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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.
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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
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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.
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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
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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
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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
[
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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
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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)
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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
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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
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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$B (5.4b)
Assuming a short length of transmission line (Fl
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0$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
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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
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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.
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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
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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.
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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
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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.
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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.
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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 !
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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
!
[
[
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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.:
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.'
,'
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:
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(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.
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Table 10.1. Summary of prototype filter transformations
11.ApplicationsofMicrowave Filters:
y Any microwave Communication system
y Radar
y Test and measurement system
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