Insertion Loss Method

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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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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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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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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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.

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Insertion Loss Method