# Phase Role in the Non-Uniformity of Main-Line Couplings in Asymmetric Extracted-Pole Inline Filters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Uneven Admittance Inverters in Inline Fully Canonical Filters with Dangling Resonators

**Figure 4.**Magnitude (

**a**) and phase (

**b**) response of a 7th-order B28-Rx filter with a clearly differentiated input and output phases in the out-out-band region.

## 3. Asymmetrical Polynomial Definition

## 4. Phase Determination

#### 4.1. Odd-Order Ladder Filters

#### Hyperbola Types

**Figure 6.**Phase sweep plot when the distance to the vertex is zero. The black trace represents those combinations of $\psi $, $\varphi $ that yields $|{J}_{N+1}|=1$. For illustration purpose all $|{J}_{N+1}|$ values that fit in $1\pm 0.001$ have been plotted.

#### 4.2. Relationship between the Phase Map and the Admittance Redistribution Method

#### 4.3. Even-Order Ladder Filters

## 5. Fast Estimation of the Phase Maps

#### 5.1. Hyperbolic Model Estimation

#### 5.2. Illustrative Synthesis Example

**Table 5.**Characteristic polynomial coefficients for the filter with ${\Omega}_{tz}$ TZs distribution and $RL=$ 18 dB for different additional phase terms.

Initial Characteristic Polynomials | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

deg. | $\mathit{P}\left(\mathit{s}\right)$ | $\mathit{E}\left(\mathit{s}\right)$ | ${\mathit{F}}_{\mathbf{11}}\left(\mathit{s}\right)$ | ${\mathit{F}}_{\mathbf{22}}\left(\mathit{s}\right)$ | ||||||||

0 | 78.6542 | 0.1852 − 0.1290j | −0.0161j | −0.0161j | ||||||||

1 | 43.3847 | 0.8449 − 0.4246j | 0.1470 | 0.1470 | ||||||||

2 | 70.4446j | 1.9944 − 0.8376j | −0.2183j | −0.2183j | ||||||||

3 | 37.6407 | 3.3274 − 1.0378j | 1.0165 | 1.0165 | ||||||||

4 | 20.7380j | 3.6860 − 1.0261j | −0.5080j | −0.5080j | ||||||||

5 | 10.7200 | 3.4792 − 0.5680j | 1.8598 | 1.8598 | ||||||||

6 | 2.0000j | 1.7997 − 0.3115j | −0.3115j | −0.3115j | ||||||||

7 | 1 | 1 | 1 | 1 | ||||||||

Modified Polynomials by$\mathit{\varphi}=-{\mathbf{83.6889}}^{\circ}$, $\mathit{\psi}=-{\mathbf{36.6610}}^{\circ}$ | ||||||||||||

deg. | ${\mathit{E}}_{\mathit{m}}\left(\mathit{s}\right)$ | ${\mathit{F}}_{\mathbf{11}\mathit{m}}\left(\mathit{s}\right)$ | ${\mathit{F}}_{\mathbf{22}\mathit{m}}\left(\mathit{s}\right)$ | |||||||||

0 | 0.2040 + 0.0965j | 0.0064 − 0.0147j | 0.0064 − 0.0147j | |||||||||

1 | 0.7885 + 0.5218j | 0.1348 + 0.0586j | 0.1347 + 0.0586j | |||||||||

2 | 1.7186 + 1.3137j | 0.0871 − 0.2001j | 0.0870 − 0.2001j | |||||||||

3 | 2.5552 + 2.3705j | 0.9321 + 0.4056j | 0.9321 + 0.4055j | |||||||||

4 | 2.7234 + 2.6874j | 0.2027 − 0.4658j | 0.2026 − 0.4658j | |||||||||

5 | 2.2232 + 2.7359j | 1.7053 + 0.7420j | 1.7053 + 0.7420j | |||||||||

6 | 1.1653 + 1.4064j | 0.1243 − 0.2856j | 0.1242 − 0.2855j | |||||||||

7 | 0.4974 + 0.8675j | 0.9170 + 0.3990j | 0.9169 + 0.3989j | |||||||||

Modified Polynomials by$\mathit{\varphi}=-{\mathbf{53.51}}^{\circ}$, $\mathit{\psi}=-{\mathbf{14.18}}^{\circ}$ | ||||||||||||

deg. | ${\mathit{E}}_{\mathit{m}}\left(\mathit{s}\right)$ | ${\mathit{F}}_{\mathbf{11}\mathit{m}}\left(\mathit{s}\right)$ | ${\mathit{F}}_{\mathbf{22}\mathit{m}}\left(\mathit{s}\right)$ | |||||||||

0 | 0.2257 − 0.0040j | 0.0054 − 0.0151j | 0.0054 − 0.0151j | |||||||||

1 | 0.9382 + 0.1179j | 0.1384 + 0.0495j | 0.1384 + 0.0494j | |||||||||

2 | 2.1230 + 0.4152j | 0.0734 − 0.2055j | 0.0734 − 0.2055j | |||||||||

3 | 3.3415 + 0.9912j | 0.9572 + 0.3421j | 0.9572 + 0.3420j | |||||||||

4 | 3.6328 + 1.2007j | 0.1710 − 0.4784j | 0.1709 − 0.4783j | |||||||||

5 | 3.2060 + 1.4660j | 1.7513 + 0.6259j | 1.7513 + 0.6258j | |||||||||

6 | 1.6682 + 0.7436j | 0.1048 − 0.2933j | 0.1048 − 0.2932j | |||||||||

7 | 0.8305 + 0.5569j | 0.9417 + 0.3365j | 0.9416 + 0.3365j |

**Figure 11.**Hyperbola representation of a 7th-order filter: (

**a**) the black trace represents those combinations of $\psi $, $\varphi $ that yields $|{J}_{N+1}|=1$ and (

**b**) a comparison between the modeled and simulated hyperbola. For illustration purpose all $|{J}_{N+1}|$ values that fit in $1\pm 0.001$ have been plotted.

**Table 6.**Lowpass elements of the synthesized 7th-order ladder filter with ${\Omega}_{tz}$ TZs distribution and $RL=$ 18 dB when $\psi =\varphi ={0}^{\circ}$.

Parameters | ${\mathit{B}}_{\mathit{k}}$ | ${\mathit{b}}_{\mathit{k}}$ | ${\mathit{J}}_{\mathit{rk}}$ |
---|---|---|---|

Res. 1 | −2.0663 | −2.4 | 2.1118 |

Res. 2 | 2.9706 | 2.1 | 2.4151 |

Res. 3 | −2.4493 | −1.7 | 2.0115 |

Res. 4 | 2.7822 | 1.8 | 2.2187 |

Res. 5 | −2.9497 | −2 | 2.4700 |

Res. 6 | 2.2032 | 1.7 | 1.7362 |

Res. 7 | −1.3197 | −1.5 | 1.0718 |

Source | −0.4226 | ||

Load | −0.8955 |

#### 5.3. Ellipsoidal Model Estimation

**Figure 12.**(

**a**) Illustrative double parabolic estimation representation of the ellipsoidal model, and (

**b**) the result of applying the model to the filter with ${\Omega}_{Ek}$ (blue trace) superimposed to the result of the phase map (black trace).

## 6. Experimental Validation

**Figure 16.**(

**a**) Measured and simulated filter transmission response, and input/output reflection coefficient. (

**b**) Difference between input and output measured reflection coefficient phase $\Delta \theta ={\theta}_{22}-{\theta}_{11}$. The simulation was carried out with ANSYS

^{®}Electronics Desktop 2021 R1. The manufactured filter was characterized with the network analyzer N5242B PNA-X.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RN | Resonator Node |

NRN | Non-Resonator Node |

FIR | Frequency-Independent Reactance |

RL | Return Loss |

OoB | Out-of-Band |

## References

- Amari, S.; Macchiarella, G. Synthesis of inline filters with arbitrarily placed attenuation poles by using non-resonating nodes. IEEE Trans. Microw. Theory Technol.
**2005**, 53, 3075–3081. [Google Scholar] [CrossRef] - Macchiarella, G. Generalized Coupling Coefficient for Filters With Nonresonant Nodes. IEEE Microw. Wirel. Compon. Lett.
**2008**, 18, 773–775. [Google Scholar] [CrossRef] - Glubokov, O.; Budimir, D. Extraction of Generalized Coupling Coefficients for Inline Extracted Pole Filters With Nonresonating Nodes. IEEE Trans. Microw. Theory Technol.
**2011**, 59, 3023–3029. [Google Scholar] [CrossRef] - Tamiazzo, S.; Macchiarella, G. Synthesis of Cross-Coupled Prototype Filters Including Resonant and Non-Resonant Nodes. IEEE Trans. Microw. Theory Technol.
**2015**, 63, 3408–3415. [Google Scholar] [CrossRef] - Cameron, R.J. Advanced Filter Synthesis. IEEE Microw. Mag.
**2011**, 12, 42–61. [Google Scholar] [CrossRef] - Giménez, A.; Verdú, J.; Sánchez, P.D.P. General Synthesis Methodology for the Design of Acoustic Wave Ladder Filters and Duplexers. IEEE Access
**2018**, 6, 47969–47979. [Google Scholar] [CrossRef] - Verdú, J.; Evdokimova, I.; de Paco, P.; Bauer, T.; Wagner, K. Systematic synthesis methodology for the design of acoustic wave stand-alone ladder filters, duplexers and multiplexers. In Proceedings of the IEEE International Ultrasonics Symposium (IUS), Washington, DC, USA, 6–9 September 2017; p. 1. [Google Scholar]
- Giménez, A.; de Paco, P. Involving source-load leakage effects to improve isolation in ladder acoustic wave filters. In Proceedings of the IEEE MTT-S International Microwave Symposium (IMS), San Francisco, CA, USA, 22–27 May 2016; pp. 1–4. [Google Scholar]
- Triano, A.; Verdú, J.; de Paco, P.; Bauer, T.; Wagner, K. Relation between electromagnetic coupling effects and network synthesis for AW ladder type filters. In Proceedings of the IEEE International Ultrasonics Symposium (IUS), Washington, DC, USA, 6–9 September 2017; pp. 1–4. [Google Scholar]
- Triano, A.; Verdú, J.; de Paco Sánchez, P. A General Synthesis Technique of Mixed-Topology Including Parallel-Connected Structures for Fully Canonical Ladder-Type Acoustic Filters. IEEE Trans. Microw. Theory Technol.
**2019**, 67, 5061–5068. [Google Scholar] [CrossRef] - Triano, A.; Silveira, P.; Verdú, J.; de Paco, P. Phase Correction of Asymmetrical Chebyshev Polynomials for Extracted-Pole Fully Canonical Filters. In Proceedings of the IEEE MTT-S International Microwave Symposium (IMS 2019), Boston, MA, USA, 2–7 June 2019. [Google Scholar]
- Cameron, R.; Mansour, C.K. Microwave Filters for Communications Systems: Fundamentals, Design, and Applications; Wiley: Hoboken, NJ, USA, 2007. [Google Scholar]
- Tsutsumi, J.; Matsumoto, K. Super-Isolation Duplexer Aiming for Removing Rx Inter-stage Filter in W-CDMA Handsets. In Proceedings of the 38th European Microwave Conference, Amsterdam, The Netherlands, 28–30 October 2008; pp. 1066–1069. [Google Scholar]
- Rhodes, J.D.; Cameron, R.J. General Extracted Pole Synthesis Technique with Applications to Low-Loss TE011 Mode. Filters
**1980**, 28, 1018–1028. [Google Scholar] - Cameron, R.J. Fast generation of Chebyshev filter prototypes with asymmetrically-prescribed transmission zeros. ESA J.
**1982**, 6, 83–95. [Google Scholar] - Cameron, R.J. General coupling matrix synthesis methods for Chebyshev filtering functions. IEEE Trans. Microw. Theory Technol.
**1999**, 47, 433–442. [Google Scholar] [CrossRef] - He, Y.; Wang, G.; Sun, L. Direct Matrix Synthesis Approach for Narrowband Mixed Topology Filters. IEEE Microw. Wirel. Compon. Lett.
**2016**, 26, 301–303. [Google Scholar] [CrossRef] - Fanchi, J. Math Refresher for Scientists and Engineers; Wiley: Hoboken, NJ, USA, 2006. [Google Scholar]
- Lundsgaard, H. Shadows of the Circle: Conic Sections, Optimal Figures and Non-Euclidean Geometry; World Scientific Publishing Company: Singapore, 1998. [Google Scholar]

**Figure 2.**S-parameters of the extracted filter using TZs distributions ${\Omega}_{Ak}$ and ${\Omega}_{Bk}$.

**Figure 3.**Nodal diagram of the last three elements of the network before (

**left**) and after (

**right**) the admittance redistribution. The grey elements are actually part of the network but they are irrelevant in this scenario.

**Figure 5.**(

**a**) Horizontal hyperbola and (

**b**) vertical hyperbola representation. The black trace represents those combinations of $\varphi $, $\psi $ that yields $|{J}_{N+1}|=1$. For illustration purpose all $|{J}_{N+1}|$ values that fit in $1\pm 0.001$ have been plotted. The red traces corresponds to the asymptotes. The point $({\psi}_{0},{\varphi}_{0})$ is the origin and $({\psi}_{0}^{\prime},{\varphi}_{0}^{\prime})$ the origin of another hyperbola with a ${180}^{\circ}$ shift in both directions.

**Figure 7.**(

**a**) Horizontal hyperbola of the filter with ${\Omega}_{Bk}$ TZs and (

**b**) vertical hyperbola of the filter with ${\Omega}_{Ck}$ TZs. The black trace are those combinations of $\varphi $, $\psi $ that yields $|{J}_{N+1}|=1\pm 0.001$. The red line across the $\varphi $ axis is at $\psi =0$.

**Figure 8.**(

**a**) S-parameters of the network with ${\Omega}_{Ek}$ TZs and (

**b**) the phase map of the ellipse with a sweep between $\pm {180}^{\circ}$.

**Figure 9.**Comparison between the phase sweep in $\varphi $ for the filter with ${\Omega}_{tz1}$ TZs and the parabolic estimation model.

**Figure 14.**Network with the direct synthesis of the elements resulting in ${J}_{N+1}\ne -1$ (

**left**), and once the network transformation is done to achieve ${J}_{N+1}=-1$ (

**right**).

**Figure 15.**(

**a**) Equivalent electric circuit using the Butterworth-Van Dyke model for the designed filter, and (

**b**) the fabricated prototype using lumped elements.

**Table 1.**Extracted elements of the 5th-order filter prototype with ${\Omega}_{Ak}$ and ${\Omega}_{Bk}$ TZs distributions.

${\mathbf{\Omega}}_{\mathit{Ak}}$ | ${\mathbf{\Omega}}_{\mathit{Bk}}$ | |||||
---|---|---|---|---|---|---|

Parameters | ${\mathit{B}}_{\mathit{k}}$ | ${\mathit{b}}_{\mathit{k}}$ | ${\mathit{J}}_{\mathit{rk}}$ | ${\mathit{B}}_{\mathit{k}}$ | ${\mathit{b}}_{\mathit{k}}$ | ${\mathit{J}}_{\mathit{rk}}$ |

Res. 1 | −1.0927 | −1.8 | 1.1768 | −1.0927 | −1.8 | 1.1768 |

Res. 2 | 3.4897 | 2.0 | 2.4193 | 3.3440 | 2.0 | 2.4193 |

Res. 3 | −2.6930 | −2.5 | 2.6548 | −1.8121 | −1.8 | 1.7911 |

Res. 4 | 3.4897 | 2.0 | 2.4193 | 3.5215 | 2.0 | 2.4946 |

Res. 5 | −1.0927 | −1.8 | 1.1768 | −1.4090 | −2.5 | 1.7951 |

Source | −0.7388 | −0.7388 | ||||

Load | −0.7388 | −0.4553 |

**Table 2.**Extracted elements of the 5th-order filter prototype with ${\Omega}_{Ck}$ TZs distribution.

Parameters | ${\mathit{B}}_{\mathit{k}}$ | ${\mathit{b}}_{\mathit{k}}$ | ${\mathit{J}}_{\mathit{rk}}$ |
---|---|---|---|

Res. 1 | −0.6489 | −1.80 | 1.1761 |

Res. 2 | 1.1085 | 1.16 | 0.5833 |

Res. 3 | −2.9532 | −1.80 | 2.3379 |

Res. 4 | 2.0059 | 2.00 | 1.8959 |

Res. 5 | −2.5918 | −2.50 | 2.4198 |

${B}_{S}$ | −0.7353 | ||

${B}_{L}$ | −0.4519 |

**Table 3.**Extracted elements of the 6th-order filter prototype with a ${\Omega}_{Dk}$ TZs distribution.

Parameters | ${\mathit{B}}_{\mathit{k}}$ | ${\mathit{b}}_{\mathit{k}}$ | ${\mathit{J}}_{\mathit{rk}}$ |
---|---|---|---|

Res. 1 | −1.6233 | −2.50 | 2.091 |

Res. 2 | 1.2539 | 1.30 | 0.97845 |

Res. 3 | −2.4661 | −1.50 | 1.827 |

Res. 4 | 3.1101 | 2.64 | 2.9667 |

Res. 5 | −3.4831 | −2.00 | 2.4392 |

Res. 6 | 1.2111 | 1.86 | 1.2928 |

${B}_{S}$ | −0.4460 | ||

${B}_{L}$ | 0.6775 |

**Table 4.**Extracted elements of the 4th-order filter prototype with a ${\Omega}_{Ek}$ TZs distribution.

Parameters | ${\mathit{B}}_{\mathit{k}}$ | ${\mathit{b}}_{\mathit{k}}$ | ${\mathit{J}}_{\mathit{rk}}$ |
---|---|---|---|

Res. 1 | 0.9234 | 1.8000 | 1.108 |

Res. 2 | −2.3310 | −1.6000 | 1.6192 |

Res. 3 | 1.8854 | 2.0000 | 1.8385 |

Res. 4 | −2.4007 | −2.5000 | 2.3224 |

${B}_{S}$ | 0.7782 | ||

${B}_{L}$ | −0.4700 |

**Table 7.**Lowpass elements of the synthesized 7th-order ladder filter with phase correction parameters $\psi =-{36.66}^{\circ}$, $\varphi =-{83.6889}^{\circ}$ and $\psi =-{14.18}^{\circ}$, $\varphi =-{53.51}^{\circ}$.

$\mathit{\psi}=-{36.66}^{\circ}$, $\mathit{\varphi}=-{83.68}^{\circ}$ | $\mathit{\psi}=-{14.18}^{\circ}$, $\mathit{\varphi}=-{53.51}^{\circ}$ | |||||
---|---|---|---|---|---|---|

${\mathit{B}}_{\mathit{k}}$ | ${\mathit{b}}_{\mathit{k}}$ | ${\mathit{J}}_{\mathit{rk}}$ | ${\mathit{B}}_{\mathit{k}}$ | ${\mathit{b}}_{\mathit{k}}$ | ${\mathit{J}}_{\mathit{rk}}$ | |

Res. 1 | −2.0663 | −2.4 | 2.1118 | −2.1254 | −2.4 | 2.2058 |

Res. 2 | 2.5367 | 2.1 | 2.2317 | 2.7228 | 2.1 | 2.3122 |

Res. 3 | −2.8683 | −1.7 | 2.1767 | −2.6722 | −1.7 | 2.1010 |

Res. 4 | 2.3758 | 1.8 | 2.0502 | 2.5501 | 1.8 | 2.1241 |

Res. 5 | −3.4543 | −2 | 2.6728 | −3.2181 | −2 | 2.5799 |

Res. 6 | 1.8815 | 1.7 | 1.6044 | 2.0194 | 1.7 | 1.6622 |

Res. 7 | −0.6499 | −1.5 | 1.1598 | −0.8568 | −1.5 | 1.1195 |

Source | −0.0800 | −0.2833 | ||||

Load | 0 | −0.2696 |

**Table 8.**Lowpass elements of the synthesized 5th-order ladder filter before and after ${J}_{N+1}$ redistribution.

Parameters | ${\mathit{B}}_{\mathit{kb}}$ | ${\mathit{b}}_{\mathit{kb}}$ | ${\mathit{J}}_{\mathit{rkb}}$ | ${\mathit{B}}_{\mathit{ka}}$ | ${\mathit{b}}_{\mathit{ka}}$ | ${\mathit{J}}_{\mathit{rka}}$ |
---|---|---|---|---|---|---|

Res. 1 | −1.004 | 1.734 | 1.110 | −1.004 | 1.734 | 1.110 |

Res. 2 | 2.686 | −1.817 | 2.095 | 2.686 | −1.817 | 2.095 |

Res. 3 | −1.042 | 1.235 | 0.944 | −1.042 | 1.235 | 0.944 |

Res. 4 | 3.624 | −2.246 | 2.841 | 3.624 | −2.246 | 2.841 |

Res. 5 | −1.517 | 2.467 | 1.831 | −1.688 | 2.467 | 1.831 |

Source | −0.773 | −0.773 | ||||

Load | −0.457 | −0.713 |

**Table 9.**Bandpass elements of the 5th-order ladder filter with phase correction terms $\varphi =-{89.3}^{\circ}$, $\psi =-{27.7}^{\circ}$.

Parameters | ${\mathit{L}}_{\mathit{a}}$ (nH) | ${\mathit{C}}_{\mathit{a}}$ (pF) | ${\mathit{C}}_{0}$ (pF) |
---|---|---|---|

Res. 1 | 53.18 | 7.35 | 13.28 |

Res. 2 | 81.14 | 8.90 | 15.13 |

Res. 3 | 135.24 | 2.68 | 11.03 |

Res. 4 | 48.83 | 16.59 | 18.33 |

Res. 5 | 41.63 | 14.86 | 6.65 |

Source | 73.3 nH | ||

Load | 4.74 pF |

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## Share and Cite

**MDPI and ACS Style**

Triano, Á.; Silveira, P.; Verdú, J.; Guerrero, E.; de Paco, P.
Phase Role in the Non-Uniformity of Main-Line Couplings in Asymmetric Extracted-Pole Inline Filters. *Electronics* **2021**, *10*, 3058.
https://doi.org/10.3390/electronics10243058

**AMA Style**

Triano Á, Silveira P, Verdú J, Guerrero E, de Paco P.
Phase Role in the Non-Uniformity of Main-Line Couplings in Asymmetric Extracted-Pole Inline Filters. *Electronics*. 2021; 10(24):3058.
https://doi.org/10.3390/electronics10243058

**Chicago/Turabian Style**

Triano, Ángel, Patricia Silveira, Jordi Verdú, Eloi Guerrero, and Pedro de Paco.
2021. "Phase Role in the Non-Uniformity of Main-Line Couplings in Asymmetric Extracted-Pole Inline Filters" *Electronics* 10, no. 24: 3058.
https://doi.org/10.3390/electronics10243058