IIT Bombay Course Code : EE 611 Department: Electrical Engineering - - PowerPoint PPT Presentation

IIT Bombay

Course Code : EE 611 Department: Electrical Engineering Instructor Name: Jayanta Mukherjee Email: jayanta@ee.iitb.ac.in

EE 611 Lecture 15 Jayanta Mukherjee Page 1

Lecture 15

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Topics Covered

• General Introduction to filter synthesis
• Insertion Loss method
• Filter Prototypes
• Scaling of prototypes

EE 611 Lecture 10 Jayanta Mukherjee Page 2 EE 611 Lecture 15 Jayanta Mukherjee

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Filter Synthesis

• We have seen how narrowband bandstop or band pass

filters can be constructed using resonators

• For narrowband cases, lumped element models of the

distributed network were good enough to get good performance predictions

• For wider bandwidths this will not be the case; we need

design methods specifically for distributed circuits

• Design methods for lumped element filters exist, and can be

adapted in some cases to distributed circuits.

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Filter Synthesis (2)

• Let’s start by reviewing the “insertion loss” synthesis

method, which allows control of filter bandwidth and performance through a set of “prototype” designs.

ω 3 dB LI(ω) ωc Pass- band Stopband

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Filter Synthesis (2)

• btained.

are losses higher for while

• btained

is k 1 range in the response ripple equal an , For constant. a is k and polynomial Chebyshev a is T where 1 S 1 : filter pass

• low

Chebyshev For

• btained.

is loss dB 3 at which frequency the is where 1 S 1 : filters pass

• low

flat maximally For responses. Chebyshev

• r

th) (Butterwor flat maximally either produce designs common Most filter. the

• f

loss insertion the determines which , S 1

• f

values desired achieve to designed are Filters first. designs lumped

• n

focus will we so circuits, d distribute

• r

element lumped either for applies synthesis loss Insertion

c 2 c N 2 2 2 21 c 2 2 21 2 21

ω ω ω ω ω ω ω ω ω > ± <         + =

• 

       + =

• c

N N c

T k

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Synthesis Example for N=3 Butterworth

1ohm Z using 1 2 2 1 2 2 2 1 S 1 Z : Z impedance input to S convert and S for sign positive select the We

• 1

s and j0.866,

• 0.5

s roots 3 following the admits 1 2 2s s that can verify One 1 2 2 s S : (s) S identify to roots plane hand left select the we passive and realizable is filter the As 1. s and j0.866, 0.5 s : roots 6 admits s

• 1

) ( ) ( 1 1 1 S : in resulting 1 S have we thus : lossless is filter The 1 1 S G Hz. 1 and 1 with Z 3 N

• f

case he consider t us Let

2 2 3 11 11 in in 11 11 5 1,2 2 3 2 3 3 11 11 5,6 1,2,3,4 6 * 11 11 6 6 6 6 2 21 2 11 2 21 2 11 6 2 21 T c

= + + + + + = − + =

• =

± = = + + + + + + ± = ± = ± ± = =

• =

− − = + = − = = +

• +

= = = Ω = =

• s

s s s s S s s s s s S s S s s S S ω ω ω ω

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Cauer Expansion

• hm

1 R resistor and 1H L inductor an extracted now have We R R Z : R by R divide now us Let 2s Y since F 2 C since capacitor a extracted have We 1 2s 2s R D Y : R by D divide us Let Z since 1H L inductor an extracted have We 1 2s 2s 1 s s D R Z 1 2s 2s 1 2s 2s 2s Z : get we polynomial lower by the polynomial

• rder

higher the Dividing

2 1 in,3 2 1 ext,2 2 1 in,2 1 ext,1 2 1 ext,1 2 2 3 in,1

= =

• +

= + = =

• =

=

• +

+ = + = + + + = =

• =

= =

• +

+ + + = + = + + + + + =

• 1

1 1 1 1 2 1 .

1 2 2 ,

s s s s R R Y s s Ls

ext

g1 g2 g3 g4 1 VG Port 2 Port 1 Zin,1 Zin,3 Yin,2

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Alternate Implementation

: result will prototype different a, 1 2 2 s

• S

solution sign negative select the we If

2 3 11

3

+ + + =

• s

s s

g1 g2 g3 g4 1 VG Port 2 Port 1

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Prototypes

filters. general more

• btain

to s technique scaling" " need We 1. s frequencie " breakpoint " the also and 1 that Z assume protypes These sections.

• f

number the versus prototypes 453) (p Chebyshev and 450) (p flat maximally for the 1 / deviation frequency the n versus attenuatio the

• f

plots provides Book avoided. usually are filters Chebyshev section number Even needed. if used be can er transform impedance an case In this equal. not are impedances load and source the sections,

• f

number even an with filters Chebyshev for However equal. are impedances load and source the so 1, always is g filters flat maximally

• r the

c c 1 N

• =

=

• −
• +

ω ω ω F

g1 g2 g3 g4 1 VG Port 2 Port 1 g5

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Scaling of Prototypes

prototype. pass

• low

the

• f

ations transform suitable from filters stop band and pass band pass, high generate to possible also is It values. prototype the are g where Z g C g Z L are filter pass

• low

scaled frequency and impedance an in values C e capacitanc and L inductors new the Thus, . by capacitors

• riginal

the dividing and by inductors

• riginal

the dividing by just to 1 from frequency breakpoint the change can We . by Z capacitors

• riginal

the dividing and by Z inductors

• riginal

the g multiplyin by just 1

• f

instead Z impedance load) usually (and source with prototype pass

• low

a make can We

k c k ' k c k ' k ' k ' k c c c

• =

=

• ω

ω ω ω ω

g1 g2 g3 g4 1 VG Port 2 Port 1 C1 L2 C3 Z0 Z0 VG Port 2 Port 1

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Transformation from Low Pass to High Pass

passband. in the ripple

• f

expense at the filters flat maximally than cutoff better provide filters ripple equal case matching impedance in the As L and C then are filter pass high scaled frequency and impedance an for Values prototype. pass low in the capacitors and inductors switching up s ation wind transform This

• riginal.

in the s frequencie low to correspond l filter wil new in the s frequencie high then Note /

• filter to

pass

• low
• riginal

in the rming by transfo

• btained

is filter pass high A

' k ' k c

• =

= = =

• k

c k c k c k c

g Z C Z g Z L Z ω ω ω ω ω ω ω 1 1 .

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Transformation from Low Pass to High Pass

g1 g2 g3 g4 1 VG Port 2 Port 1 g5 Ck’ Lk’ VG Port 2 Port 1 Z0 Ck’ Ck’ Lk’ Z0

Low Pass Prototype High Pass Prototype

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Transformation from Low Pass to Band Pass

∆ π ω ω ∆ ω ∆ ω ω ω ω ω ω ω ω ω ω ω ∆ ω ω ω ω ∆ ω 2 g Q and 2 f

• r

C and L es with valu circuit, LC series a by replaced are prototype

• riginal

in the inductors filter, pass

• band

scaled impedance and frequency a For circuits. LC into formed each trans are capacitors and inductors

• riginal

the if

• nly

achieved be can ation transform This filter.

• riginal

the

• f

s breakpoint the 1, to and and prototype, pass low

• riginal

in the to maps transform This and s, frequencie passband lower and upper the define and

• where

1 with

• riginal

the replaces prototype pass

• low

a from transform the response, sided two a have filters pass band Because

k k k ' k ' k 2 1 2 1 1 2 1 2

= = = =

• ±
• =

=         −

• k

k

g Z Z g . , ,

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Transformation from Low Pass to Band Pass

∆ π ω ω ∆ ω ∆ 2 g Q and 2 f

• r

C and L th circuit wi LC shunt a by replaced are prototype

• riginal

in the apacitors

k k k ' k ' k

= = = =

• Z

g g Z C

k k

fr2 fr1 Z0 Port 1 Port 2 Z0 VG QE1 QE2 fr3 QE3

g1 g3 g5 g2 g4 1 Port 1 Port 2 gN+1 VG

Lowpass Prototype Bandpass Prototype

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Transformation from Low Pass to Band Stop

( )

equations. loss insertion pass low the into s frequencie ed transform same the ng substituti by

• btained

be can filters these

• f

behaviors loss Insertion g 2 Q and 2 f

• r

C and L th values circuit wi LC series a with capacitors shunt

• riginal

the and g 2 Q and 2 f

• r

C and L th values circuit wi LC parallel a with inductors series

• riginal

the replaces This used. is /ω ω ω/ω Δ ω ation transform the response, stop

• band

a

• btain

To

k k k ' k ' k k k k ' k ' k 1

• =

= = = = = = =

• −

←

• −

∆ π ω ω ∆ ∆ ω ∆ π ω ∆ ω ω ∆ 1 Z g g Z g Z g Z

k k k k

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Transformation from Low Pass to Band Stop

fr2 fr1 Z0 Port 1 Port 2 Z0 VG QE1 QE2 fr3 QE3 QE4 fr4 g1 g3 g5 g2 g4 1 Port 1 Port 2 gN+1 VG

Lowpass prototype Bandstop prototype