1

UNIT:I FILTERSAttenuation in neper:

α (¿N )=ln| I S

IR|=12 ln| P¿

Pout|

Attenuation in decibel:

D (¿dB )=10 log10| P¿

Pout|=20 log10| I S

IR|

Symmetrical T-Network:

Z0=√ Z12

4+Z1Z2 ;Z0=√ZOC ZSC

γ=ln [1+ Z12 Z2

+Z0Z2 ]

Z12

=Z0 tanhγ2; Z2=

Z0sinh γ

Symmetrical Π-Network:

Z0=Z1Z2

√Z12

4+Z1Z2

=Z1Z2Z0T

;Z0π=√ZOC ZSC

γ π=ln [1+ Z12Z2

+Z1Z0π ]=ln [1+ Z1

2Z2+Z0TZ2 ]

γ π=γT ;2Z2=Z0 π

tanhγπ

2

;Z1=Z0π sinh γ π

Symmetrical Lattice Network:

Z0=√Z A ZB ;Z0=√ZOC ZSC

γ=ln [ Z0+ZA

Z0−Z A]=ln [ ZB+Z0

ZB−Z0 ]ZA=Z0 tanh

γ2;ZB=Z0 coth

γ2

Filter Fundamentals:

Cut−off frequncy condition :X1+4 X2=0

Phaseangle∈ passband : β=2sin−1√ Z14 Z2

Attenuation∈stopband :α=2cosh−1√ Z14 Z2

Prototype Low Pass Filter :

Design Impedance :R0=√ LC

Cut−off frequency : f C=1

π √LC

Z0T=R0√1−( ff c

)2

;Z0π=R0

√1−( ff c

)2

α=2cosh−1( ff c

); β=2sin−1( ff c

)Design Equations :L=

R0

(πf c );C= 1

(πf c)R0

Prototype High Pass Filter :

Design Impedance :R0=√ LC

Cut−off frequency : f C=1

4 π √LC

Z0T=R0√1−( f c

f )2

;Z0π=R0

√1−( f c

f )2

α=2cosh−1( f c

f ); β=2sin−1( f c

f )Design Equations :L=

R0

(4 πf c);C= 1

(4 πf c )R0

Prototype Band Pass Filter : Prototype BandElimination Filter :

2

Design Impedance :R0=√ L2C1

=√ L1C2

Resonance frequency : f 0=√ f 1 f 2

Design Equations :

L1=R0

π ( f 2−f 1 );C1=

(f 2−f 1 )4 π R0 (f 1 f 2)

L2=R0 (f 2−f 1 )4 π (f 1 f 2)

;C2=1

π R0 ( f 2− f 1 )

Design Impedance :R0=√ L2C1

=√ L1C2

Resonance frequency : f 0=√ f 1 f 2

Design Equations :

L1=R0 (f 2−f 1 )π ( f 1 f 2 )

;C1=1

4 π R0 ( f 2−f 1)

L2=R0

4 π ( f 2−f 1 );C2=

( f 2− f 1 )π R0 (f 1 f 2)

m-derived sections :

ForT−section,Z2'=(1−m2

4m )Z1+ Z2m

;For π−section Z1'=

(mZ1) ( 4m1−m2 )Z2( 4m1−m2 )Z2+(mZ1 )

m-derived LPF sections:

f ∞=f c

√1−m2

m=√1−( f c

f ∞)2

m-derived HPF sections :

f ∞=f c √1−m2

m=√1−( f ∞

f c)2

m-derived BPF sections:

f 0=√ f 1∞ f 2∞=√ f 1 f 2

m=√1−( f 2−f 1f 2∞−f 1∞ )

2

m-derived BRF sections :

f 0=√ f 1∞ f 2∞=√ f 1 f 2

m=√1−( f 2∞−f 1∞f 2−f 1 )

2

Crystal Filters:

Resonance Frequency : f R=1

2π √LC

Anti−resonance Frequency : f A=f R√1+ CC '

f A−f R=f RC

2C '

seriesresonance Frequencies : f R1=f R2=f A √1∓ CC '