ELEC344, Kevin Chen, HKUST 1

Lecture 28: m-Derived ππππ Filter Section

Infinite cascade of m-derived T-section.

A de-embedded π-equivalent.

The T-section still havethe problem of anonconstant imageimpedances.

Now consider the π-equivalent as a piece of aninfinite cascade of m-derived T-sections. Then,

( )20

22121

21

/1

4/)1(

''

c

iTi

R

mZZZ

Z

ZZZ

ωω

π

−

−+=

=

ELEC344, Kevin Chen, HKUST 2

( )( )2

022

/1

/)1(1

c

ci

RmZ

ωωωω

π−

−−=

2021 / RCLZZ == ( )2

020

2221 /4 ωωω RLZ −=−=Since and

We have

* m provides another freedom to design Ziπ so that we canminimize the variation of Ziπ over the passband of the filter.

Variation of Ziπ in the passband of a low-pass m-derivedsection for various values ofm. A value of m=0.6generally gives the bestresults --- nearly constantimpedance match to andfrom R0.

ELEC344, Kevin Chen, HKUST 3

How to match the constant-k and m-derived T-sections to π-section?

Using bisected π-section.

It can be shown that

iTi ZZ

ZZZ =+=4

'''

21

211

πiiT

i ZZ

ZZ

ZZ

ZZZ ==

+= 21

21

212

''

'4/'1

''

ELEC344, Kevin Chen, HKUST 4

Composite Filters

• The sharp-cutoff section, with m< 0.6, places an attenuation polenear the cutoff frequency to provide a sharp attenuation response.

• The constant-k section provides high attenuation further into thestopband.

• The bisected-π sections at the ends match the nominal source andload impedance, R0, to the internal image impedances.

• The composite filter design is obtained from two parameters: cutofffrequency, impedance, and infinite attenuation frequency.

ELEC344, Kevin Chen, HKUST 5

Example 8.2 of Pozar: Low-Pass Composite Filter Design

Cutofffreq.

The series pairs of inductorsbetween the sections can becombined. The self-resonance of the bisected p-section will provideadditional attenuation.

Frequency response

ELEC344, Kevin Chen, HKUST 6

Filter Design By the Insertion Loss Method

What is a perfect filter?

Zero insertion loss in the passband, infinite attenuation in thestopband, and linear phase response (to avoid) signaldistortion) in the passband.

Not perfect filters exist, so compromises need to be made.

The image parameter method have very limited freedom tonimble around.

The insertion loss method allows a high degree of control overthe passband and stopband amplitude and phase characteristics,with a systematic way to synthesize a desired response.

ELEC344, Kevin Chen, HKUST 7

Characterization by Power Loss Ratio

2)(1

1

load todeliveredPower

source from availablePower

ωΓ−===

load

incLR P

PP

The power loss ratio (insertion loss) of a filter is defined as:

When both load and source are matched, we have

LRPIL log10=Since |Γ(ω)|2 is an even function of ω, it can be expressed as apolynomial in ω2.

)()(

)()(

22

22

ωωωω

NM

M

+=Γ

Where M and N are real polynomials in ω2. So the power lossratio can be given as

)(

)(1

2

2

ωω

N

MPLR +=

ELEC344, Kevin Chen, HKUST 8

Several Types of Filter Response:

Maximally flat: binomial or Butterworth response

Provide the flattest possible passband response. For a low-passfilter, it is specified by N

cLR kP

2

21

+=

ωω

Where N is the order of the filter, and ωc is the cutoff frequency.

At the band edge the power loss ratio is 1+k2. Maximally flatmeans that the the first (2N-1) derivatives of the power lossratio are zero at ω = 0.

ELEC344, Kevin Chen, HKUST 9