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.TXT 1 0 676 0 0
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24 \pard  
------------------------------------------------------------------------------
-----------------------}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24 \pard {\b 
Transmission and Reflection Properties of L-C Band Pass filters with 
Dissipation Loss}}
.TXT 3 0 869 0 0
Cg a75.625000,75.625000,25
{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24 \pard \tab \tab \tab       {
\fs20 Michael A. Earle}}
.TXT 2 27 1044 0 0
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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michael.earle@sympatico.ca}}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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{\fs20 October 2000}}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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------------------------------------------------------------------------------
----------------------}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24 \pard {\i General}}
.TXT 4 0 679 0 0
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24 \pard The frequency 
response of L-C band pass filters derived from low pass prototypes can 
be determined from the prototype circuit parameters using a 
transformed frequency variable that includes a term to account for the 
loss. The loss term depends upon the unloaded Q factor of each 
resonator in the band pass circuit. For convenience however, each 
resonator is assumed to have the same unloaded Q factor.\par  \par 
Transmission and reflection properties are found from the overall 
chain matrix of the cascaded  ladder elements that form the prototype. 
From the elements of the overall chain matrix, a transmission function 
is derived from which rejection, pass band insertion loss, phase and 
group delay are found. The insertion loss characteristic shows the 
impact of  the unloaded Q factor upon the pass band insertion loss. 
Also, the elements of the overall chain matrix are used to find the 
reflection coefficient function, hence the pass band return  loss.  
\par \par The development which follows is confined to conventional 
Chebyshev filters of odd or even order. In addition, only filters 
which begin with a shunt capacitance are selected rather than their 
dual circuits in which shunt capacitors are replaced with series 
inductors and vice versa. A simple modification to the expression for 
the chain matrix allows the dual cases to be considered. }
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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desired properties of the filter are specified  }
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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band edge - GHz\tab \tab :}
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{0:f.1}NAME:3.39
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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{0:f.2}NAME:3.41
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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return loss Db \tab \tab       :}
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{0:R}NAME:26
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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{0:N}NAME:7
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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resistance - Ohms      \tab :\tab }
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{0:R.o}NAME:50
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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{0:\df}NAME:.0005
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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frequency points\tab \tab :}
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{0:nf}NAME:400
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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factor\tab \tab \tab :}
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{0:Q.u}NAME:2000
.TXT 7 -47 705 0 0
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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parameters}}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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frequency and normalized pass band.}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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prototype constants.}
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prototype circuit parameters.}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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Transmission functions}}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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matrix definitions.}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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for dual circuits, the sequencing matrix }u(n){\i  is replaced with 
its transpose }u(n){\fs20\up T} {\i and termination r is replaced with } 
r {\fs20\up -1}{\fs20 .}}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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group delay.}
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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coefficient. }
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
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group delay.}
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\cf1\fs24 \pard The formulae presented above specifically apply to L-C 
band pass filters and are therefore useful up to frequencies where L-C 
elements can be usefully realized in practice. These expressions are 
also useful for narrow band microwave filters realized in transmission 
line structures where the frequency transformation is approximately 
represented by the expression for {\f1 W}{\dn q }above. This usually 
means restricting the band width f{\dn 2} -f{\dn 1} to one or two 
percent of f{\dn o.}}
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0 1 1 0 0 4 0 1 1 Frequency GHz
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0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
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0 1 1 0 1 1 NO-TRACE-STRING
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0 2 1 0 1 1 NO-TRACE-STRING
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Cg a74.625000,74.625000,11
{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24 \pard {\i References.}}
.TXT 3 0 1020 0 0
Cg a74.625000,74.625000,97
{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24 \pard Some useful 
references from the vast body of literature on the subject of filters 
are as follows.}
.TXT 3 0 1022 0 0
Cg a74.625000,74.625000,130
{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24 \pard 1]\tab 
Microwave Filters, Impedance Matching Networks and Coupling Structures 
\par \tab by Young, Mathai and Jones. Pub. McGraw-Hill. pp95-103}
.TXT 5 0 1023 0 0
Cg a74.625000,74.625000,105
{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24 \pard 2]\tab Notes 
on Microwave Circuits vol. 2, ch 8, \par \tab by Darko Kajfez. Pub. 
Kajfez Consulting, Oxford Mississipi}
.TXT 5 0 1025 0 0
Cg a74.625000,74.625000,112
{\rtf\ansi \deff0{\colortbl;\red128\green0\blue0;}{\fonttbl{\f0
\fcharset0\fnil Times New Roman;}}\plain\cf1\fs24 \pard 3]\tab 
Synthesis of Lumped Element, Distributed and Planar Filters.\par \tab 
by Joseph Helszajn. Pub. Mc. Graw-Hill. ch 12. }