Abstract— This article investigates the passive intermodulation (PIM) distortion induced by the asymmetric electrical contact. Based on the analytical model, it is found that the increased current path is the cause of the additional impedance, resulting in PIM distortion. To validate our theoretical analysis, different levels of asymmetric electrical contact are studied via simulation and experiments. The demonstrations not only confirm the validity of our theoretical findings, but also substantiate the simulations. To the best of our knowledge, this work is the first to identify the asymmetric electrical contact as a PIM source and discuss its underlying physical mechanisms of causing electrothermal (ET) coupling PIM. Index Terms— Asymmetric electrical contact, electrical contact failures, electro-thermal (ET) coupling, passive intermodulation (PIM)
Passive intermodulation (PIM) distortion is a frequently encountered issue in microwave circuits and wireless systems. PIM phenomenon is caused by slight nonlinearities stemming from nonlinear material or nonlinear metallic contact. In the past decades, many physical mechanisms, namely electro-thermal (ET) coupling, ferromagnetic materials, tunneling effect, and field emission, are considered as PIM sources –.
The ET coupling generally has not been considered as a dominant PIM mechanism in the traditional sparse frequency bands and low-power telecommunications . However, with the further development of wireless technologies (e.g., high power, high frequency, and base station supporting 3G/4G/5G simultaneously), the ET PIM distortion, especially induced by electrical contact failures, needs to be addressed carefully.
In practical scenarios, the poorly soldered joints, oxidation, and corrosion of junctions in the courses of service, and minor deformations of the connection surfaces may lead to electrical contact failures in microwave circuits . According to prior research , electrical contact failure increases the contact impedance of the connection surface. Typically, in high-power and high-frequency circuits, the increased contact impedance (additional impedance) will lead to significant self-heating. Furthermore, the material properties, such as metal resistivity, change as self-heating conditions vary. In other words, this bidirectional interaction between the electromagnetic domain (dissipated electrical power and the resultant self-heating) and the thermal domain (temperature rise and the consequent change in the metal resistivity), gives rise to temperature oscillations and resistivity variations. Therefore, the ET PIM distortion emerges due to the nonlinear contact impedance .
Specifically, electrical contact failures will increase the noncontact areas and lead to asymmetric electrical contact in the connected surface, that is, there are two important factors responsible for the ET PIM caused by electrical contact failures: one is the increased noncontact areas, while the other is an asymmetric electrical contact. In  and , PIM distortion induced by the additional impedance of the increased noncontact areas has been analyzed. In , the theories for ET PIM caused by the electrical contact failures have been studied, and a numerical expression was derived to calculate the PIM distortion related to the contact impedance. However, the impedance induced by the asymmetric electrical contact has been ignored, which renders the PIM distortion evaluated by the theoretical analysis insufficient.
In this article, we take a pair of faulty coaxial connectors as a case study to investigate the ET PIM distortion caused by the asymmetric electrical contact, and this work is organized as follows: Section II develops an analytical model of the asymmetric electrical contact, considering the case of coaxial connectors. In Section III, simulations are conducted to validate our theoretical analysis. In Section IV, experiments are conducted to check the goodness of the model and the consistency of the simulation results. Finally, Section V draws a conclusion of this article. In summary, the main contributions of this work are that it, for the first time, reveals the asymmetric electrical contact as a PIM source and discusses its underlying physical mechanisms, which will facilitate further improvements of the accuracy of PIM predictions and measurements related to the electrical contact failures.
II. ANALYSIS OF ASYMMETRIC ELECTRICAL CONTACT
A. Analytical Model for the Asymmetric Electrical Contact
To aid visualization and better understanding of the asymmetric electrical contact, a used coaxial connector sample is shown in Fig. 1, where the black solid/dashed lines enclose the actual contact areas of the female/male coaxial connectors. On the inner conductor of the coaxial connector, there are some small deformations. This makes the connected petals between the male and female coaxial connectors slightly unaligned, which increases the noncontact areas in the outer conductor and creates asymmetric electrical contacts in the inner conductor. In other words, the asymmetric electrical contact emerges and increases with the deformation levels of the connection surface.
As previously stated, the impedance is the dominant contributor to ET PIM, and a numerical relationship between the ET PIM and the impedance for lossy component has been developed by (33) in . Therefore, the key to understanding ET PIM distortion caused by asymmetric electrical contact is to analyze the increased contact impedance, that is, when the associated physical mechanisms for the increased contact impedance are well understood, the ET PIM can be accurately calculated.
Fig. 2 shows the analytical model of this work, where two cases of symmetric and asymmetric electrical contact are considered to analyze the impedance characteristics in the connection surface. The total contact areas are the same in both cases, that is, A1 = A0 = B1 = B0 = C1 = C0, but one focuses on the symmetric contact (the left side of Fig. 2), while the other studies the asymmetric contact (the right side of Fig. 2). It is evident that the total lengths of the current paths in the asymmetric contact are longer than that in the symmetric contact. For example, the red solid line in the asymmetric noncontact area A1 is longer than that in the symmetric noncontact area A0.
From the classical electrical contact theory , the contact impedance increases with the length of the current path through the connection interface. Therefore, the asymmetric electrical contact creates more contact impedance than the symmetric electrical contact. Furthermore, the increased impedance is proportional to the level of asymmetry, which means the severer the asymmetry of the electrical contact is, the more additional impedance will be induced.
B. Simulations for the Asymmetric Electrical Contact
A simulation model of a typical 7/16 DIN coaxial connector, which is built in CST Microwave Studio, is employed to analyze the asymmetric electrical contact. As shown in Fig. 3, the length of the model is 20 mm. The inside diameter of the outer conductor and the outside diameter of the inner conductor are 16 and 7 mm, respectively. The thickness of the noncontact areas is 0.3 mm, and the asymmetric contact occurs at 10 mm from the right end. Materials for the contact and noncontact areas of the simulation model are white bronze (conductivity: 5.96 × 107 S/m) and air (εγ = 1.0059), respectively.
Four cases, namely the symmetric contact, small asymmetry, medium asymmetry, and severe asymmetry, are designed and implemented to investigate the increased contact impedance in different asymmetric contact conditions. Fig. 4 shows the simulation results for these four cases. Here, the variable
Z, which is the additional impedance, is the difference between a certain asymmetric electrical contact case (e.g., severe asymmetry) and the symmetric contact case when the peak impedance occurs. It is important to notice that, as there is a symmetric contact, the additional impedance
Z emerges and increases with the asymmetric contact level. Therefore, the simulation results correlate well with the analysis in the analytical model.
C. PIM Expression for the Asymmetric Electrical Contact
The ET theory has demonstrated that the ET PIM distortion originates from the nonlinearity of the impedance , Specifically, for the coaxial connector, the ET PIM can be written as
where k belongs to the natural numbers, Z is contact impedance of the connected interface, in(t) is the current through the connected interface, α is the temperature coefficient, and Tα is the ambient temperature. Rth.eq. is the equivalent thermal resistance, which is related to the thermal capacitance Cth, thermal resistance Rth, and carrier angular frequency ω, meeting the relationship as 
Typically, the third-order PIM frequencies are very close to the fundamental frequencies and thus are difficult to be eliminated by filters. Below, we will take the third-order PIM distortion as an example to understand the generation of ET PIM by the asymmetric electrical contact. Usually, the magnitude of the temperature coefficient α is in the order of 10−3, thus, the ET PIM distortion from the number of k ≥ 2 can be ignored, and then the third-order ET PIM distortion can be approximately expressed as
Furthermore, as the current signal in(t) is composed of two carriers, n ∈ 1, 2, with amplitudes In, frequencies fn, and angular phases ϕn, mathematically
where ωn = 2π fn , then the third-order ET PIM distortion generated by the asymmetric electrical contact can be further rewritten as
where Z0 is the contact impedance of the connected interface with the symmetric electrical contact and
Z is the additional impedance caused by the asymmetric electrical contact.
It has been analyzed and simulated in Section II that the asymmetric electrical contact induces an additional contact impedance, resulting in ET PIM distortion. In this section, experiments are conducted to verify the impacts of the asymmetric electrical contact on PIM distortion, and the measurements are made in the mobile communication system of GSM1900, that is, the transmitter/receiver frequency bands of the experiments range from 1930 to 1990 MHz and 1850 to 1910 MHz, respectively.
A 7/16 DIN coaxial connector is employed as the device under test (DUT), and different levels of asymmetric electrical contacts are achieved by embedding copper on the different locations of the inner or outer conductor surfaces. Since the height of the embedded copper is a bit higher than that of the outer/inner conductor surface, and therefore, when the male and female coaxial connectors form a connection, the contact areas are only consisted of the embedded coppers, which means the asymmetry can be controlled by changing the locations of the embedded coppers.
Furthermore, three cases of different symmetry have been designed in experiments, which are the symmetric contact [Fig. 5(a)], the medium asymmetric contact [Fig. 5(b)], and severe asymmetric contact [Fig. 5(c)]. They have the same contact area, but the symmetry is different, that is, the embedded coppers for the three cases are the same dimensions, but they are embedded in different locations (the total area of the embedded coppers is one-quarter of the apparent contact area of the 7/16 DIN coaxial connector).
As illustrated in Fig. 5, the reflected PIM testing technology is used for the principle of the PIM analyzer; in other words, the PIM tester first generates a dual-tone excitation signal ( f1 = 1935 MHz, f2 = 1985 MHz) to the DUT, and then the reflected third-order PIM values from the DUT are measured and displayed on the PIM tester.
Fig. 6 shows the measured and predicted results for the three asymmetric electrical contact cases in Fig. 5. Two conclusions can be drawn from Fig. 6: 1) the third-order PIM distortion is directly proportional to the level of asymmetric electrical contact (symmetric contact → medium asymmetric contact → severe asymmetric contact) and 2) the nonlinearity of the PIM distortion is also confirmed because there is not a fixed step between all the curves.
Meanwhile, it is important to notice that the predicted PIM value is lower than in measurement. This is due to the fact that, in practical scenarios, the connected surface may have other PIM source (e.g., electron tunneling effect), which has been ignored in the theoretical analysis. Furthermore, the PIM values of both measured and predicted are greater than that of the reference because when embedding the coppers, the real contact area of the connected surface is reduced. In summary, the experiments quantitatively and qualitatively demonstrate that the asymmetric electrical contact is a PIM source.
Although it is well known that the electrical connection failures generate the ET PIM distortion, it is not very clear how it works for asymmetric connection scenarios. In this work, the underlying relationship between asymmetric electrical contact and ET PIM has been studied. It is revealed that the asymmetric electrical contact could increase the path length of the transmitted current through the connection interface, and the increased current path in turn leads to the emergence of additional impedance. Furthermore, the PIM distortion was generated in response to the ET coupling caused by the additional impedance. This conclusion will facilitate improved prediction and suppression of the ET PIM distortion from the asymmetric electrical contact and can also be extended to other types of connectors or soldered joints.