Analog Filters: Introduction

1
Analog Filters Introduction

• Franco Maloberti

2
Historical Evolution
3
Frequency and Size

• Active filters will achieve ten of GHz in
  monolitic form

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.

Input
Output
Analog Filter
x(t)
y(t)
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

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

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

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.

9
Frequency-domain Study

• Remember that the Laplace transform of
• The equation
• Becomes
• ICy(s) and ICx(s) accounts for initial conditions

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

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

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.

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

14
Minimum Phase Filters

• When the zeros of N(s) lie on or to the left of
  the jw-axis H(s) is a minimum phase function.

15
Type of Filters

• Low-pass
• High-pass
• Band-pass
• Band-Reject
• All-Pass

1
f
0
1
fc
f
1
0
fc
f
1
0
fc1
fc2
f
0
1
fc
fc2
f
0
16
Approximate Response

• Pass-band ripple ap20LogAmax/Amin
• Stop-band attenuation, Asb
• Transition-band ratio wp, ws

Amax
Amin
Asb
wp
ws
17
MATLAB

• Works with matrices (real, complex or symbolic)
• Multiply two polinomials
• f1(s)5s34s22s 1 f2(s)3s25
• clear all
• f15 4 2 1
• f2 3 0 5
• f3 conv(f1, f2)
• 15 12 31 23 10 5
• f3(s)15s512s4 31s3 23s2 10s 5

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).
• Xc1/sC Xc1/s(C/kf)1/C(s/kf)
• XLsL XLs(L/kf)L(s/kf)
• What occurs at w in the original network now
  will occur at kf w.

19
Impedance Scaling

• All elements with resistance dimension are
  multiplied by kz
• R -gt kz R wL -gtkzwL (VxaIcont) a -gt a kz
• All elements with capacitance dimension are
  divided by kz
• G -gt G/kz wC -gtwC /kz (IxbVcont) b -gt b/kz
• Impedences multiplied by kz
• Admittances divided by kz
• Dimensionless variables unchanged

20
Normalization and Denormalization

• Normalized filters use the key angular frequency
  of the filter (wp 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

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,

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

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