Analog Filters: Introduction
description
Transcript of Analog Filters: Introduction
Analog Filters: Introduction
Franco Maloberti
Franco Maloberti Analog Filters: Introduction 2
Historical Evolution1920 Passive LC1969 Discrete RC
1973 Thin Film 1975 TF-DIL 1980 SWITCHEDCAPACITORS
DIGITAL SIGNAL
PROCESSOR
Franco Maloberti Analog Filters: Introduction 3
Frequency and Size
Active filters will achieve ten of GHz in monolitic form
1920 1940 1960 1980 2000 202010 GHz
1GHz
100MHz
10 MHz
1 MHz
100 KHz
10 KHz
PASSIVE LCDISCRETERCTHIN FILMSCRFMONORF MONO & SC
Franco Maloberti Analog Filters: Introduction 4
Introduction
An analog filter is the interconnection of components (resistors, capacitors, inductors, active devices)
It has one input (excitation) and one input (response)
It determines a frequency selective transmission.
Analog FilterInput Output
x(t) y(t)
Franco Maloberti Analog Filters: Introduction 5
Classification of Systems
Time-Invariant and Time-Varying The shape of the response does not depends on
the time of application of the input
Casual System The response cannot precede the excitation€
x(t) → y(t) x(t − τ ) → y(t − τ )
Franco Maloberti Analog Filters: Introduction 6
Classification of Systems
Linear and Non-linear A system is linear if it satisfies the principle of
superposition
Continuous and Discrete-time In a continuous-time or continuous analog system
the variables change continuously with time
In discrete-time or sampled-data systems the variables change at only discrete instants of time
€
f x{ } = f αx1 + βx2{ } = f αx1{ } + f βx2{ }
€
x = x(t);y = y(t)
€
x = x(kT);y = y(kT)
Franco Maloberti Analog Filters: Introduction 7
Linear Continuous Time-Invariant
If a system is composed by lumped elements (and active devices) Linear differential equations, constant coefficients
x(t), input, and y(t), output,are current and/or voltages
For a given input and initial conditions the output is completely determined
€
bndnydt n
+ bn−1dn−1ydt n−1 +K + b0y = am
dmxdtm
+ am−1dm−1xdtm−1 +K + a0x
Franco Maloberti Analog Filters: Introduction 8
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 9
Frequency-domain Study
Remember that the Laplace transform of
The equation
Becomes
ICy(s) and ICx(s) accounts for initial conditions
€
Ldny(t)dt n
⎡ ⎣ ⎢
⎤ ⎦ ⎥= snY (s) − sn−1y(0) − sn−2 dy(0)
dt−K − d
n−1y(0)dt n−1
€
(bnsn + bn−1s
n−1 +K + b0)Y (s) + ICy (s) = (amsm + am−1s
m−1 +K + a0)X(s) + ICx (s)
€
bndnydt n
+ bn−1dn−1ydt n−1 +K + b0y = am
dmxdtm
+ am−1dm−1xdtm−1 +K + a0x
Franco Maloberti Analog Filters: Introduction 10
Transfer Function
If X(s) is the input and Y(s) the zero-state output
Input voltage, output voltage: voltage TF Inpur current, output current: Current TF Input votage output current: Transfer impedance Input current, ourput voltage: Trasnsfer admittance
€
H s( ) =Y s( )X s( )
= amsm + am−1s
m−1 +K + a0
bnsn + bn−1s
n−1 +K + b0
Franco Maloberti Analog Filters: Introduction 11
Transfer Function
Input and output ar normally either voltage or current
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) = Y (s)X(s)
€
H(s) s= jω = H( jω)e jφ(ω )
Franco Maloberti Analog Filters: Introduction 12
Magnitude and Phase
Magnitude is often expressed in dB
Important is also the group delay
When both magnitude and phase are important the magnitude response is realized first. Then, an additional circuit, the delay equalizer, improves the delay function.
€
H( jω)dB
= 20logH( jω)
€
Td (ω) = − dφ(ω)dω
Franco Maloberti Analog Filters: Introduction 13
Real Transfer Function
The coefficients of the TF are real for a linear, time-invariant lumped network.
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
€
H s( ) =N s( )D s( )
= am (s− z1)(s− z2)L (s− zm )bn (s− p1)(s− p2)L (s− pn )
Franco Maloberti Analog Filters: Introduction 14
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 jω( ) = ( jω − z1)( jω − z2)( jω − p1)( jω − p2)
α1α 2β2β1γ1γ2
€
φ=α1 +α 2 −β1 −β 2
€
′ φ =γ1 + γ 2 −β1 −β 2
Franco Maloberti Analog Filters: Introduction 15
Type of Filters
Low-pass
High-pass
Band-pass
Band-Reject
All-Pass
1
0 ffc
1
0 ffc
1
0 ffc1
1
0 ffc
fc2
fc2
1
0 f
Franco Maloberti Analog Filters: Introduction 16
Approximate Response Pass-band ripple αp=20Log[Amax/Amin] Stop-band attenuation, Asb
Transition-band ratio p, s
Amax
Amin
Asb
p s
Franco Maloberti Analog Filters: Introduction 17
MATLAB
Works with matrices (real, complex or symbolic) Multiply two polinomialsf1(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 18
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 19
Impedance Scaling
All elements with resistance dimension are multiplied by kz
R -> kz R; L ->kzL; (Vx=αIcont) α -> α 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 20
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 21
Design of Filters Procedure
Specifications Kind of network
Input network Infinite, zero load Single terminated/Double terminated
Mask of the filter Magnitude response Delay response
Other features Cost, volume, power consumption, temperature drift,
aging, …
Franco Maloberti Analog Filters: Introduction 22
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 23
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