S-parameter analysis and equivalent circuit model development

In order to investigate the electrical characteristics of RF coaxial connector before and after degradation, the male-to-female SubMiniature version A coaxial connector with a characteristic impedance of 50 Ω was selected, as shown in Fig. 1. The connector consists of an outer conductor, a dielectric layer and an inner conductor. The dielectric material is Polytetrafluoroethylene and its relative dielectric constant is 2.1. In this study, un-degraded and degraded connectors were both considered, where degraded samples with three levels were obtained through accelerated experiments with nitric acid steam. The degradation levels of the samples corresponding to different corrosion time in this experiment are presented in Table 1.

Figure 1.

SubMiniature version A coaxial connector samples.

Table 1.

The degradation level of the samples corresponding to different corrosion time

The S-parameters of connectors at un-degraded and different degradation conditions were obtained by experimental testing. As illustrated in Fig. 2, through two high-frequency test cables, the connector was connected to a vector network analyzer. The experimental process is depicted in Fig. 3. As shown in Fig. 4(a), with regard to the un-degraded case, the value of S₁₁ is below −20 dB in the whole frequency band. The S₁₁ parameter of the connectors at three different degradation levels is generally higher than that of the un-degraded connector, particularly at frequencies where the values nearly approach to 0 dB. This indicates an increase of signal reflections at the low-frequency band, affecting signal transmission. For the frequency range from 1.5 to 6 GHz, their S₁₁ values are less than −15 dB, which means that the electromagnetic reflection of the signal transmission channel is minimal in high-frequency band. It can be seen from Fig. 4(b), with regard to the un-degraded case, the value of S₂₁ is close to 0 dB. It can be regarded that there is almost no transmission loss in the measured frequency band, indicating that the un-degraded connector has good transmission performance. For the degraded, in the low-frequency band, the S₂₁ values are less than −3 dB and the largest transmission loss will increase as the degradation level increases. For example, S₂₁ values at the frequency point of 10 MHz are respectively −4.7, −10.3, and −30.2 dB for degradation levels 1, 2, and 3. This means that few electromagnetic waves are transmitted. For the frequency range from 10 MHz to 1 GHz, the S₂₁ values of the three samples increase as the frequency increases. From the analysis of Fig. 4, it can be concluded that the signal transmission quality in the low-frequency band was mainly influenced by the connector degradation. The transmission path with the connector after severe degradation is open, and the impact on the signal transmission in the high-frequency band is less than that in low-frequency band.

Figure 2.

High-frequency experimental setup.

Figure 3.

S-parameter measurement process.

Figure 4.

Measured S-parameters of connectors with un-degraded and different degradation levels. (a) S₁₁ and (b) S₂₁.

With the continuous improvement of signal transmission frequency, the electronic connector has been modeled and analyzed as a part of the transmission line. In this work, the application frequency band of the selected connector is from 10 MHz to 6 GHz, where the electromagnetic wavelength is relatively short. Therefore, the selected connectors were analyzed by using the transmission line theory in this work.

On the basis of the transmission line theory, the connector can be studied in segments. According to the structural differences between the inner conductor and the outer conductor, the connector can be divided into three segments. The sizes of each segment are shown in Fig. 5, where r_{ai (i =1-3)} and r_{bi (i =1-3)} are the outer radius of the inner conductor and the inner radius of the outer conductor of each segment, respectively. l_{i (i =1-3)} is the length of each segment. The connector can be regarded as a uniform transmission line, since the structure of the inner conductor and the outer conductor has no changes for every segment. The distributed parameter circuit is illustrated in Fig. 6. In the interest of simplicity, the transmission line is regarded as lossless, which means that influences of the resistance R and the conductance G are neglected, and the electrical properties are completely determined by the distributed parameters C and L. The capacitance and inductance per unit length of the coaxial transmission line can be expressed as

(1)\begin{equation}{C_i} = \frac{{2\pi {\varepsilon _i}}}{{\ln \frac{{{r_{bi}}}}{{{r_{ai}}}}}}\end{equation}

Figure 5.

Connector segment structure.

Figure 6.

The section lumped circuit model of microwave transmission lines with distributed elements.

and

(2)\begin{equation}{L_i} = \frac{{{\mu _i}}}{{2\pi }}\ln \frac{{{r_{bi}}}}{{{r_{ai}}}}\end{equation}

where ε_i and μ_i are the dielectric constant and permeability of the dielectric material between the conductors in each segment, respectively. In addition, the number of LC sections in the model can be calculated as [Reference Bogatin22]

(3)\begin{equation}n \geqslant 10 \times BW \times TD\end{equation}

where BW represents the bandwidth of the n-section lumped-circuit model and TD represents the time delay of the transmission line. According to (1), (2), and (3), the equivalent circuit for the un-degraded connector can be developed, using the calculation results of each segment shown in Table 2.

Table 2.

Distributed parameters of a connector

On the basis of the electrical contact theory, when two rough surfaces contact with each other, the two surfaces are not in complete contact, but form a peak contact. For the connector, the contact area will generate resistance effect, while the area that is not in contact will result in a capacitance effect. After the accelerated experiments, with the degraded contact surface, the contact area of the connector pin is covered by a layer of corrosion film, resulting in film resistance and film capacitance [Reference Ji, Gao, Flowers, Xie, Cheng and Jin1]. Since the thickness of the film is electrically small compared to the signal wavelength, it is reasonable to model it using a lumped RC network. As illustrated in Fig. 7, the equivalent circuit model of the degraded connector is constructed by adding a parallel connection of film resistance (R_f) and film capacitance (C_f) to the original circuit. Z₀ represents the characteristic impedance of 50 Ω.

Figure 7.

Equivalent circuit model of the degraded connector.

The film resistance and film capacitance were obtained by fitting the S-parameters of the three degraded connectors. As shown in Table 3, with the increase of degradation level, the film resistance increases and the film capacitance decreases. These results also indicate that the film resistance mainly affects the signal transmission in the low-frequency band, which is demonstrated by that the S₂₁ value of degraded connectors is lower in the low-frequency band than that of the un-degraded connector. The film capacitance mainly affects the signal transmission in the high-frequency band, because of the capacitive coupling effect. Moreover, it can be found that the connector degradation has little effect on high-frequency signal transmission.

Table 3.

Film parameters of equivalent circuit model of degraded connectors

It can be seen from Fig. 8(a) that the S₁₁ of the connectors at three different degradation levels is consistent with the S₁₁ trend obtained from the measurement. As shown in Fig. 8(b), measurements and simulations are in good agreement at the low-frequency band. While in the high-frequency band, the S₂₁ values obtained by the model are higher than that of the measurement, their trends are consistent and the differences are small, which proves that the model is accurate.

Figure 8.

S-parameters obtained from the equivalent circuit model. (a) S₁₁ and (b) S₂₁.

Design of passive equalization

Study of high-speed signal transmission characteristics and equalization technology has important academic value, which is of significance to practical engineering applications. In this section, non-return to zero (NRZ) signal is selected for analysis and discussion of the deleterious impact of degraded RF coaxial connectors on high-speed digital signal transmission. Based on the equivalent circuit model, a passive equalization method for degraded connectors in high-speed interconnection system is proposed. The eye diagram shows a full opening with only two lumped elements. As demonstrated in Fig. 9, the digital signal transmission system includes three components, a digital signal source, the transmission channel and the eye probe. In simulation, a specific NRZ pseudo random binary sequence (PRBS) generated by the signal source is transmitted through the selected channel model, and the output eye diagram characteristics are obtained by the eye probe.

Figure 9.

Simulation setup of digital signal transmission system.

Firstly, the simulation of high-speed digital signal transmission was conducted for the connectors before equalization. A 9th-order PRBS signal was generated by the digital signal source. In high-speed digital signal transmission, it is usually used to evaluate the transmission performance of the channel. The digital signal bit rate was 6 Gbps, the rise time and fall time were both 30 ps, and the signal amplitude levels 1 and 0 were 0.5 and −0.5 V, respectively. The internal resistance of the signal source and the signal receiver was set as 50 Ω, which matches the connector characteristic impedance.

As shown in Fig. 10, the four eye diagrams demonstrate varying degrees of eye opening, indicating that each connector can transmit signals, but the transmission performance is different. The corresponding eye diagram parameters are shown in Table 4. It can be seen that the transmission performance of the un-degraded connector is the best among all these channels. With increasing the degradation level, all eye diagram parameters deteriorate. This is mainly because the film resistance and film capacitance induced by the connector degradation. The eye height and signal-to-noise ratio (SNR) of the output signal of degraded connectors were significantly deteriorated. For degradation level 3, the eye height is only 0.071 V, and the eye opening factor is only 0.58. The degradation has little effect on the eye width. The jitter increased from 0.37 to 4.51 ps. These results are due to the fact that with the increase of degradation level, the increase of film resistance affects the transmission of low-frequency signals, thus affects the eye diagram at the receiver when transmitting digital signal. High-frequency signals are less affected.

Figure 10.

Simulated eye diagrams of signal before equalization. (a) Un-degraded. (b) Degradation level 1. (c) Degradation level 2. (d) Degradation level 3.

Table 4.

Parameters of the simulated eye diagrams before equalization

Without using any equalization, eye diagrams of the digital signal through un-degraded and various degraded connectors were measured. High-speed experimental setup is shown in Fig. 11, where the connector specimen was connected to an arbitrary waveform generator (AWG) and an oscilloscope using test cables. Consistent with the simulation, a 9th-order PRBS signal with a signal rate of 6 Gbps was generated by the AWG. The rise time and fall time were both set to 0.18 UI. The signal amplitude levels 1 and 0 were 0.5 and −0.5 V respectively. The measured eye diagrams are shown in Fig. 12, and the parameters are listed in Table 5.

Figure 11.

High-speed experimental setup.

Figure 12.

Measured eye diagrams of signal before equalization. (a) Un-degraded. (b) Degradation level 1. (c) Degradation level 2. (d) Degradation level 3.

Table 5.

Parameters of the measured eye diagrams before equalization

All four eye diagrams show open eyes. However, as the degradation level increases, indicating attenuation of the signal transmission performance. According to the values provided in Table 5, the best eye height is 570.4 mV and the worst value is 22 mV. After degradation, the eye width decreased from 151.02 to 62.5 ps, and jitter also increased. The measurements show a consistent trend with the simulations, indicating that the corrosion film will cause attenuation of the received digital signal. In addition, the simulation results are better than the measurement results, which is mainly because the connectors were treated as lossless and the test cable losses were not considered in the simulation.

As shown in Fig. 13, a passive equalization method with resistance and capacitance in series is proposed, which can perfectly compensate the connector transmission loss caused by degradation. According to the electrical contact theory, film resistance and film capacitance can be introduced at the interface of degraded connector, affecting signal transmission. Therefore, these two factors should be concerned for designing the compensation equalizer. The transfer function of the interconnect channel with an equalizer can be described as [Reference Sun, Wen and Wu13]

(4)\begin{equation}H(s) = \frac{1}{{1 + \frac{{1 + s{C_{eq}}\left( {{R_{eq}} + {Z_0}} \right)}}{{1 + s{C_{eq}}{R_{eq}}}} \cdot \frac{{\left( {1 + {R_f}/{Z_0}} \right) + s{C_f}{R_f}}}{{1 + s{C_f}{R_f}}}}}\end{equation}

Figure 13.

Equivalent circuit of the degraded connector with an RC equalizer.

where R_eq and C_eq are equalization resistance and capacitance respectively.

In pursuit of achieving an extremely flat frequency response, the transfer function of the channel should better be a constant. For the equalizer design, the resistance value should be

(5)\begin{equation}{R_{eq}} = \frac{{Z_0^2}}{{{R_f}}}\end{equation}

and the capacitance value should be

(6)\begin{equation}{C_{eq}} = {C_f}\frac{{Z_0^2}}{{\left( {{Z_0} + {R_{eq}}} \right){R_{eq}}}}.\end{equation}

Based on the accurate calculation by R_f and C_f, the resistance R_eq and capacitance C_eq in the passive equalizer can be specifically determined. Resistance and capacitance values of connector equalization designs for each degradation level are shown in Table 6. With the increase of degradation level, the equalization resistance decreases while the equalization capacitance increases.

Table 6.

Resistance and capacitance of the equalization circuit for degraded connectors

The S-parameters of three different degraded connectors after equalization are shown in Fig. 14, obtained from the equivalent circuit simulation. S-parameters become flat within a bandwidth range from 10 MHz to 6 GHz. S₁₁ values remains −0.2, −3.1, −7.3 dB and S₂₁ values are −5, −10, and −34 dB, respectively, for degradation levels 1, 2, and 3. Using this method, the transmitted signal will not be distorted but will experience amplitude attenuation.

Figure 14.

Simulated S-parameters of connectors at different degradation levels after equalization. (a) S₁₁ and (b) S₂₁.

The connectors at three degradation levels after using passive equalization technology are simulated respectively. The signal settings are the same with that before using equalization. The eye diagrams of the simulation are shown in Fig. 15. Applying the passive equalization, eye diagrams achieve full opening, it can be indicated that the channel transmission performance of digital signal has been greatly improved. Inspection of Table 7 shows that the eye width after equalization remains unchanged at approximately 166 ps. And, the eye opening factor is more than 0.97, which indicating that the signal exhibits low jitter and noise. Although the eye opening factors of the three degraded connectors after equalization are all close to 1, the corresponding eye heights have significant numerical difference. This is because the use of passive equalization led to signal amplitude attenuation. As the resistance of the equalization circuit decreases, the eye height decreases and the SNR continues to decline.

Figure 15.

Simulated eye diagrams of signal after equalization. (a) Degradation level 1. (b) Degradation level 2. (c) Degradation level 3.

Table 7.

Parameters of the simulated eye diagrams after equalization

After using the passive equalization, excellent eye openings were achieved for every condition, where the eye opening factor can be increased by 50%. Although the equalization will introduce inevitable amplitude attenuation to the system data transmission, this proposed equalizer still improves the channel transmission capability of high-speed signals with excellent performance.

To evaluate the practical equalization performance, as shown in Fig. 16, the equalizer was designed based on the equalization parameters listed in Table 6. The experiment was conducted using the same signal settings as before equalization, and the results are presented using the degradation level 1 case as an example. Figure 17 compares the eye diagrams, from which it can be seen that the eye diagram of the signal after equalization exhibits a significantly wider and clearer eye, with an eye opening similar to that of the un-degraded condition. Specifically, as shown in Table 8, the eye width increased from 148.79 to 153.65 ps, and the jitter was reduced, indicating suppressed signal distortion and improved signal quality. Although factors such as cable loss and component tolerances cause numerical deviations, the overall trend between experimental and simulation results remains consistent. This confirms that the proposed equalization method effectively compensates for the transmission loss caused by connector degradation and is both feasible and practically applicable, thereby validating its effectiveness.

Figure 16.

The equalizer.

Figure 17.

Comparison of eye diagrams. (a) Un-degraded. (b) Degradation level 1 before equalization. (c) Degradation level 1 after equalization.

Table 8.

Parameters of the measured eye diagrams