Insertion Loss Method
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Transcript of Insertion Loss Method
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8/14/2019 Insertion Loss Method
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4/27/2005 The Insertion Loss Method.doc 1/8
Jim Stiles The Univ. of Kansas Dept. of EECS
The Insertion
Loss MethodRecall that a lossless filter can be described in terms of
either its power transmission coefficient ( ) orits power
reflection coefficient ( ) , as the two values are completely
dependent:( ) ( )1 =
Ideally, these functions would be quite simple:
1. ( ) 1 = and ( ) 0 = for allfrequencies within the pass-
band.
2. ( ) 0 = and ( ) 1 = for allfrequencies within the stop-
band.
For example, the ideallow-pass filter would be:
( ) ( )
c c
1 1
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Jim Stiles The Univ. of Kansas Dept. of EECS
Add to this a linear phaseresponse, and you have the perfect
microwave filter!
Theres just one small problem with this perfect filterIts
impossibleto build!
Now, if we consider only possible (i.e., realizable) filters, we
must limit ourselves to filter functions that can be expressed
as finite polynomialsof the form:
( )
20 1 2
2 20 1 2
NN
a a a
b b b b
+ + +
= + + + +
The orderN of the (denominator) polynomial is likewise the
orderof the filter.
Instead of the power transmission coefficient, we often use
an equivalent function (assuming lossless) called the powerloss ratio LRP :
( )1
2
1
1LRP
PP
+
= =
Note with this definition, LRP = when ( ) 1 = , and 0LRP =
when ( ) 0 = .
We likewise note that, for a lossless filter:
( ) ( )
1 1
1LRP
= = T
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Jim Stiles The Univ. of Kansas Dept. of EECS
Therefore ( )LRP dB is :
( ) ( )10 1010 10 Insertion LossLR LR P dB log P log = = T
The power loss ratio in dB is simply the insertion loss of a
lossless filter, and thus filter design using the power loss
ratio is also called the Insertion Loss Method.
We find that realizable filters will have a power loss ratio of
the form:
( ) ( )( )
2
21LR MP N
= +
where ( )2M and ( )2N are polynomials with terms2 4 6, , ,etc.
By specifying these polynomials, we specify the frequency
behavior of a realizable filter. Our job is to first choose a
desirable polynomial!
There are many different typesof polynomials that result in
good filter responses, and each type has its own set of
characteristics.
The type of polynomial likewise describes the type of
microwavefilter. Lets consider threeof the most popular
types:
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Jim Stiles The Univ. of Kansas Dept. of EECS
1. Elliptical
Ellipticalfilters have three primary characteristics:
a) They exhibit very steep roll-off, meaning that thetransition from pass-band to stop-band is very rapid.
b) They exhibit ripple in the pass-band, meaning that
the value of will vary slightly within the pass-band.
c) They exhibit ripple in the stop-band, meaning that the
value of will vary slightly within the stop-band.
We find that we can make the roll-off steeperby accepting
more ripple.
2. Chebychev
Chebychev filters are also known as equal-ripplefilters, and
have two primary characteristics
a)Steeproll-off (but not as steep as Elliptical).
( )
1
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b)Pass-bandripple (but not stop-band ripple).
We likewise find that the roll-off can be made steeper by
acceptingmore ripple.
We find that Chebychev low-passfilters have a power loss
ratio equal to:
( ) 2 21LR Nc
P k T
= +
where kspecifies the passband ripple, ( )NT x is a Chebychev
polynomial of orderN, and c is the low-pass cutoff
frequency.
3. Butterworth
Also known as maximally flatfilters, they have two primary
characteristics
a)Gradualroll-off .
( )
1
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Jim Stiles The Univ. of Kansas Dept. of EECS
b)No ripplenot anywhere.
We find that Butterworth low-passfilters have a power loss
ratio equal to:
( )
2
1
N
LR
c
P
= +
where c is the low-pass cutoff frequency, and Nspecifies
the orderof the filter.
Q:So we always chose elliptical filters; since they have the
steepest roll-off, they are closest to idealright?
A: Ooops! I forgot to talk about the phase response ( )21S
of these filters. Lets examine ( )21S for each filter type
beforewe pass judgment.
Butterworth ( )21S Closeto linear phase.
Chebychev ( )21S Not very linear.
( )
1
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3. It increases filter insertion loss (this is bad).
4. It makes filter performance more sensitive to
temperature, aging, etc.
From a practical viewpoint, the orderof a filter should
typically be kept to 10N .
Q: So how do we take these polynomials and make realfilters?
A: Similar to matching networks and couplers, we:
1. Form a general circuit structure with severaldegrees of
design freedom.
2. Determine thegeneral formof the power loss ratio for
these circuits.
3. Use our degrees of design freedom to equate termsin the
general form to the terms of the desiredpower loss ratio
polynomial.