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

€

bndny

dt n+ bn−1

dn−1y

dt n−1+K + b0y = am

dmx

dtm+ am−1

dm−1x

dtm−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 −

dn−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)

€

bndny

dt n+ bn−1

dn−1y

dt n−1+K + b0y = am

dmx

dtm+ am−1

dm−1x

dtm−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( )

=ams

m + am−1sm−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 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 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