Chapter 4 Latest

7/31/2019 Chapter 4 Latest

1/41

Chapter 4:Electronic Filter SynthesisTopics: Filter background, definition and application

The properties of an ideal filter

Filter classification Filter design

Basic and characteristic filter


2/41

An introduction to filter

A filter is1. a circuit designed to pass signals with desired frequencies and

reject the others.

2. important to attenuate unwanted frequencies noise due to thebackground noise or nonlinear characteristic of some

electronic devices3. suitably used to

(a) block high pitches that cannot be efficiently broadcast.

(b) remove high frequencies which causes image blurring (vialow pass filter)

(c) remove low frequencies, sharpen and enhance edge (viahigh pass filter)

(d) In biomedical applications for instance; to filter out theEEG signals, to detect abnormal signal in ECG analysis, todistinguish the heart beat sound between mother and foetus.


3/41

Properties of an ideal filter

Must have identical gain at all frequencies

in its pass band

Should have zero gain outside its pass

band

Tolerable frequencies response curve

around +/-3dB


4/41

Ideal low-pass filter

Ideal high-pass filter

Ideal band-pass filter

Ideal notch filter

Common types of filters


5/41

Properties of fundamental filters

Low pass filter

o Pass only signals with frequencies lower than fc and

attenuate the rest

High pass filter

Pass only signals with frequencies greater than fc and

attenuate the rest

Band pass filter

Pass only signals with frequencies between f1

and f2

Band stop filter

attenuate only signals with frequencies between f1 and f2


6/41

Filter classification

Passive filterconsists only ofpassive elements(R,L,C)

Active filterconsists ofactive element (transistor, op-

amp) in addition to passive elements

Passive filter Active filter


7/41

Comparing Active and Passive filters

Aspect Passive filter Active filter

Gain Cannot generate gain >1 and is

lossy

Provide amplification and voltage gain

but need external sources

cost Expensive (because of the use

of inductors)

Cheap

Complexity Simple topology More complex topology

Flexibility - Isolation of each stage of filters from

source and load impedance effects can

be achieved by the use of buffer

amplifiers

Practicality Difficult to implement at low

frequencies (f


8/41

Designing a filter circuit

Parameters to be considered in designinga filter

1. Quality factor

-The higher the Q factor, the more (frequency)selective a circuit is, producing a high gain at thecircuit end

2. Damping ratio

- Determine the filter response characteristic3. RC pole

- A filter pole consists of a RC circuit. Each poledoubles the roll-off rate.


9/41

Passive Low-pass filters circuit(using RC network)

First order Low-pass filter

Second order Low-pass filter

Cut-off frequency:

Cut-off frequency:


10/41

Second order Low-pass filter has higher rate of roll-offin which itattenuates higher frequency more quickly.

Application where filter with high roll-off is desired:

1. to prevent crosstalk between adjacent channels on telephone FDM

(frequency division multiplexing) systems

2. Improve EEG signals

The frequency response of First and second order

low pass filter


11/41

Passive High-pass filter circuit(using RC network)

First order High-pass filter

Second order High-pass filter

Cut-off frequency:

Cut-off frequency:


12/41

Identify the types of filter (1) Given that we have a circuit as follow;

We know that

Using the voltage division, high fleads to lowXc hence low voltagedrop across capacitor, where Vout:

the otherwise is true iffis low.

where;

f= oscillating frequency

C= capacitance

Fig. 1


13/41

Identify the types of filter (2) As a rule of thumb:

This shows that the circuit shown in Fig.1 is a low pass filter

Hi f => LowXc => Low VoutLowf=>HiXc =>Hi Vout

This is true only for thecircuit shown in Fig. 1

where;


14/41

Identify the types of filter (3)Alternatively, we can also identify the types of filter for the circuit

shown in Fig. 1 by writing its transfer function, H(S) which isgiven as

RCj

CjR

Cj

V

VSH

in

out

1

1

1

1

)(

Substituting the value of =0 and = into the transfer function gives

01

1)0(

H

01

1)(

H

Showing that it is a low pass filter


15/41

Passive band-pass and band-stop filters using RC network

Band-pass filter via

cascaded RC circuits

Band-stop filter via

Twin T


16/41

Passive band-pass filter using RCnetwork

Band-pass filter via cascaded RC circuits

1. An ideal bandpass filter would have a completely flat passband(e.g. with no

gain/attenuation throughout) and would completely attenuate all frequencies

outside the passband.2. The bandwidth of the filter is simply the difference between the upper and lower

cutoff frequencies, while the centre frequency is given by;

Frequency response of a Band-pass filter

where

L is the lower -3dB cut-off frequency point

H is the upper -3dB cut-off frequency point


17/41

Band-stop filter via Twin T

The twin T provides a large degree of rejection at a particular frequency,

e.g. to filter out unwanted noise at 50 or 60 Hz that may be entering a circuit.

The response provided by the filter consists of a low level of attenuation away

from the notch frequency. As signals move closer to the notch frequency, the

level of attenuation rises, giving the typical notch filter response.

The notch frequency, fnotch, is given by

Passive band-stop filter using RCnetwork


18/41

Example of second order RLCpassive filters:

RLC low pass filter

RLC band pass filter


19/41

ExerciseIdentify the types of filter for the following circuit and give your

reason why.


20/41

Solution

At low frequency, lets assumefapproaching 0 (f> 0), XL appeared as a

short while XC an open, so that the circuit becomes

There is no current passing through the circuit hence no signal appear at

the output at low frequency

The reactance of capacitor , XC, and inductor , XL, in the circuit is given by


21/41

Whereas at high frequency, lets assumefapproaching infinity (f> ), XL

appeared as an open while XC as a short, so that the circuit becomes

Again, there is no current passing through the circuit hence no signal

appear at the output at high frequency

However, at the resonance frequency, the circuit appeared as

The reason is because:Minimum impedance at resonant

frequency for L-C connected in series

Maximum impedance at resonant

frequency for L-C connected in parallel


22/41

At the resonance frequency,fr, there is voltage drop across parallel L-C

branch, hence Vo is nonzero.

The above derivations show that this circuit is a Band-pass filter and itsfrequency response can be sketched as


23/41

ExampleBased on the filter circuit below

1. Find the transfer function.

2. Given that |H()| = , calculate the cutoff frequency. Use L = 4.5mH, C =

25F and R = 100k.

sCRsL

sCR

v

vsH

i1

1)( 0

Solution:Using the voltage division rule, H(s) is given by:

sRC

R

sCR

sCR

sCR

1/1

/1

where


24/41

RsLRLCs

R

sRCRsL

sRCRsH

2)1/(

)1/()(

hence

RLCLjR

RjH

2)(

Since s =j, so

Solution for (b)

The magnitude of H(j) is given by

210

|)(|222

2

LRLCR

RH

Squaring the |H()|, |H()|2 reduces the equation to

2

2)1(2

R

LLC cc


25/41

2

3

3

263

10100

)105.4()]1025)(105.4(1[(2

x

xxx cc

Substitute the values into the equation gives:

Solve for the quadratic equation ofc:

a

acbb

2

42

)1025.1(2

)11025.14()1025.2()1025.2(14

14277

x

xxxxx

c =

Therefore,

c= 4657 rad/s @ 4.657 krad/s


26/41

First order Active filter circuits(using RC circuit)

Active low-pass filter

Active high-pass filter


27/41

Second order Active filter circuits(using RC circuit)

Active band-pass filter


28/41

Again,

Given that we have an active filter as follows;

Iff> ,Xc> 0, the circuit becomes:

The capacitor acts as a short, the amount of current passed through R2

becomes negligible so the gain of the amplifier goes to zero

Identify the types of Active filter(part1)

Fig.2


29/41

Iff> 0,Xc

> , the circuit becomes:

The capacitor acts as an open, the circuit will act as an amplifier with gain

-R1/R2

*This shows the circuit is a low pass filter.

Identify the types of Active filter(part2)


30/41

Identify the types of Active filter(part3)In the case of Fig.3

This circuit will attenuate low frequencies ( 1/R1C1), but will pass intermediate frequencies with a gainof -R1/R2. So, it is a band-pass filter.

Fig.3

E l


31/41

ExampleFigure below shows a filter circuit

1. Prove that the transfer function is given as:

RCj

RCj

R

RH

f

11)(

1

2. Identify the type of filter


32/41

According to the golden rule of op-amp: V+ =V- , therefore op-amp appeared as

an open circuit

It can be seen from the circuit that voltage across Node A gives V+, while

voltage drop across Node B gives V- , and we know that

Using voltage division rule to give V+ and V- as

Eq. 1

Eq. 2

Eq. 3


33/41

Substitute Eq.2 and Eq.3 into Eq. 1 produces

Rearranging the equation gives

The AOL (open loop gain) of an op-amp is always approaching infinity, hence

the transfer function of this circuit can be reduced to

RCj

RCj

R

R

V

VsH

i

f

i

o

1

1)(


34/41

0)0(1

)0(1)0(

RCj

RCj

R

RH

i

f

1)(1

)(1)(

RCjRCj

RRH

i

f

Solution (Q2):

Substituting the value of = 0 and = into the equation

The above shows that this circuit is a high pass filter.


35/41

Characteristic filter: Butterworth filter(Part I)

The third-order low-pass design shown above is a simple

example of Butterworth filter


36/41

1. The distinctive characteristic of this filter is the maximally flatfrequency response which is desired for various applications.

2. The roll-off rate for this filter starts with 20n dB/decade. The

higher number of n, the flatter the response will be - hence the

filter is approaching to an ideal frequency curve

Characteristic filter: Butterworth filter(Part II)


37/41

Characteristic filter: Chebyshev filter I

1. This filter is well-known by the existence of ripple in the pass band and

rapidly increasing attenuation in the transition band.

2. Standard roll-off rate for Chebyshev filter is 40dB/decade which meansthe response will be steeper at the edge- hence it has a sharper cutoff as

compared to Butterworth filter.

3. This filter belongs to active filter group and which has irregular pass

band response.

4. The higher number of N the faster the roll off rate but also results in an

increase in the amount of ripple in the pass band


38/41

Characteristic filter: Chebyshev filter II

1. The response is monotone at the pass band but equiripple

response at the stop band.

2. This filter is not commonly in used as it gives inconsistent

gain due to fluctuations.


39/41

Sallen Key (Two-Pole) Low Pass Filter

-

+

+V

-V

R1

Rf1

Rf2

C1

vin

vout

C2

R2

Low Pass Filter

1. To obtain an nth order filter, n/2SK circuits should be cascaded

(assume K=1)

2. As K increased from 0 to 3, the

transfer function displays more

peaking3. The circuit becomes unstable

when K>3

where,

R1 = R2

C1 = C2

Rf1 = (K-2)Rf2


40/41

Sallen Key (Two-Pole) High Pass Filter

-

+

+V

-V

R1

Rf1

Rf2

C2

vin

vout

R2

C1

High Pass Filter

1. Using op amp, the SK is nottruly a high-pass filter,

because the gain of the op

amp eventually falls off.

2. However, the frequencies at

which the op amp gain is

fairly high, the circuit

behaves as a high-pass filter.

3. Since the HP SK circuit is

equivalent as the low-pass,

the empirical values for K

would be still valid in thiscase also.

where,

C1 = C2

Rf1 = (K-2)Rf2


41/41

Peaking clearly be seen in

the stopband

The typical Sallen Key frequency response:

Documents

Chapter 4 Latest