Download - SC Filters
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F. Maloberti
: Switched Capacitor Filters 1
Switched Capacitor Filters
Franco Maloberti
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F. Maloberti
: Switched Capacitor Filters 2
OUTLINE
Switched capacitor technique
Biquadratic SC filters
SC N-path filters
Finite gain and bandwidth effects
Layout consideration
Noise
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F. Maloberti
: Switched Capacitor Filters 5
Simple SC structures
Q = C
1
(V
1
- V
2
) every
t = T
1
2
1 2
I
1
2
C1
C1
I
T
T
V1 V2
V1 V2
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The two SC structures are
(on average) equivalent to a resistor
If the SC structures are used to get an equivalent time constant
eq
= R
eq
C
2
it results:
IV1 V2 Tt
I
Q itV1 V2
R-------------------T= =
ReqT
C1
-------=
eq T
C2
C1-------
=
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F. Maloberti
: Switched Capacitor Filters 8
SC INTEGRATOR
Starting from the continuous-time circuit of the Integrator, we can ob-tain a SC integrator by replacing the continuous-time resistor with the
equivalent resistances.
+
_
R1
C2
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F. Maloberti
: Switched Capacitor Filters 9
1 2
+
_
C1
C2
+
_
C1
C2
+
_
C2
C1
1
1
1
2
2
2
2
1
1
1
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F. Maloberti: Switched Capacitor Filters 10
We consider the samples of the input and of the output taken at
the same times nT (the end of the sampling period).
Structure 1:
taking the z-transform:
Structure 2:
taking the z-transform:
Vout n 1+( )T[ ] Vout nT( )
C1
C2-------Vin nT( )=
Vout
z( )
Vin z( )-------------------
C1C2-------
1z 1------------=
Vout n 1+( )T[ ] Vout nT( )C1C2-------Vin n( 1 )T ]+=
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F. Maloberti: Switched Capacitor Filters 11
Structure 3:
taking the z-transform:
Remember that for the continuous-time integrator:
Comparing the sampled-data and continuous-time transfer functions we get:
Vout z( )
Vin z( )-------------------
C1C2-------
z
z 1------------=
Vout
n 1+( )T[ ] Vout
nT( )C1
C2------- V
inn 1+( )T[ ] V
innT( )+{ }=
Vout z( )
Vin z( )-------------------
C1
C2-------
z 1+
z 1------------=
Vout s( )Vin s( )-------------------
1sR1C2------------------=
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F. Maloberti: Switched Capacitor Filters 12
Structure 1:
FE approximation
Structure 2:
BE approximation
Structure 3:
Bilinear approximation
It does not exist a simple SC integrator which implement the LD
approximation.
Note: the cascade of a FE integrator and a BE integrator is
equivalent to the cascade of two LD integrators.
R1
T
C1-------
s
1
T---
z 1( )
R1T
C1------- s
1
T
---
z 1( )
z
-----------------
R1T
2C1
---------- s2
T---
z 1( )z 1+( )
-----------------
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F. Maloberti: Switched Capacitor Filters 13
The key point is to introduce a full period delay from the input to
the output
1
2
+
_
21+
_
C2
C1
C1
C2'
'
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F. Maloberti: Switched Capacitor Filters 14
The same result is got with:
1 2
+
_2 1
+
_
C2
C1
C2
'
'
C1
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F. Maloberti: Switched Capacitor Filters 15
STRAY INSENSITIVE STRUCTURE
The considered SC integrators are sensitive to parasitics.
Toggle structure:
The top plate parasitic capacitance Ct,1 isin parallel with C1
It is not negligible with respect to C1 and
it is non linear The top plate parasitic capacitance Ct,1
acts as a toggle structure
Bilinear resistor:
1 2C1
Ct,1 Cb,1
1
2C1
Ct,1 Cb,1
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Both the parasiticcapacitances Ct,1, Cb,1 act
as toggle structures. Theirvalues are different (of afactor 10) and they are nonlinear.
Stray insensitivity can be gotfor the first two structures ifone terminal is switchedbetween points at the same
voltage.
The right-side parasiticcapacitor is switchedbetween the virtual groundand ground (note: even inDC Vv.g. must equal Vground)
1
1
2
2
C1
Ct,1
Cb,1
C1
12
12
Virtual
ground
C1
1
2
Virtualground
2
1
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F. Maloberti: Switched Capacitor Filters 17
The left side capacitor is connected, during phase 1, to a voltage(or equivalent) source.
The charge injected into virtual ground is important, not the onefurnished by the input source.
Structure A is equivalent to the toggle structure, but the injected
charge has opposite sign. Equivalent negative resistance allows to implement non inverting
integrators.
It is possible to easily realize a stray insensitive bilinear resistorwith fully differential configuration.
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F. Maloberti: Switched Capacitor Filters 18
SC BIQUADRATIC FILTERS
Consider a (continuous-time) biquadratic transfer function
If the bilinear transformation is applied, it results a z-biquadratic trans-fer function
where the coefficients are:
H s( )p0 sp1 s
2p2+ +
s2
s0
Q0-------
0
2+ +
----------------------------------------=
H s( )a0 za1 z
2a2+ +
b0 zb1 z2b2+ +
----------------------------------------=
a0 p02
T---p1
4
T2
------p2+=
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F. Maloberti: Switched Capacitor Filters 19
a1 2p08
T2
------p2=
a2 p02
T---p1
4
T2
------p2+ +=
b0 02 2
T---
0Q------
4
T2
------+=
b1 202 8
T2------
=
b2 02 2
T---
0
Q------
4
T2------
+ +=
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F. Maloberti: Switched Capacitor Filters 20
All the stable z-biquadratic transfer functions are realized by the topology:
+
-
+
-
G
D
E
C
A
B
F
I
J
H
1
F1
F2
Vin
t
V01V02
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F. Maloberti: Switched Capacitor Filters 21
Features:
Loop of two integrators one inverting and the other noninverting.
Damping around the loop provided by capacitor F or (and)
capacitor E (usually only E or F are included in the network).
Two outputs available V0,1 V0,2.
Denominator of the transfer function determined by the capacitors
along the loop (A, B, C, D, E, F).Transmission zeros (numerator) realized by the capacitors (G, H,
I, J).
Input signal sampled during 1 and held for a full clock period
Charge injected into the virtual ground during 1.
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Charge conservation equations:
DV0,1(n+1) = DV0,1(n) - GVin(n+1) + HVin(n) - CV0,2(n+1) - E[V0,2(n+1) - V0,2(n)]
(B + F)V0,2(n+1) = BV0,2(n) + AV0,1(n) - IVin(n+1) + JVin(n)
Taking the z-transform and solving, it results:
10 Capacitors
6 Equations a0, a1, a2, b0, b1, b2
Dynamic range optimization
H1V0 1,Vin-----------
IC IE GF GB+( )z2
FH BH BG JC JE IE+ +( )z EJ BH( )+ +
DB DF+( )z2
AC AE 2DB DF+( )z DB AE( )+ +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------= =
H2V
0 2,Vin-----------
DIz
2
AG DI DJ( )z DJ AH( )+ +DB DF+( )z
2AC AE 2DB DF+( )z DB AE( )+ +
-------------------------------------------------------------------------------------------------------------------------------------------= =
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F. Maloberti: Switched Capacitor Filters 24
Scaling for minimum total capacitance in the groups of capacitorsconnected to the virtual ground of the op-amp1 and the op-amp2.
Since there are 9 conditions, one capacitor can be set equal tozero
E = 0 F type
F = 0 E type
Firstly the 6 equations are satisfied. Later capacitors D and A
are adjusted in order to optimize the dynamic range. Finally all
the capacitor connected to the virtual ground of the op-amp arenormalized to the smaller of the group.
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F. Maloberti: Switched Capacitor Filters 25
Scaling for minimum total capacitance
Assume that C3 is the smallest capacitance of the group. In order to makeminimum the total capacitance C3 must be reduced to the smallest value al-lowed by the technology (Cmin)
Multiply all the capacitors of the group by
+
_
C2
C1
C3
C4
Cn
kCminC3
------------=
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F. Maloberti: Switched Capacitor Filters 27
SC LADDER FILTERS
Orchards observationDoubly-terminated LC ladder network that are designed to effect max-
imum power transfer from source to load over the filter passband fea-
ture very low sensitivities to value component variation.
Syntesis of SC Ladder Filters:
Symple approach
Replace every resistance Ri in an active ladder structure with aswitched capacitor Ci = T/Ri.
Use a full clock period delay along all the two integrator loop (itresults automatically verified in single ended schemes).
It results an LD equivalent, except for the terminations.
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F. Maloberti: Switched Capacitor Filters 28
Quasi LD transformation:
Prewarp the specifications using sin(T/2)
Attenuation wwsbwpbApb Asb DESIRED SPECIFICATION Attenuationw
ww
Apb
Asb
PREWARPED SPECIFICATION
sin( pb T/2) sin( sb T/2)
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F. Maloberti: Switched Capacitor Filters 29
Effect of the terminations:
if R1 = T/ C1 and R3 = T/C3 we get:
+
_
R1
R
C2_
+
_
C2C1
C3
HDI s( )R3
sC2
R1
R3
R1
+---------------------------------------= HDI s( )
C1sTC
2
C3
+----------------------------=
Vout n 1+( ) C2 C3+( ) Vout n( )C2 C1Vin n( )+=
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F. Maloberti: Switched Capacitor Filters 30
Taking the z-transform we get:
along the unity circle z=ejT
The half clock period delay will be used in the cascaded integrator in
order to get the LD transformation
The termination is complex and frequency dependent.
The integrating capacitor C2 must be replaced by C2 + C3/2.
zVout C2 C3+( ) C2Vout C1Vin+=
HDI z( )C1
C2 z 1( ) zC3+-----------------------------------------
C1z1 2
C2 z1 2
z1 2
( ) z1 2
C3+---------------------------------------------------------------------= =
HDI ejT
( )C1e
j T 2
C2 ejT 2
ej T 2
( ) ejT 2
C3+-------------------------------------------------------------------------------------
C1ej T 2
2j C2
C3
+( )T
2-------- C
3
T
2--------cos+sin
---------------------------------------------------------------------------------= =
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F. Maloberti: Switched Capacitor Filters 31
Complex termination:
Note: the output voltage changes during
Taking the z-transform:
+
_
C2C
1
C3
F1
Vout n 1+( )C2 Vout n( )C2
2
C2 C3+-------------------- C1Vin n( )+=
zVoutC2 Vout C2C2C3
C2 C3+--------------------
C1Vin+=
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F. Maloberti: Switched Capacitor Filters 32
along the unity circle z=ejT
The imaginary part of the contribution of the termination isnegative
The integrating capacitor must be replaced by
HDI z( )C1
C2 z 1( )C
2
C3
C2 C3+--------------------+
----------------------------------------------------
C1z1 2
C2 z1 2
z 1 2
( ) z 1 2
C2
C3
C2 C3+--------------------+
-------------------------------------------------------------------------------------= =
HDI ejT( ) C1e
j T 2
2j C21
2---
C2C3C2 C3+--------------------
T2
--------
C2C3C2 C3+--------------------
T2
--------cos+sin
----------------------------------------------------------------------------------------------------------------=
C2 C21
2---
C2C3C2 C3+--------------------
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F. Maloberti: Switched Capacitor Filters 53
FINITE GAIN AND BANDWIDTH EFFECT
If the op-amp has finite gain A0 the virtual ground voltage is V0/A0
z-transforming:
+
_
C2
C1
C2V0 n 1+( ) 11
A0------+
C2V0 n( ) 11
A0------+
C1 Vin n 1+( )V0 n 1+( )
A0-------------------------+=
H z( )Vo z( )
Vin z( )----------------
C1z
C2 11
A0------+
z 1( )C1A0-------z+
----------------------------------------------------------------= =
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F. Maloberti: Switched Capacitor Filters 54
Comparing H(z) with the transfer function with
Substituting z = esT, on the imaginary axis
Magnitude error
Phase error
A0
Hid z( )
C1z
C2 z 1( )------------------------=
H z( )H
idz( )
11
A0
-------+
C
1
C2A0
---------------
z
z 1
------------+
----------------------------------------------------------
Hid
z( )
11
A0
-------+
C
1
C2A0
---------------
1
z 1
------------
1
2
---
1
2
---+ +
+
-----------------------------------------------------------------------------------
Hid
z( )
11
A0
-------
1
2
---
C1
C2A0
---------------+ +
C1
2C2A0
-------------------
z 1+
z 1------------+
----------------------------------------------------------------------------------------= = =
H ejT
( )H
idejT
( )
11
A0
-------
C1
2C2A0------------------- j
C1
2C2
A0
T 2( )tan---------------------------------------------------+ +
-----------------------------------------------------------------------------------------------------
Hid
ejT
( )
1 m ( ) j ( )-------------------------------------------= =
m ( )1
A0------ 1
C1
2C2-----------+
=
( )C1
2C2
A0
T 2( )tan------------------------------------------------
C1C
2A
0T
-----------------------=
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F. Maloberti: Switched Capacitor Filters 55
For the noninverting integrator
z-transforming and solving
Same magnitude and phase error result
+
_
C2
C1
C2V0 n 1+( ) 11
A0
------+ C2V0 n( ) 1
1
A0
------+ C1 Vin n( )
V0 n 1+( )
A0
-------------------------++=
H z( )Vo z( )
Vin z( )----------------
C1
C2 1
1
A0------
+
z 1( )
C1
A0-------
z+
----------------------------------------------------------------= =
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F. Maloberti: Switched Capacitor Filters 66
FULLY DIFFERENTIAL CIRCUITS
Fully differential configurations reduce the clock feedthroughnoise and increase the dynamic range.
They allow an increase design flexibility
Simple integrator (inverting and non inverting)
+
_
C2
C1
1
2
2
12
(2)
(1)
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F. Maloberti: Switched Capacitor Filters 67
Immediate sampling (inverting and non inverting) integrator:
Delayed sampling (inverting and non inverting) integrator:
1
+
_
122
1
1 22
Vin
-Vin
-Vin
Vin
1
1
1
+
_
122
1122
Vin
-Vin
-Vin
Vin2
2
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F. Maloberti: Switched Capacitor Filters 69
The peaking in the frequency response due to the phase error is
strongly reduced
It is easy to realize bilinear integrators
1
+
1
2
1
1
2
Vin
Vin
2
2
C2C1
C2
C1
_
_
NOISE IN SC CIRCUITS
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F. Maloberti: Switched Capacitor Filters 74
NOISE IN SC CIRCUITS
The noise sources in a SC network are: Clock feedthrough noise
Noise coupled from power supply lines and substrate
kT/C noise Noise generators of the op-amp
The first two sources are the same as in mixed analog-digital circuits.
kT/C noise:
C id th i l t k
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F. Maloberti: Switched Capacitor Filters 75
Consider the simple network:
In the on state the switch can bemodeled with a noisy resitor
Noise equivalent circuit:
The white spectrum of the on resistance is shaped by the low pass
vin
CS1
4kTR fC
S1
on
Ron
ti f th R C filt
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F. Maloberti: Switched Capacitor Filters 76
action of the RonC filter.The noise voltage across the capacitor C has spectrum:
When the switch is turned off the noise voltage vn,c is sampled andheld onto C
Sn,c vn c,2
4kTRon H f( )2
f4kTRonf
1 2fRonC( )2
+-----------------------------------------= = =
f
S
The folding of the spectrum in band base gives a white spectrum
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F. Maloberti: Switched Capacitor Filters 77
The folding of the spectrum in band-base gives a white spectrum.
It power (the dashed area) is equal to the integral of Sn,c
Procedure for the noise calculation in SC networks:
f
v n,c
fCK/2
*
vn c,2 4kTRonf
1 2fRonC( )2
+-----------------------------------------
0
df4kT
2C----------- xatan( )0
kT
C-------= = =