Passive Filter Solutions
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Passive Filter Solutions |
Filter Solutions supports a wide variety of passive filter designs. Filters may be single terminated, or double terminated with equal or unequal load and source resistances. Diplexers are supported for all filter types. Finite Q analysis is supported for all passive filters. All-pass sections are supported to create delay equalizers. Finite Q compensation is supported for single terminated filters and double terminated filters with unequal load and source resistances. Amplitude compensation is provided for doubly terminated filters with finite Q. All synthesized filters are checked for a proper frequency response prior to being displayed. All filters are displayed in graphical easy to read formats with a net list selection that displays the filter net list in a textual format that is ready for an AC or transient analysis. Filter Solutions creates both balanced and unbalanced filters.
To generate single terminated filters, enter a source resistance of "0" in the source resistance entry box. To generate a equally terminated filter, enter the same number for source resistance and load resistance. If you enter a different source resistance and load resistance, Filter Solutions will create both single and double terminated filters that fit the filter characteristics selected in the Control Panel.
Filter Solutions computes up to eight filters internally that attempt to meet your design specification. To prevent multiple filters from being displayed, switch settings are provided to narrow the number of displayed filters. You may select only first element shunt filters, first element series filters, voltage source filters, or current source filters. You are also provided a switch to display only the filters with the minimum inductor count.
All Filter Solutions graphical filter displays are user interactive. It is possible to change the value of any component, add new components, or delete components, and then generate a frequency, time, reflection, or impedance analysis on the modified filter. Simply pass the cursor over the component you desire to change. When it highlights, left click the mouse to change the component, or right click to add a component as shown: Components may be set to specific values, changed by a fixed percentage, or set to the nearest value in a standard parts list, nearest 1%, 5%, 10%, or 20% industrial standard part.
After a component is changed, it appears in blue for easy visual reference. If a frequency value changes as a result of the new element values, it also is displayed in blue.
Changing, Adding, or Deleting a Filter Solutions Component
Adding components is useful when modeling parasitics or evaluating the effect of other attached circuits. LC pairs have the option to automatically update the other component in the pair such that resonant frequency is maintained. All pass sections may be added by right clicking an element adjacent to the load or source resistor, or by adding all pass poles and zeros to the pole zero plot.
Changing element values is useful to evaluate the effect of real component values in the filter
Frequency, Reflection, Impedance, and Time Analysis
Each filter generated by Filter Solutions may have a frequency, reflection, impedance, or time analysis performed by selecting the appropriate control on the circuit window. Frequency, reflection, and impedance analysis include magnitude, phase, and group delay. Time analysis includes step, ramp, and impulse response. Depressing the left mouse key at any location brings up a cursor tracer with the frequency and trace information in the cursor window. Depressing the right mouse key adds and deletes permanent cursor tracers. These analysis include all user modifications made to the filter. When a filter has been modified by changing, adding, or deleting an element, or if any elements contain a finite Q, the "Ideal" analysis trace appears in dark blue for quick easy comparison purposes. The attenuation due to the source resistor may be included or removed as desired by checking or unchecking the "Inc Gen Bias" box in the passive control panel. Examples are below:
Frequency Analysis
Time Analysis
Reflection Analysis
Filter Solutions allows you to install components with a finite Q into your filter. The specific filter frequency, reflection, impedance and time analysis will use the Q you have provided in the analysis, and will use a dark blue background trace to show the ideal filter with infinite Q for quick and easy comparison purposes. The degradation effects of finite Q may be compensated for manually by moving the pole locations in the pole/zero plot. The compensation is done in real time if the RTC box on top of the pole/zero plot is checked. The poles may be "Stretched" along the real axis, and flattened slightly along the imaginary axis. This works reasonably well for single terminated filters and largely unequal terminated filters. For filters whose source to load ratio is greater than 0.2 and less than 5.0, finite Q compensation is marginal. Equally terminated filters may not be compensated for in this manner, but may be amplitude equalized. Finite Q compensation is done automatically by Filter Solutions if the "Q Comp" box is checked in the passive control panel. With some practice and patience, you can do a better compensation than Filter Solutions can.
Since the effect of finite Q is unique to each individual passive circuit, the effects of finite Q are only included in the specific passive circuit being analyzed. The control panel analysis functions assume an infinite Q.
Below is an example of an 8th order Elliptic filter with uncompensated, manually compensate, and automatically compensated finite inductor Q of 20. The cutoff frequency of the filter is 100MHz.
Finite Q Elliptic Filter
Effect of Finite Inductor Q of 20
Below are the pole/zero plot that manually compensates the filter
Manual Finite Q Compensation. Dark blue indicates uncompensated pole/zero positions.
Below is the resulting frequency response.
Manual Q Compensation frequency Response.
If the "Q Comp" is selected in the passive control panel, Filter Solutions will automatically compensate for finite Q. Below is the frequency response of the automatic compensation.
Automatic Finite Q Compensation
Coupled resonator band pass filters are narrow band approximations of band pass filters. The advantages to using coupled resonators over classical band pass filters are more desirable element values at high frequencies and flexible element value selections.
Coupled resonator filters consist of a series of LC resonant pairs coupled together with capacitors or inductors. Usually capacitors are chosen due to the lower cost and higher performance. Filter Solutions allows you to select one of the LC resonant element values, and will calculate the other element values needed to create the desired filter. Filter Solutions supports coupled resonators with and without end coupling elements. This allows greater flexibility in element value selection. Coupled resonators without end coupling elements require two less elements than those with end coupling elements.
To create band pass resonant filters, you MUST select Gaussian, Bessel, Butterworth, or Chebyshev Type I filters. Coupled resonant filters may not be created from filters with stop band zeros. Select Band Pass filter class, and then create your passive circuits. The passive circuit will have a check box in the tool bar for "Resonator". Check this box, and a coupled resonant band pass filter will be displayed. The first element of the filter will match the series/shunt selection in the Passive Control Panel. If your selected pass band is too wide, you will receive an error message informing you that you need to reduce the width of the pass band.
When your resonant filter is displayed, the resonator tool box is displayed in the upper right of the circuits window. You are offered the choice of inductor or capacitor coupled resonators, and you may select new resonator element values. Capacitor coupled resonators may have their inductors changed. Inductor coupled resonators may have their capacitors changed. Enter your desired inductance or capacitance, and select "Recalc" to update the filter.
More information is available on the derivation of coupled resonator filters in "Microwave Filter, Impedance-Matching Networks and Coupling Structures", by Matthaei, Young, and Jones, ISBN 0-89006-099-1, pages427-434.
Coupled Resonator Example:
Below is a classical 4th order Butterworth band pass filter with center frequency 500MHz and a band width of 40MHz. Notice some capacitor values are less than 1 pF, and some inductor values are below 1 nH.
Classical Band Bass Filter
Below is the series resonance capacitor coupled band pass filter: Capacitor values are now more reasonably in the low pF range, and all inductor values are 10 nH. However, more capacitors are required to build this filter.
Capacitor Coupled Series Resonator Filter
The magnitude frequency response error is shown below. Dark blue baseline traces depict the true Butterworth response. Yellow depicts the coupled resonator filter response. Note that the error is only significant at frequencies of very high attenuation.
Resonator Coupled Frequency Error
Tubular band pass filters are narrow band approximations of band pass filters. The advantages to using tubular over classical band pass filters are: more desirable element values at high frequencies, flexible element value selections, and all nodes are attached to a grounded capacitor. Any parasitic node capacitance that results in the physical construction of the filter may be subtracted from the design value of the grounded capacitor connected to that node. In most cases, the inductance values adjustable.
Tubular filters consist of an alternating series of inductors and capacitors, with each node containing a grounded capacitor. Nuhertz Filter permits tubular with or without a leading shunt capacitor, and with the first series element being either an inductor or a capacitor. A “Minimum Elements” selection also exists that minimizes the number of elements by skipping series capacitors. This “Minimum Elements” selection has the disadvantage of requiring a wider range of required capacitor values.
The magnitude frequency response error is shown below. Grey baseline traces depict the true Chebyshev response. Blue depicts the tubular filter response. Note that the error is only significant at frequencies of very high attenuation.
In order to obtain a more symmetric tubular frequency response, an option of selecting outer shunt inductors exists.
Tubular with outer shunt inductors
Band pass filters with stop bands (Chebyshev II, Hourglass and Elliptic) may have their inductor count reduced through the use of a zigzag filter implementation. A disadvantage to zigzag filters is the source resistance of the original design changes, and equal resistance terminations and single terminations are not possible in the design of the zigzag filter. However, optimal reflections occur when the design source resistance is set equal to the design load resistance and then using the source resistance that is calculated. Odd order zigzag filters are more efficient than even order in that they require fewer indictors.
To create a zigzag filter, select a band pass Chebyshev II, Hourglass, or Elliptic filter; nonzero source resistance, and check the "Zigzag" check box in the passive control panel.
Zigzag Filter Example
The following is a 4th order classical Elliptic Filter. Note that there are five inductors.
Classical Band Pass Filter
The following is an example of an equivalent 4th order zigzag filter. Note that there are only
four inductors, and the source resistance is no longer 50 ohms.
Even order zigzags require only N indictors, where N is the order of the low
pass prototype.
Even Order Zigzag Filter
The following is an example of a 5th order zigzag filter. Note that there are
still only four inductors. Odd order zigzags require only N-1
indictors, where N is the order of the low pass prototype. A classical 5th
order band pass Elliptic would require seven inductors.
Odd Order Zigzag Filter
For the price of one additional capacitor, any odd order or even order zigzag
filter may be designed with arbitrary source and load resistances. The
filter below has an identical response as the filter above, but is designed with
equal 50 ohm terminations. Note the addition of the 42.26 pF capacitor.
The filter remains a 5th order band pass filter with only four
inductors.
Equally Terminated Zigzag Filter
Zigzag filters may be constructed with one or two inductor values for the price of one or more capacitors. However, high stop band attenuation or ratio is generally required to avoid negative element values. Odd order zigzags may be constructed with one user selected inductor value and arbitrary source and load. Below is a 3rd order zigzag with equal terminations and two user selected 50nH inductors.
Equal Inductor, Equally Terminated Zigzag Filter
Equal inductor zigzag filters have an additional advantage in that all the nodes may contain a capacitor ground, making it easy to absorb any parasitic node capacitance. See the section on parasitic node capacitance for more information.
Filters with equal or near equal termination cannot be compensated for finite Q by altering the pole/zero plot. Since most of the degradation due to finite Q is at the break frequency, it is possible to reduce the magnitude of the frequency response everywhere except the transition to produce an evenly attenuated frequency response. To amplitude equalize your filter, simply select "Series", "Shunt", or "Const" in the amplitude Equalization Selection box. This box are displayed whenever a passive filter has elements with finite Q.
The series and shunt amplitude equalizer have fewer components, but change the load impedance slightly, and therefore degrade the frequency response of the filter slightly. The constant resistance equalizer contains more components, but the load resistance is maintained and the performance of the filter is not degraded.
The amplitude equalizer element values are changeable, just like all other element values in the filter. If you do not like a particular amplitude equalization effect, you may adjust the values in the equalizer to obtain a more desirable effect.
The addition of any of the following RLS circuit at the end of the filter approximately achieves amplitude equalization:
 Shunt Amplitude Equalizer |
 Series Amplitude Equalizer |
 Equal Resistance Amplitude Equalizer |
| Low Pass and High Pass Amplitude Equalizers |
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If the resonant frequency of the LC tanks are set to the filter break frequency, then attenuation is minimized at the break frequency
and inserted everywhere else. If the Q of the LC tank is set properly, an approximation of an attenuated infinite Q frequency trace will result.
For band pass and band stop filter, there are two break frequencies, therefore, two LC tank sections are required as shown below:
 Shunt Amplitude Equalizer |
 Series Amplitude Equalizer |
 Equal Resistance Amplitude Equalize |
| Band Pass and Band Stop Amplitude Equalizers | ||
The effect on a low pass Elliptic filter with inductor q's of 40 and capacitor Q's of 150 is seen below. The dark blue trace is the ideal response. The yellow is the equalized finite Q response.
The following frequency graphs depicts the equalization effect of finite Q on a sixth order Elliptic low pass filter with and without amplitude equalization. The dark blue trace is the ideal filter. The yellow trace is the actual filter with finite Q effects. The filter generating the curve has inductor Q's of 40 and capacitor Q's of 150.
Finite Q Effects With and Without Amplitude Equalization
Filter Solutions supports both balanced and unbalanced passive filters. Many applications require balanced circuits due to the better noise immunity characteristics. To translate an unbalanced circuit to a balanced circuit, cut the series inductors in half and place an identical inductor in the bottom, and double series capacitors, and place an identical capacitor on the bottom. All parallel elements should be left alone. All pass elements have different topologies for balanced and unbalanced. See the All Pass section for these topologies.
The entire passive circuit may be balanced by checking the "Balance" box above each passive filter schematic. Individual elements, LC tank pairs, or all pass sections may be balanced or unbalanced with the use of the right mouse key.
Example:
Balanced and Unbalanced Passive Filters
Modified Filter Transfer Functions
When you modify filter elements, in most cases Filter Solutions will calculate the new transfer function for you to view or export to the Windows clipboard. The Q of the elements is linearized about the frequency displayed next to the LC tank elements, or the cutoff or center frequency of the filters for other elements. The F(S) key above the circuit display will perform this function.
The format of exported transfer functions are in a form readable by Matlab and Matrix-x. For details, see the documentation for Other Applications.
When "Net List" is selected in the circuits control bar, the filter net list is shown in a textual window. The net list is setup for an AC and a transient analysis on the load resistance, and is ready to plug into any application that uses net lists. When finite Q is selected, the parasitic resistances are included in the net list. Resistance values used are based upon the resonant frequency of LC pairs, and the cutoff or center frequency for all other components. Coupled coils are modeled with a set of "T" inductors with a negative inductor in the center. The pulse source is comments out to prevent source conflicts with the AC source. Select "Netlist" again to remove the net list window.
The net list may be printed, copied to the windows clipboard, or saved to a text file. Individual element values may be selected and copied to the windows clipboard for ease in retrieving component values for other applications. Select nothing to copy the entire contents to the clipboard.
Below is the Filter Solutions net list for filter shown in "Graphical Displays" above and shown again to the right:
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MULTIPLE ANALYSES |
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Filter Solutions provides you with a flexible means to update your Passive Filter elements values with the nearest value from your available standard parts, or to the nearest 1%, 5%, 10%, or 10% standard industrial value. Up to three standard parts databases are maintained by Filter Solutions. If you need more than three, they may easily be maintained in another text or word processing document and copied or pasted in and out of Filter Solutions.
To update filter parts to the nearest value in the parts list, left click on an element in a filter display, select "Parts" in the selection box at the bottom of the Change Control Panel, then select OK.
Passive filters may have individual element or all like elements set to the nearest standard value from the selected database or industrial parts list. The checkbox at the bottom of the Change control panel determines if one or all elements are updated.
The upper part of the Filter Solutions standard parts window is shown below with some sample standard parts.
| Parts List Formats: | |
| Comments |
Precede the comment line with a “%” character. Up to 10 comments are allowed. |
| Numerical Entries | Integer, floating, and engineering notation are all acceptable. Six significant digits are stored in the database. Example:12, 0.00047, 470n, 50pF are all acceptable numerical entries. All numerical entries are sorted in ascending order in the database. Up to 500 numerical entries are allowed per element type. |
| Capacitors | Precede the capacitance numerical entry with a “C” or “c” character. |
| Inductors | Precede the inductance numerical entry with a “L” or “l” character. |
| Resistors | Precede the capacitance numerical entry with a “R” or “r” character. |
Passive all pass elements are created with one of four element combinations. Element values are all functions of Q and Wo of the all pass section. All passive implementations in Filter Solutions have the pole in the left half plane. Passive all pass sections must be placed adjacent to a termination resistor, or another all pass that is adjacent to a termination resistor. Some all passes require coupled coils.
The passive implementations for first and second all pass stages are:
Unbalanced and Balanced All Pass Sections |
Of the two solutions for Q>1, the solution to the right is normally used because it has fewer inductors. Occasionally, the shunt capacitor is excessively large. When this happens, the solution to the left is used.
Filter Solutions also provides balanced all pass lattice sections. The translation from unbalanced T's to balanced lattice all pass sections are shown below. You may easily switch between balanced and unbalanced by right clicking an element in the all pass section and selecting the "Balance 1", "Balance 2", or "Unbalance" checkboxes.
Balanced All Pass Lattice Sections |
A Monte Carlo statistical analysis may easily be performed visually with Filter Solutions too see the effect of statistical element value error on filter frequency, reflection, impedance, or time response. Graphical traces may be overwritten or retained as desired. Both Uniform and Gaussian distributions are provided for inserting element value error. Individual elements may be assigned specific tolerance values, or the default tolerance for all elements may be specified.
Below is an example of the effect of random error from 5% capacitors has on the group delay of a 5th order Bessel Filter.
Random Error Due to 5% Capacitors
Filter Solutions provides a powerful tool to study the effect of each individual element sensitivity. Each element is measured and tabulated for its sensitivity effect on magnitude, phase, and group delay at critical or user defined frequencies. In addition, each element may be individually plotted in a element value sweep for its effect on the frequency response at critical or user defined frequencies.
When ideal capacitor values are changed to nonideal real part values in a filter, the filter frequency response is expectedly degraded. It is possible to translate this error in the pass band of the filter to the stop band, where errors are less critical, by tuning the inductor values in the filter. Filter Solutions does this task automatically by clicking the "L tune" button that appears in the circuit display tool bar whenever one or more capacitor values has been updated. Inductors elements may then be custom designed for the filter.
Likewise, when ideal inductor values are changed to nonideal real part values, capacitor values may be tuned by selecting the "C tune" button. Real capacitors values generally come in a broader range of selections than real inductor values.
The example below shows the result of the frequency magnitude response of a filter with manually altered capacitors before and after an automatic inductor tuning process.
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Classical filter design methods synthesis filters around known resistive terminations. Unfortunately, actual terminations are frequently complex rather than resistive. Filter Solutions uses RMS error reduction methods to synthesis filters around such complex terminations. The source and load terminations of passive LC and transmission line filters may be defined with the use of impedance tables where the real and imaginary portions of the impedance is defined as a function of frequency, as shown below. Note that impedance may be entered in either polar, cartesian or parallel format, and that reactance may be entered directly in Ohms, or through equivalent capacitance or inductance.
 Termination Impedance Definition |
The impedance tables contains two compensation option, "Element Tune" and "Impedance Compensate".
Element Tune
This option simply tunes all the reactive elements to minimize the RMS error. The default status is "Checked".
Impedance Compensate
This option attempts to improve the synthesis accuracy by adding reactive elements to the complex termination in an attempt to make it more resistive in nature. Sometimes this has the effect of improving the filter performance, but frequently it degrades the performance. It is very important for the user to carefully examine the effect of this option on the desired filter prior to accepting the results. The default status of this option is "Unchecked".
A simple right mouse click on any PI, I or L combinations of like elements permits the user to perform a Norton transformation, sometimes referred to as a capacitor transformer since most operations are performed on capacitors. A PI may be converted to a T, and a T may be converted to a PI. PI, T and L combinations frequently appear in band pass filters, making this feature a strong tool for custom band pass filter design. Tools exist to add, delete, or change element values with no effect on the shape of the frequency response.
Combinations of elements that are candidates for a Norton transformation
by the use of a right mouse click
The theory behind Norton transformations is well known and shown below.
Norton Transformation Equivalent Circuits
Equal Inductor Band Pass Filters
All pole and zigzag filters may be synthesized with only one inductor value. For odd order zigzags, the single inductor value is selectable by the user within a specified range. The operation changes the value of the source resistance for even order all poles, but it may be reset by requiring two inductor values instead of one. Equal inductor all pole filters have an additional advantage in that all the nodes may contain a capacitor ground, making it easy to absorb any parasitic node capacitance.
Classic All Pole Band Pass Filter
Equal Inductor All Pole Band Pass Filter
See the section on ZigZag filters for more on equal inductor zigzag designs.
Band pass filters with stop bands (Elliptic, Hourglass, and Chebyshev) generally consist of combinations of two series LC tanks in a notch configuration. The node between these tanks tend to contain a parasitic capacitance that can degrade the frequency response of the filter. Filter Solutions permits the user to select a parasitic capacitance value, and will then synthesize around this value such that no degradation of performance occurs. The operation changes the value of the source resistance, but even order filters may Norton the source back to its original value. Below is an example of Filter Solutions compensation for 2 pF parasitic node capacitance.
Classic Design with Parasitic Effects
Filter Solutions Design with Parasitic Compensation
All pole band pass and zigzag filters may absorb parasitic node capacitance with the use of equal inductor designs and zigzag designs. However, equal inductor zigzag designs generally require large stop band attenuation or ratios, making the translation above more desirable for filters requiring small stop band attenuations when parasitic capacitance is an issue.
Chebyshev I and Elliptic filters may be implemented with sets of LC resonators with the inductors coupled together. Chebyshev I filters only require coupling of adjacent resonators. Elliptic filters require cross coupling. However, an Elliptic string of resonators may be folded in half so that all the coupled resonators are immediately adjacent with each other to simply the physical construction. Filter Solutions supports sensitivity analysis, Monte Carlo analysis, group delay equalization, amplitude equalization, and manual editing analysis for all couplings and other elements of the filter. Coupled values may be displayed in units of mutual inductance or inductive coupling coefficient.
Chebyshev I cross coupled filters also support real and quadruplet zeros delay equalization.
All Filter Solutions cross coupled filters are synthesized with a minimum possible number of couplings for the given design requirement.
Three views are provided for each filter designed by Filter Solutions. Shown below is each view for a 6th order Elliptic 1MHz band pass filter centered on 10MHz, with 0.1 dB pass band ripple and 40 dB stop band attenuation:
View 1, Full Coupling Matrix:
View 2: Specific Coupling Matrix:
View 3: Adjacent Coupling Matrix:
All cross coupled filters displayed above exhibit the same nice frequency response:
Smith charts, Jones charts, and polar plots are provided for frequency and reflection responses for easy to read informative graphical feedback. Left and right mouse keys provide easy to read curser data from the impedance grid.
Smith Chart Display
Phase angle and group delay may be altered by the presence of dual and
quadruplet off axis zeros. Unlike all pass stages, the mere addition of dual
and quadruplet off-axis zeros also effects the pass band magnitude response,
so additional calculations are needed to adjust the pole locations as needed
to restore the pass band. However, it is generally more efficient to use
quadruplet zeros in LC passive filters in that there are no canceling poles
and zeros that are inherent in LC all pass stages.
Filter Solutions offers a fast and easy approach to real and quadruplet
delay equalization for low pass, high pass, and band pass LC passive
filters. Poles and group delay are updated in real time in response
user zeros manipulation to flatten the pass band back into an equiripple
(Chebyshev I) or maximally flat (Butterworth) shape, and LC passive filters
are calculated instantly with the positioned zeros.
Quadruplet Zero Equalized Low Pass Chebyshev Passive Filter, Frequency
Response and Pole/Zero Plane
