High-Frequency Filter Design - haxr.org

High-Frequency Filter Design

Omar X. Avelar, Omar de la Mora & Diego I. Romero

RADIO FREQUENCY AND WIRELESS ELECTRONICS (ESI 039AA) Instituto Tecnológico y de Estudios Superiores de Occidente (ITESO) Departamento de Electrónica, Sistemas e Informática (DESI)

1. OBJECTIVES

To apply the insertion loss method to design a lumped prototype of a low-pass filter starting from its design specifications .

To implement an ideal lumped prototype of a low-pass filter as a realis-tic microstrip filter with open stubs using Richard’s transformation, Kuroda’s identities and Gupta’s formulas .

To implement a lumped prototype of a low-pass filter as a stepped im-pedance microstrip filter .

To observe the limitations of these microstrip filters using a high-fre-quency circuit simulator and a full-wave EM simulator, contrasting the main differences in filter responses.

To get familiar with the main characteristics of the high-frequency cir-cuit simulator APLAC

To get familiar with the main characteristics of the full-wave EM simula-tor Sonnet

2. COMPONENTS AND INSTRUMENTATION A numerical software tool: Matlab, Octave, Scilab, or something similar. The student version of the high- frequency circuit simulator APLAC. The

EM simulator Sonnet Lite.

3. THEORETICAL PROCEDURE1. Using the insertion loss method, design a low-pass filter prototype with

the following specifications: a) cutoff frequency at 5 GHz; b) minimum insertion loss of 20 dB at 8 GHz; c) Chebyshev response with ripple of 0.5 dB; d) reference impedance of 50 Ω.

2. Implement the filter prototype with ideal lumped components. Simulate the filter in APLAC, plotting |S11| and |S21| from 1 MHz to 10 GHz in dB as well as in linear scale. Check that the filter satisfies all the design specifications.

3. Using Richard’s transformation, implement the filter prototype with ide-al lossless transmission lines on air (εr = 1). Using Kuroda’s identities, transform the filter such that only shunt-connected open stubs are used. Simulate the filter in APLAC, plotting |S11| and |S21| from 1 MHz to 10 GHz in dB as well as in linear scale. Check that the filter satisfies all the design specifications.

4. Neglecting losses and metal thickness, implement the previous filter in microstrip technology using Gupta’s formulas [ 1 ]. All microstrip lines are on a dielectric substrate with relative dielectric constant εr = 4.12. The substrate height is H = 0.25 mm. Simulate the filter in Sonnet Lite, plotting |S11| and |S21| from 1 MHz to 10 GHz in dB as well as in lin-ear scale. Use a sufficiently low resolution such that you do not exceed the 16 MB limit of this version of the EM simulator. Compare in the same figure APLAC’s responses with ideal transmission lines (from pre-vious step) and Sonnet’s responses, for both |S11| and |S21| (in two separated figures).

5. Simulate again the previous filter in Sonnet Lite, but now considering that the loss tangent of the dielectric substrate is tan δd = 0.01, and that all metallic traces (including the ground plane) use copper with conduc-tivity σCu = 5.8×107 S/m. Plot |S11| and |S21| from 1 MHz to 10 GHz in dB as well as in linear scale. Compare in the same figure APLAC’s re-sponse with ideal transmission lines (from step 3) and new Sonnet’s re-sponses, for both |S11| and |S21| (in two separated figures).

6. Using the stepped impedance technique, implement the original filter prototype (from Step 1) with ideal lossless transmission lines on air (εr = 1) using a HighZ-LowZ configuration. Simulate the filter in APLAC, plotting |S11| and |S21| from 1 MHz to 10 GHz in dB as well as in lin-ear scale. Check that the filter satisfies all the design specifications.

7. Neglecting losses and metal thickness, implement the previous stepped impedance filter in microstrip Simulate again the previous filter in Son-net Lite, but now considering that dielectric losses (tan δd =

8. 0.01) and metallic losses (σCu = 5.8×107 S/m). Plot |S11| and |S21| from 1 MHz to 10 GHz in dB as well as in linear scale. Compare in the same figure APLAC’s responses with ideal transmission lines (from step 6) and new Sonnet’s responses, for both |S11| and |S21| (in two sepa-rated figures). technology using Gupta’s formulas [1 ]. Use the same di-electric substrate as before (εr = 4.12, H = 0.25 mm). Simulate the filter in Sonnet Lite, plotting |S11| and |S21| from 1 MHz to 10 GHz in dB as well as in linear scale. Compare in the same figure APLAC’s responses with ideal transmission lines (from previous step) and Sonnet’s respons-es, for both |S11| and |S21| (in two separated figures).

MAY 2010 1

ITESO High-Frequency Filter DesignBy: Omar X. Avelar, Omar de la Mora & Diego I. Romero

4. DESIGN & SIMULATIONSWe start to calculate the low-pass filter prototype by choosing the order based on the specifications of (1) in the Theoretical Procedure section by using the fol-lowing figure (Fig. 1).

Since , ww c

−1=0.6 we zoom in the previous table and locate that point.

(Fig. 2)

Which means that by choosing an order greater than 4 we cover the minimum insertion loss of 20 dB at 8 GHz.

We will choose a fifth order filter to keep the last impedance the same as the characteristic impedance. Fig. 4 displays the table of a Chebyshev prototype val-ues with 0.5 dB [2].

As the circuit components are defined as:

for a capacitor: gkw cZ o

and for an inductor: Zo g kw c

The following MATLAB / GNU Octave script calculates the components for both prototype topologies (source series and shunt source) with a characteristic im-pedance o Z o f 50-Ω.

% Low-Pass Filter Prototype % a) Cutoff frequency at 5GHz % b) Minimum Insertion Loss of 20dB at 8GHz % c) Chebyshev response with ripple of 0.5dB % d) Reference impedance 50ohms

clc; clear all; close all;

fc = 5e9; Zo = 50; ILmin = 20; f_IL = 8e9; epsilon_r = 1; c = 0.3e9; lambda = c/(sqrt(epsilon_r)*fc); lambda_8 = lambda/8;

N = 5; % Filter Order g = [1.7058 1.2296 2.5408 1.2296 1.7058 1]; % Chebyshev Coefficients

% Ideal Lumped Components Series Source

C1 = g(1)/(2*pi*fc*Zo); L2 = (Zo*g(2))/(2*pi*fc); C3 = g(3)/(2*pi*fc*Zo); L4 = (Zo*g(4))/(2*pi*fc); C5 = g(5)/(2*pi*fc*Zo);

% Ideal Lumped Components Shunt Source

L1 = (Zo*g(1))/(2*pi*fc); C2 = g(2)/(2*pi*fc*Zo); L3 = (Zo*g(3))/(2*pi*fc); C4 = g(4)/(2*pi*fc*Zo); L5 = (Zo*g(5))/(2*pi*fc);

Which values end up to the following circuits (Fig. 5 and Fig. 6)

MAY 2010 2

Fig. 1: Attenuation in Chebyshev filter with 0.5dB ripple of n-th order. (D.M. Pozar, Microwave Engineering, Wiley, 2005).

Fig. 2: Zoomed in 0.6.

Fig. 4: Normalized Chebyshev element values, 0.5 dB ripple.

Fig. 5: Shunt-source prototype.

2 . 7 1 4 9 n H

7 8 2 . 7 9 f F

5 0 5 0

7 8 2 . 7 9 f F

S w e e p " S - P a r a m e t e r A n a l y s i s "$ # p t s t y p e r a n g eL O O P 1 0 0 0 F R E Q L I N 1 M E G H z 1 0 G H zW I N D O W = 0 g r i d Y " " " " 0 1W I N D O W = 1 g r i d Y " " " " - 8 0 0

S h o w W = 0 Y = M a g ( S ( 2 , 1 ) )+ Y = M a g ( S ( 1 , 1 ) )

S h o w W = 1 Y = M a g d B ( S ( 2 , 1 ) )+ Y = M a g d B ( S ( 1 , 1 ) )E n d S w e e p

2 . 7 1 4 9 n H 4 . 0 4 3 8 n H

Fig. 6: Series-source prototype.

5 0 5 0

1 . 0 8 5 9 p F 1 . 0 8 5 9 p F

1 . 9 5 7 0 n H 1 . 9 5 7 0 n H

1 . 6 1 7 5 p F


S h o w W = 0 Y = M a g ( S ( 2 , 1 ) )+ Y = M a g ( S ( 1 , 1 ) )


Fig. 3: Low-pass prototype.

Order g1 g2 g3 g4 g5 g6 g7

2 1.4029 0.7071 1.9841

3 1.5963 1.0967 1.5963 1

4 1.6704 1.1926 2.3662 0.8419 1.9801

5 1.7058 1.2296 2.5409 1.2296 1.7058 1

6 1.7254 1.2478 2.6064 1.3136 2.4759 0.8696 1.9841


By utilizing APLAC HF Simulator, we get the following curves (Fig. 7 & Fig. 8).

We can appreciate there is no difference in a lumped elements simulation and hence choose the series source topology as it becomes easier to implement.

The design characteristics are therefore also met by both topologies.

We can appreciate the -0.5 dB cutoff point in the transmission |S21|, which was expected to be 5 Ghz (Fig. 9).

And an attenuation greater than 20dB at 8GHz is also satisfied.

MAY 2010 3

Fig. 7: (Left Column)Series-source topology

Top: Linear plot of |S11| and |S21|.Bottom: Semi-logarithmic plot of |S11| and |S21|.

Fig. 8: (Right Column)Shunt-source topology

Top: Linear plot of |S11| and |S21|.Bottom: Semi-logarithmic plot of |S11| and |S21|.

Fig. 9: Filter characteristics.


The next step is to change the filter with lossless transmission lines, by using Richard's transformation on air (εr = 1).

By replacing the capacitors as transmission lines ended as open circuits, and in-ductors as transmission lines ended in short-circuit.

Whose length is

8, to normalize the frequency.

We calculate the transmission lines characteristic impedances by running the fol-lowing MATLAB / GNU Octave (having followed this report and the previous workspace open).

% Richard's Transformation Series Source

Zo1_series = Zo/g(1); Zo2_series = Zo*g(2); Zo3_series = Zo/g(3); Zo4_series = Zo*g(4); Zo5_series = Zo/g(5);

And now we apply Kuroda's identities (Fig. 11) to replace the transmission lines so that only shunt-connected open stubs are used.

Two of the four different identities were used.

MAY 2010 4

Fig. 10: Richard's transformation summarized.

Fig. 11: An example of Kuroda's identities conversion.

Fig. 13: Kuroda's Identitiy (B).Series inductor to shunt capacitor.

Fig. 12: Kuroda's Identitity AShunt capacitor to series inductor.


We start by applying Kuroda's Identity A to the input and output ( 1 /g1 and

1 /g5 ) in parallel with the normalized input and output transmission lines (Fig. 14).

This was done to make it possible to add a transmission line to the inductor formed by g2 and g4 (Fig. 15).

After this is done to both sides and applying Kurodaś Identity B we end up with-the circuit shown in Fig. 16.

Z g1=129.3118 Z g2=81.5212 Z g3=24.0329Z g4=79.9588 Z g5=19.6788

And simulating this circuit in APLAC gives us the following transmission and re-flection plots (Fig. 17).

According to such simulation plots (Fig. 17), the cut-off frequency remains at 5 GHz, with greater than 20 dB attenuation at 8 GHz, 0.5 dB ripple and connected to a reference system of 50-Ω.

One thing to notice is that this implementation has a higher roll-off than the one with lumped components.

MAY 2010 5

Fig. 16: Transmission line implemented filter with shunt connections only after Kuroda's Identities.

5 0 5 0

Z = 1 2 9 . 3 1 1 8L E N G T H = 7 . 5 m m

Z = 8 1 . 5 2 1 2L E N G T H = 7 . 5 m m

Z = 2 4 . 0 3 2 9L E N G T H = 7 . 5 m m

Z = 7 9 . 9 5 8 8L E N G T H = 7 . 5 m m

Z = 7 9 . 9 5 8 8L E N G T H = 7 . 5 m m

Z = 1 9 . 6 7 8 8L E N G T H = 7 . 5 m m

Z = 2 4 . 0 3 2 9L E N G T H = 7 . 5 m m

Z = 8 1 . 5 2 1 2L E N G T H = 7 . 5 m m

Z = 1 2 9 . 3 1 1 8L E N G T H = 7 . 5 m m


$ S h o w W = 0 Y = M a g ( S ( 2 , 1 ) )$ + Y = M a g ( S ( 1 , 1 ) )


Fig. 14: First step.

Fig. 15: Second step.

Fig. 17: Transmission Lines – After Richard's and Kuroda's TransformationsTop: Linear plot of |S11| and |S21|.

Bottom: Semi-logarithmic plot of |S11| and |S21|.


The next step is to implement such design in microstrip technology, at first ne-glecting losses and metal thickness during the simulation.

Using APLAC microstrip calculator we fill the following table (Fig. 18) with their corresponding physical lenghts by taking into account εr = 4.12 and the sub-strate height of H = 0.25 mm.

And proceed to draw and implement the circuit on Sonnet Lite (Fig. 19).

The electro-magnetic simulation gives us the following transmission and reflec-tion plots.

And for easier comparison we look at the simulation from the lossless transmis-sion lines from APLAC (Fig. 21).

The bandwidth is identical and remains at 5 GHz.

An easy to spot difference would be the retransmission bump at 9.5 GHz in the microstrip implementation (Fig. 20) compared to the lossless transmission line but this is considered as noise floor in the numerical calculations of the software as it is in the range of -90 dB approximately.

MAY 2010 6

Fig. 18: Microstrip parameters for theRichard's – Kuroda's transformations.

Width[m] Length [m]

Z1 129.3118 5.45E-05 2.8 3.59E-02 4.482107E-03

Z2 81.5212 1.98E-04 2.95 3.49E-02 4.366669E-03

Z3 24.0329 1.43E-03 3.46 3.23E-02 4.032025E-03

Z4 79.9588 2.07E-04 2.96 3.49E-02 4.359286E-03

Z5 19.6788 1.84E-03 3.54 3.19E-02 3.986205E-03

Z [Ω] εe λ

Fig. 19: Physical implementation of a filter using Richard's – Kuroda's transformations in Sonnet Lite.

Fig. 20: Lossless microstrip filterTop: Linear plot of |S11| (Blue) and |S21| (Red).

Bottom: Semi-logarithmic plot of |S11| (Blue) and |S21| (Red).




To compare the results even further, we will now take into account the loss tan-gent of the dielectric substrate ( d=0.01 ) and all the metallic traces with

cooper conductivity cu5.8x107 S /m .

Both can be found as Sonnet Lite simulation parameters. The simulation plots of such settings are in Fig. 22.

The loss tangent and metal losses produced almost no radical changes in the electro-magnetic simulation.

The bandwidth is reduced from 5 GHz to 4.983 GHz, and the attenuation at 8 GHz is higher in the lossy simulation; 83.14 dB vs 76.75 dB. It can also be seen that the transmission only reaches 1 (linear) at relatively low frequency and only ap-proaches 1 after that and never reaching it.

MAY 2010 7

Fig. 22: Dielectric and metal losses on the microstrip filterTop: Linear plot of |S11| (Blue) and |S21| (Red).


Fig. 23: Lossless microstrip filterTop: Linear plot of |S11| (Blue) and |S21| (Red).





The prototype done in step 1 (Fig. 25) will be implemented as a stepped imped-ance filter, where:

• Series inductors are replaced by high-impedance transmission lines (Z high ).

• Parallell capacitors are replacedby low-impedance transmission lines (Z low ).

• Lengths of the transmission line sections are calculated:

l=g k RoZ high

(for an inductor).

l=g k Z lowRo

(for a capacitor).

The calculations are summarized in the following table (Fig. 26).

Where according to the schematic:

Z_low = g1=g 5Z_high = g2=g4Z_low = g3

Drawing this in APLAC results in the following schematic (Fig. 27).

And simulating such implementation in APLAC, we obtain the following plots.

The roll-off of such implementation is way less steep than the Richard's – Kuro-da's transformation from lumped components.

But the 20 dB of attenuation at 8 GHz still comply with our design requirements.

MAY 2010 8

Fig. 26: Impedance calculations for thestepped impedance filter.

g Length [m]

1.7058 0.34116 1.682340E-03

1.2296 0.40987 2.351658E-03

2.5408 0.50816 2.505857E-03

βl

Zlow

Zhigh

Zlow

Fig. 25: Low-pass prototype.

Fig. 27: Stepped impedance filter.

5 0 5 0L E N G T H = 3 . 2 5 7 8 m mZ = 1 0

Z = 1 5 0L E N G T H = 3 . 9 1 3 9 m m

L E N G T H = 4 . 8 5 2 6 m mZ = 1 0

Z = 1 5 0L E N G T H = 3 . 9 1 3 9 m m

L E N G T H = 3 . 2 5 7 8 m mZ = 1 0


$ S h o w W = 0 Y = M a g ( S ( 2 , 1 ) )$ + Y = M a g ( S ( 1 , 1 ) )


Z = 5 0L E N G T H = 1 0 m m

Z = 5 0L E N G T H = 1 0 m m

Fig. 28: Stepped Impedance with Transmission LinesTop: Linear plot of |S11| and |S21|.


Z [Ω] Width[m] εe λ β

Zo 50 5.02E-04 3.15 3.38E-02 185.86

Zlow 10 4.06E-03 3.75 3.10E-02 202.79


To implement the filter in microstrip technology we also make use of APLAC mi-crostrip calculator and fill the following table (Fig. 28).

And the implementation using Sonet Lite (Fig. 30) with the following parameters: εr = 4.12 and the substrate height of H = 0.25 mm.

And by simulating it in Sonnet Lite, we obtain the following plots (Fig. 31).

Comparing Sonnet Lite (Fig. 31) vs APLAC (Fig. 32) simulations we can observe that they are practically the same with no radical changes between them.

MAY 2010 9

Fig. 29: Microstrip calculations for the stepped impedance filter.

Z [Ω] Width[m] Norm. Width [mm]

50 5.02E-04 3.15 3.38E-02 185.86 0.50229

10 4.06E-03 3.75 3.10E-02 202.79 4.06000

150 3.13E-05 2.77 3.61E-02 174.29 0.03134

εe λ β

Zo

Zlow

Zhigh

Fig. 30: Physical implementation of a filter using stepped impedance in Sonnet Lite.

Fig. 31: Lossless microstrip filter (Stepped Impedance)Top: Linear plot of |S11| (Blue) and |S21| (Red).





And finally we will now take into account the loss tangent of the dielectric sub-strate ( d=0.01 ) and all the metallic traces with cooper conductivity

cu5.8x107 S /m .

The dielectric and metal losses simulation has a notable overall attenuation dur-ing the pass-band. It only reaches 1 (linear scale) at practically DC (Fig. 33).

The reflection is even worse at this point (considering that the reflection coeffi-cient is higher than the Richard's – Kuroda's transformation implementation.

The cut-off frequency remains almost the same in the three cases. And the atten-uation at 8 GHz remains higher than 20 dB.

MAY 2010 10

Fig. 33: Dielectric and metal losses (Stepped Impedance)Top: Linear plot of |S11| (Blue) and |S21| (Red).


Fig. 34: Lossless microstrip filter (Stepped Impedance)Top: Linear plot of |S11| (Blue) and |S21| (Red).





5. CONCLUSIONSThe design of filters by utilizing prototypes is so convenient. It can be seen that the method of pysical implementation of high frequency filters can greatly deter-mine subtle but particular characteristics that can be used as an advantage in the design.

Both designs complied with our requirments but the Richard's – Kuroda transfor-mmed filter had a steeper roll-off and was more robust to dielectric and metal losses.

A bigger window in the filter response simulations clearly shows the cyclic be-havior of the filters. (Fig. 36) even if the base design was merely a low-pass.

Electro-magnetic simulations software such as APLAC is very powerful once you get the hang on it. It is of great recommendation to start working and simulating with various kind of software to compare your results and make sure you are go-ing in the right direction of the design.

6. REFERENCES[ 1 ] K. C. Gupta, R. Garg and R. Chadha, Computer-Aided Design of Mi-crowave Circuits. Norwood, MA: Artech, 1981.

[ 2 ] RF Cafe – Chebyshev Filter Prototype Values, 1989. [Link]

[ 3 ] Microwave Fiters, D. B. Leeson, 1994-1999. [Link]

MAY 2010 11

Fig. 36: Behavior repeating at four times the cut-off frequency.

http://www.rfcafe.com/references/electrical/cheby-proto-values.odt

http://home.sandiego.edu/~ekim/e194rfs01/filterek.pdf

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