Microwave Filters

Prof. D. Kannadassan,

School of Electronics Engineering

ECE102 - Microwave Engineering Fall 2012_13 1

Microwave Filters - Implementation

Richard’s Transformation

Kuroda’s Identities

General Design Procedure

N, fc, MF or ER,

IL at pass band

and Stop band

To LP,

BP, HP,

BS filters

Maximally Flat Low-pass filter – Prototype design

• Consider a two-element (2nd order) low-pass filter

• The input impedance:

• And the input reflection coefficient:

• And, Power loss ratio

• From both equation,

• Now compare this equation with “Maximally flat response” with ωc=1

• By simple algebra, [available in Pozar- pp. 392-393]

• This same procedure has been extended to calculate for higher order filters and the

lumped elements (normalized to source impedance) are tabulated.

• Two possible approached

1. First element is capacitor

2. First element is Inductor

C

IL

N

10

2010

log2

110log

ECE102 - Microwave Engineering Fall 2012_13

7

Equal Ripple LPF - prototype

• For Equal Ripple following specifications are important

– Stop band IL (dB) - IL

– Ripple Magnitude (dB) - R

– Cut-off frequency- fc

• From these, the order of the filter can be calculated:

C

R

IL

N

1

20

201

cosh

110

110cosh

Example

• Solution:

– Actually we have to calculate the order of the filter

• IL=20dB at 11GHz, fC=8GHz

C

IL

N

10

2010

log2

110log

4

44.3

811log2

110log

10

20

20

10

N

N

ECE102 - Microwave Engineering Fall 2012_13

10

Scaling of fC and Z0

• Scaling of cut-off frequency and source impendence is given as

• For capacitance:

• For Inductance

C

kk

LRL

0'

C

kk

R

CC

0'

Example

• Design a maximally flat low-pass filter with a cut off frequency of 2 GHz,

impedance of 50 Ω, and at least 30 dB insertion loss at 3 GHz.

• Solution:

• Order of the filter for the given specification

521.4

log2

110log

10

2010

C

IL

N

Filter Implementation

• Lumped elements can’t be suitable for

Microwave Frequencies, because:

– Lumped elements loose it original value at

higher frequencies

– The sizes are comparable with wave length

of operation which will give spurious

response

– Accurate designs are much complicated.

• For these reasons, and more, we are going

for “Equivalent Transmission Line design”

• Two Implementation procedures:

– Richard’s Transformation and Kuroda’s Identities

– Stepped Impedance method

Richard’s Transformation

• The frequency transformation: (ω to Ω domain)

For example

Z0=L Z0=L

Z0=1/C Zero length

Kuroda’s Identities

• These identities are introduced to reduce the practical difficulties in circuit

implementations

• Physically separate transmission line stubs

• Transform series stubs into shunt stubs, or vice versa

• Change impractical characteristic impedances into more realizable ones

Example

• Design a third order LPF with equal Ripple of 3dB at the pass band.

g1=3.3487=L1

g2=0.7117=C2

g3=3.3487=L3

• Prototype Structure

• Apply Richard’s Transformation:

• Now,

• In order to use “Kuroda’s identities”, we need series lines. (Don’t confuse the small

series line, they are actually to separate the stubs, have zero length)

• Adding “Unit line of λ/8 length” – will give series line and improve the perforance

Kuroda’s identities Next step

n2=1+Z2/Z1=1.299

Finally:

Multiply the lines with Z0

Typical

Problem-1 • Design a LPF with only shunt stubs, for the cut-off frequency of 10GHz. Calculate

the insertion loss at 13GHz. Assume the normalized filter elements as follow:

Problem-2

Problem-3