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Analog Filters: Introduction
Franco Maloberti
Historical Evolution
1920
Passive LC
1969
Discrete
RC
1980
1973
Thin Film
1975 TF-DIL
SWITCHED
CAPACITORS
DIGITAL
SIGNAL
PROCESSOR
Franco Maloberti
Analog Filters: Introduction
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Frequency and Size
10 GHz
RF
MONO
1GHz
RF MONO & SC
100MHz
10 MHz
1 MHz
PASSIVE LC
THIN FILM
100 KHz
10 KHz

SC
DISCRETE
RC
1920
1940
1960
1980
2000
2020
Active filters will achieve ten of GHz in monolitic form
Franco Maloberti
Analog Filters: Introduction
3
Introduction


An analog filter is the interconnection of
components (resistors, capacitors, inductors, active
devices)
It has one input (excitation) and one input
(response)
Input
Output
Analog Filter
x(t)

y(t)
It determines a frequency selective transmission.
Franco Maloberti
Analog Filters: Introduction
4
Classification of Systems

Time-Invariant and Time-Varying
 The shape of the response does not depends on
the time of application of the input
x(t)  y(t) x(t   )  y(t   )

Casual System
 The response cannot precede the excitation

Franco Maloberti
Analog Filters: Introduction
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Classification of Systems
Linear and Non-linear
 A system is linear if it satisfies the principle of
superposition
f x  f x1  x 2   f x1  f x 2 
 Continuous and Discrete-time
 In a continuous-time or continuous analog system
 the variables change continuously with time

x  x(t); y  y(t)

In discrete-time or sampled-data systems the
variables change at only discrete instants of time

Franco Maloberti
x  x(kT);y  y(kT)
Analog Filters: Introduction
6
Linear Continuous Time-Invariant

If a system is composed by lumped elements (and
active devices)
 Linear differential equations, constant coefficients
dn y
d n1 y
bn n  bn1 n1 
dt
dt



dm x
d m1 x
 b0 y  am m  am1 m1 
dt
dt
 a0 x
x(t), input, and y(t), output,are current and/or
voltages
For a given input and initial conditions the output
is completely determined
Franco Maloberti
Analog Filters: Introduction
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Responses of a linear system

Zero-input response
 Is the response obtained when all the inputs are
zero.



Depends on the initial charges of capacitors and initial
flux of inductors
Zero-state response
 Is the response obtained with zero initial
conditions
The complete response will be a combination of
zero-input and zero-state.
Franco Maloberti
Analog Filters: Introduction
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
Frequency-domain Study

Remember that the Laplace transform of
d n y(t)  n
dy(0)
L  n  s Y(s)  sn1 y(0)  sn2

dt
 dt 



The equation
dn y
d n1 y
bn n  bn1 n1 
dt
dt

dm x
d m1 x
 b0 y  am m  am1 m1 
dt
dt
 a0 x
Becomes
(bn sn  bn1sn1 

d n1 y(0)

dt n1
 b0 )Y(s)  ICy (s)  (am sm  am1sm1 
 a0 )X(s)  ICx (s)
ICy(s) and ICx(s) accounts for initial conditions
Franco Maloberti
Analog Filters: Introduction
9
Transfer Function

If X(s) is the input and Y(s) the zero-state output
Y s am sm  am1sm1   a0
H s 

X s bn sn  bn1sn1   b0
Input voltage, output voltage: voltage TF
 Inpur current, output current: Current TF

 Input votage output current: Transfer impedance
 Input current, ourput voltage: Trasnsfer admittance

Franco Maloberti
Analog Filters: Introduction
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Transfer Function




Input and output ar normally either voltage or current
Y (s)
H(s) 
X(s)
Where Y(s) and X(s) are the Laplace transforms of
y(t) and x(t) respectively.

In the frequency domain the focus is directed toward
Magnitude and/or Phase on the j axis of s
H(s) s j  H( j)e j ( )
Franco Maloberti
Analog Filters: Introduction
11
Magnitude and Phase

Magnitude is often expressed in dB
H( j) dB  20log H( j)


Important is also the group delay
d ( )
Td ()  
d

When both magnitude and phase are important the
magnitude response is realized first. Then, an
additional
 circuit, the delay equalizer, improves the
delay function.
Franco Maloberti
Analog Filters: Introduction
12
Real Transfer Function

The coefficients of the TF are
real for a linear, time-invariant
lumped network.
N s am (s  z1)(s  z2 ) (s  zm )
H s 

Ds bn (s  p1 )(s  p2 ) (s  pn )



Only real or conjugate pairs of
complex poles
For stability the zeros of D(s) in
the half left plane
 D(s) is a Hurwitz polynomial
Franco Maloberti
Analog Filters: Introduction
13
Minimum Phase Filters

When the zeros of N(s) lie on or to the left of the
j-axis H(s) is a minimum phase function.
H  j  
1
1
1

2
2
( j  z1)( j  z2 )
( j  p1)( j  p2 )
2
  1  2  1  2
 1   2  1  2


Franco Maloberti
Analog Filters: Introduction
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Type of Filters

Low-pass
1

High-pass
0

Band-pass
f
fc
1
f
Band-Reject
All-Pass
Franco Maloberti
fc
1
fc1 fc2
0
1

f
0
0

1
0
f
fc
fc2
f
Analog Filters: Introduction
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Approximate Response



Pass-band ripple p=20Log[Amax/Amin]
Stop-band attenuation, Asb
Transition-band ratio p, s
Amax
Amin
Asb
p s
Franco Maloberti
Analog Filters: Introduction
16
MATLAB

Works with matrices (real, complex or symbolic)
Multiply two polinomials
f1(s)=5s3+4s2+2s +1; f2(s)=3s2+5





clear all;
f1=[5 4 2 1];
f2 = [3 0 5];
f3 = conv(f1, f2)
15 12 31 23 10 5
f3(s)=15s5+12s4+ 31s3 + 23s2 + 10s +5
Franco Maloberti
Analog Filters: Introduction
17
Frequency Scaling




If every inductance and every capacitance of a
network is divided by the frequency scaling factor kf,
then the network function H(s) becomes H(s/kf).
Xc=1/sC; X’c=1/[s(C/kf)]=1/[C(s/kf)]
XL=sL; X’L=s(L/kf)=L(s/kf)
What occurs at ’ in the original network now will
occur at kf ’.
Franco Maloberti
Analog Filters: Introduction
18
Impedance Scaling





All elements with resistance dimension are
multiplied by kz
 R -> kz R; L ->kzL; (Vx=Icont) -> kz
All elements with capacitance dimension are
divided by kz
 G -> G/kz; C ->C /kz; (Ix=Vcont) -> /kz
Impedences multiplied by kz
Admittances divided by kz
Dimensionless variables unchanged
Franco Maloberti
Analog Filters: Introduction
19
Normalization and Denormalization


Normalized filters use the key angular frequency of
the filter (p in a low-pass, …) equal to 1.
One of the resistance of the filter is set to 1

or

One capacitor of the filter is set to 1

Frequency scaling and impedance scaling are
eventually performed at the end of the design
process
Franco Maloberti
Analog Filters: Introduction
20
Design of Filters Procedure

Specifications
 Kind of network




Mask of the filter



Input network
Infinite, zero load
Single terminated/Double terminated
Magnitude response
Delay response
Other features

Cost, volume, power consumption, temperature drift,
aging, …
Franco Maloberti
Analog Filters: Introduction
21
Design of Filters Procedure (ii)

Normalization
 Set the value of one key component to 1
 Set the value of one key frequency to 1

Approximation
 To find the transfer function that satisfy the
(normalized) amplitude specifications (and,
when required, the delay specification.
 Many transfer functions achieve the goal. The
key task is to select the “cheapest” one
Franco Maloberti
Analog Filters: Introduction
22
Design of Filters Procedure (iii)

Network Synthesis (Realization)
 To find a network that realizes the transfer
function




Many networks achieve the same transfer function
Active or passive implementation
The behavior of networks implementing the same
transfer function can be different (sensitivity, cost, …
Denormalization
 Impedance scaling
 Frequency scaling
 Frequency transformation
Franco Maloberti
Analog Filters: Introduction
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