GENESYS S/FILTER Copyright 1986-2000 Eagleware Corporation 635 Pinnacle Court Norcross, GA 30071 USA Phone: (678) 291-0995 FAX: (678) 291-0971 E-mail: eagleware@eagleware.com Internet: http://www.eagleware.com Printed 10/2000 Printed in the USA
Table of Contents Chapter 1: Introduction... 5 Overview...5 General Information...5 Feature Overview...5 S/FILTER Documentation...6 Chapter 2: Starting... 7 Overview...7 First Example...10 Second Example...14 Third Example...17 Chapter 3: Filter Design Concepts... 21 Overview...21 Transmission Zeros...22 Element Extractions...23 Extraction Rules...24 Extraction Examples...25 Symmetry Versus Asymmetry...28 Electrical Symmetry In Lowpass Filters...29 Electrical Asymmetry In Lowpass Filters...30 Electrical Symmetry In Highpass Filters...30 Electrical Asymmetry In Highpass Filters...30 Electrical Symmetry In Bandpass Filters...30 Electrical Asymmetry In Bandpass Filters...31 Chapter 4: Transforms... 33 Overview...33 Examples Of When To Use Transforms...34 Norton Series Transform...38 Norton Shunt Transform...41 Norton Scale Series Element Of Pi...44 Basic Equations... 45 Norton Scale Shunt Element Of Tee...46
L-Left To L-Right Transform...48 L-Left To Pi...50 Description...50 L-Left To Tee Transform...53 L-Right To L-Left Transform...56 L-Right To Pi Transform...58 L-Right To Tee Transform...61 Pi To L-Left Transform...64 Pi To L-Right Transform...66 Pi To Pi Transform...68 Pi To Tee Transform...72 Pi To 2-Pi Transform...74 Symmetric Pi To Symmetric 2-Pi Transform...76 Tee To L-Left Transform...78 Tee To L-Right Transform...80 Tee To Pi Transform...82 Tee To Tee Transform...84 Tee To 2-Tee Transform...87 Move TRF Left One Transform...89 Move TRF Leftmost Transform...90 Move TRF Right One Transform...91 Move TRF Rightmost Transform...92 Absorb TRF Into Source Transform...93 Absorb TRF Into Load Transform...94 Combine All TRFs Transform...95 Split Series Element Into 2 Parts Transform...96 Split Shunt Element Into 2 Parts Transform...98 Delete Element Transform...100 Swap Element With The One To The Right Transform...101 Chapter 5: Filter Synthesis Overview...103 Overview...103 Filter Synthesis...103 Foster's Reactance Theorem...104 Extraction of Finite Zeros...107 Chapter 6: Examples...111 2
How To Design...111 Equal Termination Example...112 Maximum Realizability Example...114 All Series Resonators Example...116 All Parallel Resonators Example...118 Response Symmetry Example...121 Equal Inductor Example...124 Physical Symmetry Example...125 Termination Coupling Example...130 Distributed Elements Example...132 Parametric Bandpass Example...136 Chapter 7: Error Messages... 141 Chapter 8: Reference... 147 The Specifications Tab...147 The Extractions Tab...149 The Transforms Tab...152 The History Tab...154 The Extraction Goals Dialog...156 The Customize Permutation Table Dialog...158 The Shape Wizard...161 3
Chapter 1: Introduction Overview Thank you! Eagleware is proud of a tradition of highperformance, high-quality engineering software. Any suggestions you have are important to us, so please tell us about your experience with our products. General Information This manual describes Eagleware's direct filter synthesis program, S/FILTER. For installation/starting information on all GENESYS products including S/FILTER, please refer to the Installation Guide or to the Getting Started guide. Feature Overview S/FILTER is a powerful tool for filter design. Principle features include: Lowpass, highpass, and bandpass filter design Directly synthesize multiple solutions for each design Tune zero locations to optimize a filter design Design classic transfer function types using the Shape Wizard (Butterworth, Chebyshev, Elliptic) Asymmetrically place DC and infinite zeros to increase high- and/or low-side filter selectivity for overall response symmetry Enter criteria for design solutions - based on component values, ratios, etc. Completely customize the solution table - display solutions and sort them by any parameter Control extraction sequence with series or shunt first element Over 100 circuit transformations available including Norton, Pi and Tee
Introduction Apply transforms to create component symmetry Apply transforms to create equal-inductor filters Design exact solutions for narrow- and wide-band filters Create arbitrary stopbands by placing and tuning finite transmission zeros Remove transformers automatically by applying the "Remove Transformer" macro Design "inexact" filters TIP: Some filter extraction sequences result in inexact transfer function approximations. S/FILTER has an option to allow these extractions, which may exhibit some nonexact response behavior. These filters often have fewer parts or more favorable topologies than the exact solutions have, and can be optimized or tuned to exhibit more exact behavior. S/FILTER Documentation This manual describes the use of S/FILTER to design filters. S/FILTER works in conjunction with the SuperStar circuit simulator and SCHEMAX, the schematic editor. Details concerning these programs are given in the Simulation manual. Described in the Examples manual are example filters which have been designed with S/FILTER. Files for these examples are provided with GENESYS. 6
Chapter 2: Starting Overview S/FILTER is launched by clicking S/FILTER on the Synthesis menu within the GENESYS environment, as shown below. The S/FILTER screen appears. The screen should now look similar to the figure below:
Starting S/FILTER is fully integrated into the GENESYS environment so that graphs and schematics are fully customizable. The S/FILTER window is shown in the upper-left of the figure above. S/FILTER has four tabs inside its window: 1. Specifications Tab Initial electrical design parameters are entered into this tab, such as cutoff frequencies and transmission zero locations. 2. Extractions Tab 8
Overview Physical realization parameters are entered in this tab, such as desired element values. S/FILTER fills in the table with solutions fitting the desired electrical criteria from the Design Tab, and (if specified) the physical criteria from the Extractions Tab. 3. Transforms Tab Circuit transformations, such as Norton and Pi/Tee transforms, are applied on this tab. 4. History Tab This tab contains any actions performed on the filter since a solution was picked on the Extractions Tab. Any transforms applied to the filter are shown in the History 9
Starting Table. The History Table also functions as an "undo" list, since you can see everything done to the filter. By clicking on any row in this table, S/FILTER switches the schematic to reflect the selected point in the design process. This makes it very easy to return to any point in the filter design. First Example Illustrates: Using the Shape Wizard Generating schematics with series or shunt elements first The following filter is to be designed: 50 Ω source and load terminations Lowpass filter with 3 db cutoff at 100 MHz Butterworth (maximally flat) response Third order filter Minimum number of inductors Start S/FILTER by clicking "S/FILTER" in the GENESYS Synthesis menu. Enter '50' into the Source and Load boxes on the Specifications Tab. This sets the terminations to 50 Ω each. Click the Shape Wizard button on the Specifications Tab as indicated in the figure below: 10
First Example The following dialog appears. Fill in the prompts as shown: Click OK. The Shape Wizard fills in the S/FILTER prompts based on the desired response shape. This tool is used when a classic response shape is needed. The S/FILTER specifications tab should now look like the figure below. 11
Starting Notice that the Shape Wizard has filled in several prompts. The Lower Cutoff frequency has been set to 100 MHz, as requested. The filter type has been set to lowpass, and the shape has been set to maximally flat. Also, notice the ripple value is set to 3 db. In a maximally flat filter, S/FILTER uses this value as the cutoff frequency attenuation. The Zero Table contains information on the order of the filter. The DC row has been disabled because by definition, lowpass filters cannot have a transmission zero at DC. Since Butterworth filters are all-pole, there are no finite zeros defined in the grid; only zeros at infinity. The three zeros at infinity make this a third order filter. The response for this filter is shown below. If your response is scaled differently, click the Rescale Plot button on the Specifications Tab. 12
The attenuation at 100 MHz is 3 db, as expected. First Example Next, click the Extractions Tab to select a schematic for this filter. The Extractions Tab is shown below: Make sure the Series Element First checkbox is selected, as shown above. This indicates that the filter should have a series element as the leftmost component (adjacent to the source termination). The Total Permutations prompt indicates that there is only one unique way to arrange the three transmission zeros in this filter: Also, notice that the Solution Table only contains one entry. The S/FILTER schematic always represents the highlighted row in the Solution Table. The schematic should look like the one shown below: This filter has two inductors. Uncheck the Series Element First checkbox to generate a filter with a shunt element first. The schematic below appears: 13
Starting This schematic only contains one inductor, so it is the minimum inductor solution. Although the two schematics are unique, they both represent the same zero extraction order ( ). In other words, a shunt capacitor realizes a transmission zero at infinity, as does a series inductor. So, the number of solutions reported for a filter in either configuration can generate the same number of schematics for the alternate topology. Second Example Illustrates: Unique and non-unique extractions The following filter is to be designed: 50 Ω terminations Chebyshev (equiripple) response with 0.25 db passband ripple Bandpass filter with passband from 50 to 75 MHz Second order response Enter '50' into the Source and Load boxes on the Specifications Tab. This sets the source and load terminations to 50 Ω each. Enter the following data into the Shape Wizard: 14
Second Example Click OK. The S/FILTER window should look like the figure below: The Shape Wizard has set the cutoff frequencies and passband ripple to the values requested. Notice that the Zero Table contains 2 zeros at DC, and 2 zeros at f=. Conventional bandpass filters are symmetric with respect to the number of DC and infinite zeros. This corresponds to a 2nd order (4th degree) bandpass filter. Next, click the Extractions Tab. The Extractions Tab should look like the figure below. 15
Starting Make sure the Series Element First checkbox is selected. The total number of permutations is reported as 6, but only 3 unique solutions are found. There are six unique permutations of the 4 zeros: 1. DC DC 2. DC DC 3. DC DC 4. DC DC 5. DC DC 6. DC DC These six permutations correspond to the following six schematics: 16
Third Example Note: If the "Series Element First" option is unselected, six more schematics are generated for the six transmission zero permutations shown above, each schematic having a shunt element adjacent to the source. Notice that schematic #2 and schematic #3 are nearly identical, except for the series resonator element ordering. Electrically speaking, these two schematics are identical since simply changing the resonator element order does not change either the resonant frequency or the impedance presented to surrounding elements. Also, notice that schematic #4 and schematic #5 are identical to #2. Therefore, only schematics 1, 2, and 6 are unique. S/FILTER automatically detects these redundant solutions, and only includes 1 occurrence in the Solution Table. See the Filter Design Concepts chapter for a discussion of extraction theory. Third Example Illustrates: Finite Zeros The following filter is to be designed: 50 Ω terminations Equiripple response with 0.1 db passband ripple Bandpass filter with passband from 100 to 150 MHz 17
Starting Insertion loss greater than 30 db below 75 MHz and above 200 MHz Enter the following data into the Specifications Tab: One zero at DC and at f= has been specified. This gives a minimum order bandpass filter. The finite zeros at 75 MHz and 200 MHz give nearly infinite attenuation at their resonant frequencies. The response for this filter is shown below: 18
Third Example The insertion loss is greater than 30 db at 75 MHz and at 200 MHz, but returns to about 26 db below 75 MHz and above 200 MHz. Tuning the finite zero frequencies improves the far-out insertion loss slightly, but it also degrades the insertion loss at 75 and 200 MHz. So, two more transmission zeros are added at f=. The plot below shows the new filter response with two more transmission zeros at f= : The insertion loss now remains greater than 30 db below 75 MHz and above 200 MHz. 19
Chapter 3: Filter Design Concepts Overview Filters are circuits used to shape the frequency response of networks in which they are placed. S/FILTER synthesizes lossless filters. Losses of practical elements can be incorporated after synthesis by replacing ideal elements with their practical models. This chapter presents a short review of basic concepts and definitions related to classical filter synthesis. Some commonly used approximation types for filters that are available in S/FILTER are as follows: A. Maximally flat behavior in both passband and stopband (Butterworth filters) B. Maximally flat passband, equal ripple stopband (Inverse Chebychev filters) C. Maximally flat passband, general stopband D. Equal ripple passband, maximally flat stopband (Chebyshev filters) E. Equal ripple passband, equal ripple stopband (Elliptic filters) F. Equal ripple passband, general stopband
Filter Design Concepts Transmission Zeros Transmission zeros are critical frequencies where signal transmission between input and output is stopped. S/FILTER uses the transmission zero frequencies together with the passband edge frequencies and passband ripple to form the transfer function between the input and output of the filter, and for shaping the response of the filter. Transmission zeros must always be placed in the stopband(s) of a filter. Placement of transmission zeros into passbands is automatically prevented by S/FILTER. The following transmission zero types are available: 1. Transmission Zero at DC A transmission zero at f=0 (DC) adds one degree to the filter transfer function. These transmission zeros are needed in highpass and bandpass filters. Increasing the number of transmission zeros at DC increases the selectivity (slope) of the filter in the lower stopband more than it increases the upper stopband selectivity. During extraction, each transmission zero at DC produces a series capacitor or a shunt inductor. 2. Transmission Zero at Infinity A transmission zero at f= adds one degree to the filter transfer function. These transmission zeros are needed in lowpass and bandpass filters. Increasing the number of transmission zeros at f= increases the selectivity of upper stopband more than it increases the lower stopband selectivity. During extraction, each transmission zero at f= produces a series inductor or shunt capacitor. 3. Transmission Zero at Finite Frequency A finite-frequency transmission zero adds two degrees to the filter transfer function. During extraction, finite transmission zeros are realized either as parallel LC resonators in series or series LC resonators to ground. The parallel resonator in series stops signal flow by being an open circuit at the resonant frequency while the series resonator to ground becomes a short circuit at the resonant frequency. 22
Element Extractions Element Extractions In S/FILTER, elements are extracted from either Z(s) (the network impedance function) or Y (s) (the network admittance function). These functions are purely reactive because the output port is either open or short circuited, eliminating the only resistive component of the circuit: the load termination. Therefore, S/FILTER is able to design a filter as a one-port circuit, deferring load matching until the final step. See Appendix A, Filter Synthesis Overview, for information on synthesis basics. Element extraction produces a circuit that possesses the transmission and reflection zeros immersed in the desired gain function. The simplest practical structures are obtained by extracting groups of elements called "transmission zero sections" in cascaded form. Each section is responsible for realizing a transmission zero. A transmission zero is formed either by creating an open circuit in the series arm or by creating a short circuit in the shunt arm to stop signal flow between input and output. The three kinds of transmission zeros on s=jw axis are realized as follows: Transmission zeros at DC can be realized by either a series capacitor (becoming a series open circuit at DC) or a shunt inductor (becoming a shunt arm short circuit at DC), as shown below: Transmission zeros at f= can be realized by either a series inductor (becoming a series open circuit at f= ) or a shunt capacitor (becoming a shunt short circuit at f= ), as shown below: Finite-frequency transmission zeros can be produced either by a parallel resonator in series (becoming a series open circuit at resonance) or by a series resonator to ground (becoming a short circuit at resonance), combined with a partial extraction at DC or f= (see Appendix A), as shown below: 23
Filter Design Concepts Extraction Rules The partial extraction process that S/FILTER uses on finite transmission zeros requires that a transmission zero at DC or infinity be extracted after the finite zero to absorb the remaining partial zero. S/FILTER has an option to allow extractions that violate this rule, at the expense of non-ideal responses. Permutations that violate this rule are called inexact. The following rules apply to the extraction of transmission zeros: 1. The final zero at infinity cannot be extracted until all finite zeros in the upper stopband have been extracted. For example, if a lowpass filter has 2 transmission zeros at infinity and one at 500 MHz, there are 3 unique permutations. They are shown below: 1. 500 2. 500 3. 500 The first two permutations are exact, since zeros at infinity are extracted last. Permutation #3, however, is inexact because the finite zero is extracted last. 2. The final zero at DC cannot be extracted until all finite zeros in the lower stopband have been extracted. For example, a highpass filter with a finite stopband zero must have DC as the final extraction. In general, lowpass filters must obey rule #1, whereas highpass filters must obey rule #2. Bandpass filters have both upper- and lower-stopbands, and therefore must obey both rules. For example, consider a bandpass filter with 2 zeros at DC, 2 zeros at infinity, and 2 finite zeros: one at 100 MHz (lower stopband), and 1 at 500 MHz (upper stopband). The following permutations would be inexact: 24
Extraction Examples 1. DC 100 DC 500 (Violates Rule #1) 2. DC DC 100 500 (Violates Rule #2) 3. DC 500 DC 100 (Violates Both Rules) 4. 100 DC 500 DC (Violates Rule #1) 5. 500 DC DC 100 (Violates Rule #2) 6. DC DC 500 100 (Violates Both Rules) If an extraction sequence violates either of these rules, it is considered inexact since it does not exactly represent the intended transfer function. The severity of the response distortion is related to where (what position) in the extraction sequence a rule is violated. If it occurs at the last zero extraction, as in example #1 in the table above, the partial extraction only affects the last elements extracted, and usually has a minimal impact on the filter response. However, if the violation occurs early in the extraction sequence, as in example #2 in the table above, impedance differences will ripple from the first violation to the end of the structure. In most cases, this results in a poor filter response, which often requires tuning or optimization. Extraction Examples In the examples below, elements are labeled with the type of transmission zero they represent, rather than actual values. Example 1 25
Filter Design Concepts 200-250 MHz bandpass filter 2 Zeros at DC 2 Zeros at Infinity There are 3 unique permutations with a series element first: Example 2 200-250 MHz bandpass filter 1 zero at DC 1 zero at Infinity 1 finite zero at 125 MHz There are 4 unique permutations with a series element first: Example 3 60 MHz lowpass filter 26
Extraction Examples 2 zeros at f= 1 finite transmission zero at 100 MHz 1 finite transmission zero at 200 MHz There are 12 unique solutions with a series element first: 27
Filter Design Concepts Symmetry Versus Asymmetry Since all transmission and reflection zeros in S/FILTER are chosen to be on the s=jw axis, the resulting filters will have either electrical symmetry (S 11 =S 22 ) or electrical asymmetry (S 22 =-S 11 ). Electrical symmetry of filters does not guarantee physical symmetry. Also, the nature of symmetry / asymmetry and transformer ratio is different in lowpass, highpass and bandpass filters. Electrical symmetry and asymmetry are illustrated in the figures below: Electrical Symmetry: S 11 and S 22 for a 5th degree lowpass Filter 28
Electrical Symmetry In Lowpass Filters Electrical Asymmetry: S 11 and S 22 for a 4th degree lowpass filter Physically asymmetric filters have the following property: where the Z i are element impedances. Even degree lowpass filters are always physically asymmetric if all transmission zeros are at f=. Electrical Symmetry In Lowpass Filters If the number of transmission zeros at f= is odd, electrically symmetric filters are obtained. In other words, the input and output ports of these filters may be interchanged without causing any change in the response. No transformer will be needed in synthesis of such filters if the source and load impedances are equal. However, these filters are physically symmetric only under special cases. If all transmission zeros are at f=, all extraction orders will produce physically symmetric filters. If finite transmission zeros exist, physical symmetry is only obtained for a careful choice of finite transmission zero frequencies and zero quantities at those frequencies. 29
Filter Design Concepts Electrical Asymmetry In Lowpass Filters Lowpass filters with an even number of transmission zeros at f= (even degree lowpass filters) are electrically asymmetric filters (S 22 = S 11 ). For these filters, the required load resistance (to get the specified filter characteristics) is always different from the source. Therefore, a transformer is almost always inserted at the load end of the synthesized structure. This transformer may be removed by clicking the "Remove Transformer" button on the Transforms tab. Electrical Symmetry In Highpass Filters If the number of transmission zeros at DC is odd, electrically symmetric highpass filters are obtained. No transformer will be needed in synthesis of such filters if the load impedance is equal to the source. However, similar to the electrically symmetric lowpass filters, these structures come out to be physically symmetric only for special cases. If all transmission zeros are at DC, resulting structures will always be physically symmetric. If finite transmission zeros exist, physical symmetry can be obtained only for a careful choice of finite transmission zero frequencies and zero quantities at those frequencies. Electrical Asymmetry In Highpass Filters Highpass filters with an even number of transmission zeros at DC (even degree highpass filters) are always electrically asymmetric (S 22 = S 11 ). For these filters, the required load resistance (to get the specified filter characteristics) is always different from the source resistance. Therefore, a transformer is almost always added to the load end of the filter. This transformer may be removed by clicking the "Remove Transformer" button on the Transforms tab. Electrical Symmetry In Bandpass Filters Bandpass filters with an odd number of transmission zeros at DC are electrically symmetric. If there are no finite transmission zeros, physically symmetric structures can be obtained by alternately extracting transmission zeros at DC and f=. Random extraction of these transmission zeros will lead to physically asymmetric structures with transformers at the load 30
Electrical Asymmetry In Bandpass Filters end for most cases. If there are finite transmission zeros, physical symmetry is possible only if finite transmission zero frequencies and zero quantities permit a symmetric zero extraction process. For example, all finite transmission zeros are located at the same frequency or all finite transmission zeros have an even quantity, etc. Electrical Asymmetry In Bandpass Filters Bandpass filters with an even number of transmission zeros at DC always result in asymmetric structures, and almost always require a transformer. Filters resulting from synthesis procedures are rarely in desired topologies. Furthermore, the element values are often unrealizably large or small. Therefore, it is often necessary to apply some circuit transforms to get realizable circuits. In most of these transforms, ideal transformers are produced. These transformers can be used to eliminate the transformers resulting from synthesis, as will be seen in the applications presented in the Transforms Chapter. 31
Chapter 4: Transforms Overview Often, directly synthesized filters do not have realizable element values, or may not be in a desired topology. For this reason, transforms must be applied to filters to realize the desired form with desirable element values. A proper transform can turn a seemingly unrealizable filter into a form that is very easy to realize. For example, a synthesized filter may have a transformer that can be absorbed into neighboring components by scaling impedances. This is accomplished by using one or more Norton transforms, thus eliminating the transformer. Norton Transforms Norton transforms are the basis of most of the circuit transforms that S/FILTER supports. The two basic Norton transforms are the Norton Series transform and the Norton Shunt transform. In these transforms, a single series (shunt) element is replaced by a Tee (Pi) network together with a transformer. Depending on whether the transformer ratio N is greater or less than one, either the left or the right element of Tee and Pi sections is negative. If the transform is carefully applied, negative elements are absorbed by the neighboring positive elements. As an example, consider the filter below. Elements are labeled with "+" or "-" to indicate whether the element's value is greater or less than zero. To scale the shunt resonator impedances (C2 and L2), a Norton Series transform is applied to the series capacitor (C1). In this transform, "N" is a selectable parameter. The circuit below shows the result of the transform, with dashed components indicating new elements added by the transform.
Transforms If N is chosen less than one, C1 is negative and C3 is positive. In this case, there is no other element in the network that can absorb the negative capacitance, since there is no capacitor in parallel with C1. On the other hand, if N is chosen greater than one, C1 is positive and C3 is negative. In this case, if C4 > C3, C4 can absorb the negative element by combining the two parallel capacitances. By carefully choosing where a Norton transform is applied and by carefully choosing parameters, element values can be scaled and new filter topologies can be explored. In fact, many popular topologies, such as Top-C Coupled bandpass filters, are designed by applying transforms to directly synthesized filters. Note: In the individual transform descriptions listed in this chapter, gray boxes in examples indicate elements affected by the transform being applied. Examples Of When To Use Transforms The list of examples below is meant to illustrate situations where transforms are used rather than as a tool for learning transforms. See the transform descriptions later in this chapter for information on particular transforms, and applications of each transform. Create series resonators 34
Examples Of When To Use Transforms The first and second series branches (L1 and C2) are converted to series resonators by performing a Norton Shunt transform on C1 and L2. The network shown below is the result of applying these two transforms. The transformer generated in the second transform is chosen to cancel the existing transformer. Create shunt resonators By applying a Norton Series transform to L2 and C4 in the schematic above, all shunt elements are converted to resonators. This makes realization with transmission lines (e.g. coaxial resonators) much more convenient. The schematic below is the result of applying these transforms. The transformer created by the second transform is chosen to cancel the existing transformer. Scale element impedances (values) up or down 35
Transforms Element values in the schematic above are extreme. In general, a maximum capacitor or inductor value ratio over 100 is hard to realize due to parasitic concerns. In this filter, the maximum capacitor ratio (C2/C1) is 109. The maximum inductor ratio (L2/L1) is 111. The schematic shown below is the result of applying a Norton Shunt transform to C1. The maximum capacitor ratio (C2/C1) is now 14. The maximum inductor ratio (L2/L1) is now 2. Remove components Often during transforms, elements may be produced that do not affect the filter's performance, and can be removed. The schematic above is the result of successive transforms. The series capacitor at the input blocks DC but is essentially a short for all other frequencies. Therefore, it can be removed from the schematic without affecting the filter's frequency response. Introduce components Often it is useful to apply a transform to a component without disturbing that component's placement within a resonator. Single elements can be split to facilitate this need. The schematic above is to have a transform applied to C1. The schematic 36
Examples Of When To Use Transforms shown below is the result of splitting this capacitor. Now, a transform can be applied to the new capacitor (C2) without removing the series resonator. Create equal inductor or equal capacitor filters The schematic above is converted to equal inductors by applying a Norton Series transform to the series capacitor C1. The final schematic is shown below. In this example, the transformer required to scale L2 to be equal to L1 is the inverse of the existing transformer. So, with one transform the filter is converted to equal-inductor and no transformer! 37
Transforms Norton Series Transform Description This transform takes a series element, and replaces it with a Pi structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the Pi structure are of the same type as the original element. Supported element types are shown below: Before Transform: After Transform: Note: When using this transform on a series resonator in series, both elements within the resonator must be selected. Otherwise, a single element transform is applied. 38
Norton Series Transform Example In this example, the transformer ratio is chosen as 4. Basic Equations The basic equations for the Norton Series Transform are shown below: Options Options available for this transform are: Specify Left (Za) Specifying Z a causes S/FILTER to calculate the transformer ratio using the following equation: 39
Transforms Note: This option is not available when this transform is applied to a resonator. Specify Right (Zc) Specifying Z c causes S/FILTER to calculate the transformer ratio using the following equation: Note: This option is not available when this transform is applied to a resonator. Specify Transformer Ratio (N) When the transformer is specified, S/FILTER uses the three basic equations given above to calculate Z a, Z b, and Z c. Calculate N To Remove Existing Transformers If this option is chosen, S/FILTER calculates the inverse of the existing transformer for use as N in the new transformer. When the filter schematic is simplified, transformers are combined so that no transformer exists in the simplified schematic. 40
Norton Shunt Transform Norton Shunt Transform Description This transform takes a shunt element, and replaces it with a Tee structure plus transformer, as shown below: Before Transform: After Transform: The elements in the Tee structure are be of the same type as the original element. Supported element types are shown below: Before Transform: After Transform: Note: When using this transform on a parallel resonator to ground, both elements within the resonator must be selected. Otherwise, a single element transform is applied. 41
Transforms Example In this example, the transformer ratio is chosen as 4. Basic Equations The basic equations for the Norton Shunt Transform are shown below: Options 42
Options available for this transform are: Choose The Left Element (Za) Norton Shunt Transform Specifying Za causes S/FILTER to calculate the transformer ratio using the following equation: Note: This option is not available when this transform is applied to a resonator. Choose The Right Element (Zc) Specifying Zc causes S/FILTER to calculate the transformer ratio using the following equation: Note: This option is not available when this transform is applied to a resonator. Choose The Transformer Ratio (N) When the transformer is specified, S/FILTER uses the three basic equations given above to calculate Za, Zb, and Zc. Calculate N To Remove Existing Transformer(s) If this option is chosen, S/FILTER calculates the inverse of the existing transformer and use that as N for the new transformer. This causes the transformers to cancel each other when the filter schematic is simplified so that no transformer exists in the resulting schematic. 43
Transforms Norton Scale Series Element Of Pi Description This transform takes an existing Pi of like components, and replaces it with a Pi structure plus series coupling elements, in which the user can specify the center series element. This result of this transform is shown below: Before Transform: After Transform: The elements in the scaled Pi structure are be of the same type as the original element. Supported element types are shown below: Before Transform: After Transform: Example In this example, the series inductor (L2) is scaled to 10 nh. 44
Norton Scale Series Element Of Pi Basic Equations Options There are no options for this transform. 45
Transforms Norton Scale Shunt Element Of Tee Description This transform takes an existing Tee of like components, and replaces it with a Tee structure plus shunt coupling elements, in which the user can specify the center shunt element. This result of this transform is shown below: Before Transform: After Transform: The elements in the scaled Tee structure are of the same type as the original element. Supported element types are shown below: Before Transform: After Transform: Example In this example, the shunt capacitor (C2) is scaled to 50 pf. Basic Equations The equations for the Norton Scale Series Element of Pi Transform are shown below: 46
Norton Scale Shunt Element Of Tee Options There are no options for this transform. 47
Transforms L-Left To L-Right Transform Description This transform takes a structure in the form of L-Left and gives a L-Right structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the L-Right structure are of the same type as the L-Left. Supported L-Left element types are shown below: Before Transform: After Transform: Example Basic Equations The equations for the L-Left To L-Right Transform are shown below: 48
L-Left To L-Right Transform Options There are no options for this transform. 49
Transforms L-Left To Pi Description This transform takes a structure in the form of L-Left and gives a Pi structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the Pi structure are of the same type as the L- Left. Supported L-Left element types are shown below: Before Transform: After Transform: Example In this example, "Force Symmetry" is chosen, forcing L1 equal to L3. Basic Equations The basic equations for the L-Left To Pi Transform are shown below: 50
L-Left To Pi Options The options for this transform are: Choose Left Element (Za) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Middle Element (Zb) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: 51
Transforms Choose Right Element (Zc) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Transformer Ratio (N) If this option is selected, the following restriction is used: Force Symmetry (Za=Zc) If this option is selected, the following equation is used to calculate the transformer ratio: 52
L-Left To Tee Transform L-Left To Tee Transform Description This transform takes a structure in the form of L-Left and gives a Tee structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the Tee structure are of the same type as the L- Left. Supported L-Left element types are shown below: Before Transform: After Transform: Example In this example, the center element is chosen as 10 nh. Basic Equations The basic equations for the L-Left To Tee Transform are shown below: 53
Transforms Options The options for this transform are: Choose Left Element (Za) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Middle Element (Zb) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: 54
Choose Right Element (Zc) L-Left To Tee Transform If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Transformer Ratio (N) If this option is selected, the following restriction is used: 55
Transforms L-Right To L-Left Transform Description This transform takes a structure in the form of L-Right and gives a L-Left structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the L-Left structure are of the same type as the L-Right. Supported L-Right element types are shown below: Before Transform: After Transform: Example Basic Equations The equations for the L-Right To L-Left Transform are shown below: 56
L-Right To L-Left Transform Options There are no options for this transform. 57
Transforms L-Right To Pi Transform Description This transform takes a structure in the form of L-Right and gives a Pi structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the Pi structure are of the same type as the L- Right. Supported L-Right element types are shown below: Before Transform: After Transform: Example In this example, "Force Symmetry" is chosen, forcing C1 equal to C3. Basic Equations The basic equations for the L-Right To Pi Transform are shown below: 58
L-Right To Pi Transform Options The options for this transform are: Choose Left Element (Za) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Middle Element (Zb) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: 59
Transforms Choose Right Element (Zc) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Transformer Ratio (N) If this option is selected, the following restriction is used: Force Symmetry (Za=Zc) If this option is selected, the following equation is used to calculate the transformer ratio: 60
L-Right To Tee Transform L-Right To Tee Transform Description This transform takes a structure in the form of L-Right and gives a Tee structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the Tee structure are of the same type as the L- Right. Supported L-Right element types are shown below: Before Transform: After Transform: Example In this example, C1 is chosen as 150 pf. Basic Equations 61
Transforms Options The options for this transform are: Choose Left Element (Za) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Middle Element (Zb) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: 62
Choose Right Element (Zc) L-Right To Tee Transform If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Transformer Ratio (N) If this option is selected, the following restriction is used: 63
Transforms Pi To L-Left Transform Description This transform takes a structure in the form of a Pi and gives a L- Left structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the L-Left structure are of the same type as the Pi. Supported Pi element types are shown below: Before Transform: After Transform: Example Basic Equations The equations for the Pi To L-Left Transform are shown below: 64
Pi To L-Left Transform Options There are no options for this transform. 65
Transforms Pi To L-Right Transform Description This transform takes a structure in the form of a Pi and gives a L- Right structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the L-Right structure are of the same type as the Pi. Supported Pi element types are shown below: Before Transform: After Transform: Example Basic Equations The equations for the Pi To L-Right Transform are shown below: 66
Pi To L-Right Transform Options There are no options for this transform. 67
Transforms Pi To Pi Transform Description This transform takes a structure in the form of a Pi and gives a Pi structure plus a transformer, where the user is able to choose one of the elements in the resulting network. The result of this transform is shown below: Before Transform: After Transform: The elements in the scaled Pi structure are of the same type as the original Pi. Supported Pi element types are shown below: Before Transform: After Transform: Example 68
Pi To Pi Transform In this example, "Force Symmetry" is chosen, forcing L2 equal to L4. Basic Equations The basic equations for the Pi To Pi Transform are shown below: Options If this transform is applied to a single element (L or C), the following dialog appears: 69
Transforms If this transform is applied to a Pi structure with shunt parallel resonators, the following dialog appears: The options for this transform are: Choose Left Element (Za) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Middle Element (Zb) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Right Element (Zc) If this option is selected, the following restriction is used: 70
Pi To Pi Transform The following equation is used to calculate the transformer ratio: Choose Transformer Ratio (N) If this option is selected, the following restriction is used: Force Symmetry (Za=Zc) If this option is selected, the following equation is used to calculate the transformer ratio: Equal Shunt Inductors (L1=L3) Uses successive Pi transforms to equate shunt inductors. Note: This option is only available if shunt parallel resonators are being transformed. Equal Shunt Capacitors (C1=C3) Uses successive Pi transforms to equate shunt capacitors. Note: This option is only available if shunt parallel resonators are being transformed. 71
Transforms Pi To Tee Transform Description This transform takes a structure in the form of a Pi and gives a Tee structure, as shown below: Before Transform: After Transform: The elements in the Tee structure are of the same type as the Pi. Supported Pi element types are shown below: Before Transform: After Transform: Example Basic Equations The equations for the Pi To Tee Transform are shown below: 72
Pi To Tee Transform Options There are no options for this transform. 73
Transforms Pi To 2-Pi Transform Description This transform takes a structure in the form of a Pi and gives a 2 Pi structure, as shown below: Before Transform: After Transform: The elements in the 2 Pi structure are of the same type as the original Pi. Supported Pi element types are shown below: Before Transform: After Transform: Example Basic Equations The equations for the Pi To 2 Pi Transform are shown below: 74
Pi To 2-Pi Transform Options There are no options for this transform. 75
Transforms Symmetric Pi To Symmetric 2-Pi Transform Description This transform takes a structure in the form of a Symmetric Pi and gives a Symmetric 2 Pi structure, as shown below: Before Transform: After Transform: The elements in the final Pi structure are of the same type as the original Pi. Supported Pi element types are shown below: Before Transform: After Transform: Example Basic Equations The equations for the Symmetric Pi To Symmetric 2 Pi Transform are shown below: Note: Z3 must be equal to Z1 before applying this transform. 76
Symmetric Pi To Symmetric 2-Pi Transform Options There are no options for this transform. 77
Transforms Tee To L-Left Transform Description This transform takes a structure in the form of a Tee and gives a L-Left structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the L-Left structure are of the same type as the Pi. Supported Pi element types are shown below: Before Transform: After Transform: Example Basic Equations The equations for the Tee To L-Left Transform are shown below. 78
Tee To L-Left Transform Options There are no options for this transform. 79
Transforms Tee To L-Right Transform Description This transform takes a structure in the form of a Tee and gives a L-Right structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the L-Right structure are of the same type as the Tee. Supported Tee element types are shown below: Before Transform: After Transform: Example Basic Equations The equations for the Tee To L-Right Transform are shown below. 80
Tee To L-Right Transform Options There are no options for this transform. 81
Transforms Tee To Pi Transform Description This transform takes a structure in the form of a Tee and gives a Pi structure, as shown below: Before Transform: After Transform: The elements in the Pi structure are of the same type as the Tee. Supported Tee element types are shown below Before Transform: After Transform: Example Basic Equations The equations for the Tee To Pi Transform are shown below: 82
Tee To Pi Transform Options There are no options for this transform. 83
Transforms Tee To Tee Transform Description This transform takes a structure in the form of a Tee and gives a Tee structure plus a transformer, as shown below: Before Transform: After Transform: The elements in the scaled Tee structure are of the same type as the original Tee. Supported Tee element types are shown below: Before Transform: After Transform: Example In this example, "Force Symmetry" is chosen, forcing L 1 equal to L 3. Basic Equations The equations for the Tee To Tee Transform are shown below: 84
Tee To Tee Transform Options The options for this transform are: Choose Left Element (Za) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Middle Element (Zb) If this option is selected, the following restriction is used: 85
Transforms The following equation is used to calculate the transformer ratio: Choose Right Element (Zc) If this option is selected, the following restriction is used: The following equation is used to calculate the transformer ratio: Choose Transformer Ratio (N) If this option is selected, the following restriction is used: Force Symmetry (Za=Zc) If this option is selected, the following equation is used to calculate the transformer ratio: 86
Tee To 2-Tee Transform Tee To 2-Tee Transform Description This transform takes a structure in the form of a Tee and gives a 2 Tee structure, as shown below: Before Transform: After Transform: The elements in the 2 Tee structure are of the same type as the original Tee. Supported Tee element types are shown below: Before Transform: After Transform: Example Basic Equations The equations for the Tee To 2 Tee Transform are shown below: 87
Transforms Options There are no options for this transform. 88
Move TRF Left One Transform Move TRF Left One Transform Description This transform moves the selected transformer left by one element as shown below: Before Transform: After Transform: Example Basic Equations This transform scales the element being shifted. The equations for this scaling depend on the element type: Elements: Equation(s): Options There are no options for this transform. 89
Transforms Move TRF Leftmost Transform Description This transform moves the selected transformer leftmost (adjacent to the source) as shown below: Before Transform: After Transform: Example Basic Equations This transform scales all elements between the transformer and source. The equations for this scaling depend on the element type: Elements: Equation(s): Options There are no options for this transform. 90
Move TRF Right One Transform Move TRF Right One Transform Description This transform moves the selected transformer to the right by one element, as shown below: Before Transform: After Transform: Example Basic Equations This transform scales the element being shifted. The equations for this scaling depend on the element type: Elements: Equation(s): Options There are no options for this transform. 91
Transforms Move TRF Rightmost Transform Description This transform moves the selected transformer rightmost (adjacent to the load), as shown below: Before Transform: After Transform: Example Basic Equations This transform scales all elements between the transformer and the load. The equations for this scaling depend on the element type: Elements: Equation(s): Options: There are no options for this transform. 92
Absorb TRF Into Source Transform Absorb TRF Into Source Transform Description This transform removes the selected transformer from the filter and adjusts the source impedance to match the missing transformer, as shown below: Before Transform: After Transform: Example Basic Equations The equation for this transform is shown below: Options There are no options for this transform. 93
Transforms Absorb TRF Into Load Transform Description This transform removes the selected transformer from the filter and adjusts the load impedance to match the missing transformer, as shown below: Before Transform: After Transform: Example Basic Equations The equation for this transform is shown below: Options There are no options for this transform. 94
Combine All TRFs Transform Combine All TRFs Transform Description This transform shifts all transformers within the filter to the rightmost position and combines all transformers to form one equivalent transformer, as shown below: Before Transform: After Transform: Example Basic Equations The equation for this transform is shown below: Options There are no options for this transform. 95
Transforms Split Series Element Into 2 Parts Transform Description This transform splits a series element into two equal series elements. The combined value of the two series elements is equivalent to the original element value. This transform is shown below: Before Transform: After Transform: Example Basic Equations The equations for this transform depend on the element being split. The supported elements and their equations are shown below: Before Transform: After Transform: Equation(s): 96
Split Series Element Into 2 Parts Transform Options Note: Although this transform will split parallel resonators in series into two resonators, these separate resonators cannot be recombined with the Simplify button. This is because although a single finite zero can be realized with two separate resonators, a single resonator cannot realize more than one finite zero. Therefore, S/FILTER will not combine separate resonators into one even if they have the same resonant frequency. There are no options for this transform. 97
Transforms Split Shunt Element Into 2 Parts Transform Description This transform splits a shunt element into two equal shunt elements. The combined value of the two shunt elements is equivalent to the original element value. This transform is shown below: Before Transform: After Transform: Example Basic Equations The equations for this transform depend on the element being split. The supported elements and their equations are shown below: Before Transform: After Transform: Equation(s): 98
Split Shunt Element Into 2 Parts Transform Options Note: Although this transform will split series resonators to ground into two resonators, these separate resonators cannot be recombined with the Simplify button. This is because although a single finite zero can be realized with two separate resonators, a single resonator cannot realize more than one finite zero. Therefore, S/FILTER will not combine separate resonators into one, even if they have the same resonant frequency. There are no options for this transform. 99
Transforms Delete Element Transform Description This transform deletes the selected element from the filter schematic, as shown below: Before Transform: After Transform: Example Basic Equations There are no equations for this transform. Options There are no options for this transform. 100
Swap Element With The One To The Right Transform Swap Element With The One To The Right Transform Description This transform swaps the selected element with the element located to the right, as shown below: Before Transform: After Transform: Example Basic Equations There are no equations for this transform. Options There are no options for this transform. 101
Chapter 5: Filter Synthesis Overview Overview This Appendix gives a brief overview of the techniques that S/FILTER uses to synthesize filters. During the extraction procedure, S/FILTER uses the desired filter's impedance and admittance functions. These functions are obtained by open-or short-circuiting the load impedance, which means that S/FILTER essentially ignores the load impedance during element extraction. For this reason, the filter is considered to be a oneport device during the extraction process. When extraction is complete, S/FILTER analyzes the synthesized filter at a passband reflection zero frequency. Any loss detected at that frequency is compensated with a transformer. Filter Synthesis Network characteristics can be studied in terms of voltage and current parameters defined at the network ports. A one port network (shown in the figure below) can be characterized, in frequency domain (s=σ+jω), by its input impedance Z(s) or input admittance Y(s) defined in terms of port voltage V(s) and port current I(s): Analytical properties of input immittances of passive, lossless one ports are of fundamental importance in the synthesis of
Filter Synthesis Overview filters since S/FILTER alternately uses both Z(s) and Y(s) in the extraction procedure. Some basic properties of these immitance functions are summarized below: Driving point impedance Z(s) (and Y(s)) of a linear, time invariant, passive and lumped one-port is a positive real function: 1. Z(s) is real for "s" real 2. Re[Z(s)]>0 for Re[s]> 0 This theorem establishes both necessary and sufficient conditions for realizability of a given Z(s) (or Y(s)). These properties affect the polynomials forming Z(s) as follows: Let Then: 1. N(s) and D(s) are Hurwitz polynomials (i.e. they have no roots in the right half of the s=σ+jω plane). 2. The roots on the s=jω axis, including those at s=0 and s=, if they exist, must have unit multiplicity with positive residues. 3. Condition 2 implies that the degrees of N(s) and D(s) may differ at most by unity. Foster's Reactance Theorem further details some properties of lossless networks (networks consisting of pure reactances): Foster's Reactance Theorem For a positive real rational function Z(s)=1/Y(s) to be realizable as the driving point impedance of a lossless one-port, the necessary and sufficient condition is that it should be expressible in the form where a n and b m are constants and 104
Foster's Reactance Theorem 1. 0 w1 < w2 < w3 (Interlacing poles and zeros, all on jω axis) 2. Foster's Theorem further restricts the degrees of the numerator, n, and denominator, m, by requiring that they must differ by unity. In other words, if the numerator is an even degree, the denominator is odd, and vice versa. From these conditions, the following properties can be deduced: 1. Unity degree difference between numerator and denominator implies that Z(s) must have either a single pole or a single zero at both s=0 and s=. Therefore the function Z(s) or Y(s) will belong to one of the four types: 1. Pole at s=0 and pole at s= 2. Pole at s=0 and zero at s= 3. Zero at s=0 and pole at s= 4. Zero at s=0 and zero at s= 2. Z(jω) is purely reactive. Therefore it can be written as where X(ω) is the input reactance with Alternation of poles and zeros leads to the property In other words, the reactance X(w) is always an increasing function of frequency. The rational functions satisfying these requirements are called Foster functions. 3. Since all poles of Z(s) and Y(s) are on the s=jω axis, they can always be expanded as and 105
Filter Synthesis Overview where the constants "k" and "h" are residues of the respective poles. Physically, they correspond to simple network elements, as follows: o If Z(s) has a pole at s=0, it can be extracted as a series capacitor: o If Z(s) has a pole at s=, it can be extracted as a series inductor: o If Z(s) has a pole at s=jω i, it can be extracted as a parallel resonator in series: o If Y(s) has a pole at s=0, it can be extracted as a shunt inductor: 106
Extraction of Finite Zeros o If Y(s) has a pole at s=, it can be extracted as a shunt capacitor: o If Y(s) has a pole at s=jω i, it can be extracted as a series resonator to ground: Note that if a pole of Z(s) at s=0 or s= is extracted, a zero appears at that frequency automatically in the remaining impedance function, which acts as a pole of the remaining admittance function. Hence, given Z(s), one can synthesize a variety of circuits all having the same input impedance but with different structures by extracting elements in different orders from impedance or admittance functions. Extraction of Finite Zeros Contrary to the extraction of transmission zeros at f=0 or f=, finite-frequency transmission zeros can rarely be extracted in a direct manner. This is because in order to extract these sections, Z(s) or Y(s) must have a pole at the finite transmission zero frequency. Usually this is not the case because of the existence of transmission zeros at f=0 and / or f=. However, by extracting a piece of transmission zero from either f=0 or from f= (partial extraction), it is possible to form a remainder impedance function with a pole or zero at the desired transmission zero frequency s=jω i as follows: 107
Filter Synthesis Overview 1. If Z(s) has a pole at s=0, part of that pole can be extracted (as a series capacitor C1) such that the remaining impedance function Z 1 (s) will have a zero at s=jω i : Hence, the value of capacitor is found as Now Z 1 (s) has a zero at s=jω i, which means Y 1 (s)=1/z 1 (s) has a pole at that frequency. It can be extracted as a series resonator to ground: The resulting structure is shown below. Partial extraction of the pole at s=0 shifts the zeros of Z(s) towards s=0, one of which will coincide with the desired zero at s=jω i. Finite Zero Structure Extracted in Lower Stopband 2. If Z(s) has a pole at s=, the zero shifting process may be carried out by partial removal of that pole in the same manner, yielding a series inductor followed by a series resonator to ground, as shown below: Finite Zero Structure Extracted in Upper Stopband 3. If Y(s) has a pole at s=0, the zero shifting process results in a shunt inductor followed by a parallel resonator in series, as shown below: 108
Extraction of Finite Zeros Finite Zero Structure Extracted in Lower Stopband 4. If Y(s) has a pole at s=, the same procedure gives a shunt capacitor followed by a parallel resonator in series, as shown below: Finite Zero Structure Extracted in Upper Stopband These procedures do not guarantee positive element values under all conditions. Filters with "ideal" conditions (e.g. finite zero too close to the passband edge, extremely narrow band, etc.) may yield negative values when shifting zeros as described above. Trial and error and readjustment of filter specifications are the usual methods in such cases. However, since S/FILTER reports all possible ways of extracting each sequence of transmission zeros, often even extreme cases will have at least one realizable solution. In bandpass filters, experience shows that the probability of getting negative element values decreases by extracting finite transmission zeros in lower stopband by partial pole removal from s=0 and finite transmission zeros of upper stopband by partial pole removal from s=. S/FILTER uses this approach. 109
Chapter 6: Examples How To Design This chapter utilizes examples to illustrate "How To Design" filters that meet important and practical design goals. For example, how to design filters with equal input and output terminations or how to design filters with all series resonators. These requirements often arise in the development of filters for specific applications. For example, when quartz crystals or transmission lines realize the final filter, all series or all shunt resonators are typically required. Direct synthesis creates filters with maximum economy for specific responses but achieving certain desirable topologies must be directed. One strength of S/FILTER over other synthesis programs is that it provides the user with a rich set of tools for directing the synthesis process. This chapter uses examples to illustrate how to use these tools. Once mastered, these tools are easily applied to solving your particular requirements. Note: Each example in this chapter lists two filenames. The.WSP file contains the schematic and any associated layouts or optimizations. The.SF$ file contains the S/FILTER settings used to design any filters used in the example.
Examples Equal Termination Example Files: Equal Termination.WSP Equal Termination.SF$ This example illustrates how to design filters for equal input and output termination resistance. These techniques may be used to also design filters with specific, unequal, termination resistance such as 50 Ω input and 75 Ω output. Given here is the Design tab of S/FILTER for a 50 Ω, 10.7 MHz IF filter with 300 KHz bandwidth. The Table in the Extractions tab is Customized to show the TRF Ratio, the number of Inductors and the Permutation (extraction sequence). A transformer ratio of 1:1 would result in a 50 Ω output. The Table was sorted by the transformer ratio by clicking on the TRF Ratio column header. Next the table is scrolled to find a transformer ratio of 1. Although this often occurs, in this case the closest ratios are 0.59 and 1.69 and a transform is required to remove the transformer. The Permutation sequence 10.4 DC 11 is selected as shown here. 112
Equal Termination Example The resulting schematic is Next, the transformer is removed while keeping a 50 Ω output by selecting L 2 and applying a Norton Series transform with the Calculate N option. The following schematic results 113
Examples Maximum Realizability Example Files: Maximum Realizability.WSP Maximum Realizability.SF$ The final filter schematic in the Equal Termination example has six inductors, a maximum to minimum inductor ratio of 318 and a minimum inductor value of 4.72 nh which is very low for a 10.7 MHz filter. Narrow bandpass filters often have significant realization issues but could we do better in this case? This example illustrates methods for improving the realizability of filters. To maximize realizability and resonator Q for a 10.7 MHz bandpass, we desire the minimum number of inductors, inductance between 1000 and 100,000 nh, equal 50 Ω input and output terminations, and no transformer. The Extraction table can be configured to display Permutations sorted by an error based on departure from user goals. Given here is an Extration Goals setup dialog box. Next the extractions were sorted first by Error by clicking on the Error column header and then by number of Inductors. The resulting Extraction tab is given here. 114
Maximum Realizability Example The red entries in the table indicate inexact permutations. Entry #4 with the Permutation 10.4 11 DC is selected. The minimum inductor is lower than desired but other 4-inductor extractions are less desirable. If 112.75 nh is deemed too small then the 5-inductor extraction 11 10.4 DC is an alternative. The 4 inductor permutation has a transformer. It is removed by clicking the Remove Transformer button. Fortunately this reduces the maximum inductor value. The resulting schematic is 115
Examples All Series Resonators Example Files: Series Resonators.WSP Series Resonators.SF$ This example illustrates how to design bandpass filters with all series resonators. A three-section 800 Hz bandwidth communications receiver IF filter centered at 9 MHz is realized using quartz-crystal resonators. The quartz-crystal parameters are L m = 24E6 nh (24 mh), C m = 0.01303 pf (13.03 ff, C o = 3.6 pf and R m = 31 Ω. The motional inductance and capacitance given here series resonate at 9.0 MHz. In the final filter, each crystal may series resonate at slightly different frequencies as determined by the final C m in the schematic. The filter will be designed with all inductors equal to precisely 24 mh. The Specifications tab is given here. The extraction sequence DC with a series element first is chosen because it results in a series L-C at the input. The source resistance is tuned until the first series inductor is 24 mh. The required resistance is 101.5 Ω. The load resistance is set equal to 101.5 Ω. The initial schematic is given here. The number of digits to the right of the decimal in element values has been set to 8 to view the precise values associated with high-q quartz-crystal resonators. 116
All Series Resonators Example Next two Norton Shunt transforms are applied to the shunt coupling capacitors to convert the remaining two series inductors into series L-C resonators to conform to the motional element branch of the equivalent circuit model for quartz-crystals. C 2 is selected, Apply Norton Shunt is selected and "Choose the transformer ratio (N)" is selected. We desire L 2 =L 1 so a transformer ratio equal to the square root of L 2 /L 1 = 7.587542e-5 is entered. The schematic after applying the same transform to the second shunt capacitor is given here. The filter is symmetric, after the transforms the input and output resistances are equal, and S/FILTER automatically removed the transformer. If a transformer remains with a ratio near but not exactly 1:1 the transformer may be manually deleted. Next, the design is standardized by changing the terminating resistance at ports 1 and 2 from 101.5 to 100 Ω, setting the shunt coupling capacitors to 180 pf, adding 31 Ω series resistors and adding the crystal 3.6 pf static capacitors. The inductors are set one at a time to precisely 24 mh and tuning its series capacitor to correct the response. Finally, the series capacitors are optimized to clean up the tuning. The final response and schematic are given here. 117
Examples All Parallel Resonators Example Files: Parallel Resonators.WSP Parallel Resonators.SF$ This example illustrates how to design bandpass filters with all parallel resonators. The Specifications tab of S/FILTER shows the filter specifications. The Extraction tab indicates there are 4 unique extraction sequences. Notice that Series Element First option is not selected so that the first element will be shunt. The table has been customized to show only the extraction sequence and the transformer ratio. Sequence 3 is selected since the transformer ratio is essentially 1 providing equal input and output termination resistance without requiring a transformer. 118
All Parallel Resonators Example The initial schematic for extraction sequence 3 is given here. To create a topology with all parallel resonators the circuit is transformed to place capacitors in parallel with the shunt input and output inductors. First, C 1 is selected and a Norton Series transform is applied. We require a positive capacitor on the left and the resulting negative capacitor on the right will be absorbed by C 2. The option is selected "Choose the transformer ratio (N)". To maximize realizability N=SQR(L 2 /L 1 ) or 0.2003249 is entered. This shifts the impedance of the filter to the right of C 1 up an amount which causes L 2 =L 1. The Simplify Circuit transform next combines the two transformers into one transformer at the output. 119
Examples Next we will apply a Norton Series to the remaining series capacitor to place a capacitor in parallel with the shunt output inductor. This time we want the positive capacitor on the right. Since L 3 >L 1 we choose N=SQR(L 3 /L 1 ) or 4.991998. Finally, because the transformer ratio is near unity it is absorbed into the load. The History tab with a list of the transforms and the resulting schematic are given here. 120
Response Symmetry Example Response Symmetry Example Files: Response Symmetry.WSP Response Symmetry.SF$ A symmetrical transmission amplitude response is often desired, particularly for IF filters. Conventional exact transform bandpass filters have a symmetrical response when plotted on a logarithmic frequency scale (geometric symmetry). It is arithmetic symmetry (plots on a linear frequency scale) that is desired in IF filters. Arithmetic symmetry also results in symmetry in the group delay (equal peak group-delay values near the lower and upper cutoffs). None of the popular lowpass to bandpass transforms possess arithmetic symmetry. In 1989 Eagleware developed a lowpass to bandpass transform that results in arithmetic symmetry. You may refer to pages 165 to 167 of HF Filter Design and Computer Simulation for more information on this transform. However, this transform is approximate and is available for all-pole (no finite transmission zeros) filters only. The direct synthesis routines in =S/FILTER= offer a more elegant solution to this problem. This example illustrates how to exactly design bandpass filters with arithmetic symmetry. Carassa [Band-Pass Filters Having Quasi-Symmetrical Attenuation and Group-Delay Characteristics, Alta Frequenza, July 1961, p. 488] proved that if the number of transmission zeros at infinity is 3 times the number at DC, the response possesses arithmetic symmetry. This can be maintained even with the addition of transmission zeros at finite frequencies. The Specifications tab for a 70 MHz IF filter with 30 MHz bandwidth is given here. Notice that the quantity of zeros at infinity are 3 times the quantity at DC. The finite zeros were tuned to 38.7 and 103.2 to achieve a minimum attenuation in the stop band of 42 db. 121
Examples The extraction sequence 103.2 38.7 DC without any transforms results in the schematic here. This sequence resulted in the lowest inductor count (4) and a low ratio of inductors (6.48888) but a transformer ratio of 0.39288. To remove the transformer the following transforms were applied. 1. C 4 was swapped with L 3 /C 5 L 4 was swapped with C 6 2. A Norton Series was applied to C 6 using the option "Calculate N to remove existing transformer" 3. Simplify Circuit was clicked The final schematic is given in the figure below. 122
Response Symmetry Example Notice that the transformer is removed and the inductor ratio was reduced to 2.56 (L 2 /L 1 ). The response for this remarkable filter is given here. Notice the excellent transmission and amplitude and group-delay symmetry. 123
Examples Equal Inductor Example Files: Equal Inductors.WSP Equal Inductors.SF$ Modern chip capacitors are small, inexpensive and have high Q. Inductors are more expensive, larger, more susceptible to parasitics and have lower Q. Because filter realizability is largely a function of the ratio of component values, many consider the ultimate realizability is filters with all equal inductors. This example illustrates how to design these filters. All transforms are selected to add capacitors and keep the number of inductors at a minimum. We will start with the IF filter created in the previous example and add Norton transforms until all inductors equal the first inductor. The following transforms are required: 1. A Norton Shunt is applied to C1 with N = SQR(L2/L1) = 1.60109 followed by Simplify 2. A Norton Series is applied to C4 with N = SQR(L3/L1) = 0.85837 followed by Simplify 3. A Norton Shunt is applied to C9 with N = SQR(L4/L1) = 0.728207 followed by Simplify At this point the output transformer turns ratio is near unity and the transformer is removed by applying the Absorb TRF Into Load transform. The final schematic is shown below. 124
Physical Symmetry Example Physical Symmetry Example Files: Physical Symmetry.WSP Physical Symmetry.SF$ Symmetry is an element of beauty. It is both appealing and practical. In a previous example we illustrated how to create filters with response symmetry. In this example we illustrate how to create filters with element value symmetry (physical symmetry). Physical symmetry reduces the number of unique element values which must be modeled, designed, purchased, stocked and picked for assembly, thus saving both design and manufacturing effort. The GENESYS electromagnetic simulator =EMPOWER= automatically detects physical symmetry. When symmetric filters are realized as distributed structures they execute as much as 16X faster and require 16X less memory in =EMPOWER=. EM simulation of large filters might not be feasible without symmetry. Symmetry results naturally, without additional user effort, in filter types that are: 1. All Butterworth 2. Odd-order Chebyshev 3. All Chebyshev coupled-resonator bandpass in =FILTER= 4. Lowpass filters with finite transmission zero pairings and an odd quantity of zeros at infinity 5. Bandpass filters with finite transmission zero pairings above or below (not both) the passband and odd plus equal quantities of zeros at DC and infinity Symmetry may be forced: 6. By optimizing the response while forcing symmetry The design of types 1 through 3 is straightforward using =FILTER=. These are restricted to all-pole filters. Types 4 and 5 with finite transmission zeros require the direct synthesis techniques of =S/FILTER=. Types 4 through 6 benefit from additional explanation and are the subject of this section. Consider the Specification tab of a type 4 2300 MHz lowpass filter shown below. The placement of zeros conforms to the rules 125
Examples of type 4: the number of zeros at infinity is odd (1) and finite zeros are paired at 3800 MHz. The schematic for the lowpass and the transmission and reflection responses are given here. Notice symmetry of the component values mirrored about the center 4.89 nh inductor. 126
Physical Symmetry Example Next consider case 6 where symmetry and other objectives are forced during optimization. Consider a 10.7 MHz IF bandpass filter with 400 khz bandwidth. The following Specification tab defines the design. The extraction sequence DC DC DC DC DC results in the schematic given here. 127
Examples Using techniques described in the All Series example, 5 Norton transforms are next applied to the shunt-coupling elements to create the series resonator filter given here. A transformer ratio equal to the square root of adjacent capacitors is used is applied to shunt inductors and a ratio equal to the square root of adjacent inductors is applied to shunt capacitors. It is tempting to select the transform option that allows choosing the new inductor right of L2 to equal the original L1, but the transform also modifies L1. This filter is approximately symmetrical but forcing values to be physically symmetric disturbs the response. The transformer is deleted and a set of optimization goals equal to the original response is added. Next physical symmetry is forced in the Equations folder by setting element values on the right and left side of the filter equal to each other and by setting all inductors values equal. Optimization is launched to adjust the values marked with "?" to correct the disturbance introduced when the values where changed. The schematic and equate variables are given below. Also the unloaded Q of all inductors are set at 160. The response after optimization illustrates the insertion loss introduced by finite inductor Q. 128
Physical Symmetry Example 129
Examples Termination Coupling Example Files: Terminaion Coupling.WSP Termination Coupling.SF$ Consider the final schematic in the earlier All Parallel Resonators example 800 to 1000 MHz bandpass filter. The shunt inductors are 1.93 nh. These are practical values if the parallel resonators are to be converted to transmission line resonators but for an L- C filter 1.93 nh is small. This example illustrates how to use the flexibility of the GENESYS environment to make any desired change to a schematic. We will use series capacitors as approximate impedance transformers at the input and output of the filter to increase the design impedance thus increasing the shunt inductors. First the filter is designed with termination resistance higher so that the shunt inductors are 10 nh. This requires a termination resistance of 50*(10/1.93)=259.067 Ω. Next the same procedures illustrated in the All Parallel example are used to create a filter with all parallel resonators and series coupling capacitors. Next, series capacitors are manually added to the schematic at the input and output. These series capacitors step the final termination resistance, Rs=50 Ω, up to the design impedance of the filter, Rp=259.067 Ω. The capacitor value required is given by where f o is the geometric center frequency equal to SQR(Lower Cutoff * Upper Cutoff). In this case C s =1.736 pf resulting in a residual capacitance, C p, that is effectively in parallel with the adjacent resonator. This capacitance in this case is 1.403 pf. This capacitance is subtracted from the initial parallel capacitance of the first and last resonators. The schematic after these manual modifications is given here. 130
Termination Coupling Example This impedance transformation is exact only at f o but for narrow bandwidth filters this process works well. The bandwidth of this filter is 200/SQR(F l *F u )=0.224=22.4% which begins to stress the accuracy of the transform. The response of the filter is given as the dashed responses in the plot given here. Optimization is then applied to correct the response as given by the solid traces. 131
Examples Distributed Elements Example Files: Distributed Elements.WSP Distributed Elements.SF$ This example illustrates how to design certain distributed filter structures beginning with L-C filters and converting L-C elements to distributed elements. A number of equivalent networks given in HF Filter Design and Computer Simulation are repeated here. The radian frequency, ω, is the cutoff frequency of lowpass or highpass filters and the center frequency of resonators. Consider the Physical Symmetry example filter. The series inductors will be replaced with 10 mil wide lines using the series inductor equivalent formula and the shunt capacitors are replaced with 280 mil wide open lines using the short open line formula. The substrate parameters are: 132 Er = 2.55 Tand = 0.0004 Rho = 1 Roughness = 0.055 Tmet = 0.71 mils H = 31 mils Using T/LINE a 10 mil wide line is found to have a characteristic impedance of 135 Ω and a full wavelength is 3733 mils at 2300
Distributed Elements Example MHz while a 280 mil wide line has Zo = 20.7 Ω and a full wavelength is 3380 mils. The equivalent distributed structure is then With the following dimensions: L1=?156 L3=?329 L4=?82.9 L6=?211 W1=10 W6=280 The tee, step and end discontinuities were added as shown. The lumped/distributed equivalents are only approximate and the discontinuities affect the response as well. All line lengths were therefore optimized to obtain the best response. The transmission and reflection responses with circular and square node points are for the original symmetric lowpass filter. The triangular and diamond responses are those of the distributed lowpass after optimization. The degraded stopband performance of the distributed filter above 5000 MHz is a natural limitation of distributed filters and is due to re-entrant modes. 133
Examples Next, LAYOUT is used to create a layout of the filter. This same layout is used to export a Gerber file for manufacture and as a complete circuit description for the electromagnetic simulator EMPOWER. The layout is given here. EMPOWER automatically performs an electromagnetic analysis of this layout. For a description of how to choose options please refer to the EMPOWER manual of GENESYS. The layout and EMPOWER options used in this example are stored in Distributed Elements.WSP. 134
Distributed Elements Example The responses of the circuit theory simulation in =SuperStar= and the =EMPOWER= simulation are given here. Notice the exceptional agreement. Strong agreement between circuit theory and EM analysis is not always achieved because =EMPOWER= often reveals limitations associated with circuit theory simulation. 135
Examples Parametric Bandpass Example Files: Parametric Bandpass.WSP Parametric Bandpass.SF$ In the All Parallel Resonators example we illustrated how to create all-pole filters with all parallel resonators. This is useful when L-C filters will be converted to structures using transmission line or ceramic resonators. In this example a similar process is illustrated for filters with transmission zeros at finite frequencies. A 850 to 950 MHz bandpass with singular zeros at DC, infinity, 751 MHz and 1068 MHz is specified. A passband ripple of 0.177 db is chosen. Again we start with extractions with a shunt element first. The extraction sequence 751 MHz, DC, 1068 MHz and was chosen. Next a Norton Series transform was applied to C 1 and L 2 with a turns ratio of 0.36 followed by a Simply circuit transform. Next a Norton Series was applied to C 4 and L 4 with the ratio automatically selected to remove the transformer. After simplification the schematic is: The first transform ratio of 0.36 was found empirically to give somewhat equal inductors. The inductor values are somewhat small for an L-C filter, particularly if high Q is desired in the inductors. If this filter is to be constructed with L-C elements, the approximate impedance transformation illustrated in the Termination Coupling example could be employed. In this case we plan to realize the parallel resonators using ceramic resonators. Therefore we desire equal shunt inductors in these resonators. This filter is found using optimization. The shunt inductors are set at 1.0 nh and the remaining element values are optimized to achieve the original response. The final element values are given in the following schematic. 136
Parametric Bandpass Example The ratio of maximum to minimum inductor value is 2.49. The ratio for a conventional cookbook realization of this filter is over 7. The parametric bandpass has all parallel resonators as well as an improved inductor ratio. These advantages come at the expense of an additional inductor and capacitor. The response after optimization is given here. Next, the conversion of the L-C resonators in this parametric bandpass to ceramic resonators is illustrated. Using the parallel L-C to quarter-wave transmission line equivalent formula given earlier, the three quarter-wave line resonators from left to right have the following parameters Zo=4.624 Ω Fo=937.033 MHz Zo=4.432 Ω Fo=898.149 MHz Zo=4.266 Ω Fo=864.424 MHz 137
Examples The low line impedance is consistent with the high dielectric constant of ceramic resonators. For other characteristic impedance the original L-C filter is designed with the appropriate shunt inductance. After replacing the L-C resonators with quarter-wave line resonators the responses are given below as dashed traces. Recall the L-C/transmission line equivalences are accurate at the resonant frequency only. Notice the passband return loss and transmission are close to the original L-C filter. However, further from the passband the rejection is less than the original filter in lower stopband and greater than the original filter in the upper stopband. The solid traces are optimization of element values in an attempt to achieve the original stopband rejection. The stopband frequencies were shifted lower to accommodate the effect of the line resonators. Also, for practical reasons, before optimization the characteristic impedance of all three resonators were set equal at 4.6 Ω. The final schematic after optimization is given below. 138
Parametric Bandpass Example 139