EFFECT
OF
NON-LINEAR
SETTLING
ERROR
ON THE
HARMONIC
DISTORTION
OF
FULLY-DIFFERENTIAL
SWITCHED-CURRENT
BANDPASS
EA
MODULATORS
Jose'
M.
de la Rosa, Bele'n Pe' ez-Ve dk Fe nando Medei o, Rocio del
Rio
and Angel Rod iguez-Vhzquez
Ins i u o
de
Mic oelec onica
de
Se illa,
IMSE
(CNM-CSIC)
Edi . CNM-CICA,
A da.
Reina
Me cedes
sln,
41012
Se illa,
SPAIN
Phone:
+34
95056666,
FAX:
+34
95056686,
E-mail:
[email p o ec ed]
ABSTRACT
This
pape p esen s
a
de ailed s udy o he e ec o he non-lin-
ea se ling on he ha monic dis o ion o Bandeass
ZA
Modula-
o s (BP-UMs) ealized using Fully Di e en ial
(FD)
& l ched-cu en
(SI)
ci cui s. Based on he analysis o building
blocks, closed- o m exp essions a e de i ed o he hi d-o de in-
e modula ion dis o ion
o
BP-XAMs due o de ec i e se ling, on
he one hand,
and
o he non-linexi ies o he sampling p ocess,
on he o he . Time-domain simula ions and measu emen s aken
om
a
0.8pm CMOS 4 h-o de BPSAh4 silicon p o o ype ali-
da e ou app oach.(*)
1.
INTRODUCTION
Up o now, he po en ial o
Sw&c/ ed-cu e i
(SI)
ci cui s has
been ba ely demons a ed h ough ac ual, p ac ical ci cui s.
Thus,
in he case o
XA
Modula o s
(CAMS),
pe o mances ea u ed by
epo ed
SI
silicon p o o ypes a e well below han hose o
-
Swi ched-Capaci o
(SC)
coun e pa s, e en i he la e a e eal-
ized in s anda d echnologies wi hou good passi e capaci o s.
Such poo e pe o mances a e pa ly due o he la ge in luence o
SI
non-ideali ies,
as
well
as
o he incomple e modeling o hei
in luence. Pa icula ly, o &md&ss
CaMs
(BP-XAMs), and due
o he necessi y o cope wi h he equency speci ica ions equi ed
o modem digi al wi eless communica ion sys ems
[I],
ha monic
dis o ion due o non-linea se ling becomes one o he dominan
limi ing ac o s.
The e ha e been se e al a emp s o model he non-linea an-
sien esponse o
SI
memo y cells
[2][3][4][5][6].
Rega ding ha -
monic dis o ion,
a
p ecise s udy
o
he isola ed
SI
memo y cell
was p esen ed in
[5],
bu i s ma hema ical complexi y p ecludes o
ex end i s usage o ci cui s con aining hea ily coupled memo y
cells, as i happens o
ZAMs.
The simpli ied model o Fully Di -
e en ial
(FD)
memo y cells p esen ed in his pape enables hie a -
chical sys ema ic analysis o
SI
ci cui s composed o memo y
cells, such
as
in eg a o s and esona o s. This analysis p o ides
closed- o m exp essions o he hi d-o de in e modula ion dis-
o ion o BP-Us caused by wo non-linea i ies: he incomple e
se ling and he sampling p ocess a he modula o on -end. The
la e causes la ge ha monic dis o ion le els e en o
a
low se -
ling e o ,
as
con i med
by
measu emen s om
a
0.8pm
CMOS
4 h-o de BP-XAM
[7].
2.
FD
MEMORY CELLS
WITH
NON-LINEAR
SETTLING ERROR
.'
+
-
'3
I.
"1-
s
Figu e
1.
FD
memo y cell wi h non-linea se ling e o .
a)
Sche-
ma ic.
b)
Equi alen ci cui du ing he sampling phase.
sis ance
is
much smalle han ha due o he ga e-sou ce capaci-
ance,
C,,
,
and he small-signal anscbnduc ance,
gHIQ.
In
such
a case, he cell can be modelled
by
he equi alen ci cui in
Fig.l(b) du ing he sampling phase,
$,.
In his ci cui , he la ge.-
signal beha iou is modelled by
g,,,,
and
g,,,-,
which e esen
he ansconduc ances o
M+,-.
gi en by
g,,,-=
g ,lQ,/k-,
whe e
mi+,.
=
ij+.-/Ibia,,
ii+.-
=
ii/2,
and
ij
is he inpu cu .-
en
[3].
Assuming ha
ii
keeps s a iona y du ing he sampling
phase, and ha he swi ch becomes
OFF
a
(n
-
1/2)T,
(T,
is
he sampling pe iod) he di e en ial d ain cu en ,
i,
=
id+-id-
,
can be calcula ed by sol ing he ci cui in Fig.l(b) o he ini ial
condi ion
I'
=
19
+I,
gi ing:
s+..
SI
...
11
-
1
'd.
11
-
I/'
=
'j,
11
-
1
/Z
-
y(nlj.
11
-
1
/')
1
+
(1:)
+
'd.
i
-
lY("'i,
i
-
I/?)
whe e
wi h
T
=
C
/x,,,~
and
mi, l
=
ii~lI/(21b as).
cu en ,
io
=
io+ -io-,
is gi en
by:
3,
A
he end o he i- h hold phase,
Q2,
he di e en ial ou pul
(3)
'a
I1
=
-'d.
I1
-
1/1
F om
(1)-(3)
and conside ing
id.
,I
-
I
=
i,
-
3/2
,
one ob ains:
To calcula e he ha monic dis o ion, he unc ion
Y(.)
mus be
app oxima ed by
a
polynomial inside
a
gi en in e al. Fo ha pu -
pose, we ha e combined Taylo se ies expansion o
mi
((
l
and
nume ical i ing o
-0.5
5
inj
5
0.5
,
0.01
%
<E,~
<
10%
,
o
ob ain he ollowing app oxima ion:
Fig.l(a) shows a
FD
second-gene a ion memo y cell. In wha
(5)
ollo& i will be assumed ha he incomple e se ling is he dom-
inan non-ideali y. The e o e, he e ec o he cha ge injec ion e -
o and he ini e ou pu conduc ance, analysed elsewhe e
[SI,
will
whe e
E,
=
exp[-k,l
is
he
line,.
se ling
e o ,
k,
=
T,,(2 )
,
=
a,EJks[(
I
+
k, (321 a, 1,
and
a,
=
3/2
is
a
i ing
no be conside ed. Besides, in mos p ac ical cases he ime con-
s an o med by he d ain-sou ce capaci ance and he swi ch-on e-
'*'This wo k has been suppo ed by he Spanish
CICYT
P ojec
pa ame e .
.I
1,
The no a ion
is
used
o
ep esen
, R(ji~,,)
.
TIC
97-0580
0-7803-6685-9/01/$10.0002001
IEEE
1-340
Subs iN ing
(5)
in o
(4),
yields:
whe e
Thus he analysis o
a
FD
memo y
cell
wi h non-linea se ling
e o can
be
accomplished conside ing
a
memo y cell wi h linea
e o ,
E,,
which has an inpu cu en equal o
(7).
I
i
is
a
shewa e
o ampli ude
Zi
and equency
i
,
io
will con ain ha monics o
i
.
In
FD
ci cui s, he To al & "nnic Dis o ion
(THD
)
is
app oxima ely equal o he hi d-o de ha monic dis o ion,
HD3
.
The analysis o
HD3
can
be
simpli ied i
io
is
app oxima ed by
i s i s -o de ha monic, such ha
io,
E
-Zjsin(2s
pT,)
.
Pe -
o ming
a
Fou ie se ies expansion o
(7)
i can be shown ha he
ampli ude o he hi d-o de ha monic is app oxima ely gi en by:
whe e
,
=
1/T, is he sampling equency, and
HD3
is:
wi h
Mi
=
z;/(2Zbjo,).
Fig.' compa es he heo e ical model wi h HSPICE by plo ing
HD3
s.
;/
,
,
using he same example
as
in
[3]
[5]
wi h
le el47
MOS
models o
a
0.8pm s anda d CMOS echnology.
In
ha example,
gnIQ
=
82.8pAN,
C,,
=
22.lpF,
Zbias
=
20pA,
Mi
=
0.5
and
,
=
5
I2kHz. No e ha p edic-
ions gi en by
(9)
ag ee
wi h
HSPICE and
wi h
hose made
by
he
model in
[5].
Howe e ,
as
a
di e ence o his la e model, he new
one
can
also be used o p edic ing he ha monic dis o ion o
highe -le el
SI
blocks, such
as
in eg a o s and esona o s, and
i-
nally, comple e BP-ZAMs.
3.
HARMONIC DISTORTION IN SI RESONATORS
Resona o s a e he basic building blocks o BP-UMs, playing
he same ole
as
in eg a o s in LP-CAMS. Mos o BP-XAMs e-
po ed in he li e a u e ob ain hei a chi ec u e by applying he
ans o ma ion
:
+
-z
o he co esponding LP-ZAMs
[I].
As
a
consequence o his ans o ma ion, he o iginal in eg a o s
become esona o s wi h
a
ans e unc ion
Fa/(
1
+
I-')
,
whe e
0
<
a
5
2
.
This unc ion can be ealized by se e al il e s uc u es
[I].
Fig.3(a) shows he block diag am o one based on LD In eg a-
o s
(LDI's).
This
s uc u e is ad an ageous
as
compa ed o he
-1
-2
HD3
Figu e
models.
o he s because i emains s able unde changes
in
he loop coe i-
cien s. Le
us
conside ha he in eg a o s a e ealized
as
shown in
Fig.3(b) and analyse hei isola e ope a ion, assuming ha memo-
y cells
a e
desc ibed by
(4).
A e clock phase
,
he di e en ial
d ain cu en o cell
2,
is:
L1
-
y(i. ,
n)lix,
n
+
'ds,
,,,-
I (ix,
TI)
(lo)
(idsl+,,,
-
'ds,
.,,,)
ep e-
idsz.,,
E
ids2+,,,
-
whe e
'x,
i
=
'i,
I,
-
'ds
,,,,-
and
'ds
],,,
sen s he di e en ial d ain cu en o cell
1
.A e clock phase
$?
,
i
dsl,,,+1,2 -y(-ids2,,,)]id~2.,,
+
idsl~~,~ly(-ids2~,J)
(I1)
Assuming ha he ou pu s age ( ep esen ed in Fig.3(b)
as
a
simple
cu en mi o ) is ideal, he
ou pu
cu en o he in eg a o is:
F om
(5)
and
i,
=.
wi h
Thus,
he analysis o
a
SI
FD
in eg a o o med by memo y cells
wi h non-linea se ling e o can
be
accomplished conside ing
an
in eg a o wi h linea se ling e o whose inpu is equal o
(14).
Le
us
conside he esona o o Fig.3(a). Assuming ha he in-
eg a o s can be modelled by (13) and
(14),
he ini e-di e ence
equa ions ha go e n he beha iou o he esona o a e:
(15)
(16)
..
ll,,l
=
l;,,,-'o,,,
;2,,l=-(l
-&,)jl.,,_1/2+&,i,.,,-3/2+i2
,,,-
1
+i?H.,,
~O.,l~-(~-~,~~2,,~-I/2+~~~2.,,-3/2+~0,,1-1
+iOH,I,
(17)
whe e
i,
and
i2
a e espec i ely he inpu and he ou pu o he
i s in eg a o
in
he loop and
iZH
and
io,
a e non-linea e ms,
espec i ely gi en by:
(19)
2,012
N
1
,
sol ing o
i?.,,
in
(17)
and
'.
'OH,
.
,I
=
-E
sz
c2
'0.
,112.
.
,I
-
I/?
+
i,.
,I
-
1'2.
,I
-
3/2)
Assuming ha
E,,
subs i u ing i in
(16),
ob ains:
io.
,,
(1
-
2&,Vj,
,,
+
4&,jo,
,,
-
I
-
(1
-
4&,)iO,
,I
-
(20)
Figu e
3.
LDI-loop Resona o .
a)
Block diag am. b)
SI
FD
LDI.
1-341
Figu e
4.
HD,
a he ou pu o
a
LDI-loop esona o , wi h
;
:
a)
0.001
,/4
.
b)
0.002
,/4
.
c)
0.003
,/4
.
whe e
yi,
,I
=
;;,,I
-
I
+
(ioH,
,I
-
;, ,
,I
-
1
-
& ,
,,
-
,,2)
'
The ha monic dis o ion e e ed o he esona o inpu can be cal-
cula ed by analysing he ha monic con en o he abo e exp ession.
The e o e, assuming ha
ii
is
a
sinewa e o ampli ude
I;
and e-
quency
,
he ou pu
o
he esona o will be
a
quasi-sinusoidal
signal, wi h an ampli ude app oxima ely gi en by:
Pe o ming
a
Fou ie se ies expnnsion
o
i,
,I
,
i can be shown ha
he ampli ude o he hi d-o de ha monic a he esona o inpu is
gi en by:
(22)
whe e,
,'
=
,
-
,/4
and
,'T,
((
1
has been assumed.
Mul iplying
A,
by he esona o gain (e alua ed a 3 ,
)
and
di iding he esul by
I,,
ob ains
HD3
a he esona o ou pu .
This analysis has been alida ed by ime-domain simula ion using
he beha iou al simula o o
SI
ci cui s epo ed in
[9].
Figd
shows
HD,
a he ou pu
o
he esona o
as
a
unc ion
o
E,
,
o
di e en alues
o
,'.
This simula ion
was
done by changing
glI Q,
o
Cs5
=
IpF,
I,
=
0.5yA and
Ibios
=
200yA, when
clocked a
,
=
lOMHz
.
No e ha
HD,
does no inc ease wi h
E,
because he open loop gain o he esona o is also a enua ed
by he linea e o . Al hough simula ed beha iou is well p edic ed
by heo y, hei di e ences become la ge when he condi ion
,'Ts
<(
I
is no sa is ied
as
assumed in heo y.
4.
HARMONIC DISTORTION IN SI BANDPASS
XA
MODULATORS
Le us conside he 4 h-o de
BP-ZAM
shown in Fig.5, whe e
A,,,?
=
2A,,,,A,,,
,
and assume ha LDI's a e ealized
as
in Fig.3(b). Fo he analysis o he ha monic dis o ion, he ollow-
ing conside a ions ha e been aken in o accoun :
*The ha monic dis o ion e e ed o he modula o inpu is equal
o he ha monic dis o ion e e ed o he modula o ou pu
because he signal ans e unc ion is uni y in he signal band.
Figu e
5.
Block diag am o he 4 h-o de BP-ZAM.
*The con ibu ion
o
he second esona o will no
be
considesed
because i is a enua ed by he gain o he i s esona o in he
signal band.
*The quan iza ion e o , modelled
as
an addi i e whi e noise
sou ce, does no con ibu e o
he
ha monic dis o ion.
The analysis o he modula o in Fig.5 e eals ha
he
ampli ude
o
he i s esona o ou pu is gi en by:
(23)
whe e
A,
is he inpu ampli ude. Subs i u ing (23) in o (22),
assuming ha
AD,,,
=
-1
and di iding by
A,,
ob ains ha
HD,
a he ou pu o he modula o is:
HD,
G
4..&A:( 1
+
5n
,'T,)
E
4&,,A: (24)
No e ha due o he o e sampling, i is
,'T,
(<
I
,
and hence
HL),
p ac ically does no depend on
,
.
A
mo e app op ia e pa ame e
o cha ac e izing he ha monic dis o ion in BP-ZAMs is he
hi d-o de in e modula ion dis o ion, gi en by:
whe e
I,,,
is he DAC ou pu cu en and
nb
.=
Ibios/ DAC.
This exp ession has been alida ed by ime-dom un simula ion
as
shown in Fig.6(a) by ep esen ing
IM,
s.
E,
2
o di e en al-
ues o
I,,,
wi h A
~
=
IDAC/2,
Ibias
=
200pA,
C,,
=
1
pF
and
,
=
IOh4Hz. The inpu signal consis ed on wo ones o
ampli ude
A
'I/
4
and equencies
,
I
s
0.247
,
and
l?
=
0.248 S. As an illus a ion, Fig.G(b) shows
he
ou pu spec-
um co esponding o
E,
=
1
%
and
I,,,
=
50pA. No e ha ,
o he in e modula ion p oduc s appea s
-
no c i ical since hey
a e
ou side he signal band.
-
'Iheo y
.Simula ion
F equency
(MHz)
Figu e
6.
a)
IM,
s.
E,
o di e en alues o
I,,,
.
b) Ou pll
spec um o
E,
=
1
%
and
IDAC
=
50pA.
2.
O.I%<E,<l%.
The
sin~ula ion
was
canied
ou by
a ying
gn p
such
ha
1-342
5.
EFFECT OF
S/H
PROCESS
AT
THE FRONT-END
OF
SI
BANDPASS
ZA
MODULATORS
In he p e ious analysis i has been assumed ha he inpu cu -
en is cons an du ing he sampling phase. Howe e his assump-
ion does no apply o he memo y cell connec ed a he inpu
node
o
a
BP-EAM.
In his case, he a io be ween
i
and
,
is close o
uni y
( i/ s
=
1/4 in he case o he modula o in Fig.5) and
hence, he e will be la ge a ia ions o he inpu cu en du ing he
sampling phase.
As
a
consequence, addi ional ha monic dis o ion
will appea e en i
E,
(<
I
,
which can no
be
explained by
(9).
The ha monic dis o ion in memo y cells connec ed o con inu-
ous- ime sinewa e cu en s was analysed in
[6].
He e, we will ex-
end ha analysis o BP-XAMs. Le
us
conside i s he cell in
Fig.I(a) and assume
a
sinewa e inpu cu en o equency
i
and
ampli ude
Ii.
In
his case,
id
can no be sol ed
as
an
s ep- e-
sponse. In o de o ind an explici solu ion, he ci cui in Fig.l(b)
was sol ed
o
gm+,-
=
gnIp.
wi h he ini ial condi ion
17
=
g+
pkced by
g,,p
JKi,
yielding o
3:
Once he solu ion was ound,
gmQ
was e-
io*
ll
z
-
‘i, ?-l/~~l~~l/~~‘i. l-l/~~
+
+
[‘i. 7-1~~~-,(’i, 1-1/~)y(’i,11-1,~)
+‘~,i~-l~(’i,i~-~,~)l
(26)
being,
1
-
2x izco (2x inT,) 6n izco (2x i~1Ts)
,2
qll(ii)
E
1;
(27)
1
+
(2x
32I;;,,
whe e
ii/Ihias
<(
I
has been assumed. Following he same p oce-
du e
as
in Sec ion
2,
i can be shown ha , o
x iz
(<
1
,
371
iT
16( 21,,i0,)-
AH.
3
71;
(28)
In
BP-XAMs,
only he i s memo y cell connec ed o he inpu sig-
nal will con ain he abo e ha monic. To calcula e
IM,
a he ou -
pu o he modula o , i is necessa y o exp ess
Ii
as
a
unc ion o
A.,
.
The analysis
o
Fig.5 gi es
Ii
3
2&A, .
Subs i u ing his
exp ession in (28), and di iding by
A.,
i can be shown ha :
(29)
whe e
i
z ,/4
has been assumed. Fig.7(a) compa es (29) wi h
ime-domain beha iou al simula ion by plo ing
IM,
s.
z
o
IM3(dBi
c
=
SO
MHz
-60
-70
AA
Figu e
7.
IM,
due o he
S/H
p ocess a he on -end.
a)
IM3
s.
z
.
b) Compa ison wi h
IM,
due o non-linea
E,.
~ ~~
3.
A
mo e igo ous analysis
can
be
done
by
using
he
Vol e a se ies
me hod
as
we demons a ed
in
[SI.
yielding
o
simila
esul s.
O
,IM3
=
-54m
do
=2MHz
-en .
-%23
023 02s
Figu e
8.
Measu ed ou pu spec a o di e en alues o ,.
.
Rela i e
equency
( o&,
Rela i e
equency
( o ,)
di e en alues o ,,
nb
=
4
and
AX/IDAg
=
1/2.
The heo-
e ical model accu a ely
p edic s
he simula lon esul s excep o
some
cases
whe e
a
maximum
e o
o
4dB
occu s.
In
hese
cases
a mo e exac analysis using he Vol e a
se ies
me hod should
be
used.
To
conclude
his
s udy, Fig.7(b) compa es
ZM,
caused by he
non-linea se ling e o and he
S/H
p ocess
o
s
=
loMHz
and
A,/IDA,
=
1/2.
No e
ha , o
E,
>
3%
,
bo h exp essions
app oxima ely con e ge. Howe e ,
o
p ac ical designs, i.e, o
<
0.1
%
,
IM,
due o he
S/H
p ocess domina es, limi ing he
pe o mance o SI BP-UMs unless
a
S/H
ci cui will
be
used a
he on -end.
This
ac has been con i med by expe imen al
e-
sul s om
a
0.8pm CMOS 4 h-o de BP-CAM
[7].
Fig.8 shows
wo measu ed ou pu spec a o
A,/l,,,
=
0.42
when clocked
a
,
=
2MHz
and
,
=
IOMHz,
ob aining
IM,
=
-61dB and
-54dB espec i ely. In his case,
g,
-
360pAN
and
Cgs
=
2.8pF
(E,
=
0.16%
a
,
=
10MI- ~;
which acco ding
o (29) gi es
IM,
=
-64dB and
IM,
=
-55dB espec i ely.
6.
CONCLUSIONS
The e ec o non-linea dynamic
SI
e o s
on
he ha monic dis-
o ion
o
FD
BP-ZAMs
has been analysed
in
de ail. Closed- o m
exp essions, alida ed h ough ime-domain simula ion, ha e been
de i ed o
IM,
due o he non-linea se ling and he sampling
p ocess. The la e cons i u es he main sou ce o ha monic dis o -
ion in p ac ical designs
as
demons a ed by expe imen al esul s.
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