Based on the simulation and test, a physical basis for the current investigation was provided to consider and quantify the heat-generating part of the connector. For central conductor, this paper analyzed the maximum steady-state temperature of contact junction from the perspective of changes in current and voltage caused by connection resistance and magnetic resistance effect.
4.1. Electrical–Thermal Model Under DC Condition
The surface morphology measurements of the connector indicate that, under identical electroplating processes, variations in the thickness of plating material have minimal impact on the surface roughness. The contact between the pin and receptacle is essentially the compression of micro asperities on the contact surface. The average surface separation, d, can be modeled as a linear function of the contact roughness and contact force [
17]
where
F is the contact pressure. The relationship between the actual contact area,
Ac, and the visual contact area,
A, can be described as
where Φ(
h) is the standard deviation of the surface heights, which is considered to conform to the Gaussian distribution. Based on the Greenwood and Williamson model [
18], the surface asperity density,
η, can be calculated by using the total number of asperities on the contact surface and actual contact area:
where
Nc is the total number of asperities on the contact surface and was gained by surface topography instrument. For a given region,
Aa, the relationship between the number of asperities,
n, and
is defined as follows:
Based on Holm’s electrical contact theory [
1] and multiple contact
a-spots theory [
19], in the DC cases, the presence of a-spots reduces the volume of material for electrical conduction and restricts the flow of current, thus increasing resistance. The constriction resistance,
Rc, can be expressed by
where
H is the hardness of the material and
ρ is the resistivity of the contact surface material. Relevant parameters of the contact surface are listed in
Table 4. Notably, the contact pressure values presented in
Table 4 were derived from experimental measurements of insertion and extraction forces, combined with the corresponding friction coefficient. At the same thickness level, the contact pressure of silver plating is lower than that of nickel plating, and the contact pressure for both plating materials slightly increases as the thickness of the contact surface increases. Silver plating, being relatively soft, exhibits a consistent hardness regardless of increasing thickness, while nickel plating, being harder, shows a significant increase in hardness with increased thickness. Additionally, the surface asperity density and root mean square height of silver plating are higher than those of nickel plating.
According to Equation (8), the effects of contact pressure and contact asperity density on the contact resistance are shown in
Figure 11. Within the selected thickness range, the contact resistance of the silver plating shows minimal variation, remaining between 1.08 mΩ and 1.14 mΩ. In contrast, the contact resistance of the nickel plating exhibits greater variability.
The simplified circuit of the connector is shown in
Figure 12a. The bulk resistance and the constriction resistance together impede the flow of direct current (DC), and the relation can be expressed as
where
RT is the connection resistance and
Rbulk is the bulk resistance. The contact resistance of the entire contact surface is obtained by paralleling the resistance of the contact points, and the contact resistance is determined by the constriction resistance. The number and spatial distribution of a-spots are often not critical for assessing contact resistance in many practical applications, as electrical contact tends to occur fairly uniformly across the nominal contact area [
20]. Based on the measured asperity density per unit area, the mean spacing between adjacent asperities at level 1 thickness was calculated to be 78.32 μm for the nickel plating and 67.26 μm for the silver plating. The characteristic asperity size is significantly smaller than the average spacing between contact points. To enable a reasonable simplification of the analytical model, the following assumptions were made in this study: (1) the contact spots are uniformly distributed over the entire contact surface and (2) each asperity is idealized as a cylindrical structure with identical dimensions.
The span of each asperity is generally in the range of tens of microns, the voltage–temperature relation is valid under this contact condition [
21]. The relationship between the voltage drop
V across the asperity and the maximum steady-state temperature
TM in the contact can be calculated by [
19]
where
T0 is the room temperature and
λ1 and
λ2 are the thermal conductivity of the two conductors in contact, respectively.
ρ1 and
ρ2 are the resistivity of the two conductors in contact, respectively. Consider the electroplating material of the pin and the receptacle are same, so
λ1 =
λ2 =
λ,
ρ1 =
ρ2 =
ρ. The
V–T relation can be reduced to the form
Based on inspection and calculation of the properties of the materials presented in
Table 4, silver and nickel conform to the Wiedemann–Franz law [
1]
where
L0 = 2.44 × 10
−8 (
V/
K)
2 is the Lorenz constant. Thus, Equation (15) can be simplified as
Equation (13) illustrates the relationship between the temperature change at the contact point and the applied voltage. Within the measured temperature range, the resistivity increases with rising temperature, while the thermal conductivity decreases with temperature; these effects tend to approximately neutralize each other. Therefore, within the temperature variation range of the connector studied in this paper, the change in the parameter L0 is negligible and is treated as a constant. Furthermore, Equation (13) is derived based on traditional contact theory, which assumes that the boundaries of the contact conductor are thermally insulated, and Joule heating is assumed to propagate only to the colder regions within the contact point. Under this assumption, the temperature of each heat-generating contact point changes synchronously, with no heat transfer occurring between contact points. This assumption implies that the highest temperature at any single contact point can represent the maximum temperature across the entire contact surface.
4.2. Derivation of the Maximum Steady-State Temperature of Connectors with Non-Magnetic Plating Material
Under high-frequency conditions, the skin effect confines the current flow to the surface of the contact, thereby increasing both the bulk resistance and the contact resistance of the connector relative to the direct current (DC) condition. The current density becomes concentrated and follows curved paths within the narrow regions of the contact interface, leading to the emergence of contact inductance. Furthermore, the non-contact regions within the contact surface contribute to the formation of parasitic capacitance [
22]. For the silver-plated coaxial connector, the impedance network associated with a single contact spot between the pin and the receptacle is illustrated in
Figure 13, and its corresponding network impedance parameters are as follows:
where
ω =
f/2
π and
f is the signal frequency. The connection resistance at an electrical interface is the sum of the contact resistance and the bulk resistance near the contact area, while the connection impedance of a single contact point of the silver plating layer is
As inductance and capacitance primarily function as energy storage and signal transmission elements within the network impedance, their magnitudes are typically in the order of pico-Henries and pico-Farads, respectively [
23]. At a frequency of 3 GHz, the impedance relationships 1/
ωC >>
Rc and
ωL <<
Rc hold, indicating that the capacitive path behaves approximately as an open circuit, while the inductive path behaves effectively as a short circuit, respectively. Accordingly, the connection impedance can be approximated as:
As shown in Equation (16), the current flowing through the contact resistance of a connector with a non-magnetic plating material is nearly identical to that through the bulk resistance in the vicinity of the contact point. The connection resistance of the connector impedes current flow, thereby dissipating a portion of the signal’s input power and causing an increase in the temperature at the contact interface. In contrast to the DC condition, where the system gradually reaches a maximum steady-state temperature, the degree of current obstruction caused by resistance is more pronounced under high-frequency excitation. As a result, the temperature variation at the contact point becomes more significant. The connection resistance in high frequency of two end-butted cylinders was estimated as follows [
10]:
where
g =
ρ/2
Rc-Ag is the constriction radius of silver plating,
rcylinder = 100 μm is the dimension of the region of the contact point, and
aAg is the average widths of the micro asperities of silver plating. It is important to note that, under high-frequency conditions, significant current constriction occurs in the vicinity of the contact region. Consequently, the bulk resistance considered here is limited to the material volume within a localized radius around the contact point. As indicated by Equation (17), the connection resistance at high frequencies is governed by multiple factors, including the contact resistance under DC conditions, the depth of the skin effect, the resistivity of the plating material, and the average widths of the micro asperities.
To assess the power dissipation due to the connection impedance in the contact point region, it is essential to consider the power dissipated by the bulk resistance in the non-contact areas of the N-DIN connector. Based on the skin effect and the law of resistance, the total bulk resistance of the conductor in the non-contact region of the connector is given by
where
l2 is the length of the central conductor of the connector and
ra is the radius of the central conductor; the values of
l2 and
ra are shown in
Figure 1b. The thickness of the plating will affect
ra, but the variation among different
ra values is relatively minor compared to the overall radius of the central conductor.
The area of the plated contact region under high-frequency conditions is given by
The contact points are connected in parallel, so the voltage on a single contact point can be expressed as
where
PT is the power dissipation of the connector in experimental test and
n represents the number of contact points within the region through which the current flows across the contact surface. Consequently, the maximum steady-state temperature at contact point within the silver plating layer can be expressed as
Based on Equations (5)–(7), (11), and (21), the relationship between the microscopic structure of the connector contact surface and the temperature at the conductor contact points within the connector is established. By measuring the surface roughness, hardness, and contact pressure of the contact area, the contact resistance under direct current (DC) conditions can be theoretically calculated, which in turn allows for the estimation of the maximum steady-state temperature at the contact points of the non-magnetic material connector under high-frequency conditions.
4.3. Derivation of the Maximum Steady-State Temperature of Connectors with Magnetic Material Plating
When a connector with a ferromagnetic material electroplating is subjected to a magnetic field, the charge carriers (electrons or holes) within the material experience a Lorentz force. This force alters the trajectory of the charge carriers, thereby influencing the resistance of the ferromagnetic material. High-frequency and high-power signals propagate through the connector in transverse electric and magnetic (TEM) mode. The external bias magnetic field induces magnetic resistance in the material. Both magnetic resistance and contact resistance together impede the flow of current, resulting in a voltage drop across the contact area that exceeds the voltage drop generated by the contact resistance alone. According to the voltage–temperature (V–T) relationship, the actual temperature rise in the magnetic material contact area is theoretically greater than the temperature rise predicted by considering only the contact resistance.
Figure 14 illustrates the derivation and verification scheme for determining the maximum steady-state temperature of connectors with magnetic material plating.
In the contact region formed by the pin and receptacle of a coaxial connector with a ferromagnetic material plating, the impedance network associated with a single contact point is illustrated in
Figure 12c. The network impedance parameters can be expressed as follows:
The total connection impedance of the nickel-plated connector in a magnetic field is
Similarly to the calculation for non-magnetic materials, the capacitance can be approximated as an open circuit, while the inductance can be treated as a short circuit at 3 GHz. The total connection impedance of the nickel plating in the magnetic field is approximately
where
ZT-Ni represents the resistance in magnetic field,
Rbulk-Ni is the body resistance near the contact point,
RM1 represents the magnetic resistance of the bulk resistance region, and
RM2 represents the magnetic resistance at the contact point. In this paper, the sum of body resistance and contact resistance, 2
Rbulk-Ni +
Rc-Ni, is called the conventional connection impedance, and the total magnetic resistance generated in the contact area, 2
RM1 +
RM2, is called the connection magnetic resistance.
The resistivity of nickel plating is significantly higher than that of the base metal, and its thickness is comparable to the radius of the contact point. Consequently, the current injected at the contact tends to diffuse preferentially into the base material rather than spreading laterally along the plating. This results in a lower voltage drop in the base metal near the contact point compared to the plating layer. To characterize this behavior, the concept of effective resistivity is introduced. Williamson and Greenwood demonstrated that the ratio of effective to intrinsic resistivity in nickel plating is influenced by plating thickness, contact size, and the resistivity contrast between the plating and base material [
24].
where
ρeff-Ni is the effective resistivity of nickel plating,
ρNi is the resistivity of nickel plating,
d is the thickness of the plating material, and
aNi is the average width of the micro asperities of nickel plating. The
f1 values corresponding to plating thicknesses of 3 μm, 6 μm, and 9 μm are 3.5, 4.0, and 4.0, respectively.
The concept of effective resistivity helps to explain the different power loss trends with plating thickness for nickel and silver observed in
Table 2. For silver, its low hardness allows thicker coatings to form better contacts, reducing resistance and thus lowering power loss. In contrast, the effective resistivity of nickel increases with thickness under high-frequency conditions, which dominates over contact improvement and leads to higher power loss. Although level 2 and level 3 nickel coatings have similar resistivity, the thicker level 3 results in a longer current path and greater power loss.
Similarly to Equation (13), the total connection impedance of the nickel plating without magnetic field can be calculated by
where
g =
ρeff-Ni/2
Rc-Ni is the constriction radius of nickel plating.
To accurately assess the impact of connection magnetic resistance on the temperature rise of the connector, it is crucial to understand the proportion of magnetic resistance within the total connection impedance. The ratio of connection magnetic resistance to conventional connection impedance is proportional to the square of the magnetization intensity [
25]:
A hyperbolic tangent function can be used to describe the
H–M relation [
12]:
where
k1,
k2, and
k3 are parameters related to the structure, dimensions, temperature, and other properties of the magnetic material. Both experimental and simulation approaches were integrated to calculate the parameters within the N-DIN coaxial connector. The direction of the magnetic domains within the nickel-plated contact surface (
) aligns with the direction of the applied bias magnetic field (
). The magnetic flux density within the nickel plating layer is given by
It is worth noting that the maximum temperatures observed during RF operation are well below the Curie temperature of nickel (631 K), ensuring that the nickel plating retains its ferromagnetic properties. This supports the stability of internal magnetic domains and allows for persistent magnetoresistive behavior under excitation.
The magnetoresistance formulation employed in this study is based on magnetic hysteresis and the magnetic field dependence of resistance, which is theoretically supported by findings from manganite systems [
14]. Although our study does not observe phase transitions, the underlying mechanism of magnetoresistance—modulation of electrical resistivity by magnetic domain interactions—applies similarly to nickel plating. Thus, our use of a simplified equivalent circuit with hysteresis-based resistance modeling remains physically meaningful for predicting the thermal response of RF connectors under high-power conditions.
Under high-frequency conditions, metallic nickel behaves as an excellent conductor, with its conduction current significantly greater than the displacement current. According to Ampere’s circuital theorem, the line integral of the magnetic field strength
H along any closed curve
L is equal to the algebraic sum of the currents enclosed by the closed curve. The relationship between the conduction current and the magnetic field strength can be calculated as
According to
Figure 1c, the relationship between the distribution of magnetic field intensity and current density of the pin-receptacle in the cross section of the contact area can be quantified as
where
represents the magnetic field strength in the base material, while
and
represent the magnetic field strength in the nickel plating of pin and receptacle, respectively. The
represents the magnetic field strength around the contact point between the pin and the receptacle.
To facilitate the study of the electromagnetic environment at the contact surface, the integration path for the current in Equation (30) is chosen along the outer surface of the PIN. By combining the simulations of the surface current density for nickel plating layers of varying thicknesses, the external magnetic field strength at the pin-receptacle contact surface can be further calculated. Similarly to Equation (18), the body resistance of the non-contact area of the conductor in the nickel-plated connector is
The total power dissipated by the N-DIN connector in the transmission line is composed of the connection magnetic resistance in the contact point area, the conventional connection resistance, and the power consumed by the body resistance in the non-contact area. In the circuit simulation, the heat generated by magnetic resistance was not accounted for, and the total power dissipation of the connector can be expressed as
where
PS is the dissipated power in circuit simulation.
In the experimental test, a significant magnetic resistance effect was observed. The power dissipation in the experimental test can be given by
where
PT is the dissipated power in experimental test. When the input signal is characterized by a frequency of 3 GHz and a power of 100 W, the observed discrepancy between the experimentally measured power and the simulated circuit output can be attributed to the power losses incurred within the magnetic material coating. The current can be calculated by Equations (30) and (31):
The contact points are connected in parallel, so the voltage on a single contact point can be expressed as
where
PT is the total power dissipation of the connector contact surface. Therefore, the maximum steady-state temperature of the conductor contact point in the nickel plating layer can be derived as
Based on the power difference between the experiment measurement and the circuit simulation, the parameters of the magnetic material with different plating thickness levels can be obtained, as shown in
Table 5.
The results presented in
Table 5 indicate that connectors with different plating thicknesses exhibit similar
H–M relations. This further suggests that, under the same high-frequency and high-power input signal, the magnetic resistance of connectors with varying plating thicknesses is primarily governed by the signal and remains unaffected by the contact surface environment. The significant variation in connection impedance among plating materials of different thicknesses leads to corresponding differences in maximum steady-state temperature.
4.4. Comparison of Calculation and Measurement
The thermal behavior of coaxial connectors under different plating materials exhibits distinct trends with respect to plating thickness.
Table 6 presents the calculated and measured temperature variations of connectors with different plating materials and thicknesses. For silver-plated connectors, the maximum steady-state temperature decreases monotonically with increasing plating thickness. This reduction is primarily attributed to improved contact conditions, including enhanced contact pressure, diminished interfacial gaps, and an enlarged real contact area. These factors collectively lower the contact resistance, thereby reducing the voltage drop and associated temperature rise at the contact interface. This effect becomes increasingly pronounced at higher coating thicknesses. Both experimental observations and numerical simulations, as illustrated in
Figure 15, consistently confirm this behavior.
In contrast, nickel-plated connectors display a non-monotonic trend in maximum steady-state temperature with increasing plating thickness. Initially, the temperature rises due to the dominant influence of increasing effective resistivity, which offsets the effects of enhanced contact pressure and results in elevated connection resistance. However, beyond a critical thickness (from Level 2 to Level 3), the effective resistivity stabilizes, and the continued increase in contact pressure becomes the prevailing factor, leading to a reduction in connection resistance and, consequently, a decrease in the maximum steady-state temperature. This complex interplay between mechanical and electrical parameters highlights the material-specific nature of thermal responses in plated coaxial connectors.
In the traditional approach, the contact resistance is first calculated using Holm’s law and the Greenwood–Williamson (G–W) model. Then, the maximum steady-state temperature of the N-DIN RF coaxial connector is estimated based on Ohm’s law and the voltage–temperature (V–T) relationship. The predicted maximum temperatures for connectors with different nickel plating thicknesses are shown in
Figure 15. As can be seen from the figure, the temperature rise predicted by the traditional method is significantly lower than the measured values. This discrepancy arises because the traditional model does not account for the skin effect or additional losses due to magnetic reluctance. These findings highlight that the proposed prediction framework markedly reduces the relative error in temperature estimation under high-frequency and high-power excitation. This improvement is attributed to the fact that classical models underestimate the contact voltage across rough surfaces at high frequencies, and consequently the resulting thermal effects.
Furthermore, in the case of connectors with magnetic plating materials, the presence of magnetic resistance under high-frequency conditions leads to an additional increase in connection resistance at the contact point. This elevated resistance contributes to a higher voltage drop and, consequently, a rise in the maximum steady-state temperature. The value of magnetic resistance varies with plating thickness, as different thickness levels alter the electromagnetic environment at the contact interface. These variations stem from changes in surface contact conditions, which affect the local magnetic field distribution. However, by calculating the characteristic parameters (k₁, k₂, k₃) of magnetic plating materials, the relationship between magnetic field strength (H) and magnetic resistance (M) at the contact interface can be quantitatively established, providing a foundation for analyzing the thermal behavior of connectors with magnetic plating materials. The calculated maximum steady-state temperature of the non-magnetic plating material model exhibits strong agreement with experimental measurements. In contrast, for magnetic plating materials, the predicted maximum temperature is slightly higher than the measured values. Nonetheless, this deviation remains within an acceptable range and does not compromise the engineering applicability of the model in evaluating the maximum steady-state temperature of RF coaxial connectors.