Continuous-Time Filters

1.0 Operational Transconductance Amplifier (OTA)

+

-

+

-Vi Gm Vo

Io

GmViVi

Zin Zout

(a) (b)

+

-

+

-

Io

Io

I1

I2

Vo+

Vo-

Vi+

Vi-

2Gm

2Gm

+

-

+

-GmVi Vo

Io

Io

Io

Vo

Io=GmVi

Io=GmViVi = Vi+ - Vi-

Vo = Vo+ - Vo-

(c) (d) Figure 1. Ideal small signal equivalent circuit of Single ended OTA and Fully differential OTA Implementation Using Single ended OTAs. Figure 1(a) shows the symbol of single ended OTA. Its ideal equivalent circuit is shown in Figure 1(b). Its operation is given by:

∞=∞=

== +

out

in

-imo

ZZ

) v- G(vvGi

1

That is, its output impedance is ∞ which is 0 for ideal opamp. A fully differential OTA shown in Figure (c) can be implemented using two single ended OTAs with twice the transconductance. Its connection is shown in Figure 1(d). The equivalency can be shown as follws:

im-iimo

-iimooo21

imo2

imo1

VG)V-(VGI

)V-(V2G2I)(-I-II-I

VG2-II

VG2II

==

===

==

==

+

+

−

+

2. OTA Simple Building Block Circuits

+

-VoGm

+

-Vo

+

-

+

-+

-

+

-

+

-

Gm2

Gm1

Zi

Gm Gm

Gm1Gm2

Vi Vi Vi

Zi1

Gm=

Vi

Vo ZL

(a) (b) (c)

(d) (e)

R R

IoIo

Ii

1

Zi1=Gm2

Gm1Gm2ZL1=

VoVi

IiI1

I2

Figure 2 OTA simple building blocks

Figure 2(a) shows a simple voltage positive gain amplifier. Its gain is derived as follows: I imo VG=

imoo RVGRIV ==

RGVV

mi

oV == A

Zi=∞ , since the input impedance of OTA is ∞, and Zo=R, since the output impedance of OTA is ∞. In Figure 2(b) is the same as Figure 2(a) except for the input inversion resulting in negative gain. In Figure 2(c),

2

io

i

oV VV since ; 1

VV

=== A I iimo -IV-G ==

omi

ii Z

G1

IV

=== Z Figure 2(d) is the same as in Figure 2(a) with a load R=1/Gm2.

m2

m1m1V G

GRG == A As in Figure 2(a), the input and output impedance of Figure 2(d) are:

m2oi G

1 Z; =∞= Z In Figure 2(e), I im11 VG= I om22 V-G= V im1L1Lo VGZIZ ==

Lm1i

oV ZG

VV

== A I iiLm2m12 -IVZG-G ==

Lm2m1i

ii ZGG

1IV

== Z That is, the input impedance is the reciprocal of the load impedance. If the load impedance is capacitive, then the input impedance is inductive. This circuit is an impedance converter. Since the output impedance of OTA1 and the input impedance of OTA2 are both ∞, the circuit output impedance is: Z Lo Z=

3

These are summarized in the following table. Table 1: OTA Simple Building Block Circuits. Circuit Type Av Zi Zo Voltage amplifier with positive gain, Fig 2(a)

GmR ∞ R

Voltage amplifier with negative gain, Fig 2(b)

- GmR ∞ R

OTA Resistance , Fig 2(c) 1 1/Gm 1/Gm Voltage amplifer using OTA resistance, Fig 2(d)

Gm1/Gm2 ∞ 1/Gm2

OTA impedance converter, Fig 2(e)

Gm1ZL 1/( Gm1Gm2ZL) ZL

3. First Order Filter Building Block Circuits

+

-Gm

C

Vo

(b) LP1(s)

Io

(c) HP1(s)

Io

+

-Gm

Vi

Vi

+

-+

-Vi

Vo

Gm1Gm2

C2

IC2

I1I2

(d) TF1(s)

IC1 C1

+

-

C

GmVi Vo

V1 V2(e)

Ftype V1 V2LP Vi 0HP 0 Vi

(f)

+

-+

-Vi

Vo

Io

GmGm

C

I1I2

IC

VoC

Io

Figure 3. First Order Filter Single-Ended OTA Building Blocks.

4

Figure 3 shows the first order filter building blocks. The two lowpass filter implementations in Figure 3(a) and 3(b) will be shown to be the same. Also, only Figure 3(a) can be converted to fully differential ota iimplementation shown in Figure 4(a). In Figure 3(a),

o

o

m

m

m

m

i

o

imom

omim21oC

om2

im1

wsw

C/GsC/G

GsCG

VV

VG)VG(sCVGVGIIsCVI

VGIVGI

+=

+=

+=

=+−=+==

−==

In Figure 3(b), I ooimo sCV)V-(VG == V V C/Gs

C/GGsC

G

m

m

m

m

i

o

+=

+=

That is, Figure 3(a) and 3(b) implement the same transfer function. In Figure 3(c), I )V-sC(VV-G ioomo == V V ommi

o

wss

C/Gss

GsCsC

+=

+=

+=

In Figure 3(d),

o

01

21m2

21m1212

m221

m12

i

o

oi2om2im1C221o1C1

oi2C2

om22

im11

wsasa

)]C/(C[Gs)]C/(C[G)]C/(Cs[C

G)Cs(CGsC

VV

)V-(VsCVGVGIIIVsCI)V-(VsCI

VGIVGI

++

=++

+++=

+++

=

+−=++===−=

=

5

Table 2: OTA First Order Filter. Circuit Type Transfer Function Wo First order Lowpass, Fig 3(a),(b)

First order Highpass, Fig 3(c)

General First order Fig 3(d)

/CGs/CG

m

m

+/CG m

/CGss

m+/CG m

[ ] [ ][ ])C/(CGs

)C/(CG)C/(CCs

21m2

21m1212

+++++ )C/(CG 212m +

A universal first order filter can be implemented as shown in Figure 3(e). The universal filter has four terminals Vi, Vo, V1, and V2. The lowpass filter can be selected by connecting the input Vi to V1 and V2 to ground. The highpass filter is selected by connecting the input Vi to V2 and V1 to ground. These are summarized in Figure 3(f).

Gm2

+

-

Vo

2C1

2C1

2C2

2C2

Gm1+

-Vi

Gm

+

-

Vo+

-Vi

Gm Vo

2C+

-Vi

I2

I22C

2C2C

(a)

(b)

(c)

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-Gm

I1

I1

IC'

IC"

Io

IoIo

Io

Figure 4. Differential OTA Implementation of First-Order Filters.

6

Figure 4 shows the fully differential OTA implementation of the first order filters. It shows the number and value of capacitors doubled. Consequently, it will consume more VLSI real estate. However, a fully differential circuits implementation have the advantage of better noise immunity and distortion properties. The equivalentness of the two implementations will be illustrated for the first order lowpass and highpass filters. That is, the transfer function of Figure 4(a) will be shown to be identical to that of Figure 3(a) or 3(b). In Figure 4(a):

C

s

/GsC/G

GsCG

VV

VG)VG(sCVGVGsCV

)V-(VG)V-(VGI-I)V-sC(V

2I-2I)V-C(V2"I'I

II2sCV"I

II2sCV'I

)V-(VGVGI

)V-(VGVGI

m

m

m

m

i

o

imom

omimo

-oom

-iim21

-oo

21-ooCC

21-oC

21oC

-oomom2

-iimim1

+=

+=

=++−=−

+−=+=−

+=−=+

+−==

+−=−=

==

==

+++

+

+

+

+

The high pass filter shown in Figure 4(b) with one fully differential ota implementation will be shown to have the same transfer function as the single-ended ota in Figure 3(c). The derivation follows:

C/Gss

GsCsC

VV

VG]VsC[-V)V-(V2G)]V-(V)V-2sC[-(V2I

equation,first with theequating and equations twoabove theAdding)V-2sC(VI

)V-2sC(VI

)V-(VGVGI

mm

omio

-oom

-ii

-ooo

-i

-oo

oio

-oomomo

+=

+=

=+=+=

=

=

==

+++

++

+

The general first order filter shown in Figure 4(c) can similarly be derived. Note that lowpass filter needs two fully differential otas, while for highpass filter requires only one.

The calculation is independent of whether using lowpass or highpass transfer function. The lowpass filter transfer function will be used for illustration. The transfer function to be implemented is given by:

o

o

wsw

wswL1(s)

+=

+=

7

In all the subsequent examples, it will assumed that the value of w=6.2832*105 corresponding to a frequency of 100kHz.

The capacitance value for a given Gm=85.44u is obtained as follows:

pF98.1355E2832.6

6-E44.85wGC

ww

o

m

o

=+

==

=

The general filter transfer function to be implemented has a pole at fp=100kHz, and zero at fz=200kHz and with DC gain of 0.5. That is,

)]C/(C[Gs)]C/(C[G)]sC/(CC[

wsasa

5)1e2(s5)1e2(0.5s

5)1e2(s5)2e20.5(sTF(s)

21m2

21m1212

o

o1

+++++

=++

=++++

=++++

=ππ

ππ

644.8551e26e44.855)1e2(

wG

a)C(CaG

67.99pFpF)99.67(0.5-1

0.5Ca-1

aC

pF99.6751e26e44.85)5.01(

wG)a-(1C

wGC

a11C

a-1aCCC

given is G Assume

wCC

G

aCC

G

Ca-1

aC

aCC

C

o

m20210m1

11

12

o

m211

o

m21

11

1

1121

m2

021

m2

021

m1

11

12

121

2

−=+−

+==+=

=

=

=

=+−

−==

=−

=

+=+

=+

=+

=

=+

eπ

π

π

8

3.1 First Order Filters Simulation First Order Lowpass Filter Transfer Function *Filename = "lp1.cir" *First order lowpass filter fo=100k .PARAM w=6.2832e+5 Vin 1 0 DC 0V AC 1V R1 1 0 1E+20 E2 2 0 LAPLACE {V(1)}={w/(s+w)} Rout 2 0 1k * Analysis .AC DEC 10 1Hz 100MegHz .PROBE .END First Order Highpass Filter Transfer Function *Filename = "hp1.cir" *First order lowpass filter fo=100k .PARAM w=6.2832e+5 Vin 1 0 DC 0V AC 1V R1 1 0 1E+20 E2 2 0 LAPLACE {V(1)}={s/(s+w)} Rout 2 0 1k * Analysis .AC DEC 10 1Hz 100MegHz .PROBE .END General First Order Filter Transfer Function *Filename = "gen1.cir" *General first order filter p=100kHz, z=200kHz .PARAM wp=6.2832e+5 *.PARAM wz=2*wp .PARAM wz=12.5664e+5 Vin 1 0 DC 0V AC 1V R1 1 0 1E+20 E2 2 0 LAPLACE {V(1)}={0.5*(s+wz)/(s+wp)} Rout 2 0 1k * Analysis .AC DEC 10 1Hz 100MegHz .PROBE .END

9

Ideal First Order Lowpass Filter Response

Ideal First Order Highpass Filter Response

10

Ideal General First Order Filter Response Universal First Order Filter Single-ended OTA Implementation *Filename = "univ1_i1.cir", see Figure 3e *Universal First order filter. fo=100K *Using Single-ended OTA Vin 1 0 DC 0V AC 1V *LP *Xs1flt 1 0 2 S1FLT *HP Xs1flt 0 1 2 S1FLT R2 2 0 1E+20 .SUBCKT S1FLT v1 v2 v0 .PARAM C=135.98pF Xota1 v1 v0 v0 WSOTA C1 v0 v2 {C} IC=0V .ENDS .SUBCKT WSOTA in+ in- out G1 0 out in+ in- 85.44U .ENDS * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END General First Order Filter Single-ended OTA Implementation *Filename = "gen1_i1.cir", see Figure 3d *General First order filter. p=100kHz, z=200kHz *Using Single Ended OTA

11

.PARAM C=67.99pF Vin 1 0 DC 0V AC 1V Xota1 1 0 2 WSOTA Xota2 0 2 2 WSOTA C1 2 0 {C} C2 1 2 {C} R2 2 0 1E+20 .SUBCKT WSOTA in+ in- out G1 0 out in+ in- 85.44U .ENDS * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END

First Order Lowpass Filter Response

12

First Order Highpass Filter Response

General First Order Filter Response Fully Differential OTA Implementation of First Order Filter Universal implementation is not possible using fully differential OTA. Lowpass filters require two fully differential OTAs, while highpass filters only need one. Fully Differential OTA Implementation of First Order Lowpass Filter

13

*Filename = "lp1_id.cir", see Figure 4a *First order filter fo=100K *Using fully differential OTA Vin 1 2 DC 0V AC 1V .PARAM C=135.98pF Xota1 1 2 3 4 DIFFOTA Xota2 3 4 4 3 DIFFOTA C1 3 0 {2*C} IC=0V C2 4 0 {2*C} IC=0V R1 1 0 1E+20 R2 2 0 1E+20 R3 3 0 1E+20 R4 4 0 1E+20 * Fully Differential OTA Implementation * by definition .SUBCKT DIFFOTA in+ in- out+ out- G1 0 out+ in+ in- 85.44U G2 0 out- in+ in- -85.44U .ENDS * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END Fully Differential OTA Implementation of First Order Highpass Filter *Filename = "hp1_id.cir", see Figure 4b *First order filter. fo=100K *Using Fully Differential OTAs Vin 1 2 DC 0V AC 1V .PARAM C=135.98pF Xota1 3 4 4 3 DIFFOTA C1 1 3 {2*C} IC=0V C2 2 4 {2*C} IC=0V R1 1 0 1E+20 R2 2 0 1E+20 R3 3 0 1E+20 R4 4 0 1E+20 * Fully Differential OTA Implementation * by definition .SUBCKT DIFFOTA in+ in- out+ out- G1 0 out+ in+ in- 85.44U G2 0 out- in+ in- -85.44U .ENDS * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END Fully Differential OTA Implementation of General First Order Filter *Filename = "gen_id.cir", see Figure 4c *First order filter fp=100kHz, fz=200kHz *Using fully differential OTA

14

Vin 1 2 DC 0V AC 1V .PARAM C=67.99pF Xota1 1 2 3 4 DIFFOTA Xota2 3 4 4 3 DIFFOTA C11 3 0 {2*C} IC=0V C12 4 0 {2*C} IC=0V C21 1 3 {2*C} IC=0V C22 2 4 {2*C} IC=0V R1 1 0 1E+20 R2 2 0 1E+20 R3 3 0 1E+20 R4 4 0 1E+20 * Fully Differential OTA Implementation * by definition .SUBCKT DIFFOTA in+ in- out+ out- G1 0 out+ in+ in- 85.44U G2 0 out- in+ in- -85.44U .ENDS * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END

First Order Lowpass Filter Response

15

First Order Highpass Filter Response

General First Order Filter Response

16

4. Second Order Filter Building Block Circuits

+

-Gm1 +

-Gm2C1

C2

Vi Vo

(a) LP(s)

+

-Gm1 +

-Gm2C1

C2

Vi

Vo

(b) HP(s)

+

-Gm1 +

-Gm2C1

C2

ViVo

(c) BP(s)

+

-Gm1 +

-Gm2C1

C2Vi

Vo

(d) BR(s)

V1I1I2

I2

Figure 5. Second Order Filters Implementation with fixed Q and 2 OTAs each. There are two second order filter building blocks. One requiring two OTAs and the other 3 OTAs. The selection depends on whether an adjustable Q is required or not.

17

Fixed Q Implementation Figure 5 shows the second order filter implementation using 2 OTAs. The implementation has a non-adjustable Q, once the capacitive components had been determined. The tranfer function derivation will be demonstrated for the lowpass circuit. In Figure 5(a):

)V-(VGI oim11 = )V-(VGI o1m22 =

)V-V(

sCG

sCI

V oi1

m1

1

11 ==

== ooi

1

m1

2

m2

2

2o V-)V-V(

sCG

sCG

sCIV

21

m2m1

2

m22

21

m2m1

m2m1m21212

m2m1

i

o

CCGG

CG

ss

CCGG

GGGsCCCsGG

VV

+

+

=++

=

Comparing the transfer function with the standard lowpass transfer function given in Table . The wo and Q are obtained as follows:

m2m1m21

m

21

m2m1o GGG if ;

CCG

CCGG

==== w w 2

m2o

CG

Q=

Solving for Q,

1

2

m

2o

CC

GCw

== Q Note that for a given values of C1 and C2, Q is fixed but wo can be adjusted by changing Gm. The expressions for the other circuits can be derived in a similar way. The results are summarized in Table 4.

18

Table 3. Second Order Filter Transfer Function. Circuit Type Transfer Function Lowpass

Highpass

Bandpass

Bandreject

2oo

2

2o

w/Q)s(wsw

++

2oo

2

2

w/Q)s(wss

++

2oo

2o

w/Q)s(ws/Q)s(w

++

2oo

2

2o

2

w/Q)s(wsws

+++

Table 4. Second Order 2 OTAs Implementation Filter Parameters. Circuit Type Transfer Function wo* Q Lowpass wo Adjustable Q Fixed, Fig 5(a)

Highpass wo Adjustable Q Fixed, Fig 5(b)

Bandpass wo Adjustable Q Fixed, Fig 5(c)

Bandreject wo Adjustable Q Fixed, Fig 5(d)

)C/CGG()/CG(ss)C/CGG(

21m2m12m22

21m2m1

++ 21

m

CCG

1

2

CC

21

m

CCG

)C/CGG()/CG(sss

21m2m12m22

2

++ 1

2

CC

21

m

CCG

)C/CGG()/CG(ss)/CG(s

21m2m12m22

2m2

++1

2

CC

21

m

CCG

)C/CGG()/CG(ss)C/CGG(s

21m2m12m22

21m2m12

+++

1

2

CC

* Assume Gm=Gm1=Gm2 The capacitances can be computed from any of the four transfer functions. The lowpass transfer function will be used for this purpose. The transfer function to be implemented is:

2oo

2

2o

22

2

w/Q)s(wsw

ws*w*61803.0swL2(s)

++=

++=

19

The capacitances are computed as follows:

12-E02.22012)-E98.135(61803.1QCCQC

12-E04.8412-E98.1351.61803

1CQ1C

pF98.13512-E98.13556.2832E

6-85.44Ew

GC

Cw

GCQ

CQC

CCQ

61803.1w*61803.0

ww*61803.0

wQ

w*61803.0Q

ww

GCC

CCGww

ww

12

2

1

m

22

m21

2

12

2

1

2

o

o

2m

21

21

mo

22o

====

===

==+

==

=

=

=

=

===

=

=

==

=

20

Adjustable Q Implementation

+

-Gm1 +

-Gm2C1

C2

ViVo

(a) LP(s)

+

-Gm3

+

-Gm1 +

-Gm2C1

C2

Vi

Vo

(b) HP(s)

+

-Gm3

+

-Gm1 +

-Gm2C1

C2Vi

Vo

(c) BP(s)

+

-Gm3

+

-Gm1 +

-Gm2C1

C2

Vi

Vo

(d) BR(s)

+

-Gm3

I1I2

I3

V1

Figure 6. Second Order Filters Implementation with adjustable Q and 3 OTAs each. Figure 6 shows the second order filter implementation using 3 OTAs. The implementation has adjustable Q, its value is determined by the third OTA. The tranfer function derivation will be demonstrated for the lowpass circuit. In Figure 6(a):

21

)V-(VGI oim11 =

1m22 VGI =

)V-V(

sCG

sCIV oi

1

m1

1

11 ==

)V-V(sC

GGI oi1

m2m12 =

om33 V-GI =

=+= om3oi

2

m2m1

232

2o VG-)V-V(

sCGG

sC1]II[

sC1V

21

m2m1

2

m32

21

m2m1

m2m1m31212

m2m1

i

o

CCGG

CG

ss

CCGG

GGGsCCCsGG

VV

++=

++=

The wo and Q can be derived by comparing this transfer fucntion with the standard lowpass transfer function given in Table.

21m2m1mm

21

m2m1o CCC and GGG if ;

CG

CCGG

====== w w 2

m3o

CG

Q=

Solving for Q

m3

m

m3

o

GG

GCw== Q

NOTE for a given capacitor value, Q and wo can be adjusted independently of each other. Q can be adjusted by Gm3 and wo by Gm. The expressions for the other circuits in Figure 6 can similarly be derived. The results are summarized in Table.

22

Table 5. Second Order 3 OTAs Implementation Filter Parameters. Circuit Type Transfer Function wo* Q Lowpass wo Adjustable Q Adjustable, Fig 6(a)

Highpass wo Adjustable Q Adjustable, Fig 6(b)

Bandpass wo Adjustable Q Adjustable, Fig 6(c)

Bandreject wo Adjustable Q Adjustable, Fig 6(d)

G

G

G

G

m3

m

GG

)C/CGG()/CG(ss)C/CGG(

21m2m12m32

21m2m1

++ Cm

)C/CGG()/CG(sss

21m2m12m32

2

++m3

m

GG

Cm

)C/CGG()/CG(ss)/CG(s

21m2m12m32

2m3

++ m3

m

GG

Cm

m3

m

GG

)C/CGG()/CG(ss)C/CGG(s

21m2m12m32

21m2m12

+++

Cm

* Assume Gm=Gm1=Gm2; C=C1=C2 The 3 OTAs implementation will itllustrated using the same transfer function L2(s) as in the 2 OTAs implementation.

6-E8.5261803.1

6-E44.85Q

GG

61803.1w*61803.0

ww*0.61803

wQ

w*61803.0Q

w

pF98.1355E2832.6

6-E44.85wGC

wwww

mm3

o

o

o

m

o

22o

===

===

=

=+

==

==

As in the first order filter, universal filter implementation of second order filter in Figure 5 and figure 6 are shown in Figure 7(a) and 7(c) respectively. The corresponding programming are shown in Figure 7(b) and 7(d) respectively.

23

+

-Gm1 +

-Gm2C1

C2

ViVo

V1I1I2

V1 V2 V3

+

-+

-+

-

Vo

Vi

V1 V2 V3

Gm1Gm2

Gm3

C1C2

Ftype V1 V2 V3LP Vi 0 0HP 0 0 ViBP 0 Vi 0BR Vi 0 Vi

Ftype V1 V2 V3LP Vi 0 0HP 0 Vi 0BP 0 0 ViBR Vi Vi 0

(a)

(c)

(d)

(b)

Figure 7. Universal Second Order Filter Implementations.

24

4.1 Second Order Filters Simulation Second Order Lowpass Filter Transfer Function *Filename= "LP2.cir" *Second order lowpass filter .PARAM w=6.2832e+5 Vin 1 0 DC 0V AC 1V * third biquad R1 1 0 1E+20 E2 2 0 LAPLACE {V(1)}={(w*w)/(s*s+0.61803*w*s+w*w)} *output Rout 2 0 1k * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END Second Order Highpass Filter Transfer Function *Filename= "HP2.cir" *Second order highpass filter .PARAM w=6.2832e+5 Vin 1 0 DC 0V AC 1V * third biquad R1 1 0 1E+20 E2 2 0 LAPLACE {V(1)}={(s*s)/(s*s+0.61803*w*s+w*w)} *output Rout 2 0 1k * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END Second Order Bandpass Filter Transfer Function *Filename="BP2.cir" *Second order bandpass filter .PARAM w=6.2832e+5 Vin 1 0 DC 0V AC 1V * third biquad R1 1 0 1E+20 E2 2 0 LAPLACE {V(1)}={(0.61803*w*s)/(s*s+0.61803*w*s+w*w)} *output Rout 2 0 1k * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END Second Order Bandreject Filter Transfer Function *Filename="BR2.cir"

25

*Second order bandreject filter .PARAM w=6.2832e+5 Vin 1 0 DC 0V AC 1V * third biquad R1 1 0 1E+20 E2 2 0 LAPLACE {V(1)}={(s*s+w*w)/(s*s+0.61803*w*s+w*w)} *output Rout 2 0 1k * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END

Ideal Second Order Lowpass Filter Response

26

Ideal Second Order Highpass Filter Response

Ideal Second Order Bandpass Filter Response

27

Ideal Second Order Bandreject Response Universal Second Order Filter Using 2 Ideal OTAs per Filter *Filename = "univ2_i2.cir", see Figure 7a *Biquad 2nd order filter. fo=100K Vin 1 0 DC 0V AC 1V *LP *Xs2flt 1 0 0 2 S2FLT *HP *Xs2flt 0 0 1 2 S2FLT *BP *Xs2flt 0 1 0 2 S2FLT *BR Xs2flt 1 0 1 2 S2FLT R2 2 0 1E+20 .SUBCKT S2FLT v1 v2 v3 v0 .PARAM C=135.98pF .PARAM Q=1.61803 Xota1 v1 v0 n2 WSOTA Xota2 n2 v0 v0 WSOTA C1 n2 v2 {C/Q} IC=0V C2 v0 v3 {Q*C} IC=0V Rn2 n2 0 1E+20 .ENDS .SUBCKT WSOTA in+ in- out G1 0 out in+ in- 85.44U .ENDS

28

* Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END Universal Second Order Filter Using Ideal 3 OTAs per Filter *Filename = "univ2_i3.cir", see Figure 7c *Biquad 2nd order filter. fo=100K Vin 1 0 DC 0V AC 1V *LP *Xs3flt 1 0 0 2 S3FLT *HP *Xs3flt 0 1 0 2 S3FLT *BP *Xs3flt 0 0 1 2 S3FLT *BR Xs3flt 1 1 0 2 S3FLT R2 2 0 1E+20 .SUBCKT S3FLT v1 v2 v3 v0 .PARAM C=135.98pF Xota1 v1 v0 n2 WSOTA Xota2 n2 0 v0 WSOTA Xota3 v3 v0 v0 WSOTA1 C1 n2 0 {C} IC=0V C2 v0 v2 {C} IC=0V Rn2 n2 0 1E+20 .ENDS .SUBCKT WSOTA in+ in- out G1 0 out in+ in- 85.44U .ENDS .SUBCKT WSOTA1 in+ in- out G1 0 out in+ in- 52.8U .ENDS * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END

29

Second Order Lowpass Filter Response

Second Order Highpass Filter Response

30

Second Order Bandpass Response

Second Order Bandreject Response

31

5. Butterworth Filters Implementation Table 6. Butterworth D(sn) (where H(sn)=K/D(sn); K=1) N D(sn) 1

2

3

4

5

1sn +

1s2s n2n ++

1)s1)(s(s n2nn +++

1)s84776.11)(ss76537.0(s n2nn

2n ++++

1)s61803.11)(ss61803.01)(s(s n

2nn

2nn +++++

The fifth order normalized Butterworth low pass filter is obtained from the above table:

1).61803s1)(s1.61803s1)(ss(1)B5(s

n2nn

2nn

n +++++=

To scale the transfer function to the desired frequency of operation, the transformation s is applied to the normalized transformation.

s/wn =

T3(s)T1(s)T2(s) ws*w*0.61803s

wws*w*1.61803s

wws

wB5(s) 22

2

22

2

=

++

++

+

=

This is a cascade design based on the simple tandem interconnection on N first-order and biquad stages. If the stages are non-interacting, the overall transfer function is the product of the individual stage transfer functions. Non-interacting means that at any stage, say the ith stage, the output impedance Zout(i) is not loaded by the input impedance Zin(i+1) of the succeeding stage, i.e., |Zout(i)|<<|Zin(i+1)|. For op amp-based active filters, this is inherently satisfied, since op amp has high input impedance and low output impedance. But OTA based active filters, due to high input and output impedances require inter-stage buffering. These buffers are usually realized by unity-gain source follower stages. In addition, output buffering is required to drive external loads. The capacitance and transconductance of each OTA will be computed for each cascded filter. For T1(s):

pF98.1355E2832.6

6-E44.85wGC

ww

o

m

o

=+

==

=

For T2(s):

32

6-E25.138618.0

6-E44.85Q

GG

618.0w*61803.1

ww*1.61803

wQ

w*61803.1Q

w

pF98.1355E2832.6

6-E44.85wGC

wwww

mm3

o

o

o

m

o

22o

===

===

=

=+

==

==

For T3(s):

6-E8.5261803.1

6-E44.85Q

GG

618.1w*61803.0

ww*0.61803

wQ

w*61803.0Qw

pF98.1355E2832.6

6-E44.85wGC

wwww

mm3

o

o

o

m

o

22o

===

===

=

=+

==

==

5.1 Fifth Order Butterworth Filter Simulation Fifth Order Butterworth Filter Transfer Function *Filename ="butter5.cir" *Fifth order butterworth filter, fo=100k .PARAM w=6.2832e+5 Vin 1 0 DC 0V AC 1V *first biquad R1 1 0 1E+20 E1 2 0 LAPLACE {V(1)} = {(w)/(s+w)} * second biquad R2 2 0 1E+20 E2 3 0 LAPLACE {V(2)}={(w*w)/(s*s+1.61803*w*s+w*w)} * third biquad R3 3 0 1E+20 E3 4 0 LAPLACE {V(3)}={(w*w)/(s*s+0.61803*w*s+w*w)} *output

33

Rout 4 0 1k * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END Fifth Order Butterworth Filter Implementation Using Ideal 3 OTAs per Filter

V1V2V3

Vo

S2FLT

(3)

(0)

V1

V2Vo

V1V2V3

Vo

(0)S3FLT

(4)

S1FLT

(1)(2)

(0)

(5)(6)

*Filename = "but5_i3.cir" *Butterworth 5th order filter. fo=100K Vin 1 0 DC 0V AC 1V Xs1flt 1 0 2 S1FLT Xbuf1 2 3 BUF Xs2flt 3 0 0 4 S2FLT Xbuf2 4 5 BUF Xs3flt 5 0 0 6 S3FLT R2 2 0 1E+20 R3 3 0 1E+20 R4 4 0 1E+20 R5 5 0 1E+20 R6 6 0 1E+20 .SUBCKT S1FLT v1 v2 v0 .PARAM C=135.98pF Xota1 v1 v0 v0 WSOTA C1 v0 v2 {C} .ENDS .SUBCKT S2FLT v1 v2 v3 v0 .PARAM C=135.98pF Xota1 v1 v0 n2 WSOTA Xota2 n2 0 v0 WSOTA Xota3 v3 v0 v0 WSOTA2 C1 n2 0 {C} IC=0V C2 v0 v2 {C} IC=0V Rn2 n2 0 1E+20 .ENDS .SUBCKT S3FLT v1 v2 v3 v0 .PARAM C=135.98pF Xota1 v1 v0 n2 WSOTA Xota2 n2 0 v0 WSOTA Xota3 v3 v0 v0 WSOTA3 C1 n2 0 {C} IC=0V C2 v0 v2 {C} IC=0V Rn2 n2 0 1E+20 .ENDS .SUBCKT WSOTA in+ in- out G1 0 out in+ in- 85.44U

34

.ENDS .SUBCKT WSOTA2 in+ in- out G1 0 out in+ in- 138.25U .ENDS .SUBCKT WSOTA3 in+ in- out G1 0 out in+ in- 52.8U .ENDS .SUBCKT BUF in out E1 out 0 in 0 1 .ENDS * Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END

Ideal Fifth Order Butterworth Filter Response (in DB) Ideal Fifth Order Butterworth Filter Response (in V)

35

Implemented Fifth Order Butterworth Response (in DB) Implemented Fifth Order Butterworth Response (in V)

36

6. Chebyshev Filters Implementation Chebyshec D(sn) for 1dB ripple (Where H(sn)=K/D(sn); K selected to yield unity DC gain) N D(sn) 1

2

3

4

5

1.96523sn +

1.10251s09773.1s n2n ++

0.9942)s49417.0049417)(s(s n2nn +++

0.2794)s67374.00.9865)(ss27907.0(s n2nn

2n ++++

0.4293)s4684.00.98831)(ss17892.00.28949)(s(s n

2nn

2nn +++++

The fifth order normalized Chebyshev low pass filter is obtained from the above table:

0.4293).46841s0.98831)(s.17892s0.28949)(ss(.4293)0.98831)(0(0.28949)()Ch5(s

n2nn

2nn

n +++++=

To scale the transfer function to the desired frequency of operation, the transformation s is applied to the normalized transformation.

s/wn =

T3(s)T1(s)T2(s) w*0.4293s*w*0.46841s

w*4293.0w*.988310s*w*0.17892s

w*0.98831w*0.28949s

w*0.28949Ch5(s) 22

2

22

2

=

++

++

+

=

The capacitance and transconductance of each OTA will be computed for each cascded filter. For T1(s):

pF73.4695E2832.6*28949.0

6-E44.85wGC

w*28949.0w

o

m

o

=+

==

=

For T2(s):

37

6-E37.1556.5

6-E44.85Q

GG

56.5w*17892.0

w*994.0w*0.17892

wQ

w*17892.0Q

w

pF8.1365E2832.6*994.0

6-E44.85wG

C

w*0.994ww*98831.0w

mm3

o

o

o

m

o

22o

===

===

=

=+

==

==

For T3(s):

6-E03.614.1

6-E44.85Q

GG

4.1w*46841.0w*6552.0

w*0.46841w

Q

w*46841.0Q

w

pF5.2075E2832.6*6552.0

6-E44.85wGC

w*0.6552ww*4293.0w

mm3

o

o

o

m

o

22o

===

===

=

=+

==

==

6.1 Fifth Order Chebyshev Filter Simulation Fifth Order Chebyshev Filter Transfer Function *Filename= cheby5.cir *Cheby5 Fifth order chebyshev filter *First order filter .PARAM w=6.2832e+5 Vin 1 0 DC 0V AC 1V *first biquad R1 1 0 1E+20 E1 2 0 LAPLACE {V(1)} = {(0.28949*w)/(s+0.28949*w)} * second biquad R2 2 0 1E+20 E2 3 0 LAPLACE {V(2)}={(0.4293*w*w)/(s*s+0.4684*w*s+0.4293*w*w)} * third biquad

38

R3 3 0 1E+20 E3 4 0 LAPLACE {V(3)}={(0.98831*w*w)/(s*s+0.1789*w*s+0.98831*w*w)} *output Rout 4 0 1k * Analysis .AC DEC 100 1Hz 100MHz .PROBE .END Fifth Order Chebyshev Filter Implementation Using Ideal 3 OTAs per Filter *Filename = "chb5_i3.cir" *Chebyshev 5th order filter. fo=100K Vin 1 0 DC 0V AC 1V Xs1flt 1 0 2 S1FLT Xs2flt 2 0 0 3 S2FLT Xs3flt 3 0 0 4 S3FLT R2 2 0 1E+20 R3 3 0 1E+20 R4 4 0 1E+20 .SUBCKT S1FLT v1 v2 v0 .PARAM C=469.73pF Xota1 v1 v0 v0 WSOTA C1 v0 v2 {C} .ENDS .SUBCKT S2FLT v1 v2 v3 v0 .PARAM C=136.8pF Xota1 v1 v0 n2 WSOTA Xota2 n2 0 v0 WSOTA Xota3 v3 v0 v0 WSOTA2 C1 n2 0 {C} IC=0V C2 v0 v2 {C} IC=0V Rn2 n2 0 1E+20 .ENDS .SUBCKT S3FLT v1 v2 v3 v0 .PARAM C=207.5pF Xota1 v1 v0 n2 WSOTA Xota2 n2 0 v0 WSOTA Xota3 v3 v0 v0 WSOTA3 C1 n2 0 {C} IC=0V C2 v0 v2 {C} IC=0V Rn2 n2 0 1E+20 .ENDS .SUBCKT WSOTA in+ in- out G1 0 out in+ in- 85.44U .ENDS .SUBCKT WSOTA2 in+ in- out G1 0 out in+ in- 15.37U .ENDS .SUBCKT WSOTA3 in+ in- out G1 0 out in+ in- 61.03U .ENDS

39

* Analysis .AC DEC 100 1Hz 100MegHz .PROBE .END

Ideal Fifth Order Chebyshev Filter Response (in DB) Ideal Fifth Order Chebyshev Filter Response (in V)

40

Implemented Fifth Order Chebyshev Filter Response (in DB)

41

Implemented Fifth Order Chebyshev Filter Response (in V)

7. General Biquad Implementation of a General Second Degree Transfer Function.

+

-Gm1 +

-Gm2

C2C1

Vi

Vo+

-Gm3

+

-

+

-

C3

Gm4

Gm5

I1

I4

I5

I2I3

IC3IC2

V1

Figure 8. General Second Order Filter Implementation. Figure 8 shows the single-ended OTA implementation of general second order filter. The transfer function is derived as follows:

42

om11 V-GI =

1m22 VGI =

om33 V-GI =

im44 VGI =

im55 VGI =

)V-(VsCI oi3C3 =

im4om141C1 VGVGIII +−=+=

)VGVG(sC1I

sC1V im4om1

1C1

11 +−==

o2oi3im5om31m2C3532C2 VsC)V-(VsCVGVG-VGIIIII =++=+++=

43

2o

o2

012

2

321

m4m1

32

m32

321

m4m2

32

m5

32

32

i

o

wQ

wss

asasa

)C(CCGG

CCG

ss

)C(CCGG

CCG

sCC

Cs

VV

+

+

++=

+

+

+

+

+

+

+

+

+

=

m4m1m313212

m4m2m51312

i

o

GGGsC)C(CCsGGGsCCCs

VV

+++++

=

It will be shown that general implementation of 5 OTAs reduces to 3 or 2 OTAs for the standard filters such as lowpass filter. First it will be shown for lowpass transfer function general implementation reduces to 4 OTAs. Comparing the two transfer functions,

2o

o2

2o

321

m4m1

32

m32

321

m4m2

32

m5

32

32

i

o

wQ

wss

w

)C(CCGG

CCG

ss

)C(CCGG

CCG

sCC

Cs

VV

+

+

=

+

+

+

+

+

+

+

+

+

=

Equating the coefficients,

0G0CC

G

open0C0CC

C

m532

m5

332

3

=⇒=+

⇒=⇒=+

That means that both capacitor C3 and Gm5 can be deleted, resulting in the circuit shown in Figure 9(a). The 4 OTAs implementation can be reduced to 3 OTAs if Gm1=Gm4. This is shown below:

+

-Gm1 +

-Gm2

C2

C1Vo

+

-Gm3

Gm4

I1

I4

I2I3

IC2

V1

Vi +

-

+

-Gm1 +

-Gm2C1

I1I2

V1

ViC2 Vo

+

-Gm3

I3

IC2

+

-Gm1 +

-Gm2

C2

IC2

C1

I1I2

V1

Vi Vo

(c)

(b)

(a)

IC1

IC1

IC1

Figure 9. General Biquad Lowpass Filter Transformation: (a) 4 OTAs, (b) 3 OTAs, and (c) 2 OTAs.

44

m4m1oim1im4om141C1

im44

om11

GG; )V-(VGVGV-GIIIVGIV-GI

==+=+===

That is, Gm1 and Gm4 can be combined as shown in Figure 9(b).

)V-(VGII oim11C1 == The 3 OTAs can be reduced to 2 OTAs if Gm2=Gm3. This is shown as follows: m2=Gm3. This is shown as follows:

m3m2o1m2om31m232C2

om33

1m22

GG; )V-(VGVGVGIIIVGI

VGI

==−=+=−=

=

That is, Gm2 and Gm3 can be combined as shown in Figure 9(c). That is, Gm2 and Gm3 can be combined as shown in Figure 9(c).

)V-(VGII o1m22C2 ==

2C2

2C2

Gm3

2C1

2C1

Gm2

2C3

2C3

Gm1

Gm4 Gm5+

-Vi

+

-

Vo

(b)

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

Figure 10. Differential OTA Implementation of General second-order filter.

45