Download - UNIT 1 Equations
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 '