LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE...

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LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters RLC prototype low sensitivity One form of this technique is called “Leapfrog Technique” Fundamental Building Blocks are - Integrators - Second-order Realizations Filters considered - LP - BP - HP - BE - Zeros ω j ECEN 622 (ESS) TAMU-AMSC

Transcript of LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE...

Page 1: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

LEAP FROG TECHNIQUE

Operational Simulation of LC Ladder Filters

• RLC prototype low sensitivity • One form of this technique is called “Leapfrog Technique” • Fundamental Building Blocks are - Integrators - Second-order Realizations • Filters considered - LP - BP - HP - BE - Zeros ωj ECEN 622 (ESS)

TAMU-AMSC

Page 2: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Ladder Networks

Elements connected in series and in parallel

Zeros are easily recognized: Zseries=infinity, or Zshunt=zero

Problem: How to design active ladder filters? How to design inductors?

Ladder Filters

Page 3: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

434

3423

2312

1211

Z)0i(vY)vv(i

Z)ii(vY)vv(i

−=−=−=−=

i1 i3 v2 v4

v1

gnd

Y1 Y3

Z2 Z4

By means of a simple example it is illustrated how to implement an active filter based on the equation of a passive RLC prototype.

Page 4: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

434

3423

2312

1211

Z)0i(vY)vv(i

Z)ii(vY)vv(i

−=−=−=−=

- +

R

Zx

v1

-v2

RZvvv xx /)( 21 −−=

Next we match the equations coefficients with the implementations.

Vx’=i1 Rdummy For R dummy=1 one can make v’x=i1

Two key transformations: 1. If needed converter current equations to voltage equations by multiplying by a dummy (artificial) resistor value of 1. 2. Express the relation of current and voltage of equations always as integrator. The integrator is the basic building block.

Page 5: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

General Principles 3i − 1i − 1i + 3i +

2i −i

2i +

3iI −

3iV =

2iV − −

2iI −

+

1iI −

1iV −−

iIiV−

+1iI + + 1iV +

2iV ++

2iI +

3iI +

3iV ++ −

Interior Portion of a General LC Ladder Network • Interior components are reactive elements only. • • Branch voltages are voltage and current • How to select the proper “STATE” variable ? a) b) c)

)finite(0Rand0Ror0R Lss ≠=≠

o o+ −cv

Ci

o o+ −v

LLi

o o+ −v

Li C

∫=⇒= dt)t(iC1vi

C1

dtdv

cc

∫=⇒= dt)t(vL1iv

L1

dtdi

LL

∫ ∫∫ +−

=⇒+= dt)t(vL1dt)t(i

LC1i

Ci

dt)t(idL

dtdv 2

2

2

Page 6: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Systematic approach by writing voltage (KVL) and current (KCL) equations

3i − 1i − 1i + 3i +

2i −i

2i +

3iI −

3iV =

2iV − −

2iI −

+

1iI −

1iV −−

iIiV−

+1iI + + 1iV +

2iV ++

2iI +

3iI +

3iV ++ −

] I I [ Y

1 V 1 3 i 2 i

2 i i− − −

− − =

]VV[Z

1I i2i1i

1i −= −−

]II[Y1V 1i1i

ii +− −=

]VV[Z

1I 2ii1i

1i ++

+ −=

]II[Y

1V 3i1i2i

2i +++

+ −=

- Immittance Functions - Voltage Transfer Functions, convert I to V functions

RIV.,e.i k'k =

Page 7: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Terminations of LC Ladder Filters Source a) b)

1Z

2V 2Y+

3I

−+

inV

oR

1I

]VVR

RV[ZRRIV 2

'1

oin

11

'1 −−==

]VV[RY

1V '3

'1

22 −=

1Y 1V+

−−

+inV

oR2I

2Z

3V+

−−=⇒−

−= '

21ino1

12o

1in11 V)VV(

RR

RY1VI

RVVVY

]VV[ZRV 31

2

'2 −=

Page 8: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Load termination

2nV −

+

+

outn VV =nR

1nI −1nZ −+

a) or b)

'1n

nnout V

RRVV −==

]Z/)VV[(RV 1nout2nnout −− −=

)VV(R

RZ

RV out2nn

1nout −= −

1n

n

2nn

out

ZR1

VR

R

V

+=

Lossy Integrator

1nV −

+

outn VV =nR2nI −

2nZ −

−1nY −

]VRRV[

RY1V out

n

'2n

1nout −= −

Page 9: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

The approach to map a passive RLC prototype is to pick the state-variables which can be expressed as integrators, since integrators are the basic building block. By applying KCL, KVL, KCL, …, as many times as the order of the filter, one can write the state-equations that can be implemented by active filters. an example is shown below:

2IoV

LRinV 2C1C 1V+

−−+

SR

A Typical Passive RLC Filter

( )

−−= − '

21inS1

1 VVVRR

SRC1V ( )a15

( )b15

( )16

( )17

,RIV 2'2 = R is an arbitrary value

KCL

where

KVL

KCL

( )o1'2 VV

SLRV −=

−= o

'2

2o V

RLRV

SRC1V

Page 10: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

SRR

1S

SRR

1SRC1

2S

3S 4S

5S

2SRC1 LR

R−

1−

1−1

1'2V

SLR

Signal Flow Graph of RLC Prototype of Fig. Shown in previous page.

Page 11: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Low-Pass Ladder Filters (Zeros at Infinity) (All Poles)

Example: A Fifth-Order LP Filter • 5 State Variables • 5 State Equations

]I,V,I,V,I[ 54321

•1I→

onRinV+

n1L

2V+

n2Cn3L3I→

+

4Vn4C

n5L5I→

onR outV+

−KVL-KCL-KVL-KCL-KVL (Sequence) * 02)1(1 =−+− VnsLonRIinV

RIV 1'

1 =

][ '121

'1 V

RRVV

sLRV on

nin

−−=

Current Analogs

)1(

Page 12: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

0VsCII 2n231 =−−

]VV[sRC

1V '3

'1

n22 −=

0VIsLV 43n32 =−−

]VV[sL

RV 42n3

'3 −=

0IVsCI 54n43 =−−

]VV[sRC

1V '5

'3

n44 −=

0RIIsLV n655n54 =−−

]VR

RV[sL

RV '5

n64

n5

'5 −=

n6

'5

n65out RRVRIV ==

]RRV

RRV[

sLR

RRV

n6out

n64

n5

n6out −=

But

)2(

)3(

)4(

)5(

Page 13: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Active RC Building Blocks

o

inV

r

+−

r

+−•

o

o

o

kV

jV

jC

jV

Thus, eq(1) is implemented as follows

o

inV

r

+−

r

+−•

o

o

o

2V

'1V

1C

'1V

1R

2R

inR

on

in11

in12

in1in

RLCR

RLCR

RLCR

=

=

=

Cj/j1τ

Cj/j2τ

Cj/j3τ

][ '121

'1 V

RRVV

sLRV on

nin

−−=

Page 14: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

o'

1V

r

+−

r

+−•

o

o'3V

2C

2V3R

'3R

o

2V

r

+−

r

+−•

o

o

4V

3C

'3V4R

'4R

o'3V

r

+−

r

+−•

o

o'5V

4C

4V5R

'5R

o

4V

r

+−

r

+−•

o

o'5V

5C

'5V6R

'6R

2 3 n 2

2 ' 3 n 2

C R RC C R RC

= =

R / L C R R / L C R

n 3 3 4

n 3 3 ' 4

= =

n 4 4 5

n 4 4 ' 5

RC C R RC C R

= =

n 6 n 5 5 6

n 6 n 5 5 ' 6

R / L C R R / L C R

= =

]VV[sRC

1V '3

'1

n22 −=

]VV[sL

RV 42n3

'3 −=

]VV[sRC

1V '5

'3

n44 −=

]RRV

RRV[

sLR

RRV

n6out

n64

n5

n6out −=

Page 15: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

'5V

ino

+−

+−

+−

+−

+−

+−

+−

+−

+−

+−

rr

inR 1C

1R

rr

'1V

2R

'3R

2C

r

r

3R

'4R

3C

r

r'3V

2V

4R

'5R

4C

r

4Vr

5R

'6R

5C

6R

Low-Pass 5th-Order Active –RC Leapfrog Filter

Page 16: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

o

inV+

R/Ron

n1SLR

+n2SRC

1+

n3SLR

+n4SRC

1+

n5SLR

'1V '

3V '5V

2V 4V

R/Ron− 1− 1−

o

oV

R/R n6−1−1−

o

inV+

R/Ron

1n1

onR/SL

R/R+

−+

n2SRC1

− +n3SL

R− +

n4SRC1

− +1R/S

R/R

n6n5L

n6+

1− 1−

o

oV

1−1−

Page 17: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

OTA-C Implementation +

−mjg

+

−mjg

o

o

o

o jV

jCjVjV

'inV

+

−1mg

+

−2mg

o

o

o

o '1V

1C'1V

2V

'inV

+

−6mg

+

−7mg

o

o

o '5V

5C'5V

4V+

−3mg

o

oo 2V

2C

'3V

'1V

+

−4mg

o

oo '

3V

3C4V2V

+

−5mg

o

oo 4V

4C

'5V

'3V

RL

Cg n1

1

1m =

on

n1

2

2mRL

Cg

=

Thus, EQ (1) is implemented as

)2(EQ

n22

3m RCCg

=

)3(EQ

RL

Cg n3

3

4m =

)4(EQ

n44

5m RCCg

=

)5(EQ

on

n5

5

6mRL

Cg

=

on

n5

5

7mRL

Cg

=

][ '121

'1 V

RRVV

sLRV on

nin

−−=

Page 18: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

'1V

+

−1mg

o+

−2mg +

−3mg +

−4mg +

−5mg +

−6mg

+

−7mg o

o'5 VV =

5C4C4V

••

'3V

3C2C

2V•

1CinV

5th-Order OTA-C Leapfrog Implementation

Page 19: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Voltage Scaling of OTA-C All Pole Leapfrog Filters.

+ −1mg

o

aV

+ −2mg

•o

cV• o

dV

+ −3mg

o

bV•

2C

xV

1C 3C

2dc2mx sC/)VV(gV −=

+ −3mg

o

aV

+ −k/gm2

o

cV• o

dV

+ −3mkg

o

bV•

2C

k/Vx

1C 3C

+ −1mkg

+ −3mg

Changing without changing k

VV xx → dcba Vand,V,V,V

Page 20: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

+ −1mg

o

inV

+ −3mg o

oV

+ −4mg

+ −2mg

1C

2C

3C

4C+ −

6mg

+ −5mg

+−7mg5C

Page 21: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

o

+inV

+ +

+ +

o

RRon

RRon−

'1V oV

RRon−n5SL

R

n3SLR

n1SLR

n2SRC1−

n4SRC1−

System Level Representation: 5th-Order LP Leapfrog Filter

Page 22: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

SCALING MAXIMUM DYNAMIC RANGE

4max4

maxo

2max2

maxon6n54

1max1

maxo3

max3

maxon6n5o4

in

kVV

kVV

;R

RR

Ls)s(TV

kVV

;kVV

;R

RR

LsVV

1V

=

=

+=

==

+=

=

o

+inV

+ +

+ +

o

4321on kkkkR

R

1k

RRon−

'1V '

3V '5V oV

4V2V

2k

3k

4k1k1

2k1

3k1

4k1

RRon−n5SL

R

n3SLR

n1SLR

n2SRC1−

n4SRC1−

Page 23: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Current Level Scaling Technique Case 1

oV

+

−mg o

C R

oIinIo

R represents input resistance of a current-mirror

If needs to be constant, then R must be decreased by k. oV

→→→→

k/RR&k/CC2optionk/RR&kgg1option

summaryIn mm

Case 2

oV

+

−mg o

1C 2C

oIinIo

represents input impedance of next block 2C

sC/gI mo −=

sCRgRIV m

oo −==

→→

−=k/CC2option

kgg1optionsCgkkI mmm

o

RkIV oo =

1

mo sC

gI −=

2oo sC

1IV =

Page 24: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

LC Ladder Filters with Finite Zeros

'1iY −

'1iY +

1iZ − 1iZ +oo o2i − i

'1iI −↑

3iI −→

2iY − iY 2iY +

2i + 3iI +→+

2iV −

+

iV−

+

2iV +

1iI −→

1iI +

1iZ − 1iZ +oo o2i − i3iI −

2iY − iY 2iY +

3iI +→+

2iV −

+

iV−

+

2iV +

1iI +→1iI −→

+i2i,i VK −

+2ii,2i VK ++

−+

2ii,2i VK −−−

+i2i,i VK +

− − −

i1i,2iY −− 1i,i,1iY +− 2i,1iY ++

Page 25: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Dependent Sources to Replace the Bridging Admittances

In a similar for KCL for node I+2 and solving for results 2iV +

2iatKCL −

]VV[YVYII i2i'

1i2i2i1i3i −+=− −−−−−−or

'1i2i

1i3ii'

1i2i

'1i

2i YYIIV

YYYV

−−

−−

−−

−−

+−

+

+=

1i,2i

1i3ii2i,i2i Y

)II(VKV−−

−−−−

−+=

+−+−−=− +−−+− ii'

1ii2i'

1iii1i1i VV(Y)VV(YVYIIor

'1ii

'1i

1i1i2i'

1ii'

1i

'1i

2i'1ii

'1i

'1i

i YYYIIV

YYYYV

YYYYV

+−

−++

+−

+−

+−

++−

+++

+++

=

1i,i,1i

'1i

'1i

2ii,2i2ii,2ii RYVVVKVKV

+−

−+++−−

−++=

'2i,i

'3i

'1ii2i,i2i RY/)VV(VKV +++++ −+=

KCL @ NODE i

Page 26: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

LC Ladder Simulation 1. Start from the desired LC LP prototype network satisfying SPECS.

2. Eliminate shunt elements in the series branches or series element in the shunt branches.

3. Normalize and transform the network if the realization is not low-pass. 4. Select the state variables such that

5. Solve for each state variable in terms of the other states

and output variable including the proper scaling.

6. Synthesize each equation with its corresponding building block.

)s,x(fdtdx

µ=

Page 27: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Example: Low-Pass With Finite Zeros ωj

n2L

2I→

oonR

n2L

n2C n4C

4I→

o

n5Cn1C n3C onR outV

+

3V

+

1V

+

iV

+

oonR

n2L n4L

4I→

o

n45Cn12Cn234C

onR

outV

+

3V

+

1V

+inV

−+

2I→

+

− 331VK

+

−553VK

+

− 113VK

+

−335VKn

n

n

n

n

n

n

n

nnn

nnnn

nnn

CCK

CCK

CCK

CCK

CCCCCCC

CCC

45

435

234

213

234

453

12

231

5445

432234

2112

=

=

=

=

+=++=

+=

Page 28: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

n45

n435

n234

n213

n234

n453

n12

n231

n5n4n45

n4n3n2n234

n2n1n12

CCK

CCK

CCK

CCK

CCCCCCC

CCC

=

=

=

=

+=++=

+=

Page 29: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

)1(VKRV

RV

RV

SC1V 331

'2

on

1

on

in

n121 +

−−=

r3V •CK31

ro

+

− 4R

+

−o

1C

1V2R

o1V3R

o'2V

inV

]VV[SL

RV 31n2

'2 +=

]VV[SL

RV 31n2

'2 −= )2(

)2( '

r ro

+

− 4R

+

−o

2C

'2V

3V

ro1V

Page 30: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

)3(VKVK]VV[SRC

1V 131553'4

'2

n2343 ++−=

131553'2

'4

n2343 VKVK]VV[

SRC1V −−−= )3( '

r

5V 353CK

ro

+

5R

+

−o

3C

3V

6R̂o'

2V

'4V

o

1V 331CKo

]VV[SL

RV out3n4

'4 −−= )4(

o

6R

+

−o

4C

'4VoutV

3V o

7R

R/LCRR/LCR

n447

n446==

Page 31: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

)5(VKRV

RV

SC1VV 335

n6

out'4

n456out −

−==

r ro

+

8R

+

−o

5C

outV

9RooutV

'4V3V 535CK

o

Page 32: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

inV o

+−

+−

+−

+−

+−

+−

+− +

r r

1C

1R

r r

1V

3R

4R2C

6R̂•

r

6R•

3C

r'2V

8R5C

CK31

'2V

8R

3V

7R

r

9R

+−

1V

inVo

353CK 331CK

r

r

Active RC Fifth-Order Elliptic Filter

Page 33: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

OTA-C Elliptic Implementation

)1(VKRV

RV

RV

SC1V 331

'2

on

1

on

in

n121 +

−−=

+− 1mg

+

−1mg

o

o

o

o

1V1C'

2V1VinV

3V o

+− 31g

+

−rg o

=

=

31r

31

onn121r

1m31

Kgg

RCCggg

131

onn121m

r3131

CK

RCg

gKg

=

=

]VV[SL

RV 31n2

'2 += )2(

+− 2mg

+

−2mg

o

o

o

2C

3V

1V'2V

RL

Cg n2

2

2m =

Page 34: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

131553'2

'4

n2343 VKVK]VV[

sRC1V −−−= )3(

+− 3mg

+

−5mg

o

o

o

3V3C

'2V

5V

4V

1V

o

+− 4mg

+

−rg o

]VV[sL

RV out3n4

'4 −−= )4(

+

−6mg

+

−6mg

o

o

o

4CoutV

3V'4V

R/LCg

n44

6m =

Page 35: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

)5(VKRV

RV

sC1VV 335

on

out'4

n455out −

−==

+− 7mg

+

−mxg

o

o

outV5C

3V

'4V

outV

o

+− 35g

+

−rg o

35r

35

n455

7m

r

35

Kgg

RCC

ggg

=

=

Page 36: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

−+o

−+ −

+

−+

−+

−+

−+

−+

−+

−+

−+

−+

−+

−+

o

in V1 C

1 V

2 C

'2 V

3 C3 V

4 C

'4 V

5 V5 C

out V

Fifth-Order OTA-C Elliptic Filter

Page 37: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

i1 -vi

R1

C1 L2/R2

R

- v2

i2

C4 - R

- vout

R1

R R5

R

R1 vi

C3

L2

R5 C4 C1

V2 vout

i1 i2

i3

-1

-1

i2

C3

C3

i3

i3 C3 C3 -1

Simulating Floating Capacitors (Elliptic Filters)

3sC)vout(3sC)2v(3i −=

Page 38: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

Finite Zeros: An Alternative Approach ωj

n4Lo

onR

2I

4I

o

n5Cn1C n3C onR oV

+

3V

+

1V

+

inV−+

n2L•••

•n2C n4C

5V

+

)1()VV(sCRVI

RVVsC 31n2

on

12

on

in1n1 −−−−=

RVI;RIV

'2

22'2 ==

)VV(sCRV

RV

RVVsC 31n2

on

1'2

on

in1n1 −−−−= )1( '

Page 39: LEAP FROG TECHNIQUEs-sanchez/622-Leapfrog-2011.pdf · 2014. 9. 8. · LEAP FROG TECHNIQUE Operational Simulation of LC Ladder Filters • RLC prototype low sensitivity • One form

31

'2

n2 VVRVsL −=

)VV(sC)VV(sCRV

RVVsC 53n431n2

'4

'2

3n3 −−−+−=

53'4

n4 VVVR

Ls −=

)VV(sCRV

RVVsC 53n4

n6

5'4

5n5 −+−=

)2(

)3(

)4(

)5(

Component Calculations for an OTA-C implementation onn1

1m

1 RCgC

= n34m

3 RCgC

=

onRR =R

LgC n4

5m

7 =

R1g 6m =

on2m R

1g =

RL

gC n2

3m

6 =n6

7m R1g =

Note that if then the OTA # 7 can be eliminated and the output of OTA # 6 can be simply connected to its negative terminal.

RR n6 =