NBPFCG models represent lumped-element Generalized Chebyshev (or "Quasi-Elliptic") narrowband bandpass filters. The applicability of this filter type is not limited to narrow ( i.e., less than 5%) bandwidths, as the name would appear to imply. The group delay is flatter than that of a "regular" Generalized Chebyshev bandpass filter of the same bandwidth, especially for wideband filters. And the passband magnitude displays arithmetic, rather than geometric, symmetry. The insertion loss ripples between zero and a specified maximum in the passband. The stopband attenuation is defined by arbitrarily specified transmission zeros. Real-frequency finite transmission zeros can be specified to improve selectivity at the expense of ultimate stopband attenuation and passband group delay, while complex-plane finite transmission zeros can be specified to provide passband group delay equalization at the expense of selectivity and ultimate stopband attenuation. Generalized Chebyshev filters represents a compromise between the simplicity of Chebyshev filters, the optimum amplitude response of more complicated Elliptic filters, and the phase linearity of Bessel and Gaussian filters. Because this type of filter allows one to make explicit design trade-offs between complexity, selectivity, and group delay equalization, it is often used to meet the demanding requirements of modern communications systems.
Name | Description | Unit Type | Default |
---|---|---|---|
ID | Element ID | Text | NBPFCG1 |
N | Order of filter's lowpass prototype | 3 | |
FP1 | Passband corner frequency (when Qu is infinite). | Frequency | 0.5 GHz |
FP2 | Passband corner frequency (when Qu is infinite). | Frequency | 1.5 GHz |
*PPD | Passband parameter description:- Maximum Insertion Loss,- Minimum Return Loss, or- Maximum VSWR. | Enumerated | Maximum Insertion Loss |
PPV | Passband parameter value (when Qu is infinite) | dB or Scalar | 0.1 dB |
TZF | Real frequency, finite transmission zeros | (Real) Frequency | {2} GHz |
*TZR | Real parts of complex finite transmission zeros | (Imaginary) Frequency | {0} GHz |
*RS | Source resistance | Resistance | 50 ohm |
*RL | Load resistance | Resistance | 50 ohm |
*QU | Average unloaded Q of the lumped bandpass resonators. | 0 |
* indicates a secondary parameter
N. In mathematical terms, N is defined as the highest exponent of the complex frequency variable s in the transfer function,S_{21}(s), of the filter's normalized lowpass prototype, or, equivalently, half of the highest exponent of s in the transfer function of the bandpass filter. In terms of the number of circuit components, N corresponds to the total number of resonances in single-mode or multi-mode direct-coupled-resonator microwave filters; while, for lumped-element filters, N corresponds to the number of lumped-resonant (LC) circuits that produce zeros of attenuation at finite frequencies. And, in terms of a measurable electrical characteristic, N corresponds to the number of positive-frequency passband reflection (S|_{11}|) zeros.
PPD & PPV. Parameters PPD and PPV work together to specify the characteristic of the filter's passband. PPD is used to indicate what the value of PPV represents. The flexibility these parameters provide eliminates the need to manually convert from the passband specification parameter of one's preference into whatever specific parameter the software was written to accept.
TZF & TZR. List parameters TZF and TZR are used to specify the complex transmission zeros, Z, of the bandpass filter response. Of the N positive-frequency complex transmission zeros, Z_{i} = TZR_{i}+jTZF_{i}, the model allows up to (N-1) to be specified. Each consists of a real part, TZR_{i}, and an imaginary part, the real frequency TZF_{i}. If TZR_{i} is not specified, it is assumed to be zero. You must provide transmission zeros with nonzero real parts in pairs; that is, for each zero A+jB transmission, zero -A+jB must be present. Each unspecified transmission zero is mapped to a normalized lowpass prototype frequency of infinity. Each unspecified transmission zero is mapped to a normalized lowpass prototype frequency of infinity.
32 > N > 1.
FP1 > 0.
FP2 > 0.
FP1 ≠ FP2.
If PPD = "Maximum Insertion Loss", then PPV > 0.
If PPD = "Minimum Return Loss", then PPV > 0.
If PPD = "Maximum VSWR", then PPV > 1.
TZF_{i} > 0 .
If TZR_{i} is specified, then TZF_{i} must be specified.
If TZR_{i} is zero, then FP1 >TZF_{i}>FP2 .
If TZR_{i}≠0 , there must be a TZR_{k}=-TZR_{i} and a TZF_{k}=TZF_{i} .
RS > 0.
RL > 0.
QU > 0 specifies a finite unloaded Q (recommend QU<1e^{12}).
QU = 0 specifies an infinite unloaded Q.
The model is implemented as a short-circuit admittance matrix,
, whose equivalent normalized lowpass prototype transfer function, S_{21}(s), is [1, 2]:
where F_{N} and E_{N} are normalized lowpass prototype polynomials of order N, and
A specified narrowband bandpass transmission zero, Z[i] = TZR[i] + jTZF[i] is mapped to a normalized lowpass prototype transmission zero, z[ i ], using [5]:
And, _FREQ (the variable that represents the project frequency) is mapped to the normalized lowpass prototype radian frequency,ω, using [5]:
while the specified narrowband bandpass resonator unloaded Q, QU, is converted to an equivalent lowpass prototype element unloaded Q, q_{u}, using [5][6]:
Polynomial
is constructed using a doubly recursive algorithm [1][3]:
where i = 2 to N and, employing the normalized lowpass prototype transmission zero, z_{k}, for k = 1 to N:
And, polynomial E_{N}(s) is found by applying the "alternating singularity principle" [1][2][4] to the roots of
. Then, E_{N} and F_{N} are split into complex-even and complex-odd polynomials [2] such that E_{N}= E_{e}+E_{0} and F_{N}= F_{e}+F_{0}, where
and e_{i} and f_{i} (i = 0 to N) are the complex coefficients of E_{N} and F_{N}. Finally [3]:
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.
The transmission zeros can be tuned or optimized by assigning variables to the elements of the TZF and/or TZR lists and then tuning or optimizing these variables.
Note that 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).
This filter model is non-causal and not usable in transient simulations. An error message is issued if a transient simulation of a circuit containing this model is attempted. (Causality is defined as the response of a circuit following a stimulus--not preceding a stimulus. Non-causal models do not correspond to a physically realizable device.)
[1] Richard J. Cameron, "Fast generation of Chebyshev filter prototypes with asymmetrically-prescribed transmission zeros," ESA J., vol. 6, pp. 83-95, 1982.
[2] Richard J. Cameron, "General coupling matrix synthesis methods for Chebyshev filtering functions," IEEE Trans. Microwave Theory Tech., vol. 47, no. 4, pp. 433-442, April 1999.
[3] Douglas R. Jachowski, unpublished notes, 1995 and 2002.
[4] J. D. Rhodes and A. S. Alseyab, "The generalized Chebyshev low pass prototype filter," Int. J. Circuit Theory Applicat., vol. 8, pp. 113-125, 1980.
[5] H. J. Blinchikoff and A. I. Zverev, Filtering in the Time and Frequency Domains, (Robert E. Krieger Publishing Co., 1987), pp. 178-186, 272.
[6] George L. Matthaei, Leo Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, (Artech House, 1980), pp. 149-156.