# Low Pass Stepped Impedance Microwave Filter: Quick design procedure

*(article under continuous revision, as time allows!)*

This article will outline the steps required to design quickly a microwave low pass filter based on stepped impedance realization. A significant number of sources on the literature present how to do this, but this article will outline the minimum necessary calculations needed to design such a filter rapidly and with confidence.

The design procedure will take us from the definition of an appropriate low pass normalized prototype into the final geometry of the actual PCB stepped impedance implementation to be fabricated. Simulation results of the final structure with and without material losses will be compared against the actual lossless filter response and discrepancies justified.

In order to give to the reader the option for an easy fabrication, a design example will be described based on the use of the classic FR4 PCB (εr=4.2) of thickness h=1.5mm. FR4 boards are widely and cheaply available from the electronics suppliers, so it would be possible to fabricate the final stepped impedance microwave filter, even at home!

An understanding of the design methodology is best understood by giving a practical example. It is assumed that the reader is familiarized at least with the fundamental terminology of filter circuits.

**Design a 5th order stepped impedance Chebyshev filter with cut-off frequency fc=1.2GHz and a ripple of 0.5dB to be connected with a 50 Ohm source (Rs) and a 50 Ohm load (Rl).**

**Design a 5th order stepped impedance Chebyshev filter with cut-off frequency fc=1.2GHz and a ripple of 0.5dB to be connected with a 50 Ohm source (Rs) and a 50 Ohm load (Rl).**

To start with, it is necessary to generate the low pass prototype filter circuit with a cut-off frequency ω=1rad/sec which, consists of elements with normalized values. We will then apply frequency and impedance transformation to bring the filter response to the required actual specifications.

There are many textbooks and internet sources that provide the normalized element values for Chebyshev filters, such as in [1] or [2] and [4]. This type of filter is a good option when some ripple is allowed in the passband and a sharp out of band attenuation is desired.

We first need to generate a prototype filter which corresponds to the 5th order Chebyshev filter with 0.5dB ripple, using the available standard tabulated tables. The attenuation at the normalized cutoff frequency (ω=1rad/sec, or fc=0.159 Hz) of the Chebyshev prototype filter is the value of the ripple (0.5dB) or -3dB, depending on the table used as a reference. Make a note of this, as there are prototype filter element tables that give different element values depending on this assumption. In addition, two different circuit topologies can be selected depending on whether the first component starting from the source is a (series) inductor or a (shunt) capacitor. The selection of all these parameters is left to the designer to consider depending on the application under consideration.

Based on the above, our design example will consider the first element to be an inductor and pick up the prototype element values that will result an attenuation equal to the ripple value, that is -0.5dB, at cut-off frequency.

For the curious reader, a comparison between the normalized prototype filter responses of a 5th order Chebyshev 0.5dB ripple, between equal source and load resistances, based on two different tables, is shown below:

The response for each of those circuits is shown next:

These are the normalized frequency responses, that is the cut-off frequency is at fc=0.159 Hz. It can be seen that at 0.159 Hz the attenuation can be different depending on the normalized attenuation point taken as a reference. As stated previously, in our design example we will assume that attenuation is 0.5dB at cut-off.

The next step would be to apply frequency and impedance transformation into a LP filter, to bring our filter response to the required specifications, at fc=1.2 GHz and Ro=Rs=Rl=50 Ohms. This is achieved with the use of the following equations [2]:

*[A way to remember these equations is to realize that the impedance of an actual component at fc must be equal to the impedance of the normalized component at 0.159Hz (1rad/sec) taking into consideration the resistance transformation, that is:*

*Which simplify into (1) and (2). This applies for a LP to a LP filter transformation.]*

In equations (1) and (2) L and C are the actual component values of our wanted filter, Ln and Cn are the normalized (prototype) element values, fc is the cut-off frequency of our actual filter and Ro is the value required to bring the prototype resistances (1 Ohms) into 50 Ohms. Doing the calculations we get the element values which result in the following circuit:

To check the accuracy of our calculations, we perform a circuit simulation. The S-parameter response of the circuit up to 2.5 GHz, together with a close-in look at the ripple region, is shown below:

At 1.2 GHz, the transformed circuit has a 0.5dB attenuation verifying the accuracy of our calculations. Notice that the 5th order pattern is shown from the number of peaks and dips at the response, which are five. At DC the filter has no attenuation, a characteristic for odd order Chebyshev filters. If it was an even order Chebyshev filter then at DC the filter would have an attenuation equal the ripple value. Also, bear in mind that lossless components were considered during the simulation.

Now the next step, would be to relate these lumped element values with a distributed stepped impedance filter based on microstrip lines. It is essential to consider the following points before proceeding to this step:

- A series microstrip inductor is formed by introducing a short length of a transmission line of a high characteristic impedance (Zh).
- A shunt microstrip capacitor is formed by introducing a short line of a transmission line of a low characteristic impedance (Zl).
- The values of Zh and Zl are referenced against the port impedances of the transmission line defined at the input and output sections of the filter, that is Zo=50 Ohm in our example.

Based on the above we immediately see that Zl<Zo<Zh. Also, keep in mind that a high Z (to make the series inductor) means a narrow microstrip line, while a low Z (to make the shunt capacitor) means a wide microstrip line.

So, effectively what we need to do now is to calculate the length and width for each of these distributed elements that will result in the same reactance values as those defined in the lumped circuit. Let’s see how this can be done.

To calculate the length of the distributed (series) inductor we use the following equation [3]:

To calculate the length of the distributed (shunt) capacitor we use the following equation:

Where, lH and lL is the length of the microstrip line for the high (inductor) and low (capacitor) impedance transmission lines respectively, L and C are the actual filter circuit values (not normalized),c is the speed of light, and εeffH and εeffL are the effective dielectric constants of the microstrip transmission lines for the HIGH and LOW characteristic impedances respectively.

In order calculate the width (w) and εeff of our microstrip lines we use the standard graphs provided in many microwave engineering textbooks, which relate the ratio w/h (width of microstrip line/thickness of PCB board) to the impedance of the microstrip line. When such graphs are used make sure that the appropriate dielectric constant is considered (er=4.2 for FR4 PCBs). Alternatively you can calculate w and εeff by using the standard microstrip line equations which are also available from many sources.

For easiness, the graphs of Zo versus w/h for er=4.2 (FR4) are shown below [3]:

Now, what’s left to select is the value of Zh and Zl. In theory, the ratio of Zh/Zl should be as high as possible to realise distributed component values as nearer as possible to the lumped values. But, there is a practical limitation to this, as these values are bounded by the feasible resulting widths of the microstrip lines (i.e. extremely high Zh results in a very narrow width microstrip line, while an extremely low Zl results in a very wide microstrip line. Can you really fabricate such a filter?!)

To proceed with our filter design, we assume that Zh=125 Ohms and Zl=25 Ohms. We will also have to consider microstrip lines for the input and output of the filter which should have Zo=50 Ohms. Based on the above graphs, we calculate the widths and εeff for each section of these microstrip lines are:

So, now we have sufficient information to calculate the length of each distributed component based on equations (3) and (4).

Inductor L1 is 11.311nH. Using (3) we get length of L1 (lL1)= 16.1mm, which is the same as length of inductor L3.

Capacitor C1 is 3.262pF. Using (4) we get the length of C1 lC1=13.1mm, which is the same as length of capacitor C3.

In a similar manner, we perform the calculation for L2, to get lL2=24.0mm.

The table below, summarizes these calculations:

So, we have now completed all the necessary steps required to realize the 5th order stepped impedance microstrip Chebyshev filter .

The final geometry of the filter is shown graphically in the following figure:

This completes the design of our filter.

## Extra section: Losses and practical limitations

If we were to fabricate this filter, we would have to remember that some unavoidable dielectric (tand) and conductor (S/m) losses are present in the distributed PCB circuit. In addition, equations (3) and (4) are only approximations (good though), which are used to realise the inductors and capacitors into a stepped impedance filter. As, explained before, the selection of Zh and Zl values affect the response of the final circuit too. In addition, Chebyshev type filters are filters with sharp cut-off responses: This characteristic is extremely dependent on the losses (Q) of the element values, which in practice must be as high as possible, otherwise the passband response is greatly affected (Ref: [2]). Furthermore, a stepped impedance realization does not result in sharp responses, although it is easy to fabricate it. On the other hand, for example, a Bessel type filter can perform closest to the specifications when implemented as a stepped impedance microstrip topology.

Keeping these issues in mind, the stepped impedance filter with losses was simulated using the calculated values as shown before and cross-checked with the ideal (lossless) lumped circuit response. The distributed circuit, together with the S-parameters up to 2.5 GHz are shown below:

The lossy responses consider the dielectric loss with tand=0.02 and the conductor loss of the ground plane and that of copper tracks with 5.8.10^7 S/m. It can be seen that the finite Q effect of the dielectric and conductor materials affect the filter response significantly, increasing the overall attenuation at the passband and reducing the out of band rejection.

The response of the (i) ideal lumped circuit,(ii) stepped impedance circuit (lossy and lossless) and (iii) a lossy 3D structure, were considered next. The 3D 5th order Chebyshev filter structure with the design values as calculated previously are shown next:

All responses (4xS11 and 4xS21) are merged into the same graph to compare filter performance. This is shown next:

## Summary

This article presented the steps necessary to design a LP microwave filter based on stepped impedance realisation. The design steps followed are outlined below:

- Define the low pass prototype filter element values using the standard tables
- Apply frequency and impedance transformation
- Proceed with the microstrip line realization: Select the value of Zh for the series inductor and Zl for the shunt capacitor.
- Calculate the widths and effective dielectric constants for each section of the transmission line (filter inductors/capacitors and input/output transmission lines) according to the PCB specifications (εr and h).
- Calculate the lengths for each section of the transmission line.

The effect of losses was discussed too. For an accurate microstrip transmission line realization, it is evident that the RF designer has to select a PCB board with the lowest possible losses delivering a substrate with an accurate dielectric constant. This will ensure a good agreement between the calculated and the actual laboratory responses.

## References

[1] Chris Bowick 'RF Circuit Design' 1982, Newnes

[2] Web source: http://www.rfcafe.com/references/electrical/cheby-proto-values.htm

[3] Word document article: F. Kung ‘Microstrip Filter Design’, May 2007 (can’t remember the website of this article, sorry!)

[4] D. Pozar ‘Microwave Engineering’ 1997, 2nd edition, John Wiley & Sons