Switched Capacitor Filter(SC filters)

By student : EE562Minh Anh Nguyen

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

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

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.

Basic building blocks• The ideal operational amplifier is a voltage-

controlled voltage source with:• Infinite gain and input impedance• Zero output impedance.

• Vo=A(Vi)

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

Opamps Vs. OTA

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

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

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.

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

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

Charge injection (Const.)• For nMOS charge during the on state• Charge stored in the channel

• Charge during the off state;

Qch CoxWL VDD Vth Vi( ) Vi

charge due to overlap capacitance Vi VoVo

Qgsov Cgsov VDD Vi( ) Vi

Qgdov Cgdov VDD Vi( ) Vi

Qch 0

charge due to overlap capacitance Vi VoVo

Qgsov Cgsov VDD Vi( ) Vi

Qgdov Cgdov VDD Vi( ) Vi

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

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

Resistor equivalence to a switched capacitor (Const.)

• The charge transferred from V1 to V2 is • The average current flow from V1 to V2 is • With the current flow through the switch capacitor resistor proportional to the voltage across it, the equivalent “switch capacitor resistance is

Q C V1 V2( ) V2

IeqQ

T

T

Resistor equivalence to a switched capacitor (Const.)

T1f

f

IeqQ

T

T

Q

T

C V1 C V2T

Q

T

C V1 C V2T

Q

T

C V1 V2( )T

Q

T

C V1 V2( )T

Q

T

C V1 C V21

f

Q

T

C V1 C V21

f

Q

TC V1 V2( ) f

Q

TC V1 V2( ) f

Ieq C V1 V2( ) f f

where V V1 V2 V2ReqV

Ieq

V

ReqV1 V2

C V1 V2( ) f

fReq

1C f

f

Resistor equivalence example

• 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

What is the equivalent resistance of 10nF capacitance sample at a clock frequency of 100kHz.

Req1

C f

f

Req1

1 10 9 100 103

Req 1 104

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

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.

SC filter Wave form• The typical Waveforms

Transfer function

• The basic idea to calculated the transfer function

H s( )ZfZi

Zi

RC active filters• Calculated the transferred function for RC active

filtersV2 Vi( )

RC2

ddt

V2 Vo( ) 0V2 Vi( )

RC2

ddt

V2 Vo( ) 0

dVodt

ViR C2

dVodt

ViR C2

VoVi

s( )1

R C2

VoVi

s( )1

R C2

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

SC filters• Calculated the transferred function for SC filter

0V2 Vi( )

ReqC2

ddt

V2 Vo( )0V2 Vi( )

ReqC2

ddt

V2 Vo( )

dVodt

ViReq C2

dVodt

ViReq C2

Req1

C1 f

fVoVi

s( )1

1C1 f

C2 s

VoVi

s( )1

1C1 f

C2 s

VoVi

s( )C1 fC2 s

VoVi

s( )C1 fC2 s

SC filters (inverting)• Phase1:S1 on, S2 offVi is store in C1, S1 is driven by Vi, S2 is

maintained at 0, by the virtual ground.

• Phase2: S1 off, S2 onVi is disconnected, C1 is complete discharge for the

next cycle.

SC filters (inverting)

• Calculated the transfer function 0

V2 Vi( )Req

C2ddt

V2 Vo( )0V2 Vi( )

ReqC2

ddt

V2 Vo( )

dVodt

ViReq C2

dVodt

ViReq C2

Req1

C1 f

fVoVi

s( )1

1C1 f

C2 s

VoVi

s( )1

1C1 f

C2 s

VoVi

s( )C1 f

C2 s

VoVi

s( )C1 f

C2 s

Gm-C filter• An ideal transconductor is described by the

following expression io Gm Vi Vi

The ouput voltage of the integrator is

VoIo

sC1

sC1

VoGm VisC1

sC1

VoVi

GmsC1

VoVi

GmsC1

H s( )GmC1

C1

First order low pass filter

• Calculated the transfer functionH s( )

ZfZi

Zi

H s( )R2

1S C2

R1

R1

H s( ) Kw

s w

w

KR2R1

R1

w1

R2 C2

C2

H s( )1

R1 C2 s R2 C2 1( )

R2

First order high pas filters

• Calculated the transfer functionH s( )

ZfZi

Zi

H s( )R2

R11

sC1

sC1

H s( ) Ks

s w

w

KR2R1

R1

w1

R1 C1

C1

H s( )R2 s

R1 s R1 C1 1( )

C1

Comparison

• This is the table compare the transfer function for some of the filter

SC filter Noise• The resistance of switch M1 produce a noise

voltage on C with variance kT/C• The corresponding noise charge is

• 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•

Q2 C2 V2Q2 C2 V2

Q2 KTCQ2 KTC

Q2 2KTCQ2 2KTC

SC filters noise (const.)• The mean square noise current M1 and M2 KT/C

noise is

• The noise spectrum are single sided by convention, the distributed between 0 and f/2.The spectra density noise is

• The noise from an SC resistor is equal to the noise of physical resistor

I2 Q2 f 2I2 Q2 f 2

I2 2 KT C f 2I2 2 KT C f 2

I2

f

2 KT C f 2f

2

I2

f

2 KT C f 2f

2

I2

f4 KT C f 2

I2

f4 KT C f 2

Req1

C f

f

C1

Req f

f

I2

f

4KTReq

I2

f

4KTReq

SC resistor noise spectrum

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)

Disadvantage

• Sample data effect (noise)• Need clock circuit and anti-aliasing filters• Not suited for high frequency

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

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

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

Reference• http://www.ics.ee.nctu.edu.tw/~jtwu/publications/pdf/96isc

-lvsc.pdf• Microelectronic circuit by Sedra/Smith• Switched Capicitor Filters: Theory, Analysis and Design

by Anandmohan, Concorde University Ramachandran, Concorde University and Swamy

Thank You