**Introduction to 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:

·
Low-pass filter (LPF) that transmit all signals
between DC and some upper limit w

_{c}and attenuate all signals with frequencies above w_{c}.
·
High-pass filter (HPF) that pass all signal with frequencies
above the cutoff value w

_{c}and reject signal with frequencies below w_{c}.
·
Band-pass filter (BPF) that passes signal with
frequencies in the range of w

_{1}to w_{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(w)) 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.

# Realization of Filters

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:

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 1W and cutoff
frequency of 1 Radian/s. Figure 2.1
shows the characteristics. Impedance
transformation and frequency scaling are then applied to denormalize the LPP
and synthesize different type of filters with different cutoff frequencies.

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 known as 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.

# Brief Overview of Low-Pass Prototype Filter Design Using Lumped Elements

There 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:

·
Maximally flat or Butterworth function.

·
Equal ripple or Chebyshev approach.

·
Elliptic function.

We will not go into the
details of each approach as many books have covered them. Interested reader can refer to reference [3],
which is a classic text on network analysis or [4], a more advance
version. The basic idea is to
approximate the ideal amplitude response |H(w)|

^{2}of an amplifier using polynomials such as Butterworth, Chebyshev, Bessel and other orthogonal polynomial functions. This is usually given as:
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 ladder
LC network with the number of reactive elements corresponding to the order of
the polynomial PRLC lumped circuit using distributed elements such
as microstrip and stripline in microwave region.

_{N}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 [3] and [4]. It is suffice to say that for each approach, values of g_{1}, g_{2}, g_{3}… g_{N}for an Nth order LPP have been tabulated by many authors (For instance see [2]). Here we will demonstrate the design of a low-pass filter and a band-pass filter using the insertion-loss method and illustrate the implementation of the
The Table 8.3 of reference [2] is
repeated here. We will use this table to
design a LPP Butterworth filter. The
values of g

_{i}correspond to inductance and capacitance in the LPP Butterworth filter.**Designing a**Low Pass Prototype (LPP)

We will now design a 4

^{th}order Butterworth LPP and use this design for the rest of the lab. The specification of the filter is as follows: R_{S}= R_{L}= 50W. Cutoff frequency f_{c}= 1.5GHz or w_{c}= 9.4248´10^{9}rad/s.**Step 1 – Design the LPP filter with**

**w**

_{c}**= 1 rad/s.**

# Step 2 – Perform impedance and frequency scaling

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 (Kirchoff’s 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.

Suppose the impedance of an inductor is jwL. At w = 1 the
impedance is jL. Another inductor with
inductance L/w

_{o}will give similar impedance at w = w_{o}. Thus we observe that the frequency response of the inductor is scaled by w_{o}. Similarly if a capacitor C is replace with capacitance C/w_{o}, its frequency response is also scaled by w_{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 R

_{o}= 50W and w_{o}= 2p(1.5´10^{9}) on the schematic of Figure 4.1, the new schematic of the low-pass filter is shown in Figure 4.2 below.**5.0 Implementing the Low-pass Filter using 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 (Z

_{o}) 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).
The ratio Z

_{H}/Z_{L}should be as high as possible, limited by the practical values that can be fabricated on a printed circuit board. Typical values are Z_{H}=100 to 150W and Z_{L}=10W to 15W. 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 w_{c}are:**Designing with Microstrip line**

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 e

_{e}= e_{r}, 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 e_{e}falls within the range 1 and e_{r}. 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.**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 for e

_{e}and Z_{o}.#
Implementing the 4^{th} Order Butterworth
Low Pass
Filter using Step Impedance Microstrip Line

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

·
Cutoff frequency f

_{c}= 1.5GHz.
·
Required Z

_{L}= 15W.
·
Required Z

_{H}= 110W.
·
L

_{1}=4.061nH, L_{2}=9.083nH, C_{1}=3.921pF, C_{2}=1.624pF.**Implementation:**

A typical FR4 fiberglass PCB
with e

_{r}= 4.2 and H = 1.5mm is used. From Figure 6.3 the following trace parameters are obtained:**Analysis of the step-impedance low pass filter using Agilent Advance Design System (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).

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 “
components represent a short length of microstrip transmission lines used in
our low pass filter. Here MLIN 1 corresponds to transmission line section 1, MLIN 2 to transmission line section 2 and so forth
(Figure 7.3 to Figure 7.5).

**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**

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 the parameters
leave as default. The parameters dialog
box for

**MSUB**can be invoked by doubling clicking on the**MSUB**component.
7.
Set the characteristic W and L of each
components according to the table of Section 6.2

**MLIN**

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 wire to connect the components together and ground the outer terminals of the TERM1 and TERM2 components (Figure 7.7).
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.

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.
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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