# Design Of Low Pass Filter Overview Computer Science Essay

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This chapter will throw light on the conversion of the lumped element filter in to the microstrip design as it is not possible to fabricate the lumped elements due to their bulky and lossy nature. Also the lowpass filter was to be designed using the stepped impedance lines as discussed in the previous chapter and the calculations for a chebyshev response for the prototype element values were required. The stepped impedance lines are also known as high and low impedance transmission lines due to the abrupt changes in the width of the transmission line sections. The order of the filter was not known at that stage since the corresponding calculations had not been done.

This chapter will be divided into three subsections. The first part will contain the specifications of the filter, CAD design procedure, the calculations required to find out the filter order and finally the calculations for the lumped element design. The next part will be focussing on the design procedure and simulations of the lumped element filter in Agilent ADS software. The third part will be conversion from lumped element design to microstrip lines and also the design and simulation of the filter in ADS software suite.

## Part 1: Specification

Parameters

Value/Types

Filter type

Chebyshev

Stop band frequency ()

4GHz

Cut off frequency ()

2.45GHz

Pass band return loss ()

-15dB

Stop band insertion loss ()

30dB at

Source and load impedance ()

50â„¦

Characteristic impedance of inductive line ()

120â„¦

Characteristic impedance of capacitive line ()

20â„¦

Dielectric constant ()

2.2

Substrate material

RT/Duroid 5880

Substrate thickness ()

1.575mm

Conducting layer thickness ()

17Âµm

Table 4.1: The low pass filter specification

The amplitude response of the network is designed to meet a critical specification on the Pass band return loss () and the stop band insertion loss () which has been provided. Therefore in order to have maximum efficiency we need:

## CAD procedure for Low pass filter design

CAD stands for computer aided design. This is a procedure that involves synthesis and analysis of the filter design.

Figure 4.1: CAD design procedure for a low pass filter

This design procedure gives the designer an idea to obtain a strategy to meet his goal.

## Calculations for filter order

The first step would be to determine the passband ripple level from the minimum passband return loss which is defined as:

Here the value of can be left as 0.032 since it will required in the next equation for finding the filter order.

To determine the number of elements n at the designated stopband frequency the following equation was applied:

In this equation the value for ripple factor () is already known however the value for which is the degree chebyshev polynomial of the first kind is required and is defined by:

for

for

The formula given below was applied to find out the angular frequency scaling . It is defined as the ratio between the angular stop band frequency to the cut off frequency :

Now since the angular frequency scaling greater than 1, therefore the following equation should be used in order to find:

for

To find the order of the filter, the degrees of chebyshev polynomial and also the Insertion loss, a table is inserted below showing all the calculated values. Here the angular frequency remains the same for all expressions.

1

1.63

1.63

2.657

0.354

2

1.63

4.314

18.6105

2.029

3

1.63

12.433

154.579

7.742

4

1.63

36.218

1.311*103

16.33

5

1.63

105.637

11.159*103

25.54

6

1.63

308.158

94.961*103

34.83

7

## 1.63

## 898.159

## 808.128*103

## 44.13

Table 4.2: Calculations for the filter order

Although the sixth order filter calculated above meets the condition of the stop band insertion loss, however the seventh order has to be chosen due to the following factors:

First issue being the even order chebyshev filter mismatch of the characteristic impedance where input, output, load and source are not equal

Another issue is the non-symmetric nature of the inductive and capacitive value, this would result in heavy errors

Since the filter order is achieved the next step will be to calculate the element values.

## Calculation of prototype element values

Prototype element values at 0.1dB passband ripple level

1

0.3052

1.0000

2

0.8430

0.6220

1.3554

3

1.0315

1.1474

1.0315

1.0000

4

1.1088

1.3061

1.7703

0.8180

1.3554

5

1.1468

1.3712

1.9750

1.3712

1.1468

1.0000

6

1.1681

1.4039

2.0562

1.5170

1.9029

0.8618

1.3554

7

## 1.1811

## 1.4228

## 2.0966

## 1.5733

## 2.0966

## 1.4228

## 1.811

## 1.0000

Table 4.3: Showing element values for 0.1dB passband ripple highlighting 7th order

There are practically two filter type structures either "PI-type" or "TEE-type" for the use of our design. Both of them are shown below:

Figure 4.2: Seventh order PI-type structure

Figure 4.3: Seventh order TEE-type structure

The filter type to be used in this project for the lumped element lowpass chebyshev filter design will be of the TEE type configuration. Not for any tough reasons but it is usually more common in practice to have an inductor followed by the capacitor while designing a microstrip filter.

## Performing frequency and impedance scaling

The new inductor and capacitor elements can be scaled using the following formula:

- Final inductor value

- Final capacitor value

- Impedance scaling for both source and load impedance at 50â„¦

- The angular cut off frequency

## Results

For series inductors:

For shunt capacitors:

Finally the end of the design calculation is achieved and all the parameter values are shown in the table below, splitting the prototype value from the new calculated L and C.

Parameters

Normalised values

De-normalised values

1

50

L1 (nH)

1.1811

3.8363

C2 (pF)

1.4228

1.8485

L3 (nH)

2.0966

6.8098

C4 (nH)

1.5733

2.044pF

L5 (nH)

2.0966

6.8098

C6 (pF)

1.4228

1.8485

L7 (nH)

1.1811

3.8363

1

50

Table 4.4: Parameter values for the normalised and de-normalised 7th order lowpass filter

## Part 2: Design procedure in ADS

Design of a lumped element filter in ADS is very straightforward and simple. To create a new filter we just need to go to the elements library and select Lumped-Components as shown in the figure below:

Figure 4.4: Selecting components from ADS library

After this the filter can be easily designed in TEE-type configuration as illustrated below:

Figure 4.5:Lumped element lowpass filter schematic

Also in order to simulate the circuit, S-parameters need to be imported from the library. The value should be set from 0 GHz to 5 GHz since the stop band frequency is 4GHz. It is always good to have some extra bandwidth. The simulated S-Parameters response is given below:

Figure 4.6: Simulated S-Parameters of the lumped element lowpass filter

## Observation

It can be clearly see that the frequency response does not meet the specification but also provides extra guard band which is not required

This lack of a sharp frequency cut off can be due to the approximated values that were calculated using the calculator

Thus it can be concluded that lumped element designs are not very accurate and they need some tuning to achieve a more realistic response

## Part 3: Conversion of lumped element lowpass filter design to microstrip stepped impedance line

It is now time to take the L-C circuit design and transform the filter from lumped element to microstrip technology. Here stepped impedance will be used to achieve the target as mentioned before in the chapter "Discontinuities in transmission lines". The stepped impedance design technique is also referred to as the high and low characteristic impedance transmission line. Previously was being referred to as characteristic impedance for both input and output terminations but now the use of inductive and capacitive characteristic impedance should be implemented because they are useful in designing the microstrip filter.

The ratio of characteristic impedance of the inductor to the capacitor should be as high as possible but not as big that it becomes impractical for fabrication on a PCB. The impedance values as per the specification are:

Inductive impedance

Capacitive impedance

For conversion from lumped elements to microstrip stepped impedance lines the width (w) and the length (l) of the microstrip lines has to be calculated. There are many ways to do this and some of them are listed below:

## Method 1

The dimensions can be found using the ratio w/h by calculating A and B and after then finding the lengths and . The formulas for this procedure are given below:

For ,

For ,

Where A and B individually are given by the formula as depicted below:

The width can be calculated using the above formulas as well as the effective dielectric constant for inductor and capacitor impedances can be calculated. The formula to calculate the effective dielectric constant is:

The calculations are not presented here as they were too lengthy but the table below will show the final calculated values for the width and effective dielectric constants for the microstrip lines:

( or )= 20â„¦

10.1

1.575

## 16.173

2

3.1

1.575

## 4.818

1.87114

( or )= 120â„¦

0.56

1.575

## 0.852

1.7242

Table 4.5: Showing the calculated values for widths and effective dielectric constants for and

To calculate length of microstrip lines, first the value of should be attained from the following equation:

Next step will be to find the value for and :

Now the final phase is to calculate the lengths of the inductive and capacitive lines. They can be done as follows:

Using the above equations the following lengths were obtained:

However this method was found out to be long and time consuming so after doing extensive and through research, method 2 was found much simpler to put into practice and also the results were much more accurate.

## Method 2

In this method the use of a tool within ADS called "LineCalc" was employed in order to get the most accurate results for our dimensions of the microstrip lines. This is based on the concept of finding out the electrical lengths of the transmission lines and all the other parameters were provided in the specification sheet for our use within the software. A step by step procedure on how to use this tool is given below.

## LineCalc in ADS

First a new blank design should be opened in ADS. Then go to Tools->LineCalc->Start LineCalc. The procedure is depicted below:

Figure 4.7:LineCalc illustration

After this a window opens where it is required by us to put in all the values such as , , , and the calculated electrical lengths and the impedance to which that length belongs. To calculate electrical length the method is fairly simple. The following formulas were employed to get the electrical length:

Inductor:

Capacitor:

Where:

The table below represents all the calculated values for the width and length using LineCalc:

Type

Component Values

L

0.4921

0.8812

7.283

C

0.5691

16.212

7.766

L

0.8735

0.8812

12.929

C

0.6293

16.212

8.588

L

0.8735

0.8812

12.929

C

0.5691

16.212

7.766

L

0.4921

0.8812

7.283

Table 4.6: Width and Length of microstrip line using LineCalc

The image shown below shows the values of Length and width calculated for the inductive line () in LineCalc and also it points out the values that are requisite to put in before clicking on the synthesize button:

Figure 4.8: Inductive line value in LineCalc

The values for capacitive line () are given in the image below:

Figure 4.9: Capacitive line value in LineCalc

The image gives very clear evidence of filling out the correct parameters in LineCalc to get the dimensions for any microstrip line. However it should be noted that while the values are calculated for inductor the impedance should be changed to 120â„¦ and for capacitor it should be brought down to 20â„¦. Also note that the electrical length is in radians.

The values from LineCalc can also be compared to the values calculated using the formulas for the length and width of microstrip line. The values from ADS are definetly more precise and presented up to 6 decimal places.

Finishing off the conversion from the lumped element lowpass filter in stepped impedance microstrip line, it should now be implemented in the actual schematic to get a better idea of how the lowpass filter performs.

## Simulation Results

At this stage of building the circuit the substrate materials must be defined and used in all simulation analysis. The material used should be strictly kept to specification. A new component called MLIN is introduced here which represents the short length of the transmission line in the lowpass filter.

To define substrate materials a block called MSUB should be pulled out from the "Tlines-Microstrip" library. S-Parameters are also required to plot the simulation results within a certain defined frequency. The block S_Param can be retrieved from "Simualtion-S_Param". Both of these blocks are shown in the figure below with their specific values.

Figure 4.10: S-Parameters and MSUB block illustration

At this point the substrate thickness variable 'h' used in the calculation purposes should not be confused with the conductor thickness represented by variable "T". As can be seen from the above figure, ADS software defines its substrate in upper case form and so its thickness is defined as capital 'H'. Also the variable 'T' represents the conductor thickness. These variables have not been used in any other equations.

First we draw the circuit by only using MLINS and inspect the output.

Figure 4.11: 7th order lowpass filter in Microstrip technology without discontinuities

Here it is observed that there are no discontinuities, therefore the response might not be very sharp. The simulation is given below:

Figure 4.12: Simulated S-Parameters of the lowpass filter without discontinuity

## Observations

As expected an inaccurate response with close enough values is obtained. It is seen that the cut off frequency is not at -3dB point but it is coming at -3.5dB. Also the insertion loss and the return loss are deviated a lot and do not have the expected gain. Furthermore only three ripples are obtained in as this circuit is not designed to be an industry level circuit. However to get a response with 7 ripples a professional filter design will be required. Unfortunately, ADS does not allow us to design such a high performance filter.

To add more perfection to the lowpass filter circuit we need to add discontinuities in the design. MTSEP is the new element to be installed in the circuit. It is a Microstrip width element that creates a more accurate simulated result and a discontinuity between the inductive and capacitive characteristic impedance lines when applied in the circuit thereby taking their widths into consideration.

The improved circuit is shown below:

Figure 4.13: 7th order lowpass filter with discontinuities

When simulated the following response was achieved:

Figure 4.14: 7th order low pass filter simulation with discontinuities

## Observation

The condition for centre frequency is not met at -3dB point as it is 2.17GHz which is far below than what was expected

Return loss is -15.684dB

Insertion loss is -41.6dB which is close enough to what was calculated in the theory but still it does not meet the specification of -30dB

Careful tuning is required in order to achieve the perfect response

## Tuning

To realize the most practical response some tuning is required for the lengths of all the components as the widths are fixed always because of the MSTEP. Also if tuning was applied to all the widths it would end up a really cumbersome and time consuming task. So to keep the process as efficient as possible all the element lengths are selected and changed accordingly while observing the output at the same time.

The tuning process can be started by clicking on the tuning fork in the toolbar within the schematic. The figure below highlights the same:

Figure 4.15: Highlighting the tuning fork

Some tuning was applied to the circuit which ended up producing the following response:

Figure 4.16: Simulated S-Parameters of the lowpass filter with tuning

## Observations

Here it can be seen that the response is much better than what was acquired before and the specifications are met approximately

The return loss is -14.925dB whereas the insertion loss at fs is -35.496dB

This is not perfectly fine so even more tuning needs to be applied to the circuit

The final tuned circuit is shown below with the final values:

Figure 4.17:Final tuned Lowpass filter schematic

The simulation response for this circuit is given below:

Figure 4.18: Final Simulation of S-Parameters for the Lowpass filter

## Observations

This is the best possible response that can be obtained for this circuit

The cut off frequency of 2.45GHz is shown at -3dB point as well as the return loss is greater than -15dB which was required in the specifications

However obtaining insertion loss to be -30dB was not possible as it was shifting the centre frequency and also increasing the return loss by a lot

The top view layout of the low pass filter as the designer expects to see on a PCB is shown in the figure below:

Figure 4.19:Top view layout of the low pass filter for fabrication

This filter measures .

The two port network has been successfully designed and meets the specification for a seventh order lowpass filter with a ripple factor of 0.1dB. The cut off frequency is shown to be 2.45GHz for this filter. Hence it is capable of receiving and passing all frequencies less than 2.45GHz while eliminating all frequencies after 2.45GHz for a return loss performance of -35dB. This will provide sufficient bandwidth gain for the design of patch antenna at 2.4GHz which will be acting as the transmitter.