Lecture Two

Filter Design
2.1 Introduction
2.2 Types and Specification of Filters
2.3 Transfer Function of a Linear Circuit
2.4 Butterworth and Chebyshev Filters
2.5 Examples
2.6 Switched-Capacitor (SC) Filters
2.7 Appendix: Look-up Table for RLC Filter Synthesis
2.1 Introduction

Electronic Filter is a popular and important building block in electronic systems.

There are three realisation techniques for Filters:

(1) Passive LC filters:
using inductors and capacitors
Disadvantage: Inductors are very difficult to fabricate in monolithic form (IC).

(2) Active-RC filters:
using operational amplifier, resistances and capacitors

(3) Switch-Capacitor filters (SCF):
using operational amplifier, switches and capacitors and very good for realising fully integrated monolithic filters.
2.2 Types and Specification of Filters

2.2.1 The Types of Filters

The function of filter:

Performing a frequency selection: passing signals whose frequency spectrum lies within a specified range, and stopping signal whose frequency spectrum falls outside this range.

There are four major filter types:

(1) Low-pass filter
(2) High-pass filter
(3) Bandpass filter
(4) Bandstop filter

Their ideal transmission characteristics are shown in Figure 2.1.

Figure 2.1
2.2.2 Filter Specification

• Low-pass filter

Figure 2.2

Figure 2.2 shows the realistic specification of a low-pass filter. The transmission of a low-pass filter can be specified by following four parameters:

(1) The passband edge ( )
(2) The maximum allowed variation in passband transmission (Amax)
Amax(dB) is an upper bound for the deviation of the passband transmission from the ideal value in a real circuit, typically ranges from 0.05 to 3dB.
(3) The stopband edge ( )
(4) The minimum required stopband attenuation (Amin)
The zero transmission cannot be achieved at all stopband in a real circuit. However the stopband signal must be attenuated by at least Amin(dB). Usually, Amin can range from 20 to 100dB.

As can be seen from Figure 2.2 the magnitude response ripple throughout the passband with the ripple peaks are equal to Amax, so it is usual to refer to Amax as the passband ripple and to as the ripple bandwidth.

• Bandpass filter
Figure 2.3

• The step to design a filter

(1) Determining the filter specifications
(2) Finding a transfer function that meets given specification. This process is called filter
approximation.
(3) Realisation of the transfer function
2.3 Transfer Function of a Linear Circuit

2.3.1 General introduction

• Complex numbers

Figure 2.4

For a complex number Z=a+jb

Magnitude:
Phase:
As can be seen form Figure 2.4

We also know

Thus Z can be given by
• The transfer function

Figure 2.5
A linear circuit that can be represented by the general two-port network shown in Figure 2.5. The filter transfer function is

(2.1)

• The magnitude and phase of transfer function

Since the transfer function can be expressed in terms of its magnitude and phase as

(2.2)

Here: is the magnitude and is the phase.

• The output voltage

If input voltage is , from Eq. (2.1)

(2.3)

Magnitude:
(3.4)

Question: What is the phase of the output voltage?
As can be seen, the magnitude and phase characteristics of the input signal can be shaped as it passes through the filter.
• The complex expression for R, C and L
2.3.2 Transfer function for the 1st order circuits

• RC low-pass filter

Figure 2.6

Its transfer function is

Here is the time constant. The magnitude and phase of are as follows

* |T|=20log(Vout/Vin)

Figure 2.7

Question: Determine for the circuit in Figure 2.8 and find
(1) The type of filter
(2) The time constant

Figure 2.8

• RC high-pass filter

Figure 2.9

Transfer function:

Magnitude and phase of

Figure 2.10

Question: Determine for the circuit in Figure 2.11 and find
(1) The type of filter
(2) The time constant

Figure 2.11
Conclusions:

(1) Circuits with 1 reactive element are of the 1st order
(2) They can be LPF and HPF
2.3.3 Transfer function for the 2nd order circuits

Figure 2.12

Series LC circuit (Figure 2.12)

Transfer function:

The impedance of the series LC circuit:

Several important parameters for LC circuits:

(1) Resonant frequency:
(2) Generalised detuning:
(3) Characteristic impedance:
(4) Quality factor:

The transfer function is a band-pass filter. Its passband is

Figure 2.13
2.4 Butterworth and Chebyshev Filters
In this section we present two functions that are frequently used in approximating the transmission characteristics of filters. These function have the advantage that closed-form expressions are available for their parameters. Thus one can use them in filter design without the need for computer.
2.4.1 Synthesis of low-pass filters

The synthesis steps for low-pass filters:
(1) Choose the type of filter (Butterworth or Chebyshev) and its transmission parameter (Amax, Amin, ωp, ωs)
(2) Determine the order of the filter according to its parameters
(3) Find suitable filter diagram and element values from the synthesis table
(4) Change the normalised values to actual valus

• Butterworth filter

Figure 2.14
Transfer function:

Where n is the order of the filter.

The filter specifications: Kr, KA, ωp, ωs. You can find Kr and KA form Amax and Amin.
The order of the Batterworth filter

where
(selectivity factor)

• Chebyshev filter

Figure 2.15

Transfer function:

where is Chebyshev polynomial of the n-th order.
The order of the Chebyshev filter:
where

2.4.2 Synthesis of high-pass filters

Conversion of low-pass filter into high-pass filter with the same ωp

Figure 2.16

Figure 2.17

3.4.3 Synthesis of band-pass filters

Figure 2.18
Figure 2.19
2.5 Examples

Figure 2.20
Example 1: Design a Chebyshev low-pass filter (LPF) for RS=RL=50Ω, its parameters
have been shown in Figure 2.20.
Kr=? KA=?

Example 2: Use the Chebyshev LPF (n=3) in Example 1 as a prototype to design a
high-pass filter (HPF). Here RS=RL=50Ω.

Example 3: Use the Chebyshev LPF (n=3) in Example 1 as a prototype to design a
band-pass filter (BPF). Here RS=RL=50Ω and f0=10MHz.

2.6 Switched-Capacitor (SC) Filters

2.6.1 Advantages of switched-capacitor filters

Switched-capacitor filters: analogue sampled filters.
• Using MOS technology
• Basic components: Op-Amp, switches, capacitors

Passive LC filters:
(1) Work well at high frequencies; in low-frequency applications the required inductors are large and physically bulky, and the characteristics are nonideal.
(2) Impossible to fabricate in monolithic form and are incompatible with any of the modern techniques for assembling electronic systems.

Active RC filters:
(1) Components: Op-Amp, resistors and capacitors
(2) Difficulties for production in monolithic IC: need for large-valued capacitors, large bandwidth Op-Amps and the requirement of accurate RC time constants. Also, resistors require more chip area.

SC filters:
Using SC filters, it is possible to achieve high precision capacitor ratio values and, moreover, since MOS capacitors approach closely ideal capacitor characteristics, greater linearity and stability are possible than with diffused resistors. It is therefore possible to achieve fully integrated and stable active filter circuits.

Advantages of SC filters:

• Overcome the problems of LC filters and RC filters.

LCF: L is difficult for IC

RCF: Big R and C: big chip area and high tolerance

R: 103-106, C: 10-9-10-6F

SCF: High precision:

Time constant: capacitance ratios rather than RC products

MOS: High precision capacitance ratio values is easier to be implemented in IC
Ratio: 0.01-0.1%

• Can be used for programmable analogue array (FPAA)

• SCF: Process analogue signals directly

2.6.2 Basic principle of switched-capacitor filters

Key point: Using switches and capacitor (switched capacitor) to replace resistor in a RC circuit. Using MOS technology the switched capacitor can be implemented using two MOSFET switches, operated using a two-phase clock system in which the clock phases do not overlap.
How to realise this equivalent?

Considering a two-phase, non-overlapping MOSFET clock system (the simplest SC circuit): The switches are driven by 1 and 2. Vi is a time function f(t) and its frequency is much less than the switching frequency.
When Φ1 is high, the switch is at position 1, the capacitor is connected to Vi and charged:
q1=CVi

When Φ2 is high, the switch is at position 2, the capacitor is connected to Vo

q2=CVo
The charge transferred in TC

q=q1-q2=C(Vi-Vo)
Because

The equivalent resistance is

If the switching frequency is much larger than the signal frequencies, then the equivalence of switched capacitor with resistor is accurate enough.
Simple SC circuits

Integrator
, where R=1/fcCs
2.6.3 Building block of switched-capacitor filters

If , , then
Question: Implementation of a band reject filter:

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