report on basics of microwave filter

8/7/2019 report on basics of microwave filter

1/39

AIET/DOECE/2010-11/PTS/1

1.0 Introductionto Filter:

A filter is a network that provides perfect transmission for signal with frequencies in

certain passband region and infinite attenuation in the stopband regions. Such ideal

characteristics cannot be attained, and the goal of filter design is to approximate the ideal

requirements to within an acceptable tolerance. Filters are used in all frequency ranges

and are categorized into three main groups:

y Low-pass filter (LPF) that transmit all signals between DC and some upper limit [c

and attenuate all signals with frequencies above [c.

yH

igh-pass filter (H

PF) that pass all signal with frequencies above the cutoff value [cand reject signal with frequencies below [c.

y Band-pass filter (BPF) that passes signal with frequencies in the range of[1 to [2 and

reject frequencies outside this range. The complement to band-pass filter is the band-

reject or band-stop filter.

In each of these categories the filter can be further divided into active and passive type.

The output power of passive filter will always be less than the input power while active

filter allows power gain. In this lab we will only discuss passive filter. The characteristic

of a passive filter can be described using the transfer function approach or the attenuation

function approach. In low frequency circuit the transfer function (H([)) description is

used while at microwave frequency the attenuation function description is preferred.

Figure 1.1a to Figure 1.1c show the characteristics of the three filter categories. Note that

the characteristics shown are for passive filter.


2/39


Figure 1.1A A low-pass filter frequency response.

Figure 1.1B A high-pass filter frequency response.

Figure 1.1C A band-pass filter frequency response.

A Filter

H([)V1([) V2([)

[[

[1

2

V

VH !

!

1

21020

V

VLognAttenuatio

[c

|H([)|

[

1Transfer

function

Attenuation/dB

[

0

[c

3

10

20

30

40

[

Attenuation/dB

0

[c

3

10

20

30

40[c

|H([)|

[

1 Transfer

function

[1

|H([)|

[

1 Transfer

function

[2

[

Attenuation/dB

0

[1

3

10

20

30

40 [2


3/39


2.0 RealizationofFilters:

At frequency below 1.0GHz, filters are usually implemented using lumped elements such

as resistors, inductors and capacitors. For active filters, operational amplifier is

sometimes used. There are essentially two low-frequency filter syntheses techniques in

common use. These are referred to as the image-parameter method (IPM) and the

insertion-loss method (ILM). The image-parameter method provides a relatively simple

filter design approach but has the disadvantage that an arbitrary frequency response

cannot be incorporated into the design. The IPM approach divides a filter into a cascade

of two-port networks, and attempt to come up with the schematic of each two-port, such

that when combined, give the required frequency response. The insertion-loss method

begins with a complete specification of a physically realizable frequency characteristic,

and from this a suitable filter schematic is synthesized. Again we will ignore the image

parameter method and only concentrate on the insertion loss method, whose design

procedure is based on the attenuation response or insertion loss of a filter. The insertion

loss of a two-port network is given by:

211

loadtodeliveredPower

sourcethefromavailablePower

[+!!!

load

incI

P

PP (2.1)

Where + is the reflection coefficient looking into the filter (we assume no loss in the

filter).

Design of a filter using the insertion-loss approach usually begins by designing a

normalized low-pass prototype (LPP). The LPP is a low-pass filter with source and load

resistance of 1; and cutoff frequency of 1 Radian/s. Figure 2.1 shows the

characteristics. Impedance transformation and frequency scaling are then applied todenormalize the LPP and synthesize different type of filters with different cutoff

frequencies.


4/39


Figure2.1 A normalized LPP filter network with unity cutoff frequency (1Radian/s).

Low-pass prototype (LPP) filters have the form shown in Figure 2.2 (An

alternative network where the position of inductor and capacitor is interchanged is also

applicable). The network consists of reactive elements forming a ladder, usually knownas a ladder network. The order of the network corresponds to the number of reactive

elements. Impedance transformation and frequency scaling are then applied to transform

the network to non-unity cutoff frequency, non-unity source/load resistance and to other

types of filters such as high-pass, band-pass or band-stop. Examples of high-pass and

band-pass filter networks are shown in Figure 2.3 and Figure 2.4 respectively.

R

Figure2.2 Low-pass prototype using LC elements.

A Filter

H([)V1([) V2([)

RS =1

RL =1Attenuation/dB

[

0

[c = 1

3

10

20

30

40

L1=g2

L2=g4

C1=g1 C2=g3RL= gN+1

1

L1=g1 L2=g3

C1=g2 C2=g4RL= gN+1g0= 1


5/39


Figure2.3 Example of high-pass filter, note the position of inductor and capacitor isinterchanged as compared with low pass filter.

Figure 2.4 Example of band pass-filter, the capacitor is replaced with parallelL

Cnetwork while the inductor is replaced with series LC network.

3.0 BriefOverviewofLow-PassPrototype Filter Design Using

Lumped Elements:

are a number of standard approaches to design a normalized LPP of Figure 2.3 that

approximate an ideal low-pass filter response with cutoff frequency of unity. Among the

well known methods are:

y Maximally flat orButterworth function.

y Equal ripple or Chebyshev approach.

y Elliptic function.

We will not go into the details of each approach as many books have covered them.

which is a classic text on network analysis , a more advance version. The basic idea is to

approximate the ideal amplitude response |H([)|2 of an amplifier using polynomials such

as Butterworth, Chebyshev, Bessel and other orthogonal polynomial functions. This is

usually given as:

)(1)(

)()(

[[

[[

No

o

i

o

PC

K

V

VH

!!

L2L1

C1 CN

C2L2

L1C1 L3 C3

CNLN


6/39


Here Ko and Co are constants and PN([) is a polynomial in [ of order N. Ko and Co are

usually dependent on the type of polynomial used. A comparison of approximating the

LPP amplitude response with Butterworth, Bessel and Chebyshev polynomials is

illustrated in Figure 3.1.

0.1 1 1030

20

10

0

20 log HB [( )( )

20 log HC [( )( )

20 logH

b [( )( )

[

Figure 3.1 Amplitude response of fourth order (N=4) Butterworth, Chebyshev andBessel filters using (3.1).

Each approximation has its advantages and disadvantages, for instance the

Chebyshev approximation provide rapid cutoff beyond 1.0 radian/second. However the

user must compromise this with ripple in the pass band. The Bessel approximation has

the slowest cutoff rate, but this is offset with a favourable linear phase response, which

reduces phase distortion. A Butterworth approximation has a characteristic between the

two. A ladderLC network with the number of reactive elements corresponding to the

order of the polynomial PN in (3.1) is then compared with equation (3.1). The respective

inductance and capacitance of the reactive elements can then be obtained. An alternative

approach would be to synthesize the transfer function of (3.1) using standard techniques

as listed in references . It is suffice to say that for each approach, values of g1, g2, g3

gN for an Nth orderLPP have been tabulated by many authors .Here we will demonstrate

the design of a low-pass filter and a band-pass filter using the insertion-loss method and

Amplitude in

dB Bessel

Butterworth

Chebyshev

[


7/39


illustrate the implementation of the RLC lumped circuit using distributed elements such

as microstrip and stripline in microwave region.

We will use this table to design a LPP Butterworth filter. The values of gi correspond to

inductance and capacitance in the LPP Butterworth filter.

N g1 g2 g3 g4 g5 g6 g7 g8 g9

1 2.0000 1.0000

2 1.4142 1.4142 1.0000

3 1.0000 2.0000 1.0000 1.0000

4 0.7654 1.8478 1.8478 0.7654 1.0000

5 0.6180 1.6180 2.0000 1.6180 0.6180 1.0000

6 0.5176 1.4142 1.9318 1.9318 1.4142 0.5176 1.0000

7 0.4450 1.2470 1.8019 2.0000 1.8019 1.2470 0.4450 1.00008 0.3902 1.1111 1.6629 1.9615 1.9615 1.6629 1.1111 0.3902 1.0000

Table 3.1 Element values for Maximally flat (Butterworth) LPP (g0 = 1, [c =1).

4.0 Designinga LowPassPrototype (LPP):

We will now design a 4th

orderButterworth LPP and use this design for the rest of the

lab. The specification of the filter is as follows: RS = RL = 50;. Cutoff frequency fc =

1.5GHz or[c = 9.4248v109 rad/s.

Step 1 Designthe LPPfilterwith[c = 1 rad/s.

Using Table 3.1, the schematic of the LPP filter is as shown in Figure 4.1.

Figure 4.1 The 4th

orderButterworth LPP filter.

L1=0.7654H L2=1.8478H

C1=1.8478F C2=0.7654FRL= 1g0= 1

L1 = g1 = 0.7654H

L2 = g3 = 1.8478HC1 = g2 = 1.8478F

C2 = g4 = 0.7654F


8/39


Step2 Performimpedanceandfrequencyscaling:

The filter designed in Figure 4.1 supports load impedance of 1; and cutoff frequency of

1 radian/second. This filter can be converted into a low-pass filter, which meets arbitrary

cutoff frequency and impedance level specification using frequency scaling and

impedance transform. For a new load impedance of Ro and cutoff frequency of[o, the

original resistance Rn , inductance Ln and capacitance Cn are changed by the followings :

noRRR ! (4.1a)

o

no

[! (4.1b)

oo

n

R

[! (4.1c)

The transformation as shown in (4.1a) to (4.1c) implies that the schematic does not need

to be changed, only the element values are scaled down or up to reflect the new

specifications. Space does not permit us a detailed discussion of how equations (4.1a)-

(4.1c) achieve this. But a qualitative justification is as follows.

The transfer function of a linear two-port network is a function of the impedance

or admittance of the individual R, L and C in the network. This is because the transfer

function is derived using circuit theory rules (Kirchoffs voltage and current laws)

involving the impedance or admittance. Furthermore the numerator and denominator of

the transfer function involve combination of operations such as parallel of

impedance/admittance and addition of the impedance/admittance. These operations have

the characteristic that if each impedance/admittance is multiplied by a constant, the net

effect is equivalent to multiplying the total impedance/admittance by the constant. For

instance:

2121

21

2

21//// ZZA

ZZA

ZZAAZAZ !

! (4.2a)


9/39


2121 ZZAAZAZ ! (4.2b)

? A321321 //// ZZZAAZAZAZ ! (4.2c)

There is no non-linear operation such as square or cube of the impedance/admittance.

With this in mind the transfer function is written as:

),(

),(

21

21

n

n

ZZZD

ZZZNH

.

.![ (4.3)

If each impedance/admittance is multiplied by Ro:

[[ HZZZDR

ZZZNR

ZRZRZRD

ZRZRZRNH

no

no

nooo

nooo !!!),(

),(

),(

),(

21

21

21

21'

.

.

.

.(4.4)

However multiplying each impedance with Ro means we are scaling the impedance due

to each R, L and C by Ro as seen in the following:

nonoRoRRRRRZR !! (4.5a)

noonoo

j

j

ZR !!! [[ (4.5b)

o

n

o

oCoR

CC

R

Cj

CjRZR !

!!

[[

11(4.5c)

Frequency scaling is achieved by using the transformation

o

n [

[[ ! (4.6)

Suppose the impedance of an inductor is j[L. At [ = 1 the impedance is jL.

Another inductor with inductance L/[o will give similar impedance at [ = [o. Thus we

observe that the frequency response of the inductor is scaled by [o. Similarly if a


10/39


capacitor C is replace with capacitance C/[o, its frequency response is also scaled by [o.

The resistor being independent of frequency is not affected by frequency scaling.

Combining the frequency scaling and impedance scaling operation, one would arrive at

the equations (4.1a) to (4.1c).

Using the transformation (4.1a) to (4.1c) with Ro = 50; and [o = 2T(1.5v109) on

the schematic of Figure 4.1, the new schematic of the low-pass filter is shown in Figure

4.2 below.

Figure 4.2 The denormalized low-pass filter with cutoff frequency at 1.5GHz and

impedance of 50;.

5.0 Implementingthe Low-pass Filterusing Microstrip Line

Hi Z-Low Z Transmission Line Filter:

A relatively easy way to implement low-pass filters in microstrip or stripline is to use

alternating sections of high and low characteristic impedance (Zo) transmission lines.

Such filters are usually referred to as stepped-impedance filter and are popular because

they are easy to design and take up less space than similar low-pass filters using stubs.

However due to the approximation involved, the performance is not as good and is

limited to application where a sharp cutoff is not required (for instance in rejecting out-

of-band mixer products).

L1=4.061nH L2=9.803nH

C1=3.921pF C2=1.624pFRL= 50g0=1/50


11/39


A short length of transmission line of characteristic impedance Zo can be represented by

the equivalent symmetrical T network shown below :

Figure5.1 Equivalent T network for a transmission line with length l.

H

ere Z11 and Z12 are the Z parameters of the two port network. ljZZZ o cot2211 F!! (5.1a)

ljZZZ o cosec2112 F!! (5.1b)

and F is the propagation constant of the transmission line. For EM wave propagation that

is of TEM mode or quasi-TEM mode, the propagation constant can be approximated as:

oeoeo kIIIQ[F !$ (5.2)

where Ie is the effective dielectric constant of the transmission line structure. When Fl 1:

lZXH

F$ (5.4a)

0$B (5.4b)

Assuming a short length of transmission line (Fl


12/39


0$X (5.5a)

lZ

BL

1

F$ (5.5b)

Figure5.2 Approximate equivalent circuits for short section of transmission lines.

The ratio ZH/ZL should be as high as possible, limited by the practical values that can be

fabricated on a printed circuit board. Typical values are ZH

=100 to 150; and ZL

=10; to15;. Since a typical ow-pass filter consists of alternating series inductors and shunt

capacitors in a ladder configuration, we could implement the filter on a printed circuit

board by using alternating high and low characteristic impedance section transmission

lines. Using (5.4a) and (5.5b), the relationship between inductance and capacitance to the

transmission line length at the cutoff frequency [c are:

F

[

H

c

LZ

L

l ! (5.6a)

F

[ LcC

CZl ! (5.6b)

jX/2

jB

jX/2

X } ZoFl

B} YoFlWhen Zop 0Fl> 1

Fl


13/39


6.0 Designingwith Microstripline:

Cross section of microstrip and strip transmission line on printed circuit board (PCB) is

shown in Figure 6.1. For stripline the propagation mode is TEM since the conducting

trace is surrounded by similar dielectric material. Hence Ie = Ir, the dielectric constant of

the medium. For microstrip line the propagation mode is a combination of TM and TE

modes. This is due to the fact that the upper dielectric of a micostrip line is usually air

while the bottom dielectric is the printed circuit board dielectric. A TEM mode cannot be

supported as the phase velocities for electromagnetic waves in air and the PCB are

different, resulting in mismatch at the air-dielectric boundary. However at frequency of

6GHz or lower, the axial E and H fields are small enough that we can approximate the

propagation mode as TEM, hence the name quasi-TEM applies. For microstrip line the

effective dielectric constant Ie falls within the range 1 and Ir. At low frequency most of

the electromagnetic field is distributed in the air, while at high frequency the

electromagnetic field crowds towards the PCB dielectric. This result in the curve shown

in Figure 6.2, thus the microstrip line is dispersive.

Figure6.1 Cross section view of microstrip and strip transmission line as implementedon a printed circuit board.

Microstrip Line

Conducting trace (thickness = t)

Dielectric

Air

Ground Plane

H

W

Strip Line

Ir IrH


14/39


Figure6.2 Effective dielectric constant of microstrip and strip transmission line.

Strip line is a planar type of transmission line that lends itself well to microwave

integrated circuitry and photolithographic fabrication. The geometry of a stripline is

shown in figure. A thin conducting strip of width W is centered between two wide

conducting ground planes of separation b and the entire region between the ground plates

is filled with a dielectric. In practice, stripline is usually constructed by etching the center

conductor on a grounded substrate of thickness b/2, and the covering with another

grounded substrate of the same thickness.

Since stripline has two conductors and a homogeneous dielectric, it can support a

TEM wave, and this is the usual mode of operation. Like the parallel plate guide and the

coaxial lines, however, the stripline can also support higher order TM and TE modes, but

these are usually avoided in the practice. Intuitively one can think stripline as a sort of

flattened out coax both have a center conductor completely enclosed by an outer

conductor and are uniformly filled with a dielectric medium. A sketch of the field lines

for stripline is shown in figure the main difficulty we will have with stripline is that it

does not lend itself to a simple analysis, as did the transmission lines and waveguides.

Since we will be primarily concerned with the TEM mode of the stripline, an electrostatic

analysis is sufficient to give the propagation constant and characteristic impedance. An

exact solution ofLaplaces equation is possible by a conformal mapping approach, but

the procedure and results are cumbersome. Thus, we will present closed form expressions

that give good approximations to the exact results and then discuss an approximate

f

1

Ir

Ie

Microstrip Line Strip Line

f1

Ir

IeRegion where (6.1)applies.


15/39


numerical technique for solving Laplaces equation for a geometry similar to stripline;

this technique will also be applied to microstrip.

Formulas for propagation constant, characteristic impedance, and attenuation:

We know that the phase velocity of a TEM mode is given by

Thus the propagation constant of the stripline is

C = 3*108 m/sec is the speed of light in free space. The characteristic impedance of a

transmission line is given by


16/39


Where L and C are inductance and capacitance per unit length of the line. Thus, we can

find Zo if we know C. as mentioned above, Laplaces equation can be solved by

conformal mapping to find the capacitance per unit length of the stripline. The resulting

solution, however, involves complicated special functions,so for practical computations

simple formulas have been developed by curve fitting to the exact solution the resulting

formula for the characteristic impedance is

Where We is the effective width of the center conductor given by

These formulas assume a zero strip thickness, and are quoted as being accurate to about

1% of the exact results. It is seen that the characteristic impedance decreases as the strip

width W increases.

When designing stripline circuits, one usually needs to find the strip width, given

the characteristic impedance which requires the inverse of the formulas have beenderived as

Since stripline is a TEM type of line, the attenuation due to dielectric loss is of the

same form as that for other TEM lines. The attenuation due to conductor loss can be

found by the perturbation method or Wheelers incremental inductance rule. An

approximate result is


17/39


Filter design by the Insertion Loss method:

The perfect filter would have zero insertion loss in the pass band, infinite

attenuation in the stop band, and a linear phase response in the pass band. Of course, such

filters do not exist in practice, so compromises must be made; herein lies the art of the

filter design.

The image parameter method may yield a filter response, but if not there is no

clear-cut way to improve the design. The insertion loss method, however allows a high

degree of control over the pass band and stop band amplitude and phase characteristics,

with a systematic way to synthesize a desired response. The necessary design trade-offs

can be evaluated to best meet the application requirements. If, for example, a minimum

insertion loss is most important, a binomial response could be used; a Chebyshev

response would satisfy a requirement for the sharpest cutoff. If it is possible to sacrifice

the attenuation rate, a better phase response can be obtained by using a linear phase filter

design. And in all cases, the insertion loss method allows filter performance to be

improved in a straight forward, manner at the expense of a higher order filter. For the

filter prototypes to be discussed below, the order of the filter is equal to the number of

reactive elements.


18/39


6.1 Formulas for Effective Dielectric Constants and Characteristics

Impedance:

We will use the microstrip line to implement the low pass filter designed earlier.

Microstrip line is popular, as it is easily fabricated and low cost as compared to stripline.

There is no closed form solution for the propagation of electromagnetic wave along a

microstrip line. The solution for wave propagation is usually obtained through numerical

method. Parameters such as the effective dielectric constant, characteristic impedance

and line attenuation are then obtained from the numerical solution as a function of

frequency. Empirical formulas are obtained from the numerical solution by the methods

of curve fitting. Assuming the conductors and dielectric are lossless, and ignoring the

effect the conductor thickness t, an example of the empirical formulas forIe and Zo are

given by :

W

H

rr

e12

1

1

2

1

2

1

!II

I (6.1)

1or

444.1ln667.0393.1

120

1or4

8ln

60

"

e

!

H

W

H

W

H

W

W

HZ

H

W

H

W

r

r

o

I

T

I(6.2)

Zo and Ie as a function of W/d is plotted in Figure 6.3 using equations (6.1) and (6.2).The dielectric constant of the PCB dielectric is assumed to be 4.2 (for FR4).

1 2 3 4

6

1

11 121

2

3

4

6

2.

3

12.

2

Z

s( )

1

121 s

1 2 3 4

6

1

11 123

3.2

3.

3.

43.

31

3.

44

Ie s( )

121 s

Zo


19/39


Figure6.3 Zo and Ie versus W/H forIr= 4.2.

6.2Implementingthe 4th

Order Butterworth LowPass FilterusingStep

Impedance Microstrip Line:

Consider the schematic of Figure 4.2 again. The filter parameters are as follows:

y Cutoff frequency fc = 1.5GHz.

y Required ZL = 15;.

y Required ZH = 110;.

y L1=4.061nH, L2=9.083nH, C1=3.921pF, C2=1.624pF.

Implementation:

A typical FR4 fiberglass PCB with Ir= 4.2 and H = 1.5mm is used. From Figure 6.3 the

following trace parameters are obtained:

W/H H/mm W/mm IeZo = 15; 10.0 1.5 15.0 3.68Zo = 50; 2.0 1.5 3.0 3.21Zo = 110; 0.36 1.5 0.6 2.83

Table6.2 Dimension of various microstrip line characteristic impedance.

Therefore

19 307.60103356.32 !!! sfk ceLoeLL TIIF

19258.53103356.32

!!! sfk ceHoeHH TIIF

Using equations (5.6a) and (5.6b):

0.1 0.2 0.3 0.4 0.5 0.6 0.780

100

120

140

160

0 s( )

150

s

0.1 0.2 0.3 0.4 0.5 0.62.7

2.8

2.9

2.976

2.745

I e s( )

0.70.1 s

o


20/39


mmZ

Ll

HH

c 5.611 !!F

[

mmZC

lL

Lc 2.912 !!F

[

mml 0.153 !

mml 8.34 !

Figure6.4 The top view of the layout for the Low Pass Filter on the printed circuit

board.

7.1 Analysisofthestep-impedancelowpassfilterusing Agilent Advance

DesignSystem (ADS)software:

1. Log into the workstation.

2. Run the ADS version 2003A software (newer version may be used).

3. From the main window of ADS, create a new project folder named step_imp_LPF

under the directory D:\ads_user\default\ (Figure 7.1 and Figure 7.2).

l2l1

50; line 50; line

l4l3

0.6mm15.0mm

3.0mm

To 50;Load


21/39


Figure7.1 Opening a new project in ADS main window.

Figure7.2 The New Project dialog box.

4. The new schematic window will automatically appear once the project is properly

created. Otherwise you can manually create a new schematic window by double

clicking the Create New schematic button on the menu bar.

5. From the component palette drop-down list, set the component palette to TLines-

Microstrip. Draw the schematic as shown in Figure 7.5. The MSUB component is

the general substrate characteristics of the printed circuit board. The MLIN

components represent a short length of microstrip transmission lines used in our low


22/39


pass filter. Here MLIN1 corresponds to transmission line section 1, MLIN 2 to

transmission line section 2 and so forth (Figure 7.3 to Figure 7.5).

Figure 7.3 The Schematic Editor window of ADS (New version of ADS may beslightly different).

Figure7.4 Select the Tlines-Microstrip component palette from the Palette List.

Component Palette

Work Area

PaletteL

ist

Ground Node


23/39


Figure7.5 Insert the microstrip line component MLIN and substrate component MSUBinto the Work Area.

6. Set the characteristics of the substrate MSUB1 as to H = 1.5mm, T = 1.38mils

(typical), Er = 4.2 and Cond = 5.8E+07 (conductivity of copper). The rest of theparameters leave as default. The parameters dialog box forMSUB can be invoked by

doubling clicking on the MSUB component.

7. Set the characteristic W and L of each MLIN components according to the table ofSection 6.2.

8. Now change the component palette to Simulation-S_Param. Insert the

components S parameter simulation control S P and the termination network

Term into the schematics. The termination network components TERM1 and

TERM2 are actually a sinusoidal voltage source in series with an ideal series of

resistance as shown in the model during S parameter simulation. The S parameter

simulation control SP1 determines the start, stop and frequency stepping. Use the

Microstrip Line

Substrate

Component


24/39


wire to connect the components together and ground the outer terminals of the

TERM1 and TERM2 components (Figure 7.7).

Figure7.6 Select the S parameter component palette.

9. Set the parameters in SP1 to Start = 100MHz, Stop = 4GHz and Step = 10MHz. The

final schematic should be as shown in Figure 7.7. In Figure 7.7, since there is a step

discontinuity between the transmission line sections, this has to be modeled by

inserting a step element MSTEP at the junction between two transmission line

sections, this will make the simulated result more accurate.

Figure7.7 The final schematic for the low pass filter model.

To model step

discontinuity in

microstrip line


25/39


10.Finally run the simulation.

11.Invoke the data display window. Insert a Rectangular Plot component in the data

display.

12.Select the item to display as S21, with the dB option. The S21 represents the

attenuation from terminal 1 (input) to terminal 2 (output) of the filter as sinusoidal

signals from 100MHz to 4GHz are imposed.

13.Study the 3dB cut-off frequency of the low-pass filter. You can use the Marker

feature of the ADS display window to show the value of the attenuation at specific

frequency.m1freq=1.410GHzdB(S(2,1))=-3.051

0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0

-20

-15

-10

-5

-25

0

freq, GHz

dB

(S(2,1))

m1

Figure7.8 A sample result from the Data Display window of ADS, illustrating the S21

of the step-impedance low pass filter.

14.Adjust the parameter of TL1, TL2, TL3 and TL4 until the 3dB cutoff frequency is

within 100MHz of 1.5GHz. This can be done using the optimization feature of the

software. But as a start you can manually tune the width and length of each

transmission line section to achieve the desirable cut-off frequency at 1.5GHz.


26/39


LabProcedure:

Following the steps in Section 4 to Section 6, design a 4th

orderButterworth Low-Pass

Filter using ladderLC network with cut-off frequency at 1.8GHz. Show the steps of how

the inductance and capacitance in the network are determined from theL

ow-PassPrototype. Also show the conversion of the LC circuit into microstrip circuit, tabulating

the dimensions of each section of the transmission line. Upon completing the design,

simulate the frequency response of the low pass filter using HP ADS software, again

following the steps shown in Section 7. Use a frequency sweep from 100MHz to 5GHz,

with a step of 10MHz.

8.0 VARIOUS MICROWAVE FILTERS:

In general, most RF and microwave filters are most often made up of one or more

coupled resonators, and thus any technology that can be used to make resonators can also

be used to make filters. The unloaded quality factorof the resonators being used will

generally set the selectivity the filter can achieve

8.1 Lumped-element LC filters:

The simplest resonator structure that can be used in rf and microwave filters is an

LC tank circuit consisting of parallel or series inductors and capacitors. These have the

advantage of being very compact, but the low quality factor of the resonators leads to

relatively poor performance.

Lumped-Element LC filters have both an upper and lower frequency range. As the

frequency gets very low, into the low kHz to Hz range the size of the inductors used in

the tank circuit becomes prohibitively large. Very low frequency filters are often

designed with crystals to overcome this problem. As the frequency gets higher, into the

600 MHz and higher range, the inductors in the tank circuit become too small to be

practical. An inductor of 1 nanohenry (nH) at 600 MHz isn t even one full turn of wire.


27/39


8.2Planarfilters:

Microstrip transmission lines (as well as CPW or stripline) can also make good

resonators and filters and offer a better compromise in terms of size and performance

than lumped element filters. The processes used to manufacture microstrip circuits is very

similar to the processes used to manufacture printed circuit boards and these filters have

the advantage of largely being planar.

Precision planar filters are manufactured using a thin-film process. HigherQ factors can

be obtained by using low dielectric materials for the substrate such as quartz or sapphire

and lower resistance metals such as gold.

8.3 Coaxialfilters:

Coaxial transmission lines provide higher quality factor than planar transmission lines,

and are thus used when higher performance is required. The coaxial resonators may make

use of high-dielectric constant materials to reduce their overall size.

8.4 Cavityfilters:

Still widely used in the 40 MHz to 960 MHz frequency range, well constructed cavity

filters are capable of high selectivity even under power loads of at least a

megawatt. HigherQquality factor, as well as increased performance stability at closely

spaced (down to 75 kHz) frequencies, can be achieved by increasing the internal volume

of the filter cavities.

Physical length of conventional cavity filters can vary from over 82" in the 40 MHz

range, down to under 11" in the 900 MHz range.

In the microwave range (1000 MHz (or 1 GHz) and higher), cavity filters become more

practical in terms of size and a significantly higher quality factor than lumped element

resonators and filters, though power handling capability may diminish.


28/39


8.5 Dielectricfilters:

Pucks made of various dielectric materials can also be used to make resonators. As with

the coaxial resonators, high-dielectric constant materials may be used to reduce the

overall size of the filter. With low-loss dielectric materials, these can offer significantly

higher performance than the other technologies previously discussed.

8.6 Electroacousticfilters:

Electroacoustic resonators based on piezoelectric materials can be used for filters. Since

acoustic wavelength at a given frequency is several orders of magnitude shorter than the

electrical wavelength, electroacoustic resonators are generally smaller than

electromagnetic counterparts such as cavity resonators.

A common example of an electroacoustic resonator is the quartz resonator which

essentially is a cut of a piezoelectric quartz crystal clamped by a pair of electrodes. This

technology is limited to some tens of megahertz. For microwave frequencies, thin film

technologies such as surface acoustic wave (SAW) and, bulk acoustic wave (BAW) have

been used for filters.

8.7 Bandpassand Bandstop Filters:

A useful form of bandpass and bandstop filter consists of /4 stubs connected by /4

transmission lines. Consider the bandpass filter here


29/39


Figure8.1 bandpassandbandstopfilter

This filter can also be configured as a bandstop filter by using open rather than shorted

stubs. While it s easy to see that this filter will pass the center frequency for which the

lines are all /4, we would like to be able to design such a filter using lumped element

prototypes.

Recall that the equivalent circuit of a quarter wave transmission line resonator is, for

shorted or open circuit termination, a parallel or series tuned resonant circuit, as shown:

But it is important to note that a parallel tuned circuit is transformed through a /4 line to

the impedance of a series tuned circuit, and vice versa. This allows us to determine the

equivalent circuit of the transmission line filter.


30/39


The quarter wave sections transform the center shunt parallel resonant circuit admittance

to a series impedance that is a series resonant circuit.

Thus, the equivalent circuit of the bandpass filter using quarter wave lines is the same as

the prototype lumped element filter that is created through the customary transformation

from lowpass to bandpass prototype filters. Using the known relationships between

transmission line Zo and the L and C of the equivalent resonance, we can identify the

relationships between the required L and C of the prototype circuit and the Zo we need

for the shunt stubs.

8.8 Coupled Line Filters:

The parallel coupled transmission lines can also be used to construct many types of

filters. Fabrication of multi section band pass or band stop filters is particularly easy in

microstrip or stripline form, for bandwidths less than about 20%. Wider bandwidth filters

generally require very tightly coupled lines, which are difficult to fabricate. We will first

study the filter characteristics of a single quarter-wave coupled line section, and then

show how these sections can be found in reference.

With the added tool of the impedance or admittance inverter, we can analyze and

design a

number of transmission line filters. As we have seen in connection with directional

couplers, coupled transmission lines have frequency sensitive coupling, and can be


31/39


analyzed by the even-odd mode method.

The result of this analysis is tabulated in Table 8.8 of Pozar, and we can see that there are

among the less useful permutations several that have bandpass characteristics. In

particular, the configuration that represents coupled /2 open lines is the easiest to

construct in microstrip and stripline.

The equivalent circuit of two coupled /4 open lines can be shown to be as depicted here:

So we can see that a structure of a number of coupled lines will admit to an equivalent

circuit of alternating series and parallel resonant circuits, and the design parameters of the

prototype filter can be imposed onto the structure of parallel coupled lines.


32/39


Figure8.2A parallel-coupled lines filter in microstrip construction

Stripline parallel-coupled lines filter. This filter is commonly printed at an angle as

shown to minimize the board space taken up, although this is not an essential feature of

the design. It is also common for the end element or the overlapping halves of the two

end elements to be a narrower width for matching purposes

In microstrip or stripline, the transmission line conductors of the coupled line filter take

the form shown here, with the offsets between connected /4 sections added to permit

seeing the individual coupled line pairs.

9. FILTER DESIGN BY THE INSERTION LOSS METHOD:

We limit this tutorial to a procedure called the insertion loss method, which usesnetwork synthesis techniques to design filters with a completely specified frequency

response. The design is simplified by beginning with low-pass filter prototypes that are

normalized in terms of impedance and frequency. Transformations are then applied to

convert the prototype designs to the desired frequency range and impedance level.

The insertion loss method of filter design provides lumped element circuits. For

microwave applications such designs usually must be modified to use distributed

elements consisting of transmission line sections. The Richards transformation and the

Kuroda identities provide this step. We will also discuss transmission line filters using

stepped impedances and coupled lines; filters using coupled resonators will also be

briefly described.


33/39


The insertion loss method allows a high degree of control over the passband and

stopband amplitude and phase characteristics, with a systematic way to synthesize a

desired response. The necessary design trade-offs can be evaluated to best meet the

application requirements. If, for example, a minimum insertion loss is most important, a

binomial response could be used; a Chebyshev response would satisfy a requirement for

the sharpest cutoff. If it is possible to sacrifice the attenuation rate, a better phase

response can be obtained by using a linear phase filter design. And in all cases, the

insertion loss method allows filter performance to be improved in a straightforward

manner, at the expense of a higher order filter. For the filter prototypes to be discussed

below, the order of the filter is equal to the number of reactive elements.

9.1 Characterization by Power Loss Ratio

In the insertion loss method a filter response is defined by its insertion loss, or

power lossratio,PLR:

2)(1

1

loadtodeliveredo er

sourceromavailableo er)1(

[+!!!

load

inc

LR

Observe that this quantity is the reciprocal of |S12|2 if both load and source are matched.

The insertion loss (IL) in dB is LR

IL log10)2( !

We know that |+([)|2 is an even function of [; therefore it can be expressed as a

polynomial in [2. Thus we can write

)()(

)()()3(

22

22

[[

[[

NM

M

!+

where M and N are real polynomials in [2. Substituting this form in (1) gives the

following:

)(

)(1)4(

2

2

[

[

N

MPLR !


34/39


Thus, for a filter to be physically realizable its power loss ratio must be of the

form in , Notice that specifying the power loss ratio simultaneously constrains the

reflection coefficient, +([). We now present two practical filter responses.

Maximally flat

This characteristic is also called the binomial orButterworth response, and is

optimum in the sense that it provides the flattest possible passband response for a given

filter complexity, or order. For a low-pass filter, it is specified by

N

c

LR kP

2

21)5(

!

[

[

where N is the order of the filter, and [c,is the cutoff frequency. The passband extends

from[ = 0 to [ = [c;at the band edge the power loss ratio is 1 + k2. If we choose this

as the -3 dB point, as is common, we have k= 1, which we will assume from now on.

For[ > [c,the attenuation increases monotonically with frequency, as shown in Figure

1. For[ >> [c,PLRk2([/[c)

2N, which shows that the insertion loss increases at the rate

of 20 NdB/decade. Like the binomial response for multisection quarter-wave matching

transformers, the first (2N - 1) derivatives of (5) are zero at [ = 0.

Equal ripple

If a Chebyshev polynomial is used to specify the insertion loss of an N-order low-pass

filter as

,1)6( 22

!

c

NLR TkP [

[

then a sharper cutoff will result, although the passband response will have ripples of

amplitude 1+k2, as shown in Figure 1, since TN(x) oscillates between +1 for |x|< 1. Thus,

k2 determines the passband ripple level. For large x, TN(x) (2x)N, so for[ >> [c, the

insertion loss becomes

,2

4

22

N

c

LR

kP

!

[

[


35/39


which also increases at the rate of 20 N dB/decade. But the insertion loss for the

Chebyshev case is (22N)/4 greater than the binomial response, at any given frequency

where [ > > [c.

Figure9.1. Maximallyflatandequal-ripplelow-passfilterresponses (N=3).

10.FILTER TRANSFORMATIONS

The low-pass filter prototypes of the previous section were normalized

designs having a source impedance ofRs= 1 ; and a cutoff frequency of[c = 1. The

designs must be scaled in terms of impedance and frequency, and converted to give high-

pass, bandpass, or bandstop characteristics. Several examples will be presented to

illustrate the design procedure.

Impedance and Frequency Scaling

Impedance scaling. The network needs to be scaled from a source

resistance of 1 to R0 and a cutoff frequency of 1 to [c. If we let primesdenote impedance and frequency scaled quantities, we have the following

transformation equations for the kth element in the low-pass network.:


36/39


.'

,'

0

0

c

k

k

c

kk

R

CC

LRL

[

[

!

!

Low-pass to high-pass transformation. The frequency substitution where,

[

[[

cn

can be used to convert a low-pass response to a high-pass response. This substitution

maps [ = 0 to [ = +, and vice versa; cutoff occurs when [ = +[c. The negative sign is

needed to convert inductors (and capacitors) to realizable capacitors (and inductors).

Applying (8) and impedance scaling to the series reactances, j[Lk, and the shunt

susceptances, j[Ck, of the prototype filter gives

.

,1

0

0

kc

k

kc

k

C

RL

LRC

[

[

!

!

Bandpassand Bandstop Transformation:

Low-pass prototype filter designs can also be transformed to have the bandpass or

bandstop responses. If [1 and [2 denote the edges of the passband, then a bandpass

response can be obtained using the following frequency substitution:

0

12

0

0

0

012

0

where

1

[

[[

[

[

[

[

[

[

[

[

[[

[[

!(

(!

n

is the fractional bandwidth of the passband. The center frequency, [0, could be chosen as

the arithmetic mean of w1 and w2, but the equations are simpler if it is chosen as the

geometric mean, [0 = ([1[2)1/2

. Then the transformation of maps the bandpass

characteristics to the low-pass response giving the following new filter elements:


37/39


(1) A series inductor is transformed to a series LCcircuit with impedance and frequency scaled values

00

0

0

RL

C

RLL

k

k

kk

[

[!

!

(!

(2) A shunt capacitor is transformed to a shunt LCcircuit with impedance and frequency scaled values

00

0

0

'

'

R

CC

C

RL

k

k

k

k

(!

(!

[

[

The inverse transformation can be used to obtain a bandstop response. Thus,

.

1

0

0

(n

[[

[[[

Then series inductors of the low-pass prototype are converted to parallel LC circuits

having impedance and frequency scaled values

00

0

0

1'

'

RLC

RLL

k

k

kk

(!

(!

[

[

The shunt capacitors of the low-pass prototype are converted to series LCcircuits having

impedance and frequency scaled values

00

0

0

R

CC

C

RL

k

k

k

k

[

[

(!

(!

The element transformations from a low-pass prototype to a high-pass,

bandpass, or bandstop filter are summarized in Table 3. These results do not include

impedance scaling, which can be made by multiplying the values of L, Rs and RL by R0

and dividing the values ofCby R0.


38/39


Table 10.1. Summary of prototype filter transformations

11.ApplicationsofMicrowave Filters:

y Any microwave Communication system

y Radar

y Test and measurement system


39/39

report on basics of microwave filter

Documents