Classic filter design methods synthesize filters around known resistive Terminations. However, actual Terminations are frequently complex rather than resistive. Filter Solutions and FilterQuick use RMS error reduction methods to synthesize filters around such complex terminations.
The source and load terminations of lumped LC and transmission line filters may be defined with the use of impedance tables wherein the real and imaginary portions of impedance are defined as a function of frequency, as shown below. Note that impedance may be entered in either Polar, Cartesian or Parallel format. Reactance may be entered directly, in Ohms, or through the equivalent capacitance or inductance.
Termination Impedance Definition
The impedance tables contain two compensation options, "Element Tune" and "Impedance Compensate".
This option tunes all the reactive elements to minimize the RMS error. The default status is "Checked".
This option attempts to improve the synthesis accuracy by adding reactive elements to the complex termination, to make it more resistive. . Sometimes this technique 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".
Norton, Pi->T, and T->PI
A simple right mouse click on any Pi, T 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 Bandpass 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 the additional advantage, that all the nodes may contain a grounded capacitor, making it easy to absorb any parasitic node capacitance.
Classic All-Pole Band Pass Filter
Equal Inductor All-Pole Band Pass Filter
Elliptic Filters may have all shunt or all series inductors set to equal values. These filters are sometimes known as "parametric" filters, and are useful in minimizing inductor spread in medium and narrow band filters.
Shunt Equal Inductor Elliptic
(See the section on ZigZag filters for more on equal inductor Zigzag designs).
Parasitic Node Capacitance
Bandpass 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 tends to contain a parasitic capacitance that can degrade the frequency response of the filter. FilterSolutions and FilterQuick permit the user to select a parasitic capacitance value, and will then synthesize around this value with no degradation of performance. The operation changes the value of the source resistance, but even-order filters may be restored to their original value by a Norton transformation.
Here is an example of FilterSolutions compensation for 2 pF parasitic node capacitance.
Classic Design with Parasitic Effects
Filter Solutions Design with Parasitic Compensation
All-pole bandpass 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 stopband ratios, making the translation, shown above, more desirable for filters requiring small stop band attenuation, in cases where parasitic capacitance is an issue.
Chebyshev I and Elliptic Filters may be implemented with sets of LC resonators containing two 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 to each other, simplifying the physical construction. FilterSolutions supports sensitivity analyses, Monte Carlo analyses, amplitude equalization and manual editing analysis for all couplings and other elements of the filter. FilterSolutions supports group delay equalization. Coupled values may be displayed in units of mutual inductance or inductive coupling coefficients.
Chebyshev I Cross-coupled Filters also support real and quadruplet zeros delay equalization.
All Cross-coupled filters are synthesized with a minimum possible number of couplings for the given design requirement.
Three views are provided for each filter design. The following displays each view for a 6th order Elliptic 1MHz Bandpass filter centered at 10 MHz, 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 frequency response:
Smith Charts, Jones Charts, and Polar plots are provided in the forms of frequency and reflection responses for easy to read graphical feedback. The use of left and right mouse keys provides an integrated method for reading data from the Impedance grid.
Smith Chart Display
Real and Quadruplet Zeros Delay Equalization
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 affects the passband magnitude response. Therefore additional calculations are needed to adjust the pole locations to restore the passband. It is generally more efficient to use quadruplet zeros in LC Lumped Element Filters as there are no canceling poles and zeros that are inherent in LC All-pass stages.
FilterSolutions offers a fast and easy approach to real and quadruplet delay equalization for low pass, high pass, and bandpass LC Lumped Element Filters. Poles and group delay are updated in real time in response user zeros manipulation to flatten the pass band and restore an equi-ripple (Chebyshev I) or maximally flat (Butterworth) shape. LC Lumped Element filter responses are calculated instantly with the positioned zeros.
Quadruplet Zero Equalized Low Pass Chebyshev Lumped Filter, Frequency Response and Pole/Zero Plane