• Mathematical Background • Basic Building Blocks • SC Filters • SC Oscillators Analog and Mixed-Signal Center, TAMU Edgar Sánchez-Sinencio, ECEN 622 SWITCHED-CAPACITOR FUNDAMENTALS AND CIRCUITS
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• Mathematical Background • Basic Building Blocks • SC Filters • SC Oscillators
Analog and Mixed-Signal Center, TAMU Edgar Sánchez-Sinencio, ECEN 622
SWITCHED-CAPACITOR FUNDAMENTALS AND CIRCUITS
FUNDAMENTALS ON SC CIRCUITS
====
eqceqeq C
Cf
CCTCRRC 1
• very accurate!
• Bottlenecks: Switches and High-Speed Op Amps. Non-idealities enemies!
• The best approach in audio applications.
• Design is carried-out in the Z domain, same domain as for digital circuits.
• Conventional Programming is done via capacitor banks. Spread?
• New non-uniform sampling techniques opens practical application possibilities.
ECEN 622 (ESS)
τ
Examples of Analog and Digital Signals
Analog-to-Digital Converter
Low-Pass Sampling Circuit Quantizer Decoder
xin x1 x2
fc
y1 yo
xin
CT
x2 x1
CT1
yo 1’s and 0’s
SD
Analog and Digital Signals and Systems Concepts
Types of signals Mathematical Description • Continuous-Time (CT) Laplace --Continuous values in time --Continuous values in magnitude • Sampled Data (SD) Z-Transform --Continuous values in magnitude --Discrete values in time • Digital Z-Transform --Discrete values in magnitude --Discrete values in time
OPERATOR Z-1 UNIT DELAY
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )[ ] ( )1111
1
0
1
1
1
0
0
11
−−−=−−−=
−==
=
∑
∑∑
∑
∞
=
−−
−∞
=
+−∞
=
∞
=
−−−
ynyYyZnyZYZ
ZnyZny
ZnyZZYZ
n
n
n
n
n
n
n
n
( ) ( ) 010for 0 If =−<= y,tnTy
( ) ( )[ ] ( )[ ]TnyYnTyZZYZ 111 −== −−
In general ( ) ( )[ ]TmnyYZYZ m
−=−
STABILITY OF A SECOND-ORDER HOMOGENEOUS DIFFERENCE EQUATION
The C.E.
( ) ( ) ( ) 021 21 =−+−+ nyanyany
( )( ) 01 22
11 =++ −− ZaZaZY( )( ) 02121
2 =−−=++ ZZZZaZaZ
COMPLEX ROOTS
,a,aaaZ,Z 0a- 4
if42 2
21
2
211
21 <−±= jyxreZ,Z j ±== θ±21
,ayxr 222 =+=
2
1
2 aa
rxcos −==θ
The C. E. yields 02 22 =+θ− rZcosrZ
( )
φ+π=
fsnfcosKrny n 2
θ↔ ,ra,a 21
Conditions Inital ←φ,k
Natural Response
a2 C 1 B A
C O M P L E X R O O T S
R E A L R O O T S -2 -1 1 2 a1
-1
a2=a1-1
a2, a1 Plane
Roots Realfor
4 221 aa >
=
θ=−2
2
1 2
ra
cosra For Complex Stable Roots
Relationship between the poles in the Z-Plane and coefficients a1 and a2.
B
Im C
A Re
Poles Inside Unit Circle Are Stable Roots
Z-Plane
+j
-1 1
-j
MAPPING BETWEEN THE S- AND THE Z-PLANES FOR HIGH SAMPLING
TSp
p
pp
pp
pp
peZ
/Ts/Ts
Z
TsZ
TsZ
=
−+
=
+=
−=
4
3
2
1
Invariant Impulse
2121
Bilinear
1Forward
11
Backward ( ) ( )
( ) ( ) ( )
( ) ( )
++++≅
++++≅
+=
++++≅
<<
621
421
1
1
1s smallFor
32
32
32
p
4
3
2
1
TsTsTsZ
TsTsTsZ
TsZ
TsTsTsZ
.Ts,T
pppp
pppp
pp
pppp
p
Ignoring higher order effects
TsZZZZ ppppp +≅≅≅≅ 14321
For Very High Sampling Rate The Approximation For The
Mapping is the Same!
TsZ pp +≅ 1
RELATIONS BETWEEN POLES IN THE S- AND Z- DOMAIN
(High Sampling)
( ) 022
22
=++=
ω+ω
+
bas
sQ
s nn
jbas
jQ
s
,
,
p
nnp
±−=
−ω
±ω
−=
21
2114
222
142
2
2 −ω
=
ω=
b
Qa
n
n
TsZrZcosrZ
bZbZ
ippi +=+θ−
=++
12
022
212 ( )
( ) ( )22
2121
1
1
1
21
bTaTr
rejbTaTZ
TSZj
p
,p,p
,
+−=
=±−=
+=θ±
Im
jb
Re
jb
ωn
-a
S-Plane
Im
Re
-jbT
r
1-aT 1
Z-Plane
jbT
θθ−
( ) ( )
( ) ( )( ) ( )
( ) ( ) ( )
QQQr
rT
TrT
TrrT
TrTQ
TQ
TTQTr
n
Tn
n
n
n
n
n
nn
nn
n
21
22
then
1212Q
sQ'High
111
OR
12
1
12
222
222
1r
θ−=
θ−=
−ω
≅ω+−
ω≅
ω+−+ω
=ω+−
ω=
ω+ω
−≅ω+
ω−=
<<ω→
Also by a series expansion
21
2θ−≅θcos
Thus
Also see Table of mappings.
QQQb
rb
Qb
Qcosrb
θ−
θ+=
θ−≅
=
θ−θ−≅
θ−
θ−−≅θ=
2
22
2
22
21
2
1
41
21
2
21
2122
In fact if we let,
Qb θ
−≅ 12
then ( ) 2
21 1 θ++= bb
and 122 1 bb ++=θ
21 bQ
−θ
≅
Recall that high Q and high sampling rate conditions have been used. Thus
MAPPINGS Forward Backward Bilinear Impulse Invariant Mixed
( ) ( ) ( ) ( )2
21
122 1 −− ++=→
ω+ω
+=
ZbZbZNZH
sQ
s
SNsHo
o
where
22
1 2rb
cosrb=
θ=
(See Table of Quadratic Mappings)
X
X r θ Oω
x
x QO
2ω
−
( )22o
o sQ
s
ksHω+
ω+
= ( ) 22
111
Numerator−− ++
=ZbZb
ZHM A P P I N G
Table 2 Type of Mapping fo(HZ Q b1 b2
Forward
Backward
Bilinear a=2/T
Impluse invariant
( )T
bb 211 ++ ( )2
1
1
21
+++
bbb
QT02
ω+− ( )21 T
QT
oo ω+
ω−
Tb
bb
2
211 ++ ( )21
212
21
bbbbb
+++
−( )21
2
TQ
TQ
T
oo
o
ω+ω
+
ω+
−( )21
1
TQ
To
o ω+ω
+
21
2111
bbbba
+−++ ( )( )
( )2
2121
1211b
bbbb−
+−++ ( )22
222
oo
o
aQ
a
a
ω+ω
+
−ω
22
22
oo
oo
aQ
a
aQ
a
ω+ω
+
ω+ω
−
( )
T
blnbbcos 2
2
2
2
1141
2+
−− ( )
2
22
2
2
1141
2
bln
blnbbcos
−
+
−−
−
ω− −
ω
142
2 22 QQT
cose oQTo
QTo
eω−
22
1 2
rb
cosrb
=
θ−=
Im
Re
-
ωο
S-Plane
θ
Im
Re 1 -1
r
Z-Plane
( )2Tcota oo ωω≅
Type of Integrator
Magnitude, |H(ejωT)
Phase, Arg H(ejωΤ)
Mapping (Equivalent)
Transfer Function
R V1 Vo
V1 φ1 φ2
C1 Vo
V1 φ1 φ2 C1
Vo φ2 φ1
C2
V1 φ1
φ2
C1 Vo
φ2
φ1 C2
Inverting
(Forward)
Non-Inverting
Inverting (Backward)
ωωo 2
πIn the
S-Plane ( )
sCsRoω
−=−=21
1sH i.e.
( )22
at For 2
TsinT
V
o
o
ωω
ωω
φ
2π LDI ( ) 1
21
2
1
1 −
−
−−=
zz
CCzH
( )22
at For 1
TsinT
V
o
o
ωω
ωω
φ
22Tω
+π
Forward ( ) 1
1
2
1
1 −
−
−−=
zz
CCzH
( )22
at For 2
TsinT
V
o
o
ωω
ωω
φ
( )22
at For 1
TsinT
V
o
o
ωω
ωω
φ
2π−
22Tω
−π−
LDI
Forward
( ) 1
21
2
1
1 −
−
−=
zz
CCzH
( ) 1
1
2
1
1 −
−
−=
zz
CCzH
( )22
at For 1
TsinT
V
o
o
ωω
ωω
φ
( )22
at For 2
TsinT
V
o
o
ωω
ωω
φ
Backward
LDI
( ) 1
21
2
1
1 −
−
−−=
zz
CCzH
( ) 1
21
2
1
1 −
−
−−=
zz
CCzH
22Tω
−π
2π
Type of Filters in the Z-Domain Specifications
H(s) Mapping
S Z H(s)
i.e., consider
( )22
2
pp
po
sQ
s
HsH
ω+ω
+
ω±=
LP
Apply: A backward transformation
( )( ) 22
2
1
2
then1
ZZcosrrZKzH
,TZs
o+θ−
±=
−→ −
: A forward Transformation
( ) 22 2 ZZcosrrKzH o
+θ−
±=
: A Bilinear Transformation
( ) ( )22
2
21
ZZcosrrZKzH o
+θ−
+=
Use prewarping
S-Plane
Backward
Forward
Bilinear
b)
c)
d)
Z-Plane [ ] 2ZKzN o±=
( ) oKzN ±=
( ) ( )21 ZzN +=
1
j
-1
Another Example.
Bandpass
( )22p
p
po
sQ
s
QH
sHω+
ω+
ω±
=
i) Backward Tansformation ( ) TZs 11 −−→
( )( )
( )2
1
2
21
11
1
1pi
i
pi
po
TZ
Q
TZ
ZQT
HzH ω+
−ω+
−
−ω
±=
−
−
−
( )
( )( )( )
( )( )
2
2
2
2
11
11
1
ZT
QT
TZZT
QT
TQTQ
ZTZH
zH
pp
pp
p
pp
po
+ω+
ω+
ω+−ω+
ω+
ω+ω+
−ω
=
BACKWARD FORWARD BILINEAR
Im Im Im
Re Re Re -1 1
TYPE OF FILTER NUMERATOR
LP 20 (bilinear) ( )211 −+ zKO
2−zKO
( )11 1 −− + zzKO
( )11 −+ zKO1−zKO
( )( )11 11 −− +− zzKO
( )11 1 −− − zzKO
( )11 −− zKO
( )211 −− zKO
( )[ ]21121 −− +Θ−+ zzcosK OO
LP 02 (forward LP11 LP 10 LP 01 BP 20 (bilinear) BP 01 (forward) BP 00 (backward)
HP Notch (symmetric)
212 where
rcosrcos O+
θ=θ
212 withabove as Same
lyrespective locations, pole andzero theof angles theare and
rcosrcos O
O
+
θ>θ
θθ
212 withabove as Same
rcosrcos O+
θ<θ
( )212 2 −− +θ− zzcosrrKOAll pass Table Numerators of second-order transfer functions in the z-plane
High pass Notch
Low pass Notch
( ) 22121 −− +θ−= zrzcosrsD
Examples of Basic Implementations
• First-order Low Pass (Continuous-Time)
( ) ( )( )
( ) ( ) ( )tvKtvdt
tdv;sK
sVsV;
s1KsH inpop
o
p
p
in
o
p
ω=ω+ω+
ω=
ω+=
Continuous-time
C V1
R/K
+ Vo
_
R
pR1Cω
= ∞−O ∫∫
s-plane
pω−Re
patpoleoneatzeroone
ω−∞−
Stability implies to have poles in the left-half plane (LHP)
Im
K
( )ωjH
pωω
( ) ( )( )[ ]
ω=
ωω−∠ωω+
=ωω+
−=ω=
js
tan1
Kj1
KjHsH p212pp
Frequency Response
h(t)
t
Impulse Response
( ) tp
peKth ωω −=
• First-Order Low-Pass (Discrete-Time) Sampled-Data (i.e., Switched-Capacitor)
( ) ( )( ) ;
zK
z1K
zVzVzH 1
1p
1p1
1p11
in
o−− −ω
ω=
ω−==
Im j
Re p1 ω 1
Stability implies poles inside unity circle
[ ] ( ) ( )( ) ( ) ( )zVKzVzzV
zVKzVz
in1p1o1
o1p
in1p1o1
1p
ω=−ω
ω=−ω−
−
Taking the inverse z-transform ( ) ( ) ( )
( ) ( ) ( )nTvKnTvT1nv
nTvKT1nvnTv
in1p1opo
in1p1oo1p
ω−=ω−−
ω=−−ω
This difference equation represents the first-order low pass in the Z-domain. Note that
( ) ( )( ) ( )TbnxzXZ
T1nxzXZ
oob
oo1
−⇒
−⇒−
−
z-plane
( )TjeH ω
2fc
f f
cf23
Frequency Response (A periodic transfer function!)
( ) ( )( ) Tsinj1Tcos
TsinjTcosK1Z
ZKzH
1p
1p1
1p
1p1
ω+−ωωω+ωω
=−ω
ω=
ω= jez
( )( )
−ωωω
−ωω+−ωω
ω= −ω
1TcosTsintanT
Tsin1Tcos
KeH
1p
12122
1p
1p1j
- + Vo
C Vin
1Φ1Φ
2Φ 2Φ
2Φ 2Φ
1Φ1Φ
C1
C2
where
CC1
CCK
21p
11p1
+=ω
=ω
phase
What are the relationships between the s-plane and z-plane?
• There are a number of mappings between the two planes • The most popular and exact is the bilinear mapping.
( )( )s2T1
s2T1zor1z1z
T2
z1z1
T2s 1
1
−+
=+−
=+−
= −
−
• The commonly used for high-sampling rate is:
1sTzor
11z
T1
zz1
T1s 1
1
+=
−=
−= −
−
bilinear
forward s-plane z-plane
1 0 -1
Example. Approximate a first-order low-pass continuous-time to a discrete-time low-pass under high-sampling conditions.
( )p
p
p sK
s1KsH
ω+ω
=ω+
=
( )1zT1s −=
0 ∞−
pω−
( ) ( )T1zTK
T1zTK
zHp
p
p
p
ω−−ω
=ω+−
ω=
( )T1 pω−What is the 3dB cut-off frequency in both domains?
CTf pdB3 ω=
For H(z) is more complex than the computation of f3dB.
( ) ( )( ) ( ) 2eHeH
T1TsinjTcosTK
eH
TjTj
p
pTj
dB3ωω
ω
=
ω−−ω+ωω
=
dB3ω=ω
( )( )T12
TT22cos
T1
p
2pp1
dB3 ω−ω−ω−
=ω −
( )1ffs >>
Numerical example. Low-Pass (First Order)
2K
1.01020102T;f1T;KHz20ff
sr102
3
3
pssc
3p
=
π=××π
=ω===
×π=ω
For the continuous-time
( )
dB3
dB3
4142.12
2jH
KHz1f
ω=ω
==ω
=
For the sample-data ( )
( )KHz16.1f
T12TT22
cos2ff
dB3
p
2pp1s
dB3
≅
ω−ω−ω−
π= −
Switched - Capacitor Filters • Use Z-transform mathematics
• Are described by difference equations
• Time constants are proportional to capacitor ratios
• Best implementation for audio applications
• Originally the basic goal was to replace resistors by switches and capacitors • This design approach is one of the most popular in the industry
SC Advantages
• Reduced silicon area • Good accuracy. Time constants are
implemented with capacitor ratios (~0.1%) • Don’t require a low-impedance output stage
(OTA’s could be used) • Could be implemented using digital circuit
process technology • Very useful in the audio range
NOTATION . Switches . . . . . .
Representation Network With No Switches
S W I T C H . . . n
1
3,1φ
n,4,1φ
1 2 3 n - 1 n
Phase Period Representation
t
0 n/T n/T2 n/T32
T)2n( −2
T)1n( − T
Clock Period X X
EXAMPLE n=2 NON-OVERLAPPING
t
φ 1
t
φ 2
2/T T 2/T3 T2 2/T5
PHASE PERIODS OF A CONVENTIONAL CLOCK SEQUENCE
)t(YH
t
t
t
.5T T 1.5T 2T 3T 4T
)t(YoH
)t(Y 1Hφor
)t(YeH
)t(Y 2Hφor
S/H and its respective odd and even components
)t(y)t(y)t(y eH
oHH +=
)Z(Y)Z(Y)Z(Y eH
oHH +=
or
. . Sample Data Circuit
)Z(Vin )Z(Vo
)Z(V)Z(V)Z(V oin
einin +=
)Z(V)Z(V)Z(V oo
eoo +=
)Z(V)Z(V)Z(H
in
o=
CONVENTIONAL NOTATION FOR TRANSFER FUNCTION IS:
)Z(V)Z(V)Z(H i
in
joij =
i and j can be either “e” or “o”. i,e.,
)Z(V)Z(V)Z(H e
in
ooeo =
TWO-PHASE CLOCK GENERATOR SINGLE PHASE
. O . . . . 1φ
O O
. . . . 2φ
CLOCK SIGNALS
t 1φ
t 2φ
O
O O
O
NON-OVERLAPPING
SWITCHED-CAPACITOR EQUIVALENT RESISTOR
1I
1V+
- 2V
+
-
2IR
Continuous (Conventional) Resistor
RVVI
IVV
IVVR 12
22
12
1
21 −=⇒
−=
−=
and are constant voltage sources 1V 2V1i
1V+
- 2V
+
-
2i
C
1φ 2φ
1φ
2φ
ott −
ott −T/2 T 3T/2 2T
At time we apply the clocks. ott =
At 1φ
At 2φ
1V+
- 2V
+
- C
1V+
- 2V
+
- C
1V+
- 2V
+
- C
For the next period, at 1φ
1o CV)2/Tt(Q =+
12o CVCV)Tt(Q −=+
)VV(C)Tt(Q 12o −=+
)VV(C)2T3t(Q 21o −=+
dtdQi =
=1Q ∫2/T3to +
Tto +
dt)t(i1 = ∫2/T3to +
2/Tto +
dt)t(i1
The average current becomes )aver(1I
T
)2T3t(Q
Io
)aver(1
+=
tQ
TVV(CI 21
)aver(1 ∆∆
=−
=
T1I )aver(1 = ∫
2/T3to +
2/T3to +dt)t(i1
)aver(1
21I
VVCT −
=
or
Comparing with the continuous time resistor
Cf1
CTR
Ceq ==
EXAMPLE. KHz128f,K250R C =Ω=
32AA
C
R ≅ pF25.3110128250
1fR
1C 6Ceq
=××
==
Continuous R “SC-R” AREA 5,776 178.57 2mils
ACCURACY OF TIME CONSTANTS
CONTINUOUS TIME: . 21CR=τ
%65%40CdC
RdRd
22
11 ±→±→+=
ττ
where is interpreted as the accuracy of . ττd
τ
TEMPERATURE DEPENDANT!
DISCRETE TIME: . )
CC(TC
Cf1
12
21C
=⋅=τ
11
22
CdC
CdC
TdTd
−+=ττ
%1.0C
dCCdCd
11
22 →−≅
ττ
LKCH KIRCHOFF “CHARGE” LAW
∑=
=n
1ii 0Q
jCjj VCQ =
jC
. . a b + - xV yV
jCV
0t1 >
12 tt >
2tt =at
))t(V)t(V()t(V)t(VV 1y1x2y2xC j−−−=
=jCVVoltage across the capacitor voltage difference (at present time) across the capacitor minus initial condition (at past time).
CHARGE CONSERVATION ANALYSIS METHOD
)n(q)1n(q)n(q CML +−=
1n − 21n − n 2
1n + Tt
1V+ - 2V
+
-
1φ 2φ•
•••
••
1C 2C
•φ•EXAMPLE
for 2φ0)]2
1n(V)n(V[C)]21n(V)n(V[C o
1e21
o2
e22 =−−+−−
for 1φ)1n(V)2
1n(V0)]1n(V)21n(V[C e
2o2
e2
o22 −=−⇒=−−−
1φ 2φ
THEN
)1n(vC)1n(vC)n(vC)n(vC o11
o22
o22
o21 −+−=+
1o11
o2
12
o221 Z)Z(VC)Z(VZC)Z(V)CC( −− =−+
1
1
1
1
2
1
21
1
1
1
oo
Z1Z
ZCC
CCC
ZCC
)Z(H −
−
−
−
α−α+=
−+
=
1
1
1
1
2
1
21
1
1
1
oo
Z1Z
ZCC
CCC
ZCC
)Z(H −
−
−
−
α−α+=
−+
=
)Z(VZ)Z(V
)Z(V)Z(V)Z(H o
in
21e
ooin
oooo
−
==
)21n(V)n(V e
2e2 +=
)]21n(V)2
1n(V[C)]21n(V)2
1n(V[C o1
o21
o2
o22 −−++−−+
21o
1121o
2121
21o
22 Z)Z(VCZ)Z(VC]ZZ)][Z(V[C−−
=+−
21o
11o21
o2
12 Z)Z(VC)Z(VC)Z(V)Z1(C
−− =+−
1212
11
11
2
11
o1
o2
ZCCCZC
C)Z1(CZC
)Z(V
)Z(V−
−
−
−
−+=
+−=
21212
1
212
1o
1
o2
CST)CC()CC(C
CZ)CC(C
)Z(V)Z(V
−+++=
−+=
For high-sampling rate ST1Z +≅
1T <<ω
T)CC(C
511
T5)C
CC(1
1T5)CC(C
C)Z(V
)Z(V
121
112121
1o1
o2
+
+=
++
=++
=
ST1Z +≅
12
c12
1d3
CC1
f21
T)CC(C
21f
+⋅
π=
+⋅
π≅β
Aside:
12c
1221
d3
CCf
21
CTC
121
CR1
21f ⋅
π=⋅
π≅⋅
π≅β
Switched-Capacitor (SC) Filters How to design SC Filters ?
- Two basic approaches
Approximation Methods in the s-plane
H(s) H(z) mapping Select a RC-Active Filter Topology
Pick a SC Filter topology, and use coefficient matching to determine capacitor values
Use a high-sampling rate and substitute R’s by SC resistors
SC Filter Implementation
Filter Specifications
Systematic SC Filter Design
doesn’t meet specs does meet specs
Software tools: Fiesta, Cadence, etc.
Design specifications
Block, circuit diagrams Transfer function Design equations
SC Filter Design
Simulate Filter (SWITCAP, ASIZ, etc.)
Additional specifications
Design amplifiers,
switches, etc.
Verify design
Simulate SC at transistor level (if possible)
Layout circuit
Extract layout
Fabricate
Test
What are the advantages and disadvantages of the two filter design procedures ?
•Mapping Techniques
+ Systematic and well documented (see Matlab)
+ It can use any sampling rate, including the (minimum) Nyquist rate.
- Difficult to implement by hand calculation
•Transforming R to SC resistors
+ It is, conceptually, easier to follow for analog designer
+ Its design is straightforward
- Yields not an optimal design for area, imposed high sampling rate involves larger capacitor ratios.
Switched-Capacitor Filters Components
1
1 2
2 vi
vo -
Basic Elements
Capacitors • polysilicon • metal1-metal2 • parasitics, clock-feedthrough
Switches • N-MOS • Transmission gates • Noise and on-resistance
OTAS • DC-gain • settling time (GBW, phase margin) • noise
Non-overlapping clock phases
Typical switched-capacitor integrator
+
CR
CF
Switched-Capacitor Filters: CAPACITORS
polysilicon
metal1-metal2
C1 CP1 CP2’’
CP2’
poly2
poly1
substrate
CP1, CP2’’ are very small (1-5 % of C1) CP2’ is around 10-30 % of C1
C1
CP1 CP2
metal2
metal1
substrate
Thick oxide
C1
CP1 CP2
SWITCHES
N-MOS
• Transmission gates
G
1 2
B
•V1, V2 < VG-VT (VT ~ 1 V)
• Resistance
• Small resistance for low voltages but high resistance for large voltages
• Rail to rail operation, provided that VDD and IVSSI > VT
• Smaller resistance (fast response)
• 2 clock phases
• More parasitics
( )TGSoxn VVWCLRs
−µ≅
VDD
1 2 VSS
VDD
VSS
MOS transistors biased in triode region, ~100-100 kΩ Off resistances ~ G Ω
Switched-Capacitor Filters: SWITCHES
Transmission gates
( )Toxnn V1VVDDWC
LR−−µ
≅
VDD
1 2 VSS
VDD
VSS
RnIIRp
Rn Rp
VDD VSS
V1,V2
Rn Rp
VDD=1V VSS=-1V
V1,V2
For V1, V2=0, W/L=1, µnCOX=10-4 Rn=10k/(VDD-1) If VDD, IVSSI<VT, the resistance is extremely high!!!
Switches (continues)
•CGS and CGD are Linear polysilicon capacitors, introduce offset voltage.
•CSB,DB are non-linear capacitors, introduce harmonic distortion components.
•Mobile charge introduces gain errors and harmonic distortion components.
G
D S
B
)VV(WLCeargchmobile
C
VSB21
WLC21C
WLC21WLCC
TGSOX
jBSF
0jBD,BS
OXDOXGD,GS
−=−
+φ
+=
+=
B B
mobile charge
G D S
B
Operational Transconductance Amplifier
PRACTICAL CONSIDERATIONS: • DC-gain. The inverting input is not a real virtual ground. vinvert=vo/Adc
• Settling time. C1 must be discharged during phase 1. The main limitation is due to limited output current and phase margin.
•Clock feedthrough. Can be alleviated by using especial clocking schemes.
• Noise. In most of the practical cases the dominant noise components are due to the Switches!!!
1
1 2
2 vi
vo -
C1
Cf
Switched-Capacitor Integrator Analysis
1 2
C1
vi vo
-
Basic Stray-Sensistive Integrator
Charge conservation Principle: Charge injected by C1 is equal to the charge absorved by CF (∆QCI=∆QCF)
Phase 1 CF
2/Tttt 00 +≤<
Phase 2 Ttt2/Tt 00 +≤<+
1 1 1 1 t
Vi(t0+T/2)
t0 2 2 2
∆V0
∆V0CF = -∆QC1 )2/NTt(vCF
1Cv 0i0 +−=∆
+
Switched-Capacitor Stray-Sensitive Integrator Analysis
1 2
C1
vi vo
-
Basic Integrator
( ) ( )[ ] ( ) ( )[ ] 0C)2/Tt(v0C)t(v0C0)2/Tt(vC00 f00f010i1 =+−−−+−+−−
( ) ( ))t(v)2/NTt(v
tv)t(v)2/NTt(v)2/NTt(v
tv)t(v
0CF0CF
0CFCF
0i01C
i1C
=+=
+=+=
Charge conservation Principle: Charge injected by C1 is equal to the charge absorved by CF (∆QCI=∆QCF)
Phase 1
Solving for phase 2 (t=t0+T)
CF 2/Tttt 00 +≤<
Phase 2 Ttt2/Tt 00 +≤<+
1
2/1
f
1
z1z
CC)z(H −
−
−−=
Switched-Capacitor Integrators : Stray-Insensitive Integrators
1 1
2 2
vi vo
-
1
1 2
2 vi
vo -
Backward Integrator Forward Integrator
( ) ( ) ( )[ ] 0C0)2/TnT(vC0)nT(vC0)nT(v f0f0ini =−−−−+−
( ) ( ) f0f0 C0)TnT(vC0)2/TnT(v0 −−−−−=
Charge conservation ==> Phase 1
Phase 2
Solving for phase 1 [ ] 0C)TnT(v)nT(vC)nT(v f00ini =−−−
1f
in
z11
CC)z(H
−−−=
( ) ( ) ( )[ ] 0C)2/TnT(vC)nT(vC)2/TNT(v0 f0f0ini =−−+−−
)2/TNT(v)2/TNT(v)TnT(vC)2/TnT(vC0
icin
0f0f−=−
−−−=
Charge conservation ==> Phase 1
Phase 2
[ ] 0C)TnT(v)nT(vC)2/TnT(v f00ini =−−−−
1
2/1
f
in
z1z
CC)z(H
−
−
−=
Cin Cf
Cin
Cf
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