# Identifying and Modeling Resonance-Related Fluctuations on the Experimental Characteristic Impedance for PCB and On-Chip Transmission Lines

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## Abstract

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## 1. Introduction

_{c}) [1]. In this regard, whereas the determination of γ is straightforward, solving eigenvalue equations involving measurements of lines varying in length, obtaining Z

_{c}is cumbersome [2,3]. This is because this latter parameter is strongly affected by the return loss of the TLs [4]. In fact, even after applying the most advanced de-embedding methods to the measurements, significant fluctuations in the Z

_{c}curves are observed at microwave frequencies from experimental data [5,6,7,8]. These fluctuations are associated with the interaction of standing waves with the imperfect terminations of the TLs, which are associated with the signal launchers that serve as the interface between the lines and the test equipment. Specifically, the fluctuations occur at resonant frequencies where half the wavelength of the signal equals the physical length of the line. Furthermore, we show here for the first time—to the best of our knowledge—that additional fluctuations become apparent due to standing waves taking place within the line’s transition itself, since these transitions present a non-zero physical length.

_{c}are applied to illustrate the effect of the transitions on the experimentally obtained curves. In this regard, in the traditional method, single line measurements are used, but resonances of large magnitude affect the extraction [3,9], even after removing inductive and capacitive parasitics associated with the line terminations [10]. Alternatively, a de-embedding procedure involving measurements performed on two lines varying in length can be applied [11]. Nevertheless, this method assumes that the parasitics at the terminations can be effectively represented using a single shunt admittance, which provides acceptable results for lines on-chip but fails when the transition requires a representation considering series and shunt effects. This is the case of lines on packaging technology and also on-chip when the wavelength of the signal is comparable to the physical length of the interconnect.

_{c}, for instance by considering the relationship between these parameters and the resistance, inductance, conductance and capacitance (RLGC) per unit length elements in the model of a line operating in the transverse electromagnetic mode [12]. The advantage of this method is that relative permittivity and loss tangent data can be used to infer the complex curves for Z

_{c}from γ [13]. Nevertheless, this method can only be applied in a straightforward manner when the capacitance and conductance parameters in the RLGC model are related to well identified dielectric effects, which hampers its application to lines on lossy substrates (e.g., metal-insulator-semiconductor microstrip lines). Furthermore, as frequency increases, the wavelength of the propagating signals may become comparable to the physical length, not only of the uniform transmission line section but also to that of the signal launchers terminating the lines. In this case, extracting the characteristic impedance of the uniform section of the transmission line becomes more complicated since the measured data correspond to the concatenation of three transmission lines: the uniform section of line embedded between two transmission lines corresponding to the signal launchers.

## 2. Materials and Methods

_{c}is necessary for TL characterization, as well as for calibration procedures. Thus, determining this parameter from S-parameter measurements is desirable. In this regard, in an early paper about this topic, the transmission parameters obtained from an S-to-ABCD parameter transformation of a TL exhibiting a length l are assumed to be given by [5]:

_{c}can be straightforwardly obtained as:

**T**

_{h}matrix. Unfortunately, the transitions that interface the line with the test equipment at the measurement plane after calibration introduce effects that are typically represented by lumped capacitive (C) and inductive (L) parasitics. Figure 1a shows a schematic illustrating a way to account for these parasitics, which allows the following expression to be written for the associated transmission parameters:

_{c}, but curves that include fluctuations or glitches introduced by the effect of C and L. These glitches are observed in the Re(Z

_{c}) curves in the form shown in Figure 1b when either the inductive or capacitive effects are dominant. In fact, the periodicity and magnitude of these glitches is dependent on the length of the TL as illustrated in Figure 1c [9]. Observe, however, that the fluctuations occur around the expected value for Z

_{c}, which suggests that they can be easily removed.

_{D}and a characteristic impedance Z

_{D}. This model is shown in Figure 2a, whereas the associated transmission parameters are:

_{c}(see Figure 2b).

## 3. Results

- A
- Lines on-chip

- B
- Lines on PCB terminated with probe pad adapters

_{c}$\approx $ 51 Ω. The measurements were performed using GSG probes with a 150 µm pitch.

- C
- Lines on PCB terminated with coaxial connectors

_{c}$\approx $ 72 Ω and are terminated with 40 GHz general precision connectors (GPCs) with a 2.92 mm interface [17,18]. In this case, the VNA setup was calibrated up to the coaxial interface [19].

## 4. Discussion

_{c}| curve does not vary considerably with frequency since the losses are relatively small. Conversely, for microstrip lines made on silicon, the films of the signal traces are thin enough to have per unit length (PUL) resistances of the order of kΩ/m. This high resistance causes the characteristic impedance to have a strong variation over a wide range of frequencies [20]. For the on-chip lines, the Z

_{c}obtained using (2) for the different line lengths is shown in Figure 4. In addition to the fluctuations due to the transitions, Z

_{c}exhibits large discrepancies depending on the line length, which is unexpected. On the other hand, using the open-short method [21], the fluctuations due to discontinuities are smoothed; however, an unexpected roll-down of the Z

_{c}is noticeable as frequency increases. This variation is originated because the open and short structures consider that the transition between the pads and the lines is abrupt. In contrast, using the line–line method [11], a flat Z

_{c}is achieved from 10 to 50 GHz, which can be considered closer to the expected value. In this latter case, the extraction method relies on the fact that the transition can be represented by means of a lumped shunt admittance, which is valid provided that the pad array can be considered relatively small, a condition fulfilled for on-chip interconnects but not for PCB lines.

_{c}can be observed within a range of a few tens of gigahertz. In Figure 5, Z

_{c}extracted using (2) is shown for two lines of different length, as well as that extracted from γ [13]. As can be seen, the fluctuations due to reflections within the line length have a greater magnitude for the shorter line but occur at a higher rate on the longer line. Observe in Figure 5 that, in reference to Figure 1b, the form of the curves evidences a dominant inductive effect associated with the transition. In contrast, Z

_{c}obtained from γ is smooth, so it can be taken as a good approximation for lines with these characteristics. The disadvantage of this procedure is that it requires previous knowledge of the frequency-dependent complex permittivity. Otherwise, a constant effective permittivity and loss tangent can be assumed, but causality in the representation of the line might be compromised.

_{c}from measured S-parameters. Due to the electrical length of the coaxial-to-microstrip transitions, a distributed model for the discontinuity needs to be considered, so the interconnect plus the transitions needs to be modeled with (4). In Figure 6, Z

_{c}obtained using (2) as well as from γ is shown. Unlike the case where probes were used to measure, fluctuations not only occur with a periodicity related to the length of the lines, but Z

_{c}presents an additional fluctuation of lower periodicity, which is related to reflections along the length of the coaxial connectors. Although these reflections are present in any type of discontinuity, they become more noticeable in this case for the considered frequency ranges. Therefore, when the length of the transition is considerably long, it makes it even more difficult to determine the actual frequency dependence of Z

_{c}from experimental data. Interestingly, when comparing the curves in Figure 6 with the conceptual plots in Figure 1b, it can be observed that at frequencies below 15 GHz, the transition exhibits a dominant capacitive behavior, which becomes inductive at higher frequencies. This is indicative of the importance of implementing a broadband model for this type of microwave transition measurement.

_{c}is obtained from γ, is barely noticeable.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Representation of the effect of lumped transition parasitics in Z

_{c}curves: (

**a**) schematic of a uniform TL embedded between the transitions, (

**b**) curves showing the fluctuations in Z

_{c}when either the inductive or capacitive effects are dominant and (

**c**) Z

_{c}calculated from (2) when the parasitics are ignored.

**Figure 2.**Illustrations depicting the effect of parasitics in Z

_{c}curves when the transition includes effects that are required to be represented using a distributed model: (

**a**) schematic showing a uniform TL embedded between the transitions and (

**b**) Z

_{c}calculated from (2) when the parasitics are ignored.

**Figure 3.**Sketches of the cross sections exhibited by the prototyped lines accompanied by photographs of the corresponding measurement setup, which has been previously calibrated: (

**a**) on-chip lines terminated with probing pads, (

**b**) PCB lines terminated with probing pads and (

**c**) PCB lines terminated with coaxial connectors. All dimensions are expressed in microns (microns).

**Figure 4.**|Z

_{c}| obtained for interconnects on-chip from direct measurements, line–line method and open-short method.

**Figure 5.**|Z

_{c}| for PCB lines interfaced with probe pad adapters; curves obtained from direct measurements applying (2) to the two available line lengths and from propagation constant are shown.

**Figure 6.**|Z

_{c}| obtained for interconnects on PCB with coaxial connectors from direct measurements (Equation (2)), and also from propagation constant data using the method in [13].

**Figure 7.**RLGC parameters obtained for interconnects on PCB terminated with coaxial connectors. These parameters consider Z

_{c}extracted from direct measurements (Equation (2)), and also from propagation constant data using the method in [13].

**Figure 8.**A model for the transmission line including the transitions, represented here by distributed coaxial lines. The coaxial lines were of length Lc = 3.8 mm and relative permittivity Ec = 2.5. Homogeneous transmission line S-parameters, represented by the S2P box, were obtained from extracted γ, Z

_{c}approximated from γ, and l = 25.4 mm.

**Figure 9.**Effective impedance measurement-model comparison for a PCB line terminated with a coaxial connector. A lumped model for the coaxial connectors was added for comparison purposes with inductance L

_{COAX}= 0.995 nH and capacitance C

_{COAX}= 0.38 pF.

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**MDPI and ACS Style**

Rodríguez-Velásquez, Y.; Torres-Torres, R.; Murphy-Arteaga, R.
Identifying and Modeling Resonance-Related Fluctuations on the Experimental Characteristic Impedance for PCB and On-Chip Transmission Lines. *Electronics* **2023**, *12*, 2994.
https://doi.org/10.3390/electronics12132994

**AMA Style**

Rodríguez-Velásquez Y, Torres-Torres R, Murphy-Arteaga R.
Identifying and Modeling Resonance-Related Fluctuations on the Experimental Characteristic Impedance for PCB and On-Chip Transmission Lines. *Electronics*. 2023; 12(13):2994.
https://doi.org/10.3390/electronics12132994

**Chicago/Turabian Style**

Rodríguez-Velásquez, Yojanes, Reydezel Torres-Torres, and Roberto Murphy-Arteaga.
2023. "Identifying and Modeling Resonance-Related Fluctuations on the Experimental Characteristic Impedance for PCB and On-Chip Transmission Lines" *Electronics* 12, no. 13: 2994.
https://doi.org/10.3390/electronics12132994