Microwave Filters

1

Filter and it’s requirements

• Filter is a two port network, used to control the frequency response of the system.

• For a band of frequencies, the gain is high, called “Pass band”.

• In other part of band, it attenuates, called “Stop band”

• Ideally, A filter requires :

– Zero Insertion loss in pass band

– Infinite insertion loss at stop band

2

Design methods

• So far, at microwave frequencies, three possible design procedures been

followed

• Periodic structures

• Image parameter method

– By providing cut off frequencies and available component values, a simple design of

filter is possible. But we can’t control the pass or stop band attenuation. Once we

designed for a frequency, it can’t be explored to another frequencies. Since the inaccurate

lumped elements been utilized, the quality of design limited

– Constant-k filter

– m-derived filter

• Insertion loss method

3

Insertion loss method

• While older methods are forward design (from specification to

observation), insertion loss method provides a reverse engineering method

• Following advantages

– Design for the required “Frequency response” with right specifications

– Flexible with design frequencies/characteristics impedance

– Trade off on design and implementation complexity

– Easier

• In this method, the power loss is fundamental specification:

• Power loss ratio PLR = Power available from source

Power delivered to load

can be expanded in polynomial

series in ω2

4

As:

or

• Based on the selection of M and N, there are four possible design type been

explored, so far.

• Namely:

– Maximally flat or Butterworth or Binomial Filter

– Equal Ripple or Chebyshev Filter

– Elliptical Filter

– Linear Phase Filter

5

Maximally flat or Butterworth or Binomial Filter

• The prototype low pass response is defined as (using Binomial series)

• N is order of the filter, ωC is cut off frequency (angular)

6

Equal Ripple or Chebyshev Filter

• The prototype low pass response is defined as (using chebyshev polynomial)

• TN(x) is chebyshev polynomial, N is order of the filter

7

Butterworth Vs Chebyshev

8

General Design Procedure

N, fc, MF or ER,

IL at pass band

and Stop band

To LP,

BP, HP,

BS filters

9

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

• The input impedance:

• And the input reflection coefficient:

• And, Power loss ratio

Maximally Flat Low-pass filter – Prototype design

10

• From both equation,

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

with cut-off frequency)

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

11

• 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

12

C

IL

N

10

2010

log2

110log

13

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

10

101

cosh

110

110cosh

14

15

Example

• Solution:

– Actually we have to calculate the order of the filter

• IL=20dB at 11GHz, fC=8GHz

C

IL

N

10

1010

log2

110log

4

44.3

811log2

110log

10

10

20

10

N

N

16

C

R

IL

N

1

20

201

cosh

110

110cosh

Assume R = 0.5 dB

N = 5.5 … that’s “6”

For chebyshev response: