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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
P. Bruschi - Analog Filter Design 2
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.
P. Bruschi - Analog Filter Design 3
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"
P. Bruschi - Analog Filter Design 4
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
P. Bruschi - Analog Filter Design 5
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.)
P. Bruschi - Analog Filter Design 6
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.
P. Bruschi - Analog Filter Design 7
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)
P. Bruschi - Analog Filter Design 8
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)
P. Bruschi - Analog Filter Design 9
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)
P. Bruschi - Analog Filter Design 10
Frequency scaling rules
P. Bruschi - Analog Filter Design 11
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
P. Bruschi - Analog Filter Design 12
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
P. Bruschi - Analog Filter Design 13
An important case is when the function f(s) is a constant factor K:
K
CC
sCK
Csn
11
KLLKsLsL
KRR
Element transformations
P. Bruschi - Analog Filter Design 14
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
P. Bruschi - Analog Filter Design 15
ss N
n
CLC
C
s
Cs NNn
11
LCLL
sLs
N
N
n
1
RR
Low-pass to High-pass
Element Transformation
P. Bruschi - Analog Filter Design 16
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
P. Bruschi - Analog Filter Design 17
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
P. Bruschi - Analog Filter Design 18
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
P. Bruschi - Analog Filter Design 19
Williams & Taylor
«Electronic Filter Design
Handbook»2006, McGraw-Hill
Example: Butterworth Prototype Filter
P. Bruschi - Analog Filter Design 20
N C1 L1 C2 L2 C3 L3 C4 L4 C5 L5
radians – per - seconds
=1
Chebyshev 1 dB ripple
P. Bruschi - Analog Filter Design 21
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:
P. Bruschi - Analog Filter Design 22
3dB10.34 kHzf
3dB' 65 krad/sN
LC Filter design using Tables
P. Bruschi - Analog Filter Design 23
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