2. Outline
• Introduction
• Basic building blocks
(OTAS, Capacitor,Switches and non-overlapping).
• Basic operation and analysis
(resistor equivalence of switched capacitor filters and
integrators).
• Definition of switched-capacitor filters.
• Basics circuit for Switched-capacitor filters
• Disadvantage & advantage of switched-capacitor filters.
• Compared switched-capacitor filter circuit with other
circuit
• Summary and reference of switched-capacitor filters
3. Introduction
• There are three main types filters, in integrated analog
filters.
1. Switched capacitor filter (SC filter)
-resistor replaced by switch capacitor
-sample time but analog values
2. R-C filter
-“Standard” active filter RC and Opamp with feed
back
-Resistor often implemented with MOS, so called
MOSFET C filter
3.gm-C filter
-resistors replaced by trans-conductor used open loop
-two latter types are continuous time filter
4. Historical background
• Due to the difficulty in making fully integrated
resistors the active RC filters were not able to
fabrication in monolithic form on one silicon chip.
• Switched capacitor filters characterized in the z
domain were developed late 70s and earlier 80s.
• The origin of SC principle was first report by
Maxwell around 1873.
• The first book fully dedicated to switches
capacitor was published in 1948 by P.E.Allen and
E.SanchezSinenico “Switches capacitor circuit”
Van Nostrand Reinhold,NY,1984.
5. Basic building blocks
• The ideal operational amplifier is a voltagecontrolled voltage source with:
• Infinite gain and input impedance
• Zero output impedance.
• Vo=A(Vi)
6. Basic Building Block of OTAS
• Often realized as single-stage load compensated
OTAs since the load is purely capacitive.
• Low dc gain affect the accuracy of the transfer
function
• The unity-gain frequency should be at least five
time higher than the clock frequency.
• Dc offset can result in high output dc offset
depending on the topology chosen. The techniques
exist that can significantly reduce this offset and at
the same time reduce 1/f noise.
• Not so low output impedance
• Still used as voltage amplifier
8. Building block of capacitors
• Double poly capacitors
• A highly linear capacitance is usually constructed
between two poly-silicon layers
• Substantial parasitic with large bottom plate
capacitor (20% of C1)
• Metal-metal capacitors are used but have even
large parasitic capacitances
9. Building block of switches
• MOSFET switches are good switches
• Should have as high off resistance Roff as possible.
At T=300K, MOS switches have Roff on the order of giga
ohms. The finite value is caused by finite leakage currents
that is typically dominated by reverse biased diodes.
• Should have as low on resistance Ron as possible.
Ron can be made arbitrarily small by increasing the width
of the transistors. But parasitic capacitance and leakage
current increase with increasing width.
• MOS switches does not introduce any offset
• BJT switches does introduce offset
10. MOS Switches
• Nonlinear capacitance on each side of the
switch.
• Charge injection effects
• Capacitive coupling from the logic signal to
each side of the switch.
11. Charge injection
• An additional charge, coming from the MOS
channel when the switch is turn off, stored on the
CL
• Charge store in the channel when switch is on.
• Direct coupling capacitance Cgd. (Mainly to
overlap capacitance Cgdov).
• When phase1 switches charge injection into Vi
and Vo
12. Charge injection
• Input node vi is typical low impedance node
• When phase1 switched high(off-on) charge
injected into Vi and Vo node collected by input
impedance (in this phase the output require follow
the input voltage Vi)
• When phase1 switched low(on-off) charge
injected into Vi
13. Charge injection (Const.)
• For nMOS charge during the on state
• Charge stored in the channel
Qch
CoxWL VDD
(
Vt h
Vi)
charge due to overlap capacitance Vi
Qgsov
Cgsov ( VDD
Vi)
Qgdov
Cgdov ( VDD
Vo
Vi)
• Charge during the off state;
Qch
0
charge due to overlap capacitance Vi
Qgsov
Cgsov ( VDD
Vi)
Qgdov
Cgdov ( VDD
Vi)
Vo
14. Non-overlapping clocks
• To guarantee that charge is not lost in SC
circuits, non overlapping clocks are used.
• Both clocks are never on at the same time.
• Integer values occur at end of phase 1
• End of phase2 is ½ off integer value
15. Resistor equivalence to a switched
capacitor
• The capacitor is the
“switched capacitor”
• Non-overlapping clocks
phase1 and phase2
controlled M1and M2,
respectively.
• Vi is the sample at
falling edge of phase1
• And sample frequency is f
16. Resistor equivalence to a switched
capacitor (Const.)
• The charge transferred from V1 to V2 is Q C( V1
• The average current flow from V1 to V2 is
Ieq
• With the current flow
through the switch
capacitor resistor
proportional to the
voltage across it,
the equivalent “switch
capacitor resistance is
V2)
Q
T
17. Resistor equivalence to a switched
capacitor (Const.)
T
Ieq
1
f
Q
T
Q
C V1
C V2
T
T
Q
C ( V1
T
T
Q
V2)
C V1
C V2
1
T
f
Q
C ( V1
V2) f
T
Ieq
Req
Req
C ( V1
V
V2) f
where
V
V1
V2
Ieq
V1
C ( V1
V2
V2) f
Req
1
C f
18. Resistor equivalence example
What is the equivalent resistance of 10nF capac itance sample at a clock
f requency of 100kHz.
Req
1
Cf
1
Req
1 10
Req
1
9
3
100 10
4
10
• This equivalence is very large
• Requires only 2 transistors, a clock and relatively small
capacitance
• In a CMOS process, large resistor would normally require
a huge amount of silicon area
19. What is Switched capacitor
filter?
• The switched capacitor filter is technique based on
the realization that a capacitor switched between
two circuit nodes at a sufficiently high rate is
equivalent to a resistor connecting these two
nodes.
• Used a miller integrator circuit, replaces the input
resistor by a ground capacitor together with two
MOS transistors acting as switches.
• The switches are driven by a non-overlapping two
phase clock.
• SC filters operate on the principle of transferring
analog signal samples ( represented as charges on
capacitors) from one storage element to another
20. Switched capacitor filter
• Let built an SC filter
• We’ll start with a
simple miller integrate
circuit
• Replaced the physical
resistor by an
equivalent SC resistor.
23. RC active filters
• Calculated the transferred function for RC active
filters
( V2
Vi)
R
C2
dVo
Vi
dt
R C2
Vo
Vi
(s )
1
R C2
d
dt
( V2
Vo )
0
24. SC filters (non-inverting)
• During phase1(S1 on,S2 off)
• C1 charge up to the current of vi
• During phase2(S1 off, S2 on)
Discharge into C2 or A charge packet C1Vi is
remove from C2
25. SC filters
• Calculated the transferred function for SC filter
( V2
0
Vi)
C2
Req
dVo
dt
Vi
dt
d
Req C2
1
Req
Vo
Vi
C1 f
(s )
1
1
C1 f
Vo
Vi
(s )
C2 s
C1 f
C2 s
( V2
Vo )
26. SC filters (inverting)
• Phase1:S1 on, S2 off
Vi is store in C1, S1 is driven by Vi, S2 is
maintained at 0, by the virtual ground.
• Phase2: S1 off, S2 on
Vi is disconnected, C1 is complete discharge for the
next cycle.
27. SC filters (inverting)
• Calculated the transfer function
( V2
0
Vi)
C2
Req
dVo
dt
Vi
dt
d
Req C2
1
Req
Vo
Vi
C1 f
(s )
1
1
C1 f
Vo
Vi
(s )
C2 s
C1 f
C2 s
( V2
Vo )
28. Gm-C filter
• An ideal transconductor is described by the
following expression
io
Gm Vi
The ouput voltage of the integrator is
Vo
Vo
Io
sC1
Gm Vi
sC1
Vo
Gm
Vi
sC1
H( s )
Gm
C1
29. First order low pass filter
• Calculated the transfer function
Zf
H( s )
Zi
1
R2
H( s )
H( s )
K
w
H( s )
S C2
R1
K
w
s
w
R2
R1
1
R2 C2
1
R1 C2( s R2 C2
1)
30. First order high pas filters
• Calculated the transfer function
Zf
H( s )
Zi
R2
H( s )
1
R1
H( s )
K
w
H( s )
K
sC1
s
s
w
R2
R1
1
R1 C1
R2 s
R1( s R1 C1
1)
31. Comparison
• This is the table compare the transfer function for
some of the filter
32. SC filter Noise
• The resistance of switch M1 produce a noise
voltage on C with variance kT/C
2
2
Q
C
• The corresponding noise charge is
2
Q
2
V
KT C
• This charge is sample when M1 is open
• The resistance of switch M2 contribute to an
uncorrelated noise charge C at the end of phase 2
• The mean square of charge transfer from v1 to v2
each sample period is
•
2
Q
2KT C
33. SC filters noise (const.)
• The mean square noise current M1 and M2 KT/C
2
2 2
noise is
I
Q f
2
I
2 KT C f
2
• The noise spectrum are single sided by
convention, the distributed between 0 and f/2.The
spectra density noise is
2
I
2 KT C f
2
f
f
4 KT C f
f
2
C f
Req f
2
2
1
1
C
2
I
Req
I
f
4K T
Req
• The noise from an SC resistor is equal to the noise
of physical resistor
35. Advantage
•
•
•
•
•
Reduction of power consumption for filters IC
High integration density
Area(switches + capacitor) << area resistor
Switch capacitor integrator
R is replaced by C and 2 switched (MOS
transistor)
36. Disadvantage
• Sample data effect (noise)
• Need clock circuit and anti-aliasing filters
• Not suited for high frequency
37. Why Switched-capacitors(SC)
circuits?
• Resistors occupy inordently large amount of area
in integrated circuits
• AC resistors can be simulated by periodically
switching a capacitor between slow varying
voltages
• Area(switches + capacitor)<< Area resistor
38. Application of SC filter
• Over sampled A/D and D/A converter
• Analog front-end (CDs)
• Stand alone filter (eg. National Semiconductor
LMF100)
• Replaced by ADC and DSP in many cases
39. Summary
• A miller integrator
• Replaces the input resistor R by a ground capacitor C
together with two MOS transistors acting as switches.
• The switches are driven by a non-overlapping two phase
clock
• Pole and zero frequencies proportional to sample
frequency and capacitor ratios
• Bandwidth required less than the continuous time filter
• “analog” sample data filters
