high frequency filter design - michael tsecktse.eie.polyu.edu.hk/eie403/filterdesign.pdf · michael...

HIGH FREQUENCY FILTER DESIGN

forHigh-Frequency Circuit Design Elective

byMichael Tse

September 2003

Michael Tse: HF Filter Design 2

1. Introduction

2. Filter Design for High Frequencies

3. Gm-C Filter Synthesis

4. Realization of Transconductors

CONTENTS

1.1 Types of filters1.2 Monolithic filters1.3 Integrators1.4 Simple first-order gm-C filters

2.0 Introduction to filters (separate notes)2.1 Special requirements for HF

3.1 Cascaded biquads3.2 Signal flow graphs

4.1 BJT transconductors4.2 MOSFET transconductors4.3 Exercise


• Basic forms: LC-Ladders• Due to the advent of op-amps,

“ACTIVE RC” filters became popular.• Miniaturization leads to IC filters which

use monolithic technology for activecomponents and thin-films forfrequency determining components(C,R).

• IC monolithic filters became popular.– Advantages:

• Less components, smallervolume

• Good matching of components• Automatic tuning – correct

transfer functions forprocess/temp variations

• Smaller parasitic caps on chip• Fabricated in large quantity

19201960

1970

1980

1. INTRODUCTION


• Digital filters• Analog discrete-time filters• Analog continuous-time filters

1.1 Types of Filter Realizations

1.1.1 Digital filters:Signals need to bediscretized and digitized,i.e., sampled and convertedto digital words, and thefiltering is done in the digitaldomain.

1.1.2 Analog discrete-time filtersSignals are discretized but NOT digitized.They are called sampled-data. Filtering is donedirectly to the sampled-data.Example: switched-capacitor filters (SC filters)But SC filters are mainly for low-frequencyapplications (audio range)

Discretization(sampling)

Digitization(A/D)

Filter or Processor

0011001110010…

0011001110010…

D/A


1.1.3 Analog continuous-time filters

Continuous analog signals are directly processed without any A/D orD/A conversions, sampled-&-hold, anti-aliasing filters, etc.Because of the continuous-time nature, analog continuous-time filters arevery suitable for high-frequency and high dynamic range applications.

Disadvantages:

1. Sensitive to process and temperature variations2. Aging3. Need tunings of the frequency determining components

Since we are dealing with high-frequency design, we will focus onAnalog Continuous-time Filters in these notes


1.2 Monolithic Filters

Fully integrated analog continuous-time filters were possible whenautomatic tuning of components became available, starting 1970’s.

1.2.1 Bipolar filtersProperties1. High voltage gain2. High output drive3. High frequency (up to ~100 MHz)4. Low noise and offsets

1.2.2 MOS filtersProperties1. Low power2. High packing density3. High noise immunity4. Ease of design5. Ease of scaling6. High frequency (up to ~100 MHz)


1.3 Integrators (Building Blocks)

Integrators are needed in all active filters. [ In passive filters, integrators areprovided by inductors and capacitors, in both I and V domains. ]

However, for active filters, only C exists. Hence, we need to have integrators ofoutput/input variables are in the same voltage or current domain.

1.3.1 Active RC (Op-amp RC) IntegratorsCj

.

.

.

R i

Cint

+–

Vj

Vi

Vo

n x Rm x C

Ri

Mi

mosfet intriode regionto simulate a resistance

†

Vo = -Vi

jwCintRii=1

nÂ -

C jVjCintj=1

mÂ


Active RC Integrators (con’t)

The RC integrator shown previously in notvery suitable for monolithic realizationbecause the time constant ti = RiCintcannot be tuned after realization!Note that C and R can only be fabricatedwith an accuracy of 20% and 5%respectively.

With MOSFET-C integrators, thetuning problem can be solved byvarying the gate voltage of Mi--> Ri --> ti .

Design notes:

1. Nonlinearity of MOSFETS ismainly second-order. Thus,MOSFET-C integrators must bedesigned in BALANCED FORMin order to cancel even harmonics.

2. It is difficult to implement goodMOS op-amps. Usually, BiMOStechnology is used for MOSFET-C integrator filters.

3. It is also possible to tune thefrequency using thetransconductance instead of theMOSFET resistance.


1.3.2 Transconductance-C or gm-C Filters

BASIC CIRCUIT:

Vi mg

i = V gi m

transconductance

The general gm-C integrator:

Cj

.

.

.

Cint

Vj

Vi

Vo

m x C

mgmi

mg

n x g

We can control t i =Ceff/gmi by tuning thetransconductance.Note: The transferfunction suffers fromloading effects,which depend on thesummation cap Cj.The gain gm is adesign parameter(whereas in active-RC, the op-amp gaindoesn’t matter).

†

Vo =gmiVijwCeffi=1

nÂ +

CjV jCeffj=1

mÂ

†

where Ceff = Cint + C jj=1

mÂ


1.4 Simple First-order gm-C Filters

The basic transfer function is:

Gm-C realisation:

The nodal equation is:

†

H(s) =VoutVin

=k1s+ k0s+ w0

†

gm1Vin + sCX (Vin -Vout) - sCAVout - gm 2Vout = 0

VoutVin

=

s CXCA + CX

Ê

Ë Á Á

ˆ

¯ ˜ ˜ +

gm1CA + CX

Ê

Ë Á Á

ˆ

¯ ˜ ˜

s+gm2

CA + CX

Ê

Ë Á Á

ˆ

¯ ˜ ˜

m+g

C

Vin

A1

2m–g

Vout

CX

The parameters are adjusted by

†

CX =k1CA1- k1

for 0 £ k1 <1

gm1 = k0(CA + CX )gm 2 = w0(CA + CX )


Vam+g

m–gm–g

m+g

m+g

parasiticNOT SUITABLE FOR HF

CVb

m+g

m–g

m+g

m–g

C

Va

Vb

SUITABLE FOR HF

2.1 Special Requirements forHF

(a) No nodes with an undesiredcapacitance to ground.

In VHF, parasitic caps becomesignificant and quite similar valuesto the designed capacitances.Thus, we need to make sure thateach node in the filter MUSThave a desired capacitance togroundso that we know what it is andhow it is put in the transferfunctions.

2 FILTER DESIGN FOR HIGH FREQUENCIES

†

Vo =gmsC

(Va -Vb )

†

I = gm (Va -Vb)

†

I = gm (Va -Vb)

†

Vo =gmsC

(Va -Vb )


(b) Balanced operation for reducingeven harmonics and crosstalks.Signal inversion can be obtainedeasily in gm-C.

Balanced transconductance:

Iout,diff = Iop – Ion = gmVin

2.1 Special Requirements for HF (con’t)

V + Vc12 in

V – Vc in12

++–– I =on

g Vm in2–

I =opg Vm in

2+

gm

++–

–gm

++––gm

C

C

++–

–gm

++––gm

†

Vc +12

Vin

†

Vc -12

Vin

†

Vc +gmsC

Va -Vb2

Ê

Ë Á

ˆ

¯ ˜

†

Vc -gmsC

Va -Vb2

Ê

Ë Á

ˆ

¯ ˜

†

I = -gm2

Va -Vb( )

†

I = +gm2

Va -Vb( )

Vc is cancelled!


(c) Sensitivity must be LOW for component variations to reduce errors.

2.1 Special Requirements for HF (con’t)

In VHF filters, the capacitors are small and will have 20-100% part of parasitic cap. Hence, inaccuracy is expected incapacitance ratios. Fortunately, ratios of gm are usuallyinteger numbers, matching between gm’s should be good.Thus, sensitivities of filter transfer functions to capacitorvalues MUST BE KEPT LOW.

(d) Dynamic range is determined by• dynamic range of gm• dynamic range of filter structuree.g., if internal node signal levels have large variations (swings),then the output swing becomes restricted. This usually requirescomputer simulations for optimisation.


3. Gm-C FILTER SYNTHESIS

1. Cascaded biquad2. Signal flow graph3. State space method4. Gyrator method

BIQUAD: circuit realizing a generalfilter transfer function of second order

a2 = a1 = 0 --> LOWPASSa2 = a0 = 0 --> BANDPASSa1 = a0 = 0 --> HIGHPASS a1 = 0 --> BANDSTOP

†

H(s) = K a2s2 + a1s + a0

s2 + s woQp

+ wo


3.1 Cascaded Biquads

General biquad section using gm-C realization (VHF applications)

m+g m–g

m+g

m+g

CVa m+g

C1 2

Vb

VcC3

1 2 3

54

†

Vo =C3

C2 + C3

s2Vc + s gm 4C3

Vb +gm3gm5C1C3

Va

s2 + s gm3C2 + C3

+gm1gm2

C1(C2 + C3 )

Ê

Ë

Á Á Á Á Á

ˆ

¯

˜ ˜ ˜ ˜ ˜

where

So, K, a0, a1, wo and Qp can be chosen bychoosing gm’s and C’s.

†

wo =gm1gm2

C1(C1 + C2)

Qp =C2 + C3

C1

gm1gm 2gm 3


3.1 Cascaded Biquads (con’t)

Features:1. This biquad is suitable for very high frequencies because each node has a

known capacitance to ground.2. C2 is not essential, but is unavoidable. Hence, it must be taken into account.3. Cascading multiple biquads will cause loading effects, which must be taken

into consideration because there is no ideal buffer at high frequencies.4. Output level can be scaled for optimal dynamic range by varying K.

High order filters:

Disadvantage of cascaded biquads:Passband sensitivity to component variations tends to be too large for someapplications. (A better approach is to start with LC ladder.)

biquad1 biquad2


3.2 Signal Flow Graph Synthesis

The starting point is passive lossless LC ladder.The following is a 3rd order elliptic low-pass filter.

The state equations:C2 C3 C4

I5IC3

IL3L 3

I3I1R1

R5Vin

+

–VC2

+

–

VC4

+

–

+

–

Vout

†

sC2VC2 + IL3 + IC3 = I1

sC2VC2 + IL3 + sC3 (VC 2 -VC4 ) =Vin -VC2

R1

(sC2 + sC3 )VC 2 - sC3VC 4 +VC2R1

=VinR1

- IL3

VC2 +VC 2

s(C2 + C3 )R1=

C3VC4C2 + C3

+Vin

sR1(C1 + C3 )-

IL3s(C2 + C3 )

State VC2:


3.2 Signal Flow Graph Synthesis (con’t)

Signal flow graph for state VC2:

VC2 VC4

IL3

Vin

–1

†

1sR1(C2 + C3)

†

-1s(C2 + C3 )

†

C3C2 + C3

†

1sR1(C2 + C3)

VC4VC2Vin

R1IL3

–1

†

C3C2 + C3

†

1sR1(C2 + C3)

–11

OR

C2 C3 C4

I5IC3

IL3L 3

I3I1R1

R5Vin

+

–VC2

+

–

VC4

+

–

+

–

Vout

Combining similar factors together



State VC4:

C2 C3 C4

I5IC3

IL3L 3

I3I1R1

R5Vin

+

–VC2

+

–

VC4

+

–

+

–

Vout

†

sC4VC 4 +VoutR5

= IL3 + sC3 (VC 2 -VC4 )

s(C4 + C3 )VC 4 =-Vout

R5+ IL3 + sC3VC 2

VC4 =-Vout

sR5 (C3 + C4 )+

IL3s(C3 + C4 )

+C3VC 2

C3 + C4

VC4VC2 Vout1

–1

†

C3C3 + C4

†

1s(C3 + C4 )

†

1sR5 (C3 + C4 )

IL3

VC4VC2

†

C3C3 + C4 Vout1

1

†

1sR5 (C3 + C4 ) –1

R5IL3



State IL3 :

C2 C3 C4

I5IC3

IL3L 3

I3I1R1

R5Vin

+

–VC2

+

–

VC4

+

–

+

–

Vout

†

sL3IL3 = VC 2 -VC4

IL3 =VC2 -VC 4

sL3

VC4VC2 –1

†

1sL3

IL3

1



Combining the three sub-graph, weget the final signal flow graph:

C2 C3 C4

I5IC3

IL3L 3

I3I1R1

R5Vin

+

–VC2

+

–

VC4

+

–

+

–

Vout

VC4VC2Vin

R1IL3

–1

†

C3C2 + C3

†

1sR1(C2 + C3)

–11

†

R1sL3

1

1

–1

†

C3C4 + C3

†

1sR1(C3 + C4 )

–1

Vout1

R1 = R5



We can now synthesize the circuit with gm-C. The rules are:

1. The “1” branch is gm.2. All transconductances are 1/R1.3. 1/s branch is cap to ground.4. Gains C3/(C2+C3) and C3/(C4+C3) can be realized by

capacitor ladder.C2

C3

C4

VC2 VC4

m+gVin

m–g

m–gm+g

m–g m+g

m–g

VoutVC2 VC4

C3

C2 C4CL3

Exercise:Convert it to abalanced gm-Ccircuit.


4. REALISATION OF TRANSCONDUCTORS

Transconductors (gm blocks) can be realized in BJT form or MOSFET form.

Bipolar:

1. Fixed transconductor cascadedwith gain cell. A fixedtransconductor is usually adifferential pair linearized byresistor degeneration.

2. Differential input stage withmultiple inputs, with transistorscaling for better linearity.

MOS:

1. Fixed-bias triode MOS transistoras resistor. Multiple outputs arepossible using mirrors.

2. Varying-bias triode MOStransistor as resistor.

3. Differential input with constantdrain-source current.

To avoid confusion, in the next pages, we use Gm to stand for thetransconductance of the whole block, and gm for the transistor’s.


4.1 BJT Transconductors

I 1I 1

2I1

i o1 i o1

RE /2 RE/2Vi

+

–

Q1Q2

I 1I 1

i o1 i o1

REVi

+

–

Q1Q2

I 1 I 1

Fixed transconductance using resistor

†

io1

Vi= Gm =

12

gm+ RE

Note:Distortion due to non-constant Gm. So, linearity can beimproved if RE is much greater than 1/gm of the transistor.Moreover, if Vbe is assumed fixed, Vi appears purely acrossresistor and hence Gm = 1/RE (independent of gm).

No bias current flowsin RE. The CMvoltage is nearly zero,hence larger CMrange. (The base ofeach side must not beless than Vi/2, or thetransistor will be cutoff.)


4.1 BJT Transconductors (con’t)

I 1I 1

2I1

i o1 i o1

RE /2 RE/2Vi

+

–

Q1Q2

Finding the Gm for this fixed transconductance

†

io1

Vi

= Gm =1

2gm

+ RE

rπvbe

+

– gmvbeVi

2

+

–

RE/2

io1

Half-circuit equivalent model:

†

io1 = gmVi

2rp

rp + bRE

2

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

= Vi1

2gm

+ RE

Ê

Ë

Á Á Á Á

ˆ

¯

˜ ˜ ˜ ˜


4.1 BJT Transconductors (con’t)

Gain-cell transconductor (tunable Gm)

i o1

i o1

REVi

+

–

Q1Q2

I 1 I 1

Q 43 Q

Q5Q6

2I2

+–

+–

I 2 I 2

VLS

VLS

levelshifter

†

Gm =1

RE

Ê

Ë Á

ˆ

¯ ˜

I2

I1

Ê

Ë Á

ˆ

¯ ˜

The transconductancecan be tuned bysetting the currentratio I2/I1.


4.2 MOSFET Transconductors

Fixed-bias triode MOSFET—using a MOSFET operating in triode region tosimulate a resistor

Vi+ –

Q 1 Q2Vi

I 1 I 1

I 2 I 2

Q 3 Q 4

Q 5

Q 7 Q 8

Q 6

I + i1 o1I – i1 o1

MOSFET Q9 in trioderegion acting as resistor

Q9

Vc

†

Gm =io1

Vi+ -Vi

– = mCoxWL

Ê

Ë Á

ˆ

¯ ˜

9vgs9 -VTH( )

Transconductance is

This Gm can be easilymodified to give multipleoutputs! (using mirrors)


4.2 MOSFET Transconductors (con’t)

Varying bias triode MOSFET—improved linearity

†

Gm =io1

V1 -V2=

4k1k3 I1

k1 + 4k3( ) k1

where kn =mCox

2WL

Ê Ë Á

ˆ ¯ ˜

n

Transconductance is

V1

Q 1Q2

V2

I 1 I 1

I 1 I 1

io1 io1

Q 3

Q 4

Q3 and Q4 are in triode region andundergo varying bias conditions (becausetheir gates are not connected to fixed bias.)Why is linearity improved? Try theexercise on next page.


EXERCISE

Consider the circuit of the previous page. Suppose I1 = 100µA, µCox =96µA/V2, (W/L)1 = (W/L)2 = 20, (W/L)3 = 3, and V2 = 0.

(a) Assuming a perfectly linear transconductor, find io1 when V1=2.5mVand 250mV, using the formula given in the previous page.

(b) Assume the gates of Q3 and Q4 are connected to ground and useclassical models for both the triode and active regions. Find the truevalue of io1 when V1=2.5mV and 250mV. Compare your results withthose found in (a).

(c) Repeat (b), assuming the gates of Q3 and Q4 are connected to the inputsignals as shown in the circuit.

(d) Comment on the linearity improvement, if any, when varying biastriode transistor is used.

high frequency filter design - michael tsecktse.eie.polyu.edu.hk/eie403/filterdesign.pdf · michael...

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