# Generalized Concept and MATLAB Code for Modeling and Analyzing Wideband 90° Stub-Loaded Phase Shifters with Simulation and Experimental Verifications

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{o}, and can be and can be tailored to any user-defined frequency range. As a matter of comparison, a three-stub wideband 90° stub-loaded phase shifter is simulated using CST Microwave Studio and experimentally fabricated on Rogers RT5880 dielectric substrate with dimensions of 30 × 40 × 0.8 mm

^{3}. The comparison reveals the accuracy of the proposed computerized modeling with −10 dB impedance bandwidth equal to 90% (0.55f

_{o}–1.45f

_{o}), (90°∓5°) phase difference bandwidth equal to 100% (0.5f

_{o}–1.5f

_{o}), and negligible insertion loss. The novelty of this work is that the proposed code provides the exact modeling equations of the stub-loaded phase shifter for any number of stubs regardless the complexity of the mathematical derivations.

## 1. Introduction

## 2. Concept of Wideband Differential Phase Shifting

^{8}$\mathrm{m}/\mathrm{s}$). By substituting (3) and (4) into (2), the phase angle of the transmission coefficient can be given as in (5):

- a.
- The linear region of the phase of the main line ($\mathrm{\angle}{S}_{21}$) should be as wide as possible with respect to the frequency.
- b.
- The linear region of $\mathrm{\angle}{S}_{21}$ should be parallel with the phase of the reference line ($\mathrm{\angle}{S}_{43}$) to provide constant phase difference along the frequency range of the linear region.

## 3. Analysis of Stub-Loaded Wideband Phase Shifter

## 4. General Code for Modelling 90° Stub-Loaded Phase Shifter

#### 4.1. Part 1: Deriving the T and S Matrices

- 1.
- Enter the value of the number of steps $N$, then the symbols ${Z}_{o}$, ${Z}_{p}$, ${\theta}_{1}$, and ${\theta}_{2}$ are defined using the “$syms$” MATLAB function.
- 2.
- Write the equations of ${T}_{Line}$ and ${T}_{Stub}$, given in (7) and (8).
- 3.
- Initialize the transmission matrix of the stub-loaded filter by $T={T}_{Stub}$.
- 4.
- Multiply the previous value of $T$ by (${T}_{Line}\times {T}_{Stub}$).
- 5.
- Repeat Step (4) ($N-1$) times to obtain a multiplication of $N$ times ${T}_{Stub}$ by ($N-1$) times ${T}_{Line}$, alternately, as given in (9).
- 6.
- Use the MATLAB functions “$expand$” and “$simplify$” for simplifying the results and finding some simple trigonometric identities.
- 7.
- Apply (10) and (11) to find the reflection and transmission parameters of the S-matrix.

#### 4.2. Part 2: Specified Variable Results

- 1.
- Enter values for ${Z}_{o}$ and ${Z}_{p}$, so that the best impedance bandwidth and phase difference bandwidth is achieved.
- 2.
- Set the variable normalized frequency to any range (say 0 to 2 with 0.01 step size).
- 3.
- Set the normalized lengths of the stub and the line sections to 0.25 which is corresponding to quarter wavelength line at the center frequency.
- 4.
- In the resulted ${S}_{11}$ and ${S}_{21}$, use the MATLAB function “$subs$” at each value of the normalized frequency to substitute the values of ${Z}_{o}$ and ${Z}_{p}$. In addition, substitute the value of ${\theta}_{1}$, ${\theta}_{2}$with the aid of (12).
- 5.
- Find the magnitude of ${S}_{11}$ and ${S}_{21}$ in dB, as well as the phase of ${S}_{21}$ in degrees using the MATLAB function “$phase$”.
- 6.
- The normalized length of the reference line is equal to the length of ($N-1$) line section plus 0.25 to provide 90° phase delay at ${f}_{o}$ with respect to the main line.$${l}_{ref-norm}=\left(N-1\right){l}_{Line-norm}+0.25$$
- 7.
- Find the phase of transmission coefficient ${S}_{43}$of the reference line, which is equal to$$-{\theta}_{ref}=-2\pi {l}_{ref-norm}(f/{f}_{o})$$
- 8.
- The phase difference at the output ports is equal to ($\angle {S}_{21}-\angle {S}_{43}$).

## 5. Parametric Study for the Proposed Modelling

_{o}= 50 Ω and different number of stubs.

## 6. Simulation and Experimental Verification Example

_{o}= 5.5 GHz. The dielectric substrate of the proposed structure is Rogers RT5880, whose dielectric constant ${\epsilon}_{r}=2.2$, height $h=0.8\mathrm{m}\mathrm{m}$, with a loss tangent of $0.0009$. The length of the stubs and the line sections are equal to a quarter of the guided wavelength (${\lambda}_{go}$), corresponding to the center frequency ${f}_{o}$, which is equal to:

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

- Karimbu Vallappil, A.; Rahim, M.K.A.; Khawaja, B.A.; Iqbal, M.N. Compact Metamaterial Based 4 × 4 Butler Matrix With Improved Bandwidth for 5G Applications. IEEE Access
**2020**, 8, 13573–13583. [Google Scholar] [CrossRef] - Yu, Y.; Wu, Y.; Tang, P.; Zhao, C.; Liu, H.; Wu, Y.; Yin, W.-Y.; Kang, K. An 18-30 GHz Vector-Sum Phase Shifter With Two-Stage Transformer-Based Hybrid in 130-nm SiGe BiCMOS. IEEE Trans. Circuits Syst. I Regul. Pap.
**2023**, 1–14. [Google Scholar] [CrossRef] - Fang, C.; Wang, Y.; Chen, Y.; Lin, Y.; Xu, H. A Frequency Reconfigurable Reflection-Type Phase Shifter for Multi-Band 5G Communication. In Proceedings of the 2023 6th International Conference on Electronics Technology (ICET), Chengdu, China, 12–15 May 2023; pp. 577–580. [Google Scholar]
- Park, J.; Lee, S.; Chun, J.; Jeon, L.; Hong, S. A 28-GHz Four-Channel Beamforming Front-End IC With Dual-Vector Variable Gain Phase Shifters for 64-Element Phased Array Antenna Module. IEEE J. Solid-State Circuits
**2023**, 58, 1142–1159. [Google Scholar] [CrossRef] - Alnahwi, F.M.; Al-Yasir, Y.I.A.; See, C.H.; Abd-Alhameed, R.A. Single-Element and MIMO Circularly Polarized Microstrip Antennas with Negligible Back Radiation for 5G Mid-Band Handsets. Sensors
**2022**, 22, 3067. [Google Scholar] [CrossRef] [PubMed] - Qiu, L.-L.; Zhu, L.; Xu, Y. Wideband Low-Profile Circularly Polarized Patch Antenna Using 90° Modified Schiffman Phase Shifter and Meandering Microstrip Feed. IEEE Trans. Antennas Propag.
**2020**, 68, 5680–5685. [Google Scholar] [CrossRef] - Alnahwi, F.M.; Al-Yasir, Y.I.A.; Ali, N.T.; Gharbia, I.; Abdullah, A.S.; Hu, Y.F.; Abd-Alhameed, R.A. A Compact Broadband Circularly Polarized Wide-Slot Antenna With Axial Ratio Bandwidth Encompassing LTE 42 and LTE 43 Standards of 5G Mid-Band. IEEE Access
**2023**, 11, 2012–2022. [Google Scholar] [CrossRef] - Abbosh, A.M. Ultra-Wideband Phase Shifters. IEEE Trans. Microw. Theory Tech.
**2007**, 55, 1935–1941. [Google Scholar] [CrossRef] - Abbosh, A.M.; Bialkowski, M.E. Design of Compact Directional Couplers for UWB Applications. IEEE Trans. Microw. Theory Tech.
**2007**, 55, 189–194. [Google Scholar] [CrossRef] - Lyu, Y.-P.; Zhu, L.; Cheng, C.-H. Single-Layer Broadband Phase Shifter Using Multimode Resonator and Shunt λ/4 Stubs. IEEE Trans. Compon. Packag. Manuf. Technol.
**2017**, 7, 1119–1125. [Google Scholar] [CrossRef] - Qiu, L.-L.; Zhu, L.; Lyu, Y.-P. Generalized Topology and Synthesis Design of Balanced Wideband Phase Shifters with Common-Mode Suppression. In Proceedings of the 2019 IEEE Asia-Pacific Microwave Conference (APMC), Singapore, 9–12 December 2019; pp. 198–200. [Google Scholar]
- Marini, S.E.; Mandry, R.; Zbitou, J.; Errkik, A.; Tajmouati, A.; Latrach, M. Broadband planar 90 degrees loaded-stub phase shifter. Telkomnika Telecommun. Comput. Electron. Control
**2020**, 18, 2834–2841. [Google Scholar] [CrossRef] - Shi, J.; Nie, Y.; Han, P.; Zhang, W.; Cao, Q. Compact Filtering Phase Shifter With Simple Structure. IEEE Microw. Wirel. Compon. Lett.
**2021**, 31, 1263–1266. [Google Scholar] [CrossRef] - Dai, Y.; Wang, D.Y.; Jiang, S.; Liu, L. Wideband filtering phase shifters using vertically installed planar structure. Microw. Opt. Technol. Lett.
**2022**, 65, 753–761. [Google Scholar] [CrossRef] - Qiu, L.-L.; Zhu, L. Dual-Band Filtering Differential Phase Shifter Using Cascaded Wideband Phase Shifter and Bandstop Network With Two Same Phase Shifts. IEEE Microw. Wirel. Compon. Lett.
**2021**, 31, 261–264. [Google Scholar] [CrossRef] - Han, Y.; Li, R.; Qiao, L.; Wei, F. Balanced Wideband Quasi-Schiffman Phase Shifters Based on Slotlines. IEEE Trans. Circuits Syst. II Express Briefs
**2022**, 69, 4283–4287. [Google Scholar] [CrossRef] - Ding, X.; Xue, Y.; Liu, W.; Zhang, P.; Xu, L.; Li, R.; Wei, F. Design of broadband reconfigurable phase shifters. Int. J. RF Microw. Comput.-Aided Eng.
**2022**, 32, e23424. [Google Scholar] [CrossRef] - Ghimire, J.; Diba, F.D.; Kim, J.-H.; Choi, D.-Y. Vivaldi Antenna Arrays Feed by Frequency-Independent Phase Shifter for High Directivity and Gain Used in Microwave Sensing and Communication Applications. Sensors
**2021**, 21, 6091. [Google Scholar] [CrossRef] - Xu, Z.; Wang, Y.; Liu, S.; Ma, J.; Fang, S.; Wu, H. Metamaterials With Analogous Electromagnetically Induced Transparency and Related Sensor Designs—A Review. IEEE Sens. J.
**2023**, 23, 6378–6396. [Google Scholar] [CrossRef] - Pozar, D.M. Microwave Engineering; University of Massachusetts at Amherst: Amherst, MA, USA; John Wiley & Sons: Hoboken, NJ, USA, 2012; pp. 26–30. [Google Scholar]
- Balanis, C.A. Antenna Theory: Analysis and Design; John Wiley & Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Garg, R.; Bahl, I.; Bozzi, M. Microstrip Lines and Slotlines; Artech House: Norwood, MA, USA, 2013. [Google Scholar]
- Alnahwi, F.M.; Al-Yasir, Y.I.A.; Ali, N.T.; Gharbia, I.; See, C.H.; Abd-Alhameed, R.A. A Compact Wideband Circularly Polarized Planar Monopole Antenna With Axial Ratio Bandwidth Entirely Encompassing the Antenna Bandwidth. IEEE Access
**2022**, 10, 81828–81835. [Google Scholar] [CrossRef] - Juma’a, F.K.; Al-Mayoof, A.I.; Abdulhameed, A.A.; Alnahwi, F.M.; Al-Yasir, Y.I.A.; Abd-Alhameed, R.A. Design and Implementation of a Miniaturized Filtering Antenna for 5G Mid-Band Applications. Electronics
**2022**, 11, 2979. [Google Scholar] [CrossRef] - Liu, X.; Liu, Y.; Zhang, T.; Lu, Q.; Zhu, Z. Substrate-Integrated Waveguide Band-Pass Filter and Diplexer With Controllable Transmission Zeros and Wide-Stopband. IEEE Trans. Circuits Syst. II Express Briefs
**2023**, 70, 526–530. [Google Scholar] [CrossRef] - Ragavi, B.; Sharmila, S.; Dharani, J.; Deepthika, K. Design of Dielectric coupled Line Resonator with Defector Ground Structure for Microwave frequency with Double Band Pass filter. In Proceedings of the 2023 International Conference on Computer Communication and Informatics (ICCCI), Fujisawa, Japan, 23–25 June 2023; pp. 1–6. [Google Scholar]
- Wang, X.; Chen, X.; Sun, D. A Compact Contactless Waveguide Band-pass Filter for High Sensitivity Passive Intermodulation Measurement. In Proceedings of the 2023 IEEE MTT-S International Wireless Symposium (IWS), Qingdao, China, 16–19 May 2023; pp. 1–3. [Google Scholar]
- Zhuang, Q.; Yan, F.; Xiong, Z.; Yang, M.; Liu, M. A Novel High-Power Rotary Waveguide Phase Shifter Based on Circular Polarizers. Electronics
**2023**, 12, 2963. [Google Scholar] [CrossRef]

**Figure 4.**The modeled (

**a**) reflection coefficient magnitude; (

**b**) transmission coefficient magnitude; and (

**c**) phase difference between the output ports for $N=3$, ${Z}_{o}=50\mathsf{\Omega}$, and different values of ${Z}_{p}$.

**Figure 5.**The modeled (

**a**) reflection coefficient magnitude; (

**b**) transmission coefficient magnitude; and (

**c**) phase difference between the output ports for ${Z}_{o}=50\mathsf{\Omega}$ and different values of $N$ and ${Z}_{p}$.

**Figure 6.**Phase responses of the main and the reference lines for N = 3 stub-loaded 90° phase shifter.

**Figure 7.**Three-stub 90° stub-loaded phase shifter: (

**a**) CST microwave studio structure; (

**b**) the prototype of the proposed design.

**Figure 8.**Comparison between the modeled, simulated, and measured (

**a**) reflection coefficient magnitude; (

**b**) transmission coefficient magnitude; and (

**c**) phase difference between the output ports.

**Table 1.**Comparison between the modeled equation outcomes with the CST microwave studio results and the measured results.

CST Simulation | Measurement | Modeled Equations | |
---|---|---|---|

−10 dB impedance BW (%) | 90.01% | 90.01% | 90% |

Insertion loss (dB) @ 5.5 GHz | 0.138 | 0.156 | 0 |

$\left(90\xb0\mp 5\xb0\right)$ phase difference BW (%) | 100% | 94% | 100% |

**Table 2.**Comparison between the proposed phase shifter and other important designs, where ${\lambda}_{o}$ corresponds to the first resonant frequency.

Ref. | Dimensions | Phase Difference BW (%) | Modeling Method |
---|---|---|---|

[10] | $0.77{\lambda}_{o}\times 0.3{\lambda}_{o}\times 0.01{\lambda}_{o}$ | 102 | Conventional Derivation |

[13] | $0.32{\lambda}_{o}\times 0.22{\lambda}_{o}\times 0.01{\lambda}_{o}$ | 59 | Conventional Derivation |

[14] | $0.46{\lambda}_{o}\times 0.31{\lambda}_{o}\times 0.01{\lambda}_{o}$ | 64 | Conventional Derivation |

[16] | $0.69{\lambda}_{o}\times 0.46{\lambda}_{o}\times 0.014{\lambda}_{o}$ | 59 | Conventional Derivation |

[17] | $0.52{\lambda}_{o}\times 0.42{\lambda}_{o}\times 0.01{\lambda}_{o}$ | 77 | -- |

This work | $0.53{\lambda}_{o}\times 0.4{\lambda}_{o}\times 0.02{\lambda}_{o}$ | 100% | Computer-Based |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alnahwi, F.M.; Al-Yasir, Y.I.A.; See, C.H.; Abdullah, A.S.; Abd-Alhameed, R.A.
Generalized Concept and MATLAB Code for Modeling and Analyzing Wideband 90° Stub-Loaded Phase Shifters with Simulation and Experimental Verifications. *Sensors* **2023**, *23*, 7773.
https://doi.org/10.3390/s23187773

**AMA Style**

Alnahwi FM, Al-Yasir YIA, See CH, Abdullah AS, Abd-Alhameed RA.
Generalized Concept and MATLAB Code for Modeling and Analyzing Wideband 90° Stub-Loaded Phase Shifters with Simulation and Experimental Verifications. *Sensors*. 2023; 23(18):7773.
https://doi.org/10.3390/s23187773

**Chicago/Turabian Style**

Alnahwi, Falih M., Yasir I. A. Al-Yasir, Chan Hwang See, Abdulkareem S. Abdullah, and Raed A. Abd-Alhameed.
2023. "Generalized Concept and MATLAB Code for Modeling and Analyzing Wideband 90° Stub-Loaded Phase Shifters with Simulation and Experimental Verifications" *Sensors* 23, no. 18: 7773.
https://doi.org/10.3390/s23187773