Analog Filter Design -...

P. Bruschi - Analog Filter Design 1

Analog Filter Design

Part. 3: Time Continuous Filter Implementation

Sect. 3-a:

General considerations

Passive filters

Design approaches

• Passive LC (R) ladder filters

• Cascade of Biquadratic (Biquad) and Bilinear cells

• State Variable Filters

• Simulation of LC filters with active RC networks

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2( )S

VH s

V

Filter Parameters

• For a given transfer function H(s), a particular implementation is characterized by several FOMs (Figures Of Merit). The most frequently used are:

Dynamic Range:

Sensitivity to component variations

Component value spread, e.g.

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outn

out

v

VDR

max vn-out= output noise

min

max

C

C

dx

dQ

Q

xS

dx

dx

xdx

dS

Q

x

x

0

0

00

/

/0

LC passive filters: "The Prototype filter"

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LC ladder filters are synthetized in normalized (1 rad/s, 1 W) and low-

pass form (prototype filter)

Transformation rules are used to derive the required filter function (e.g.

band-pass) and parameters (e.g. actual operating frequencies) from the

prototype filter

2( )N

S

VH s

V

Normalized Low-Pass

Function

Passive Lossless Ladder Filters

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Doubly terminated LC ladder

network

Advantages: minimum sensitivity

to component variation in the pass-

band

The lowest sensitivity is achieved

with equally terminated networks

(R1=R2).

Can be used as starting point for

the synthesis of active RC filters

Drawback: tuning requires change

of all components.

order (N) = number of capacitors + number of inductors

Ladder networks: driving point impedance (d.p.i.)

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1

1

1( )

V

Z s ZY

1

1

2

2

3

3

4

4

1( )

1

1

1

1

1

1

Z s Z

Y

Z

Y

Z

Y

ZY

ZV2YV1

1

1

2

1( )

1

V

Z s Z

YZ

"Continued fraction"

• Driving point impedances

(or d.p. admittances)

• Transfer impedances

(or t. admittances)

Cauer synthesis approach for d.p.i.

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4 2

3

4 3( )

2

s sZ s

s s

2

3

2 3( )

2

sZ s s

s s

1

2

3

1

1 2 3

2V

Z s

s

Y s s

3

1 2 2

2 / 2

2 3 2 2 3V

s s s sY

s s

1( )

1

124

/ 6

Z s ss

ss

In the end ….. Z1Y1

Z2

Y2

Prototype Filter Configurations (all poles)

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N=2M+1 (odd order)

N=2M (even order)

Pass-band gain: 121

2

RR

Rk (0.5 for equally terminated networks)

Alternate solution (all poles)

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N=2M+1 (odd order)

N=2M (even order)

Pass-band gain: 121

2

RR

Rk (0.5 for equally terminated networks)

LC ladder network for TF with imaginary zeros(e.g. Inverse Chebyshev and Cauer Elliptic filters)

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Frequency scaling rules

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Frequency scaling allows to change the normalization frequency,

allowing transformation of the characteristic frequencies of the filter

N

n

ss

1 N

n N

CC

s C sC

n

N N

L Ls L s L

RR

Impedance Scaling Rule

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Impedance scaling is used to change component values leaving the

transfer function unaltered. The target is finding feasible component

values for the chosen technology

If the network includes only:

Two terminal impedances (L,R,C components)

Voltage Controlled Voltage Sources (VCVS) i.e Ideal voltage

amplifiers.

Current Controlled Current Sources (CCCS) i.e. ideal current

amplifiers

Then: the Vout/VS transfer function is unchanged when all

the impedances are multiplied by the same function f(s)

Impedance scaling: component transformation

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An important case is when the function f(s) is a constant factor K:

K

CC

sCK

Csn

11

KLLKsLsL

KRR

Element transformations

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s

s

Bsn

0

0

0

ss N

n

1

0

0

0

s

s

Bsn

High-Pass Band-Pass Band -Stop

Goal: to change the filter response from low-pass to the other three

possibilities (high-pass, etc.) and perform frequency scaling at the same

time.

Let us recall the following transformations:

From Low-Pass to:

Element Transformation

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ss N

n

CLC

C

s

Cs NNn

11

LCLL

sLs

N

N

n

1

RR

Low-pass to High-pass

Element Transformation

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Low-pass to Band-pass

sBB

ss

s

s

Bs

n

n

2

0

0

0

0

P

Pn

sLsC

sB

C

B

sCCs 1

1112

0

S

SnsC

sLsB

LB

sLLs

12

0

C

BL

B

CC PP 2

0

B

LL

L

BC SS

2

0

Element Transformation

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Low-pass to Band-stop

1

0

0

0

s

s

Bsn

BCL

CBC SS

12

0

2

0

1

BLL

BLC PP

Design of LC ladder passive filters

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A procedure that allows designing an arbitrary transfer function with a

ladder structure does not exist.

All-pole functions (e.g. Butterworth, Chebyshev I, Bessel) can be

designed with a standard approach, where the branches of the ladder (Z

and Y elements) are pure capacitors or inductors. Given a class of

networks, not all functions are feasible.

The rigorous design of Cauer (elliptic) filters is less straightforward.

Tables are available for the most frequently used ladder topologies and

transfer functions. Several CAD design tools are also available.

Table example

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Williams & Taylor

«Electronic Filter Design

Handbook»2006, McGraw-Hill

Example: Butterworth Prototype Filter

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N C1 L1 C2 L2 C3 L3 C4 L4 C5 L5

radians – per - seconds

=1

Chebyshev 1 dB ripple

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C1 L1 C2 L2 C3 L3 C7

Note: In these tables, generally

=1 is the -3 dB angular frequency,

regardless of ripple (and Ap)e.g. Williams & Taylor «Electronic Filter Design Handbook»

Example

• Design a LC ladder Chebyshev filter with the following characteristics:

f_pass =10 kHz, Maximum Pass-band attenuation 1 dB

f_stop = 20 kHz Minimum Stop-Band Attenuation: 40 dB

Python: cheb1ord: Order=5, P=62.8 krad/s

But: tables are normalized to -3dB . From the magnitude plot:

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3dB10.34 kHzf

3dB' 65 krad/sN

LC Filter design using Tables

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C1=2.207 F

L1=1.128 H

C2=3.103 F

L2=1.128 H

C3=2.207 F

’N=-3dB=1 rad/s

-3dB=65 krad/s

C1=33.9 mF

L1=17.35 mH

C2=47.7 mF

L2=17.35 mH

C3=33.9 mF

'N

CC

'N

LL

From table: prototype filterFrequency

scaled

filter