EEE 194 RF 1

Stepped Impedance Low-Pass Filter• Relatively easy (believe that?) low-pass

implementation• Uses alternating very high and very low

characteristic impedance lines• Commonly called Hi-Z, Low-Z Filters• Electrical performance inferior to other

implementations so often used for filtering unwanted out-of-band signals

EEE 194 RF 2

Approximate Equivalent Circuits for Short Transmission line Sections

• Using Table 4-1, approximate equivalent circuits for a short length of transmission line with Hi-Z or Low-Z are found

EEE 194 RF 3

Approximate Equivalent Circuits for Short Transmission line Sections

• The equivalent circuits are:jX / 2 jX / 2

jB

XL=Zo βl

BC=Yo βl

T-Equialent circuit for transmission line sectionβ l << π / 2

Equialent circuit for small β l and large Zo

Equialent circuit for small β l and small Zo

EEE 194 RF 4

Approximate Equivalent Circuits for Short Transmission line Sections

• Series inductors of a low-pass prototype replaced with Hi-Z line sections (Zo= Zh)

• Shunt capacitors replaced with Low-Z line sections (Zo= Zl)

• Ratio Zh/Zl should be as high as possible

( )

( )

inductor

capacitor

g

h

l

g

LRl

Z

CZl

R

β

β

=

=

EEE 194 RF 5

Stepped Impedance Low-Pass Filter• Select the highest and lowest practical line

impedance; e.g. the highest and lowest line impedances could be 150 and 10 Ω, respectively

• For example, given the low-pass filter prototype, solve for the lengths of the microstriplines:

glowLn n Cn n

g high

RZl g ; l g

R Zβ β= =

EEE 194 RF 6

6th Order Low-Pass Filter Prototype

Stepped Impedance Implementation

Microstripline Layout of Filter

L1 L2

C2C1 C3

L3

Zo Zlow Zhigh ZoZlow ZlowZhigh Zhigh

l1 l2 l3 l4 l5 l6

Stepped Impedance Low-Pass Filter -Implementation

EEE 194 RF 7

Bandstop Filter• Require either maximum or minimal

impedance at center frequency fo

• Let line lengths l = λo /4• Let Ω = 1 cut-off frequency of the low-

pass prototype transformed into upper and lower cut-off frequencies of bandstopfilter via bandwidth factor :

( )1

2 2 2U LL

o o

sbwbf cot cot ; sbw

ω ωπ ω πω ω

− = = − =

EEE 194 RF 8

Bandstop Filter: Implementation1. Find the low-pass filter prototype2. The L’s and C’s replaced by open and short

circuit stubs, respectively as in Low-Pass filter design with

ZLn = (bf ) gn and YCn = (bf ) gn

3. Unit lengths of λo /4 are inserted and Kuroda’s Identities are used to convert all series stubs into shunt stubs

4. Denormalize the unit elements

EEE 194 RF 9

Coupled Filters: Bandpass• Even and Odd mode excitations resulting in

1 1Oe Oo

pe e po od

Z ; Zv C v C

= =

EEE 194 RF 10

Coupled Filters: Even & Odd Impedances

EEE 194 RF 11

Bandpass Filter Section

( ) ( ) ( ) ( )2 2 212in Oe Oo Oe OoZ Z Z Z Z cos l

sin lβ

β= − − +

EEE 194 RF 12

Bandpass Filter Section• According to Figure 5-47, the characteristic

bandpass filter performance attained when l = λ /4 or β l = π /2 .

EEE 194 RF 13

Bandpass Filter Section• The upper and lower frequencies are

( ) 11 21 2

Oe Oo,,

Oe Oo

Z Zl cos

Z Zβ θ − −

= = ± +

5th Order coupled line Bandpass Filter

EEE 194 RF 14

Bandpass Filter: Implementation1. Find the low-pass filter prototype2. Identify normalized bandwidth, uper, and lower

frequencies

• Allowing:

U L

O

BWω ω

ω−

=

0 1 1 11 11

1 1 12 22, i,i N ,N

O O O O N Ni i

BW BW BWJ ; J ; J

Z g g Z Z g gg gπ π π

+ +++

= = =

EEE 194 RF 15

Bandpass Filter: Implementation• This allows determination of the odd and

even characteristic line impedances:

• Indices i, i+1 refer to the overlapping elements and ZO is impedance at ends of the filter structure

( )

( )

21 11

21 11

1

and

1

Oo O O i,i O i,ii,i

Oe O O i,i O i,ii,i

Z Z Z J Z J

Z Z Z J Z J

+ ++

+ ++

= − +

= + +

EEE 194 RF 16

Bandpass Filter: Implementation• Determine line dimensions and S and W of

the coupled lines from the graph on Figure 5-45 p256.

• Length of each coupled line segment at the center frequency is λ /4.

• Normalized frequency is

c c

U L c

ω ωωω ω ω ω

Ω = − −