ALL POLE FILTERS SYNTHESIS AND REALIZATION TECHNIQUES

ENEE 481 K.A.Zaki2 TWO PORT PARAMETERS RgRg I2 I2 I1I1 V 1 V 2 RLRL EgEg 2-PORT NETWORK

ENEE 481 K.A.Zaki3

4 Maximally flat Chebyshev

ENEE 481 K.A.Zaki5 LOW PASS PROTOTYPE FILTERS AND THEIR RESPONSES

ENEE 481 K.A.Zaki6 Maximally Flat Low Pass Prototype 1 L CR Low-pass filter prototype, N=2

ENEE 481 K.A.Zaki7

8 PROTOTYPE LOW PASS FILTER ELEMENT VALUES: BUTTERWORTH: ORDER n : INSERTION LOSS:

ENEE 481 K.A.Zaki9 CHEBYSHEV:

ENEE 481 K.A.Zaki10 INSERTION LOSS: Order n:

ENEE 481 K.A.Zaki11 Impedance and Frequency Scaling

ENEE 481 K.A.Zaki12 MICROWAVE BANDPASS FILTER FREQUENCY MAPPING: NORMALIZED FREQUENCY OF THE PROTOTYPE CENTER FREQUECY OF THE BPF BAND EDGE FREQUENCIES OF THE BPF 3 dB FOR BUTTERWORTH AND EQUAL RIPLE FOR TCHEBYSCHEFF FILTER RELATIVE BANDWIDTH OF THE BANDPASS FILTER

ENEE 481 K.A.Zaki13 ELEMENT VALUES: FOR SERIES ELEMENT: FOR SHUNT ELEMENTS: LP BP

ENEE 481 K.A.Zaki14 LOW PASS FILTER RESPONSE AND CORRESPONDING BANDPAS FILTER RESPONSE

ENEE 481 K.A.Zaki15 MAPPING OF LPF TO BPF BPF WITH ONE TYPE OF RESONATORS

ENEE 481 K.A.Zaki16 DEFINITIONS OF IMPEDANCE & ADMITTANCE INVERTERS ZbZb K k,k+1 Z in = K 2 k,k+1 /Z b IMPEDANCE INVERTER YbYb J k,k+1 Y in = J 2 k,k+1 /Y b ADMITTANCE INVERTER Y’bY’b L’L’ C’C’ ZbZb K Z in

ENEE 481 K.A.Zaki17 K=1 Yp()Yp() Zs()Zs() Impedance inverter used to convert a parallel admittance into An equivalent series impedance. J=1 Zs()Zs() Yp()Yp() Admittance inverter used to convert a series impedance into An equivalent parallel admittance

ENEE 481 K.A.Zaki18 K1K1 Yp()Yp() K2K2 J2J2 J1J1 Zs()Zs()

ENEE 481 K.A.Zaki19 REALIZATION OF IMPEDANCE INVERTERS -L -C C K=  L K=1/  C  Z0Z0  XZ0Z0 ; L X

ENEE 481 K.A.Zaki20 GENERAL EQUIVALENT CIRCUIT OF AN IMPEDANCE INVERTER jX A jX B  Z0Z0 Z0Z0

ENEE 481 K.A.Zaki21 Filter Implementation Richard’s transformation is used to convert lumped elements to transmission line sections. Kuroda’s identities can be used to separate filter elements by using transmission line sections.

ENEE 481 K.A.Zaki22 iX L S.C. iB C L C O.C. Z 0 =L Z 0 =1/C iX L iB C The inductors and capacitors of a lumped-element filter Design can be replaced with a short-circuited and open- circuited stubs. All the length of the stubs are the same ( ) These lines are called commensurate lines

ENEE 481 K.A.Zaki23 Kuroda’s Identities Z1Z1 1/n 2 Z 2 Z 2 /n 2 Z1Z1 Z2Z2 Z2Z2 Z1Z1 1 Z1Z1 n2n2 n2n2 Z 1 /n 2 Z1Z1 Z1Z1 1/Z 2 1/n 2 Z 2

ENEE 481 K.A.Zaki24 Low Pass Filter Using Stubs 1 1 L1L1 L3L3 C2C2 1 1

ENEE 481 K.A.Zaki25

ENEE 481 K.A.Zaki26 Stepped- Impedance Low Pass Filters Approximate Equivalent Circuits for Short Transmission Line

ENEE 481 K.A.Zaki27 SMALL SECTION OF TRANSMISSION LINE AND ITS EQUIVALENT CIRCUIT MICROWAVE LPF & ITS EQUIVALENT CIRCUIT

ENEE 481 K.A.Zaki28 jX/2 jB

ENEE 481 K.A.Zaki29 CONFIGURATION OF WAVEGUIDE FILTERS COUPLING USING RECTANGULAR SLOTS COUPLING USING INDUCTIVE WINDOWS

ENEE 481 K.A.Zaki30 INPUT AND OUTPUT CONFIGURATION INPUT/OUTPUT USING PROBES FIRST/LAST RESONATOR OUTSIDE WAVEGUIDE FIRST/LAST RESONATOR INPUT/OUTPUT USING SLOTS AND ADAPTER

ENEE 481 K.A.Zaki31 jX A2 jX B2 jX B1 jX A1 jX A2 jX A1  Z0Z0 CONFIGURATION COMBINING EQUIVALENT CIRCUITS COMBINATION OF A CAVITY AND TWO SLOTS    Z0Z0 jX A2    Z0Z0    Z0Z0 Z0Z0 jX B2 jX B1 jX A2 jX A1

ENEE 481 K.A.Zaki32 SCATTERING MATRIX CAVITY Z0Z0 SLOT Z0Z0 jX A jX B jX A Z0Z0 CAVITY SLOT 

ENEE 481 K.A.Zaki33 D Side 1 Side 2 dIdI cIcI SISI c II d II S II CASCADING MULTIPOR BLOCKS a1a1 b1b1 a2a2 b2b2 ACCURATE FILTER RESPONSE IS COMPUTED BY CASCADING THE GENERALIZED SCATTERING MATRICES OF SECTIONS OF WAVEGUIDES, DISCONTINUTIES AND COUPLING SECTIONS

ENEE 481 K.A.Zaki34 MILLIMETER WAVE SEVEN POLE FILTER EXAMPLE

ENEE 481 K.A.Zaki35 OPTIMIZED RESPONSE OF 7-POLE FILTER

ENEE 481 K.A.Zaki36 SENSITIVITY ANALYSIS OF 7-POLE FILTER TO RANDOM MANUFACTURING TOLERANCES

ENEE 481 K.A.Zaki37 MEASURED PERFORMANCE OF A MILLIMETER WAVE DIPLEXER DESIGNED BY MODE MATCHING WITH NO TUNING

ENEE 481 K.A.Zaki38

ENEE 481 K.A.Zaki39 THE ABCD MATRIX FOR A LENGTH OF TRANSMISSION LINE IS : A B cos  jZ(  ) sin  (  ) = C D jY(  ) sin  (  ) cos  (  ) FOR A COAXIAL LINE OPERATING IN THE TEM MODE,  (  ) =   /(2  0 ), Z IS CONSTANT,  l  l / v,  0 IS THE FREQUENCY FOR WHICH THE LINE LENGTH IS QUARTER WAVELENGTH REALIZATION OF PRACTICAL FILTERS

ENEE 481 K.A.Zaki40 LENGTH OF LINE: ab  l Y inoc = 1 Z 11 A = jY 0 sin  cos  = jY 0 tan  FOR A SHORT CIRCUITED LINE: Z insc = 1 Y 11 == jZ 0 sin  cos  = jZ 0 tan  B D FOR AN OPEN CIRCUITED LINE: C = = = =

ENEE 481 K.A.Zaki41 FOR A SMALL LENGTH OF TRANSMISSION LINE TAN  ~  Y inoc  j Y 0  j Y 0    j  C ‘   Z insc   j Z 0  j Z 0    j  L’ FOR A SHUNT CAPACITOR: A B 1 0 = C D j  C’ 1 FOR A SERIES INDUCTORS: A B 1 j  L’ = C D 0 1 ; =

ENEE 481 K.A.Zaki42 MICROWAVE LOW PASS FILTER ELEMENT VALUES TRANSMISSION LINE RELATION HIGH IMPEDANCE LINE: SERIES INDUCTOR LOW IMPEDANCE LINE: SHUNT CAPACITOR ELECTRICAL LENGTHS OF T.L. IN DEGREE CHARACTERISTIC IMPEDANCES SERIES INDUCTOR, SHUNT CAPACITOR