Efficient Design of Chebychev Band-Pass Filters with Ansoft HFSS and Serenade

Tutorial

Dr. B. Mayer and Dr. M.H. Vogel

Ansoft Corporation

Contents

Abstract 3

1. Introduction 3

2. Circuit Representation of the Filter 5

3. Relationships between Circuit Components and Physical Dimensions in the Microwave Filter 10

4. Initial Filter Design in HFSS 14

5. Curve Fitting in Serenade 16

6. Corrected Filter Design in HFSS 20

7. Additional Information from the 3D Field Solver 21

7a. Effects of Internal Losses 21

7b. Maximum power-handling capability 23

7c. Mechanical Tolerances 24

References 25

Appendix A Derivation of the Circuit 26

Appendix B The Physical Meanings of K and Q 43

Abstract

An efficient method is presented to design coaxial Chebychev band-pass filters. The method involves a 3D full-wave field solver, Ansoft HFSS, teaming up with a circuit simulator, Serenade. The authors show how for a practical case, a 7-pole band pass filter with a ripple of only 0.1 dB, an accurate design is obtained in a matter of days, as opposed to weeks for traditional methods.

The method described is also applicable to even more challenging designs of elliptic filters and phase equalizers realized in dielectric, waveguide or coaxial technology.

1 Introduction

In this paper, we will describe an efficient method to design a filter. The method involves a 3D full-wave field solver teaming up with a circuit simulator. The basic idea has been explored by others [1] but a different circuit was used in the circuit simulator. We will explain our procedure by presenting in detail how we design a Chebychev band pass filter with the following specifications:

Center frequency 400 MHzRipple bandwidth 15 MHzRipple 0.1 dBOut-of-band rejection 24 dB at 390 MHz and at 410 MHz

In order to achieve the out-of-band rejection, we will need seven poles. The desired filter characteristic is shown in Fig. 1.

Fig. 1 Desired filter characteristic

As the basic geometry for this filter we have chosen a cavity with seven coaxial resonators, as shown in Fig. 2. In the figure, the “buckets” have been drawn as wire frames for clarity, to show that the cylinders don’t extend all the way to the bottom.

Fig. 2 Basic filter geometry

This geometry is symmetrical with respect to the central cylinder. In this kind of filter, the walls of the cavity, the long cylinders, the buckets under the cylinders and the disk-shaped objects near the first and last cylinder are all made of metal. The long cylinders are connected to the top of the cavity; the buckets are connected to the bottom of the cavity. Cylinders and buckets don’t touch. The disk-shaped objects near the first and last resonator are connected to the input and output transmission lines and provide the necessary coupling to the source and the load. We will call these objects antennas in this document. They are near the first and last cylinders, but never touch them. Each cylinder-bucket combination is a resonating structure. At this stage, without restricting ourselves, we can choose many dimensions in the filter relatively freely. We make the following choices:

Cavity dimensions 280 x 30 x 120 mmResonator diameter 10 mmBuckets’ inner diameter 12 mmBuckets’ outer diameter 16 mmBuckets’ height 15 mmAntennas’ diameter 26 mmAntennas’ thickness 6 mm

Six dimensions remain, and these six will be crucial in obtaining the desired filter characteristic:

The length of the first and last resonating cylinder (both have equal length)The length of the five interior cylinders (all five have equal length)The distance between an antenna and its nearest cylinderThree distances between neighboring cylinders (remember the filter is symmetric)

With traditional filter design methods, obtaining the correct dimensions is a time-consuming task that commonly takes several weeks. Filter design with a circuit simulator, on the other hand, is relatively straightforward. Filter theory provides the values for the lumped inductors and capacitors that are needed to obtain the desired filter characteristic. First, we will show how to design a circuit that not only has the desired filter characteristic, but also lends itself to implementation with microwave components. In such a circuit, we use series L and C for each resonator, i.e. the cylinder-and-bucket combinations, and impedance invertors to represent the distances between adjacent resonators. Second, we will show how one can determine relationships between components in the circuit and dimensions in the physical filter. Third, we will present an iterative procedure between the electromagnetic field solver and the circuit simulator to optimize the design. The procedure converges very quickly.

2 Circuit Representation of the Filter

In order to design an order-seven band-pass filter around 400 MHz with a 0.1 dB ripple, filter theory tells us to start with an order-seven low-pass filter, normalized to 1 radian/s. The normalized filter is to have a 0.1 dB ripple, like the desired band pass filter. The source and load impedances of the normalized low pass filter are normalized to 1 Ohm. This circuit is shown in Fig. 3 and its characteristic in Fig. 4. Filter theory provides us with the values for the inductors and the capacitors, denoted by g1 through g7 in the figure. These values are in our case

g1=g7=1.1812 Hg3=g5=2.0967 Hg2=g6=1.4228 Fg4=1.5734 F.

Fig. 3 Normalized low-pass filter circuit, starting point for design procedure

Fig. 4 Filter characteristic for the normalized low-pass filter in Fig. 3

The step-by-step procedure from this normalized low-pass filter circuit to the final band-pass filter circuit is presented in detail in Appendix A. Here, we show an outline of the major steps.

An important step is the replacement of shunt capacitors by series inductors and impedance inverters. Basically, an impedance inverter transforms impedances in the same way as a quarter-wave-length transmission line, but independent of frequency. The resulting circuit is shown in Fig. 5. This is still a normalized low-pass filter with the same characteristic as the circuit in Fig. 3. The reason for this change is that at microwave frequencies it is often impossible to realize the ladder circuit consisting of series inductors and shunt capacitors. Depending on the basic structure either series elements or shunt elements are easily realizable but often not both in the same structure. Taking advantage of impedance inverters, it is possible to transform shunt capacitors into series inductors. In the physical filter these impedance inverters will be realized by couplings between the coaxial resonators.

Fig. 5 Normalized low-pass filter without shunt capacitors

Following a standard procedure, we take the following steps to derive the desired band-pass filter model:

(1) De-normalize the low-pass cut-off angular frequency from 1 rad/s to bw rad/s.(2) Transform the low-pass filter to a band-pass filter with a relative bandwidth of

bw and a center angular frequency of 1 rad/s by inserting a 1 F capacitor in series with every 1 H inductor.

(3) De-normalize the center frequency to 400 MHz by choosing

L=1/(2×π×4E8) H and C = 1/(2×π×4E8) F. (4) De-normalize the port impedances from 1 Ohm to the usual 50 Ohm by introducing impedance inverters at the input and output with coupling coefficients of √50.(5) Introduce finite quality factors to the individual resonators by adding a series resistor to each resonator.(6) Introduce individual resonant frequencies to the first and last resonators to be able to be able to take the frequency shift due to the coupling antennas into account.(7) Add a homogeneous transmission line of length ZUL between filter input/output and port 1 / port 2 to be able to adjust the phase due to the connectors.

This gives us the filter shown in Fig. 6. The procedure outlined above is presented in more detail in Appendix A.

Fig. 6 Final filter circuit, representing the desired band pass filter

In this circuit, every LC pair resonates at 400 MHz. Further K12, K23, K34 and QL have been defined as

(1)and

(2)where bw is the relative bandwidth and gi is the ith g value from filter theory.

Notice that, since the g values are known from filter theory, we still know the values of the all the components in the circuit, even through the components have changed considerably in the process.

Filter theory [2] tells us that Ki,i+1 and QL have important physical meanings. Ki,i+1 is known as the coupling constant between adjacent resonators. If we have just two resonators in the cavity, with a very light coupling to the source and the load, then the relation between coupling constant K12 and resonant frequencies f1 and f2 is given by

K12 = 2(f2-f1) / (f2+f1) . (3)

QL is known as the loaded Q of the circuit. If we have just one resonator in the cavity, coupled to source and load, the relation between QL , resonant frequency fR and 3-dB band width BW3dB is given by

QL = fR / BW3dB (4)

In the next section, we will link the components of this circuit to dimensions in the physical geometry of the filter.

3 Relationships Between Circuit Components and Physical Dimensions in the Microwave Filter

As explained in the previous section, every LC pair resonates at 400 MHz. In the microwave filter, we must choose the length of each resonator such that it resonates at 400 MHz. That will determine the length of each of them.

Further, Ki,i+1 (i=1,2,3) are the coupling coefficients between adjacent resonators. Therefore, these three coefficients are related to the distances between adjacent resonators.

Finally, QL is the loaded Q of the circuit. Therefore, in an otherwise lossless circuit, it is directly related to the distance between the first or final resonator and the antenna that couples it to the source or the load.

3a Relation Between Resonator Spacing and Coupling Coefficient

The model in Ansoft HFSS that was used to determine the coupling coefficient K as a function of resonator spacing is shown in Fig. 7. Two resonators have been placed in a closed metal cavity. This cavity has the same height and the same front-to-back depth as the cavity to be used in the real filter. The left-to-right width has been chosen large enough to make its influence on the results negligible. There are no transmission lines nor ports for signal input and output, since the resonances of this structure are to be determined through an eigenmode simulation. In the Setup Materials menu, resonators are modeled as perfect conductors; the cavity is filled with air. Further, symmetry has been exploited through the use of a Perfect-H boundary condition. As can be seen in the figure, this cuts both the resonators and the cavity in half.

Fig. 7 Model used to determine the coupling coefficient K

By embedding this HFSS project in Optimetrics, dimensions can be varied easily. First, the length of the cylinders was adjusted such that the resonances are centered at 400 MHz. Then, the distance between the resonators was varied, and for each distance the eigenmode solver in HFSS computed the two eigen frequencies and obtained K. In order to get very accurate results, more accurate than necessary, twelve adaptive passes were run in each simulation, resulting in models with 63,000 tetrahedra. Total run time for

each point was 38 minutes on a 1.2 GHz PC. The simulation of the lossless structure required 557 MB of RAM. The relation between the resonator spacing and K is shown graphically in Fig. 8.

Fig. 8 Relation between resonator spacing and coupling coefficient K

With this graph, for any coupling coefficient required by filter theory, the spacing to be applied between resonators in the physical model can be readily determined.

3b Relation Between Antenna Distance and Loaded Q

The model in Ansoft HFSS that was used to determine the loaded Q as a function of antenna spacing is shown in Fig. 9. An antenna-resonator combination has been placed in a closed metal cavity. The 50-Ω transmission line is present, but in order to perform an eigenmode analysis, it has been terminated by a Perfectly Matched Layer (PML) of absorbing material. This was done by replacing the final 20 mm of dielectric in the coaxial cable by PML material. A macro, named pmlmatsetup, in the Materials-Setup menu supplies the material parameters. This construction will give use the same resonant frequency and loaded Q as the corresponding structure with a real 50-Ω load would. The cavity has the same height and the same front-to-back depth as the cavity to be used in the real filter. Again, the left-to-right width has been chosen large enough to make its influence on the results negligible, and symmetry has been exploited through the use of a Perfect-H boundary condition.

Fig. 9 Model used to determine the loaded Q

This HFSS project has been embedded in Optimetrics. The antenna distance and the cylinder length were varied simultaneously, since both influence the resonant frequency and the loaded Q. As an example of the results, the relation between antenna spacing and loaded Q is shown graphically for a constant cylinder length of 113.4 mm. In order to get very accurate results, maybe a little more accurate than strictly necessary, fifteen adaptive passes were run for each point. This results in simulations with 50,000 tetrahedra, requiring 830 MB of RAM. Total run time per point was 50 minutes on a 1.2 GHz PC.

Fig. 10 Relation between antenna spacing and loaded Q at a resonator length of 113.4 mm

With results like these, for any loaded Q and resonant frequency required by filter theory, the antenna spacing and cylinder length to be applied in the physical model can be readily determined.

4 Initial Filter Design in HFSS

Now that we have the circuit and we know the relations between circuit components and physical dimensions, we can construct the filter in the field solver, Ansoft HFSS. Filter theory tells us we need to achieve the following parameters:

Resonant frequency of the outermost resonators fR1= 400 MHz Resonant frequency of the inner resonators fR2=400 MHz Loaded Q

Coupling coefficients K12=0.02893 , K23=0.02171 , K34=0.02065The calibration projects above tell us that the dimensions of the filter, as shown in Fig. 2, are to be

Length of the two outermost resonators = 113.399 mm Length of the five inner resonators = 114.69 mm Antenna distance = 1.879 mmDistances between resonators are 25.513 mm , 28.291 mm , 28.767 mm

This filter has been modeled and simulated in Ansoft HFSS. The model is shown in Fig. 11.

Fig. 11 Initial design in HFSS

Notice that only half the geometry is actually simulated. Symmetry has been exploited through the use of a Perfect-H boundary condition. Further, all materials and boundaries in the model are lossless for now. This requires less RAM and less CPU time. The resulting filter characteristic is shown in Fig. 12. Notice that the center frequency and the ripple bandwidth are almost perfect. We see the correct number of ripples, but the ripple is 0.3 dB rather than 0.1 dB.

Fig. 12 S21 results for the initial design in HFSS

5 Curve Fitting in Serenade

The HFSS results, shown in the previous section, have been exported to Serenade, the circuit simulator. This was done through Post Process / Matrix Data / File / Export / Touchstone. In Serenade, we can determine through curve fitting what the actual parameters of this initial design are. This curve fitting is done through the Serenade setup shown in Fig. 13. All the variables are defined in the top level schematic. Circuit model and HFSS results are defined via the sub circuits denoted as MODEL and MEASU, respectively. To utilize all the available information for the optimization process, the optimization goal is to end up in a complex S-Matrix identical for the model and the S-matrix resulting from the HFSS simulation. This is defined in Serenade via an OPT block and the goal definition S=MEASU in the sub circuit defining the circuit model as shown in Fig. 14. To match the phases of the S parameters of the Serenade and HFSS simulations, homogeneous transmission lines of length ZUL are attached to ports 1 and 2 in the Serenade model. Optimization is done by starting with the optimum filter parameters given in Fig. 6 and performing 1500 iterations with the random optimizer. The solution was found without any manual interaction.

It took 35 minutes on a PC with a clock speed of 400 MHz. Figs. 15 through 17 show curve fitting results. Notice, in Fig. 15, that there is still a few hundredths of a dB difference between the HFSS results and the best fit in Serenade. This indicates that this design method is accurate to a few hundredths of a dB.

Fig. 13 Serenade setup used for curve fitting: top level schematic

Fig. 14 Serenade setup used for curve fitting: Model definition as sub circuit

Fig. 15 Result of curve fitting, magnitude

Fig. 16 Result of curve fitting, phase

Fig. 17 Results of curve fitting, complex S11 Blue and green lines are S_11 andS_22 from HFSS, which have the same magnitude but slightly different phases; red line is the best fit.

The result of the curve fitting procedure is as follows: we have built a filter with

Resonant frequency of the outermost resonators fR1= 400.058 MHz Resonant frequency of the inner resonators fR2=399.926 MHz Loaded Q QL=30.368Coupling coefficients K12=0.02825 , K23=0.02173 , K34=0.02068

Notice that the largest discrepancies occur in K12 and QL. Apparently, the calibration project that determines the coupling coefficient by simulating two identical resonators is not quite representative of the two outermost pairs of resonators, where one resonator is coupled through an antenna to the source or the load. Also, the calibration project that determines the loaded Q by simulating one resonator-antenna combination is not perfectly representative of the real situation where this resonator is coupled to a neighboring one.

Nevertheless, the calibration projects tell us how much correction is needed to achieve the desired characteristic. For example, noticing that QL is too low by a certain amount, we will aim for a QL that is higher by this amount the second time. Caution is needed when adjusting the antenna distance, since that also changes fR1. We have to change antenna distance and resonator length simultaneously, and aim for the correct QL and fR1.

Keeping this in mind, with the aid of the calibration projects we find that the dimensions of the filter are to be

Length of the two outermost resonators = 113.44 mm Length of the five inner resonators = 114.684 mm Antenna distance = 1.928 mmDistances between resonators are 25.286 mm , 28.3 mm , 28.78 mm

Hence, the dimensions that undergo the largest changes are the antenna distance and the distance between the first and second resonator.

6 Corrected Filter Design in HFSS

The corrected filter was modeled and simulated in Ansoft HFSS. The resulting characteristic and the corresponding Smith chart are shown in Figs. 18a and 18b. Note that the ripple, which was 0.3 dB in the initial design, is better than 0.13 dB now. The target is 0.1 dB.

Fig. 18a S21 results in HFSS for the corrected design

Fig. 18b Smith chart in HFSS for the corrected design

In order to obtain this result, the mesh was refined adaptively until it had 180,000 tetrahedra. With a mesh that size, the calculation of each frequency point required 1.28 GB of RAM and 9.5 minutes real time on a 1.2 GHz PC with one processor. Seventeen frequency points were needed for an interpolating frequency sweep, bringing the total time needed for the sweep to two hours and forty minutes. An identical model with only 119,000 tetrahedra (see below) provided results within a few hundredths of a dB in the pass band and saved almost half the time.

7 Additional Information from the 3D Field Solver7a Effects of Internal Losses

All simulations thus far have been performed with lossless filters. A simulation without loss results in computations with real numbers only, as opposed to computations with complex numbers. This reduces the RAM requirement and the CPU time significantly. Once the design has been finalized, however, one can easily change the material parameters and boundary conditions to go from perfectly-conducting metals to lossy metals like copper or silver. The software enables you to select materials from a database or specify the conductivity. A plot comparing a lossless and a silver filter is shown in Fig. 19. Notice that, due to the seven consecutive resonances, even with a very good conductor like silver the insertion loss will be between 0.5 and 1 dB.

Fig. 19 Comparison lossless filter and silver-plated filter

Also note that the center frequency of the silver filter is slightly lower than the center frequency of the perfect filter. A careful inspection of the data shows that this shift is between 0.07 and 0.08 MHz. A model with just one resonator shows the same shift. Further investigation reveals that this shift is due to the imaginary part of the surface impedance of the silver. According to electromagnetic theory, the conductivity of the silver translates into an equivalent surface impedance, provided that the metal thickness is much larger than the skin depth. This surface impedance has a real and an imaginary part, which are both equal to √ (πfμ0μR/σ), where f is the frequency, μ0μR is the permeability of the material, and σ is the conductivity of the material. In the case of silver at 400 MHz the surface impedance is Zsurface = 5(1+j) mΩ/square. A simulation in HFSS with Zsurface = 5 mΩ/square shows no frequency shift at all relative to the perfect conductor case, while a simulation with Zsurface = 5j mΩ/square shows the same shift as in Fig. 19.

Curve fitting with Serenade shows that replacing perfect conductors by silver in HFSS is equivalent to introducing an unloaded Q of 2,800 in each resonator in Serenade. According to filter theory, the introduction of an unloaded Q shifts the resonant frequency downward by fr/(2Q), which in this case equals 0.07 MHz. Hence, HFSS has predicted this frequency shift very accurately.

In order to account for this shift, designers should first determine the magnitude of the shift with an HFSS simulation involving just one resonator. Then, they should design a perfectly lossless filter around a frequency that is higher by this amount. The center frequency of the filter with internal losses will thus come out just right.

The computer requirements were as follows. These computations have been performed with a model with 119,000 tetrahedra. In the lossless case, this took 810 MB RAM, 207 MB disk, and 86 minutes real time on a PC with a clock speed of 1.2 GHz and one processor. In the lossy case, it took 1,332 MB RAM, 1,600 MB disk and 396 minutes real time. The large time difference is due to the change from a real to a complex solver and to the time needed for disk access. The disk access in this case is ‘spill logic’, which is a deliberate process, performed under the software’s control. It is not to be confused with the very inefficient ‘swapping’ which is done by the operating system when a process is too large for the available RAM.

7b Maximum Power-Handling Capability

It is important to know how much power the filter can handle. The maximum power handling capability can be obtained easily with the help of a field plot. Fig. 20 shows a close up of the fields around a resonator in the region where they are strongest.

Fig. 20 Fields around a resonator

The HFSS 3D Fields Post Processor tells us that, with 1 W input power, the electric field strength between the cylinder and the bucket is 105 kV/m. You can change the input power in the post processor (Data/Edit Sources). The filter would cease to operate when the fields are strong enough to cause arcing in the air. This phenomenon occurs at 3 MV/m, although, with a wide safety margin, 1 MV/m is commonly used as the maximum acceptable field strength. Therefore, the fields can be allowed to be 9.5 times as strong as they are now, which implies that the maximum power handling capability is 9.5×9.5 = 90 W.

7c Mechanical Tolerances

Once the dimensions are known that provide a filter with the desired specifications, it is important to establish mechanical tolerances. With the HFSS model fully parameterized in Optimetrics, it is an easy task to explore the effects of small dimensional changes on the filter characteristic. An example is shown in Fig. 21. There,

the distance between two resonators was made 0.08 mm larger and 0.08 mm smaller. The original characteristic and the two modified ones are shown. In this case, a manufacturing inaccuracy of 0.08 mm in the distance between two resonators results in a change of up to 0.05 dB in the filter characteristic. This way, mechanical tolerances, depending on the accuracy requirements of the filter characteristic, can be specified.

Fig. 21 Example of the effects of manufacturing tolerances

References

[1] Daniel G. Swanson and Robert J. Wenzel, “Fast Analysis and Optimization of Combline Filters Using FEM”, presented at the IEEE MTT Society 2001 International Microwave Symposium, May 2001. [2] Randall W. Rhea, “HF Filter Design and Computer Simulation”McGraw-Hill, Inc., 1995ISBN 0-07-052055-0

Appendix A Derivation of the Circuit

In this appendix, the well known low-pass prototype method for filter designs is repeated. Only the necessary facts for the present example are given. For more details a standard book on filter designs should be used [A1].

The starting point of this method is the low-pass prototype as shown in Figs. A1 and A2. These filters are normalized to a cut-off angular frequency of 1 rad/s and a generator impedance of 1 Ohm. In the case of a Chebychev filter some care has to be taken regarding the order of the filter. For an odd number of elements the load impedance is also 1 Ohm. For an even number of elements, however, the load impedance depends on the order of the filter and the ripple. Therefore, the two cases are treated separately.

Fig. A1 Low-pass prototype for the case N=odd

Fig. A2 Low-pass prototype for the case N=even

In the past, the prototype values g i (i = 1 … N+1) were read from tables, but nowadays it is more convenient to use a filter design program. Chebychev filters are defined by the filter order N and the in-band ripple. Closed-form expressions exist for the g-values. An example is given in Fig. A3. This MathCAD program is valid for even as well as odd order filters for any ripple value. Essentially the g-values are defined by a recursive relation. Only for the last value a special treatment for the even and odd order case is necessary. This is considered by an if–statement with the mod–function as condition. Exact definitions of these functions are given in the MathCAD handbook [A3]. The response of this filter is shown in Fig. A4.

Fig. A3 MathCAD program to derive the g-values for Chebychev filter responses. Example: N = 6, ripple = 0.1 dB.

Fig. A4 Chebychev filter response, N = 6, ripple = 0.1 dB

At higher frequencies it is often impractical to realize both series elements and shunt elements in one circuit. To avoid this, a new element, the impedance inverter, is introduced. In the following a procedure is given to transform each shunt capacitor in Fig. A2 to a series inductor with impedance inverters on each side. In the practical band-pass circuit, these elements are realized by coupling structures between the individual resonators.

Fig. A5 Impedance transformation with an impedance inverter

An impedance inverter is an element with a behaviour like a quarter-wave transmission line but independent of frequency. Fig. A5 shows the symbol of an impedance inverter. An impedance at the output of the circuit appears as the inverted impedance at the input, scaled with the square of the impedance inverter constant K. In addition the phase of a signal travelling through the circuit is shifted by 90°. Both the impedance change and the phase shift are frequency independent.

In Serenade such a circuit is realized by a transmission line with adaptive electrical length. Fig. A6 shows the schematic. As the frequency f is a global variable it is an easy task to create the impedance inverter with the electrical length defined by the appropriate formula.

Fig. A6 Realization of impedance inverter in Serenade

Prior to the transformation of the shunt capacitors to series inductors some basic relationships of impedance inverter circuits are derived. Starting with the chain circuit of a first impedance inverter, an inductance and a second identical impedance inverter, an equivalent impedance is derived. This is demonstrated in Fig. A7. Effectively, the equivalent input impedance seen by the generator is a parallel circuit of a capacitor and the original load impedance as shown in Fig. A8. To derive an identical circuit, a phase shift of 180° is introduced. This phase shift may be omitted in the case of Chebychev filters as these types of filters are pure chain circuits. However, in more advanced circuit like elliptic filters employing cross couplings this could be of importance. The procedure shown here proves only the equivalence of the input impedances rather than the equivalence of the entire circuits. However, the latter could be proven easily as well by comparing the chain matrices.

Fig. A7 First identity, part 1

Fig. A8 First identity, part 2

Fig. A9 Second identity

By multiplying the series inductance by a factor of x and simultaneously multiplying the impedance inverter constant K by √x, the input impedance remains constant. This leads to the second identity shown in Fig. A9. Again, the proof is only based on an identical input impedance seen by the source. The complete proof could be derived easily by comparing the chain matrixes. With these identities it is a straightforward procedure to replace the shunt capacitors by series inductors in the prototype low-pass filters. This is illustrated in Fig. A10 for the case N = odd. Both circuits are identical.

Next, the cut-off frequency is de-normalized by dividing the inductance values by the relative bandwidth bw. According to the identity jL = j( bw)L/bw, the impedance values shift from frequency to frequency bw when we replace the inductance L by inductance L/bw as shown in Fig. A11. Fig. A12 shows the results for the cases bw = 1 and bw = 0.5.

At this point, the inductors still have different values. In order to arrive at a circuit with identical elements, the inductance values are set to one and the impedance inverter constants are adjusted according to the second identity illustrated in Fig. A13.

Fig. A10 Low-pass prototype with series elements only; N = odd

Fig. A11 Low-pass prototype with de-normalized cut-off frequency; cut-off frequency bw.

Fig. A12 Chebychev low-pass filter response: N = 7, ripple = 0.1 dB, Cut-off frequency bw = 0.5 (blue traces) and bw = 1 (red traces)

Fig. A13 Low-pass prototype with identical reactive elements; N = odd.

In the following step each inductance is replaced by a series resonator consisting of an inductance of 1 H and a capacitance of 1 F to design a band pass filter around the angular frequency of 1 rad/s and a relative bandwidth bw. Fig. A14 shows the circuit and Fig. A15 shows the filter responses before and after this change.

Fig. A14 Normalized band-pass filter with relative bandwidth bw and center angular frequency of 1 rad/s

Fig. A15 Chebychev low-pass filter response, red and blue traces: N=7, ripple=0.1 dB, cut-off angular frequency bw=0.25 rad/s andChebychev band-pass filter response, pink and green traces: N=7, ripple=0.1 dB, center frequency 1 rad/s, relative bandwidth bw=0.25.

To arrive at the final filter model the center frequency is de-normalized to get a center frequency of fr. Starting by the well know formula for the resonant frequency

the inductor values and capacitor values must be chosen properly. When scaling the inductor L1 = 1 H and the capacitor C1 = 1 F of the normalized band-pass filter by the same factor s, the relative bandwidth stays constant. This is because the normalized impedance of the individual resonators,

Z1() = j (L1 – 1/C1),

and the impedance of the scaled resonators

Zs () = j( (L1/s) –1 / (C1/s) )

have the relation

Z1 () = Zs (s),

i.e. the complex resonator impedance values of Z1 at the angular frequency are identical to the complex impedance values of the scaled complex resonator impedance at the frequency s. The resonant frequency is s times larger while the relative bandwidth is not altered. Therefore, we choose the following values for the scaled inductance Ls and the scaled capacitance Cs

Ls = 1 / (2 fr) Henry,

and Cs = 1 / (2 fr) Farad.

Additionally, the port impedance is de-normalized to the usual 50 Ohms by impedance inverters at the input and the output. To be able to compare HFSS S-parameters and model parameters, feed lines are also introduced. For the impedance converter constants the usual notations Kij = bw / (gi * gj)0.5 for the inner ones and Q1 = g1 / bw for the outer ones are introduced.

One last step is necessary: the definition of a finite unloaded quality factor Q0 of the individual resonators. This is taken into consideration by an appropriate series resistor. The value of this resistor is Rs = 2 fr Ls / Q0 [A2]. In our circuit, since Ls = 1 / (2 fr),

Rs = 1 Ohm / Q0 .

For filters, where the losses are caused by a metallic casing, there exists also a frequency shift due to the imaginary part of the surface impedance. If the normal skin effect is applicable, real part and imaginary part of the surface impedance are identical. Therefore, in this case, it is possible to express this frequency shift by the unloaded quality factor Q0. According to electromagnetic field theory the relative frequency shift Δf is

Δf = -1 / 2Q0 .

Fig. A16 shows the final Serenade filter model. Fig. A17 shows the corresponding filter response for the lossless case and Fig. A18 shows the filter response in the case when metallic losses are present and the normal skin effect is applicable. This model is not only useful to asses insertion loss and frequency shift, but could also be used to gain an equal ripple filter in the case of internal losses by optimizing the K and Q values properly.

Fig. A16 Final Serenade filter model, N = 7, ripple = 0.1 dB, center frequency fr = 400 MHz, relative bandwidth bw = 0.0375

Fig. A17 Chebychev band-pass filter response, N = 7, ripple = 0.1 dB,center frequency fr = 400 MHz, bandwidth 15 MHz

Fig. A18 Chebychev band-pass filter response, N = 7, ripple = 0.1 dB,center frequency fr = 400 MHz, bandwidth 15 MHz . No internal losses (blue trace) and Q0=2700 (red trace).

Notice that, in the circuit in Fig. A16, we have introduced quantities K and Q in the impedance inverters. K and Q are defined by

and

where bw is the relative bandwidth and gi is the ith g value from filter theory. In Appendix B we will show that K has the physical meaning of coupling between two resonators and Q has the physical meaning of loaded Q of a circuit with one resonator. These physical meanings are important parts of the overall design strategy, as they are used in the HFSS calibration projects.

The derivation of the final Chebychev band pass filter of even order out of the appropriate low-pass prototype is done by the same procedure. Special treatment is only necessary at the output. Adding an impedance inverter at the output with the impedance inverter constant of [g (N+1) ] 1/ 2 realizes a constant load impedance of one. Fig. A19 explains this special step. After this step the identical procedure as in the case of odd-order filters is used to arrive at the final Serenade filter model. After carrying out all steps it turns out that the filters of even order are also symmetric although the original g-values show no symmetry at all. The circuit is shown is Fig. A20 and the simulation results for N = 6 and a ripple of 0.1 dB are given in Fig. A21.

Fig. A19 Low-pass prototype with series elements only; N = even

Fig. A20 Final Serenade filter model, N = 6, ripple = 0.1 dB, centerfrequency fr = 400 MHz, relative bandwidth bw = 0.0375 (15 MHz)

Fig. A21 Chebychev band-pass filter response, N = 6, ripple = 0.1 dB,center frequency fr = 400 MHz, bandwidth 15 MHz

References Appendix A

[A1]G. Matthaei, L. Young, E.M.T. JonesMicrowave Filters, Impedance – Matching Networks, and Coupling StructuresARTECH HOUSE, INC. 1980, ISBN 0-89006-099-1

[A2]D.M. PozarMicrowave Engineering, second edition 1998John Wiley & Sons, Inc., ISBN 0-471-17096-8

[A3]MathCAD, User’s Guide, MathCAD 2000 ProfessionalMathSoft, Inc.101 Main StreetCambridge Massachusetts 02142USAhttp://www. mathsoft.com/

Appendix B The Physical Meanings of K and Q

In the derivation of the filter circuit in Appendix A, we introduced two quantities, K and Q, defined as

and

where bw is the relative bandwidth and gi is the ith g value from filter theory. As the derivation in Appendix A shows, K is the impedance inverter constant of an impedance inverter between two neighbouring resonators, and ( 1 / Q )0.5 is the impedance inverter constant of an impedance inverter between the first (or last) resonator and the 1 Ω source (or load).

These K and Q, seemingly introduced merely for notational convenience, have important physical meanings. K is equal to the coupling constant between the two resonators, which is defined commonly as

K = 2(f2-f1)/(f2+f1),

where f1 and f2 are the resonant frequencies of a circuit that consists of just two coupled resonators. Q is equal to the loaded quality factor QL of a circuit that consists of only one resonator coupled to a normalized load impedance.

In this appendix, we will prove that K and Q, as used in Appendix A in the impedance inverters, indeed have these physical meanings of coupling constant and loaded quality factor. These physical meanings of K and Q are crucial parts in the filter design strategy, because the calibration projects in HFSS, where relations are established between K and Q and certain physical dimensions in the filter, rely entirely on these meanings.

We start with the physical meaning of K. Consider the circuit depicted in Fig. B1, consisting of two coupled resonators without a source or a load. The resonators are coupled through the impedance inverter with constant K. We need to prove that this circuit has resonant frequencies f1 and f2 that are related to K via the equation

K = 2(f2-f1)/(f2+f1).

Fig. B1Circuit used for the determination of the relation between impedance inverter constant K and resonant frequencies f1 and f2

Applying Kirchhoff’s Voltage Law to the loop left of the impedance inverter leads to

An obvious trivial solution to this equation is I0=0. We are looking for resonances, so we require that non-trivial solutions exist. This is the case if

Rewriting this so every term obtains the same denominator leads to

To derive a solution which makes physical sense, one has to solve

Upon multiplying out the bracket term and dividing by LC a standard quadratic equation in 2 results:

The two solutions are

The application of Vieta’s theorem gives the two relations

For the resonator models the condition

has been chosen. Together with the second Vieta relation this leads to

Now the first Vieta condition is solved for the impedance inverter constant K as function of the two eigenvalues

This can be rewritten as

Replacing the geometric mean by the arithmetic mean leads to the well-known formula:

or

In practical cases, where the eigenfrequencies are close together the arithmetic mean and the geometric mean give almost identical results. This completes the proof for K.

We now proceed with the physical meaning of Q. Consider the circuit depicted in Fig. B2, consisting of one resonator, coupled to a 1-Ω load impedance through an impedance inverter. The impedance inverter has coupling constant 1/Q, denoted here for brevity as K. We need to prove that the quality factor of this circuit is equal to Q.

Fig. B2Circuit used for the determination of the relation between impedance inverter constant and quality factor

Effectively, the setup is a series resonant circuit with a series resistor of K2 Ohm, an inductance L and a Capacitor C. According to basic electrical-engineering theory, the quality factor of such a circuit is defined as

and the resonant frequency is equal to

Substituting the latter expression for fr in the equation for Q gives

For our resonators we have chosen the condition

Therefore the impedance inverter constant K, depicted in Fig. B2, is equal to

where, as said earlier, Q is the quality factor of this circuit. This completes the proof that the Q, as introduced in Appendix A, has the physical meaning of the loaded quality factor of the circuit of Fig. B2.