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Elliptic-Function Highpass Filter (Closed Form): HPFE



HPFE models represent lumped-element elliptic-function (Cauer) highpass filters. The insertion loss ripples between zero and a specified maximum in the passband. The stopband attenuation increases rapidly between the passband edge and the stopband edge, and ripples between a specified minimum stopband attenuation and infinite attenuation. This type of filter offers optimum selectivity at the expense of increased complexity and poor group delay flatness.


Name Description Unit Type Default
ID Element ID Text HPFE1
N Order of the filter.   3
FP Passband corner frequency (when Qu is infinite). Frequency 1 GHz
AP Maximum Passband Insertion Loss (when Qu is infinite). DB 0.1 dB
AS Minimum Stopband Attenuation(when Qu is infinite). DB 20 dB
FS Stopband corner frequency (when Qu is infinite). Frequency 0 GHz
DM Where to put design margin:0=AP, 1=AS, 2=FP, 3=FS.   1
RS Source resistance. Resistance 50 ohm
RL Load resistance Resistance 50 ohm
QU Average unloaded Q of reactive elements in the filter.   1e12

Parameter Details

An elliptic function highpass prototype filter is completely specified by any four of the five parameters: N, FP, FS, AP, and AS. That is, the value of any parameter is dependent on the value of the other four parameters.

  • If zero is specified for any one of these five parameters, then the model computes its value from the value of the other four parameters.

  • If a value is specified for all five parameters, then the specified value of the last parameter (AS) is ignored and is computed by the model from the values of the other four parameters.

  • It is an error to specify zero for more than one of these five parameters.

N. In mathematical terms, N is defined as the highest exponent of the complex frequency variable s in the transfer function, S21(s), of the filter's normalized lowpass prototype, or, equivalently, the highest exponent of s in the transfer function of the highpass filter. And, in terms of a measurable electrical characteristic, the number of positive-frequency passband reflection (|S11|) zeros corresponds to N/2 for N even and (N+1)/2 for N odd.

DM. If zero is specified for N, the model will determine N. But, since N must be an integer, there will typically be some design margin available, and this margin must be assigned to some parameter other than N. The value of DM determines where the model will assign this design margin.

  • DM=0. After determining N, the model will recompute AP, such that 0 < new AP ≤ AP

  • DM=1. After determining N, the model will recompute AS, such that AS ≤ new AS < ∞

  • DM=2. After determining N, the model will recompute FP, such that FS < new FP ≤ FP

  • DM=3. After determining N, the model will recompute FS, such that FS ≤ new FS < FP

Parameter Restrictions and Recommendations

  1. N can be zero if FP, FS, AP, and AS are not, otherwise 0 < N < 27.

  2. FP can be zero if N, FS, AP, and AS are not, otherwise 0 < FP, and, if FS is not zero, then FS < FP.

    FS can be zero if N, FP, AP, and AS are not, otherwise 0 < FS, and, if FP is not zero, then FS < FP.

  3. AP can be zero if N, FP, FS, and AS are not, otherwise 0 < AP, and, if AS is not zero, then AP < AS.

    Recommend AP greater than or equal to 0.001 dB.

  4. AS can be zero if N, FP, FS, and AP are not, otherwise AP < AS.

  5. 0 < RS.

    0 < RL

  6. QU > 0 specifies a finite unloaded Q (recommend QU ≤ 1e12).

    QU = 0 specifies an infinite unloaded Q.

  7. 0 ≤ DM < 4

Implementation Details

The model is implemented as a short-circuit admittance matrix,

, whose equivalent normalized lowpass prototype transfer function, S21(s), is:

where Rn is the Elliptic Rational Function, and

where cd is a Jacobian elliptic function, K is Legendre's complete elliptic integral of the first kind, and a highpass-to-lowpass frequency transformation has been applied:


_FREQ is the variable containing the project frequency. The parameters of the elliptic-function filter (N, FP, FS, AP, AS) are related by "the degree equation":



The admittances are given by:


This element does not have an assigned layout cell. You can assign artwork cells to any element. See “Assigning Artwork Cells to Layout of Schematic Elements” for details.

Recommendations for Use

This model behaves as if it has ideal impedance transformers at its ports, so there is no attenuation due to mismatched source and load impedances. The model expects that the source impedance will equal RS and that the load impedance will equal RL, but RS need not have any special relationship to RL for ideal transmission (as would normally be the case).


[1] Miroslav D. Lutovac, Dejan V. Tosic, and Brian L. Evans, Filter Design For Signal Processing Using MATLAB and Mathematica, (Prentice Hall, 2001), Chapters 6, 12, and 13.

[2] Alexander J. Grossman, "Synthesis of Tchebycheff parameter symmetrical filters," Proceedings of the IRE, pp. 545-473, April 1957.

[3] Dante Youla, "A tutorial exposition of some key network-theoretic ideas underlying classical insertion-loss filter design," Proc. IEEE, vol. 59, no. 5, pp. 760-799, May 1971.

[4] Miroslav Vleck and Rolf Unbehauen, "Degree, ripple, and transition width of elliptic filters," IEEE Trans. Circuits Syst., vol. 36, no. 3, pp. 469-472, March 1989.

[5] H. J. Orchard and Alan N. Willson, Jr., "Elliptic functions for filter design," IEEE Trans. Circuits Syst., I, vol. 44, no. 4, pp. 273-287, April 1997.

[6] Rolf Schaumann, Mohammed S. Ghausi, and Kenneth R. Laker, Design of Analog Filters: Passive, Active RC, and Switched Capacitor, (Prentice-Hall, 1990), pp. 49-51.

[7] Kendall L. Su, Handbook of Tables for Elliptic-Function Filters, (Kluwer Academic, 1990).

[8] Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. Stegun, (U. S. National Bureau of Standards, 1964), Chapters 16 and 17 by L. M. Milne-Thomson.

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