order to obtain the gate-source and gate-drain
capacitances for the f.e.t. Candler and Jordana presented
a small-signal high-frequency analysis, treating the channel
as a nonuniform transmission line in the nonpinchoff mode.
The yparameters were evaluated numerically using a digital
computer. Treleaven and Trofimenkoff5 have derived a smallsignal equivalent circuit that is valid for the pinchoff mode.
Hauser6 carried out a general analysis for both the bulk and
insulated gate types of devices,and obtained expressionsfor
the elements of a more sophisticatedequivalent circuit. However, a comparison of the results of this work with Hauser's
equivalent circuit shows that Hauser's expressionsfor the real
parts of the gate-source and gate-drain admittances, as well
as the time constant associated with the transconductance,
are incorrect for the nonpinchoff case. The results presented
here are found to be ir, agreement with those of Treleaven
and Trofimenkoff,5 for the pinchoff case.
For nonpinchoff operation, the circuit with
is called the first-order approximation. The small-signal
channel current is described in terms of modified Bessel
functions of the first kind. The first-order approximation
corresponds to the retention of terms not involving a, and
those involving the first power of a, only. A second-order
approximation then implies the retention of terms in <o2as
well. In this work, a second-order approximation is used to
obtain yt and y2 in the forms
jacr
1 I iac{r 1,.
..
'Y2
jacz
-jrDcrh
e)
|
whererl,12: gate-source and gate-drain resistancesfor
the intrinsic f.e.t. for nonpinchoff operation, while a firstorder approximation is used to obtain ys and y^ as indicated
i n e q n s .l .
2
Differential equations describing the
small-signal sinusoidal operation of the
m.o.s,f.e.t.
With reference to Fig. l, the channel current can be
written as
(3)
t:1"r.ru{
where 1 - total channel current, defined flowing out of the
drain terminal
c,,, '- oxide capacitanceper unit length of channel
IJ - total gate-channel potential at the point x
x = distance co-ordinate defined infig. I
and
Yt : jacl
Y2 : jac2
g-,,
. Y m - -
lo:
(l)
a +- J<tTo
rs i iulo
portbn
where c', c2 - gate-source and gate-drain capacitances of
the intrinsic f.e.t., respectively
g^,,: low-frequency transconductance for the intrinsic device
ro : d time constant defined by eqn. I
Paper 5796 E, first received l9th August and in revised form 29th
November 1968
Mr. Haslett and Dr. Trofimenkoff are with the DeDartmentof Electrical
E n g i n e e r i n gU
, n i v e r s i t yo f C a l g a r y .A l t a . . C a n a d a
PROC. IEE, Vol. 116, No. 5, MAY
1969
Fig.I
Schematicof m.o.s.J.e.t
It should be noted that
co' :
€o"Z
T*
(4)
699
Terms in -2 are retained, in order to obtain a second-order
approximation for the equivalentcircuit.
whereeo, : permittivity of the oxide
Z - devicebreadth
4," : oxide thickness
The charge-conservation
statementcan be written as
).
)/
:;(c,"Ul
;1
3
(5)
where t : time.
If the total gate-channel potential U and the total
asthe sum of a.c.and d.c.components,
current1 areexpressed
fJ:a+uej't\
I-Iolistar
.
t
Differentiatingeqn. l7 with respectto o and equating
it with eqn.9 yields
lB:
where U :
(6)
and
i:
du
p c--'
.,u7
ax
Q)
D(oa)
pc,,j|
(8)
l*
The constants k1 and k2 can then be evaluated from the
boundaryconditions,which occur with either the input or
output a.c. short-circuited.
are definedas
The short-circuitadmittanceparameters
- . - , c
'Ytt
,o"='o
i^l
ls
(9)
_.i-r,,,
(le)
y., z t--
If it is noted that
d
dx
I
J:Dh
or:*l'
The small-signal approximation, in which only first-order
terms in the a.c. components are retained, has been used to
obtain eqn. 8. Use of eqn. 5 results in
di
(t8)
krF, + k2F2
,t:#l'
i
eqn. 3 can be separated into time-dependent and timeindependent components, to Yield
In:
The short-circuitadmittanceparameters
l
o
d
pcoxu dL'
I
ia
_" g . l
(r 0 )
uds='O
r
t.l \
rr.1\ I ,o"-: 0
it is easy to show that
2-ioui :o
(tl)
au'
tn
w h e r eD
-p(:.
r 2
wherer" + i, + ia : O.
The ioltagesand currentsare definedin Fig. 2
(t2)
)
\ -16 .z
Eqn. 7 can be integrated between x : 0 and x ': L, to yield
r"o : - - . * ! : < o 1- u 2 ) .
2L'"
(13)
where tr, : value of u at x : L
? . - v a l u eo f o a t x : 0
In terms of the d.c. gate-source and drain-source voltages
Vr" and Va,
r ) , : V s s- , y r
(14)
)
Vr)
u4-Vrr-Va,
where V7: threshold voltage, positive for an n-channel
enhancement m.o.s.f.e.t.
Eqn. 11 is a standard Besselform with solutionT'8
i - llrlzlki{berr,r1lu3t2.,/ D) + ibeitp(3D3t2\/ D)\
i
kr{ber
D) i
r1z(?a3t21/
'i
.o,1t,+ 2r)
-,i
bei,{
ffiy#,in}{, +z,t
/ .
.Du3
r)(l -r7u-
kr, kz =- arbitrarY constants
700
I t : 1 1 " ,6 -" '
( 2 1)
Substitution of eqns. 20 and 2l into eqn. 18, and solving for
kt, k2, is and 2, yield
Ytt
'lrf
I
D2u6
\
180,.../
(20)
!
U:U, ,
(l6)
(17)
i:krtrlkzqz
.
c2-(t
at the source
and at the drain
and k't, k', are arbitrary constants.
It can then be shownthat
tor'rt
Calculation of the input admittance y11 and the
forward transadmittance Y2'1
If the drain is a.c. short-circuited to the source, and a
small-signalsinusoidal voltage z*. is applied between gate and
source, then
3.{
jbei t,t(la3t'\/ D)}l
(l 5)
:"i,Sffiffi
where
beru{
where
91 ,(l - i)( ,
sour€e
Fig.2
Generalform of equivqlentcircuit
is
,i^
-.- l
B(F,.F.,.
{(Fza
Fz,)(gr,
F2,Ft,t)
g1) -(F1,
F1)(gz,
cz)\ Q2)
and
lzt
I
' irl
ur,'- B(Ft,Fz,t
rr,rrul
- Fz,)sra t] (Fr,
Fdcz)}
Q3)
{(Fza
1969
MAY
5,
No'
l16,
Vol.
IEE,
PROC.
Second-order approximation
Retaining terms involving the second power of ar in
eqns.22 and 23 yields
3.5
wnere
values of F1 for ?J- tr" and u
Fz, Fza: values of F2 for u : u " and,u
grv 8ta - valuesof 91 for u -'u"anda
g2y g2(t: valuesof92 for a : u" anda
F1r, Fta:
: ud' respectively
: a4, respectively
- u4, respectively
: ud' respectively
Calculation of the output admittance y22 and the
3.2
reverse transfer admittance Y12
If the gate is a.c. short-circuited to the source and a
small-signal sinusoidal voltage u7. is applied between the
drain and source, then
and ln
\'
0 -u,
)
.
(24)
and at the drain
u:-uas-t,
u:Dd
+ 4u'u1L u!')
- lsuluh+ 2ur)
D' (5u2-'l2us"ad+ 20p3"p3d
_.
'b_
(u" - uo)2(2u
60
" -t ua)
Elements of the equivalent circuit
circuit
Expressions
for the elements
of the equivalent
by notingthat
shownin Fig.2 canbeobtained
4
-f Yp ),
lr : ltr
I
y2:-yt2
-j
lm:
-
lzt
l.
ltz
yo:y2z_tln
Gz)
i
)
iS
Vr, :
and
f1ll
60
since r is defined as the a.c. gate-channel voltage. Substitution of eqns. 24 and 25 into eqn. 18, and solving for i*
and ir, yield
-
i-r^,rr6[fffur',
-') . D ' U ; : W ) (u!
(2s)
.
(30)
where
at the source
u:u-.-O
(l I jar5)
- .
Jug^"7+0
ittr)
!tt
urt,
I
B(Ft,F24
- gzal
F2,F1ar{Ft'lgz'
- Fz,(rr.,
- cu)j
Q6)
'y ,1 1 - i a
uds
--l
B(F t,F2d
3.3
*---,\F
r,gza Fzrg ta)
F2,F11t
Q7)
F i s .3
Equivalentcircuit of intrinsic portion off.e.t.
Zero-order approximation (o : 0)
4.1
For the d.c. case
lrr:O
First-order approximation
yt : j@ct - iag^o(r+ - r:6)
y2:j@c2:jag^or36
Yrz:0
!2t : 8m,
-2lo
(28)
tr, + aoy
{t-y<.r(r,
6mo
(l *
lnl
o
^
lzz-9-oo:-
6 mo_
l
where z6 :
Yp ..
(1 :
- jutr2\
a
Y)2: C^od.1
+jufi
D' (u, - u)z(u?+ 3u,ad+ uh)
(us + ud\
"' z, - o ' ! o' e- "rs? - u , u u , u ) )
6
D'
-T(r,
, ao)2(2u"
", :
* u4)
,c:
2-
12 -
:.36
,url2u,*ua)
3 L ' o x L@ J
u)T
p-c^,
I
1 1a i a r )
1 <
A
ro-i:-f'"
D'
- u ) ( a ? + 4 u " u d+ a 7 t )
6(a"
D' : Dla
PROC.IEE, Vol. l16, No. 5, MAY 1969
41 El6
(34)
ls : r(s
(2e)
^ (l -l- icrrz')
r
( 33 )
u"\u, 2ua)
". -2 n ,
3'o'" (a, a uo''f
iag^or36
g^o(l
tzt-
where
Tt i
j@g-oT4
-
iaro
Using the aboveequations,it can be shownthat
If second-orderand higher terms in <r.r
are neglectedin
eqns.22, 23, 26 and 27, the admittance parametersare given by
:
gmo
|
to
First.order approximation
Ytt
-
lo-1+J.@rl
where6:u7l@,-ua).
These results are in agreementwith those derived by
Hofsteinand Heiman.r
3,4
rr8)}iarll
For
t^. : ryf {r, - ua)
the pinchoff case, v:' - Va" :
Vr so that 0a : 0. Then
)
cv: jco*L
cz:0
ro:@
/o-oo
som o
(35)
-- Y!"'' - s
T O :
L
4L2 l
t)pa,
701
Second-orderapproximationto y1 and y2
Referring to eqns. 30 and 31, it can be shown that
second-orderterms in the numerator result in additional firstorder termsin the denominator,so that
4.2
-
jt tg^o(r+
Yr -
r:6)
From eqn. 12,
D,:p:p(eff)'
and substitutionfor 16from eqn. l3 yields
(36)
D , :pai
g
\
, --(r,'-!{t- r't'6
wo-i$--rs-16)
jag^or36
and
lz:
l]_ja(r1
(3t1
-16)
retaining only first-order terms in the demoninator. These
expressionsare in the form indicated by eqns. 2, and solving
for 11 and 12yields
-T l - , T 4 T t
TlT56
-
-7. 5
Tb
T4- T3E
rl:
- ,*
f*r.r^
(43)
for the pinchoff case (0d : 0)'
Define
Y : D,Dtr
(44)
of eqn.42into eqn.44 yields
For lo.l : 2V, substitution
(45)
7-7x10-8s
Then, at pinchoff,
(38)
To:
Tl
_ v
t5
rz: 0
and
T L -7 6
12:
(3e)
' J
For the pinchoff case,
l
f
'r
-
rz :
-
-
L
l
)PCo,
U.r
-
Jg^o
v
(40)
6
v
(41)
@
30
A comparison of eqns. 38 and 39 with Hauser's6 expressions
shows that he has derived 11 and 12in the form
Tl
'r
i
'o
I
24
cutoff frequency is given by
The transconductance
g^o(r4-7361
I
r '" :
T1
g^oTij
f, =^--:34MHz
lIfT6
-
neglecting the other time constants, which result from second-
order terms in the numerators of the y parameters. Also, his
expression for y- neglects the term (r, - zr8), as given in
eqns. 33, so that the time constant fory- correspondsto z',
instead of to re.
These results yield an equivalent circuit in which the gate
-source and gate-drain admittances are simple RC networks, and the current generator y mu'" is frequency dependent.
The important point to note is that the time constant associated with y- is not the same as the product of r1c1. Indeed'
There is no need to introduce a
for pinchoff, rslr(1 :2.
second RC network in parallel with the t r - cr combination,
to account for the difference between the imaginary parts of
the forward- and reverse-transferadmittances, as has been
suggestedin the literature.ll
For an r-channel device with comparable dimensions, the
free-carrier mobility is higher, and f, will be of the order
of l00MHz.
6
Financial assistance from the National Research
Council, under Grant A-3382 and in the form of a scholarship
to J. W. Haslett, is gratefully acknowledged.
7
I
2
3
4
5
Some typical values for the intri.nsic
device
5
Numerical values for the time constants may be
obtained by substitution of typical device parameters. The
following have been obtained for a commercially available
p-channel enhancement m. o.s.f.e.t.:
Z :0'004cm
Z -- O'lcm
co, :25OOqFlcm
pc- 480cm2/Vs _l
702
o
'l
8
9
(421
Acknowledgment
l0
ll
References
'The silicon insulated--gate
, . p.:
H o F s r E I Ns, , n . , a n d H E t M A NF
field effeit traniistor', Proc. Inst. Elect. Electron- Engrs., 1963,
51. pp. I l90-l20z
i " i r l l . r . , ' C h a r a c t e r i s t i cosl M O S t r a n s i s t o r s 'I,E E E T r a n s' 1 9 6 4 ,
ED-ll, pp. 324-34s
' C h a r g e - c o n t r oa
l n a l y s i so f m ' o . s . a n d j u n - c J l o n - q q t e
ols. v.'s.:
ll-3' ( 10)..pP: 1565-1570
fielci-effecttransist6rs',Proc. I E E. 1966.
'A
S m a l l - s l g n ahl .t g n - t r e q u e n c y
O,. n . , a n d J O R D A NA,. c . :
CANDLER
analvsis of the insulated-gate field-effect transistor', Intertnt
J . E l e o r o n i c s , 1 9 6 51 ,9 , P P . l 8 l - 1 9 6
'MOS FET equiv-alent
T R E L E A V EoN. ,u . , a n d i n o r l v e N r o r n , r ' N . :
circuit at riinctr-6it', Proc. Inst. EIect. Electron. Engrs.' 1966. 54,
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s i g n a l p r o p e r t i e so f f i e l d e f f e c t d e v i c e s "
iriusrn, r. n.:
D 12.pp.605-619
t E E E T r a n . r . .1'9A6d5v.E
a n c e dm a t h e m a t i c si n p h y s i c sa n d e n g i n e e r i n g '
B R o N W E L lL. :,
(McGraw-Hill, 1953)
'Besset functions for engineers' (Oxford
N. w.:
ii.lr,isreN,
U n i v e r s i t yP r e s s .1 9 5 5 )
'F.E.T. high-tiequency
neoov. i.. and rnorttueNroFF, F. N.:
a n a l v s i s 'P
. r o c . I E E , 1 9 6 6 ,I 1 3 , ( l I t , p p . 1 7 5 5 - 1 7 6 2
s n o i x r r v . w . : ' A u n i p o l a r f i e l d - e f f 6 ci tr a n s i s t o r ' .P r o c . I t t s t .R a d i o
376
pp. 1.365-1
Engrs., 1952,4q.
'Equivalent circuit and gain of MOS field effect
rrsiuen, w.:
transistors',Soliti-StateEIectron.,1966,9, pp. 7l-81
PROC. IEE, Vol. 116, No. 5, MAY 1969