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Electro-Thermal Theory of Intermodulation Distortion in Lossy Microwave Components
Abstract
An analytic formulation of dynamic electro-thermally induced nonlinearity is developed for a general resistive element, yielding a self-heating circuit model based on a fractional derivative. The model explains the 10 dB/decade slope of the intermodulation products observed in two-tone testing. Two-tone testing at 400 MHz of attenuators, microwave chip terminations, and coaxial terminations is reported with tone spacing ranging from 1 to 100 Hz.
... PIM phenomenology has been studied for more than 40 years but still remains a nagging problem. The basic physical mechanisms of nonlinearities and PIM generation were explored in metal contacts [22][23][24][25][26][27][28][29], printed RF transmission lines [30][31][32][33][34][35][36], cable assemblies [17,18,31,[37][38][39][40] and antennas [21,[41][42][43][44][45]. The main sources of passive nonlinearities include Metal-Insulator-Metal (MIM) junctions [22][23][24][26][27][28][29], electro-thermal -Charge tunnelling and diffusion at MIM junctions of conductor asperities; -Current constriction at asperities of rough contact surfaces; -Self-heating and thermal expansion of the contact surfaces and asperities; -Electro-thermal effect due to ohmic losses in conductors and contacts; -Asperity deformations of the conductors with rough surfaces and their contacts subjected to mechanical stresses, expansion and creep. ...
... where Z c (ω n , x) is a local impedance at carrier frequency ω n , and δS is a differential area of the contact spot. An equivalent thermal resistance R th,eq (ω n ) can be defined as [37] R th,eq (ω n ) = Re R th 1 + R th jω n C th (8) where R th and C th are the thermal resistance and thermal capacitance, respectively. They form a thermal node shown in Figure 5. Simultaneous solution of (5)- (7) gives an approximation of the local resistivity ...
... Electrothermal PIM (ET-PIM) is caused by self-heating due to conductor and dielectric losses [37]. ET-PIM has a distinct signature of the nonlinear coupling of electrical and thermal domains as illustrated by Figure 5. ...
Passive intermodulation (PIM) is a niggling phenomenon that debilitates the performance of modern communications and navigation systems. PIM products interfere with information signals and cause their nonlinear distortion. The sources and basic mechanisms of PIM have been studied in the literature but PIM remains a serious problem of signal integrity. In this paper, the main sources and mechanisms of PIM generation by joints of good conductors are discussed. It is shown that the passive electrical, thermal and mechanical nonlinearities are intrinsically linked despite their distinctively different time scales. The roughness of the contact surfaces plays an important role in PIM generation by conductor joints. A review of the PIM phenomenology at the contacts of the good conductors suggests that novel multiphysics models are necessary for the analysis and reliable prediction of PIM products generated by several concurrent nonlinearities of a diverse physical nature.
Keywords:
passive intermodulation (PIM); nonlinearity; multiphysics effects; electrical contacts; surface roughness; contact deformation; electro-thermal effects
... PIM in conductor joints and connectors which is primarily caused by their nonlinear voltagecurrent relations [12], [13], [14], [15] has recently attracted particular attention. Several different mechanisms of nonlinearity may coexist and include charge tunneling [13], [16], the electro-thermal effect [17], thermionic emission [12], [13], Fowler-Nordheim electron transmission [18], [19], and mechanical deformations [20]. However, the electro-thermal effect is notable only when the high-power carrier frequencies are very close to each other and the mechanical deformations develop very slowly. ...
... When the contact surfaces are oxidized, the contact resistance of a microcontact is determined by the charge tunneling described in Section II. Then the linear tunneling conductance is approximated as σ t = g 1 A r where g 1 is defined in (10) and A r in (17). Thus, σ t can be readily taken into account at σ t ≪σ H . Fig. 3(a) illustrates the equivalent circuit of the PIM measurement setup, which contains a RF source (described by P s and Z s ), a transmission line with characteristic impedance Z c , contact resistance R t (tunneling resistance), capacitance C L , and a low PIM load with impedance Z L . ...
... When i(t) and v(t) are the current and voltage of nonlinear element R t , respectively, a n = g n A r where the nonlinearity coefficients g n are defined in (11) and (12) and the real contact area A r is obtained from (17). The voltage v(t) is the sum of the two carrier voltages of frequencies ω 1 and ω 2 with amplitudes defined in the steady-state regime in (19) ...
Passive intermodulation (PIM) by metal contacts limits the bandwidth and capacity of radio links used in mobile and satellite communications. In this work, we investigate the effect of nonlinearities in metal-insulator-metal (MIM) contacts and their effects on PIM generation. An analytical expression is obtained for the tunneling current density which has an error of
1.6% in the case of a very thin insulator and low voltages in MIM junctions. The presented analytical model of the contact surfaces with the fractal geometry is used to simulate PIM products of third-order (PIM3) and fifth-order (PIM5) versus the contact resistance and applied pressure. The simulation results are validated experimentally by an open-ended rectangular coaxial structure with a slotted enclosure. The measurement results demonstrate that the presented model predicts the PIM with a mean error of about 4.8 dB when the contact pressure varies from 0.5 to 1.7 MPa.
... The PIM phenomenology has been studied for more than 40 years but still remains a nagging problem. The basic physical mechanisms of nonlinearities and PIM generation were explored in metal contacts [22][23][24][25][26][27][28][29], printed RF transmission lines [30][31][32][33][34][35][36], cable assemblies [17,18,31,[37][38][39][40], and antennas [21,[41][42][43][44][45]. The main sources of passive nonlin-earities include Metal-Insulator-Metal (MIM) junctions [22][23][24][26][27][28][29], electro-thermal processes [37,40,46,47], surface roughness [32,48,49] and contact mechanical deformations [49][50][51][52][53]. ...
... The basic physical mechanisms of nonlinearities and PIM generation were explored in metal contacts [22][23][24][25][26][27][28][29], printed RF transmission lines [30][31][32][33][34][35][36], cable assemblies [17,18,31,[37][38][39][40], and antennas [21,[41][42][43][44][45]. The main sources of passive nonlin-earities include Metal-Insulator-Metal (MIM) junctions [22][23][24][26][27][28][29], electro-thermal processes [37,40,46,47], surface roughness [32,48,49] and contact mechanical deformations [49][50][51][52][53]. The advent of micro-electro-mechanical systems (MEMS) has sparked extensive investigations of the electrical contacts with rough surfaces and their RF performance. ...
... where Z c ( n , x) is a local impedance at carrier frequency n , and S is a differential area of the contact spot. An equivalent thermal resistance R th,eq ( n ) can be defined as [37] th th,eq th th ...
Passive intermodulation (PIM) is a niggling phenomenon that debilitates performance of modern communications and navigation systems. PIM products interfere with the information signals and cause their nonlinear distortion. The sources and basic mechanisms of PIM were studied in literature but PIM remains a serious problem of signal integrity. In this paper, the main sources and mechanisms of PIM generation by joints of good conductors are discussed. It is shown that the passive electrical, thermal and mechanical nonlinearities are intrinsically linked despite their distinctively different time scales. The roughness of the contact surfaces plays an important role in PIM generation by conductor joints. A review of the PIM phenomenology at contacts of the good conductors suggests that novel multiphysics models are necessary for the analysis and reliable prediction of PIM products generated by several concurrent nonlinearities of diverse physical nature.
... Researchers [or perhaps Steer et al.] reviewed the use of multiphysics in understanding PIM and showed that at least one source of PIM is strongly dependent on temperature [5]. Wilkerson et al. establish an analytical model for the electro-thermal effect induced PIM and build a self-heating model [6]. Rocas et al. also prove the temperature rise at steady-state will change the material properties and the linear performance of the device changes also, the intermodulation distortion caused by self-heating is proved [7]. ...
... These theories are the basis of the finite element simulation. The relationship between the torque and the axial stress of the connector is always defined as [22] ( ) (6) where, F c is the axial force of the coaxial connector, N is the torque acting on the connector, d 1 is the pitch diameter of the thread, ψ is the lift angle of the thread, φ v is the friction angle, φ v =arctan(f d ), f d is the friction coefficient of the screw, d 2 is the mean diameter of the outer conductor, f c is the friction coefficient between the nut and the contact surface. In this paper, the finite element simulation is used to assess the contact pressure value. ...
... is the additional generated heat, and F th,eq is the equivalent thermal resistance that defined as [6] ( ...
This work proposes and demonstrates a multiphysics model for the passive intermodulation effect on coaxial connectors with complex electro-thermal-mechanical coupling. The passive intermodulation effect under multiphysics is numerically modeled and solved from the basis of contact components and parameters. In the experimental demonstration, to get rid of the random test error, the design of experiment method is utilized to build an empirical model for verification. The comparison shows the max prediction differences with comparison among the proposed numerical multiphysics model, empirical model and experiment are within ±3dB, this good match between the experiment and predictions proves the theory.
... As described in [3]- [5], in electrothermally-induced PIM, ET-PIM, the non-constant envelope of one or more modulated carriers results in time-varying resistive heating of lossy but otherwise linear components. This heating leads to time-varying temperature and a resistance that changes with time in accordance with the modulated envelope of the combined carrier signal. ...
... An understanding of this behavior can be derived from the formula for PIM, Equation (33) in [3]. The voltage level of each of the ET-PIM tones (corresponding to IM3L and IM3U used here) as a function of ambient temperature T a , is ...
... An essential aspect of the PIM measurement system is the use of a cancellation algorithm that requires power measurements only, and does not rely on gradient-based techniques such as the Newton-Raphson algorithm. Instead the algorithm, described in [3], [6], [7], uses projections from three power measurements to calculate the amplitude and phase settings of the vector modulators to cancel the carriers. The problem with amplitude and phase measurements of, say, voltage is that there is considerable uncertainty of the phase measurement when amplitudes are low. ...
... Namely, RF power dissipation due to conductor and dielectric losses causes self-heating which, in turn, alters conductor resistivity. Such ET process is nonlinear and leads to ET-PIM [16], [17] in transmission lines (TLs) [3] and printed circuit boards (PCBs) [18]- [23]. ...
... The theory of ET-PIM generated by resistive elements [16] revealed a distinct signature of the nonlinearity coupling electrical and thermal domains. In essence, the heat from RF loss causes resistance variation which interacts with the carriers through Ohm's law and produces PIM. ...
... The effect of temperature gradient and heat flow on ET-PIM in thin-film coplanar waveguide with spatially inhomogeneous current distribution has been elaborated in [21]. The analyses and supporting experiments in [16], [17], and [20] focused on the baseband resistivity modulation of conductors, thinner than a skin depth. But these studies are unable to explain the strong correlation between conductor surface roughness and PIM in PCBs observed in [18]. ...
The physical mechanism of the experimentally observed dependence of passive intermodulation (PIM) in printed circuits on conductor surface roughness is studied. It is shown that electrothermal (ET) nonlinearity, arising due to heating of imperfect conductors by high-power carriers in a multicarrier system, is correlated with conductor surface roughness and has a unique signature. Carriers modulate the conductor resistivity, skin depth, and surface impedance which generate PIM products. The detailed analysis demonstrates that ET-PIM depends on the conductor resistivity, shape, and roughness profile and also on the electric and thermal properties of the substrate. Their effects on PIM are illustrated by examples of uniform microstrip lines with different conductor and substrate materials, and periodically perturbed and meandered microstrip lines.
... In [12,90,3,91,92] wurde die elektrothermische Nichtlinearität in Mikrostreifenleitungen detailliert untersucht. Der spezifische Widerstand des Metalls kann als Funktion der Temperatur in Gleichung (15) formuliert werden [93,86]: ...
... Dabei ist ∆ der Spannungsabfall und die Stromstärke. Für ein erwärmtes metallisches Volumen treibt die Verlustleistung = die Wärmeleitungsgleichung (19) an [90,8]: ...
Mithilfe des LDS®-Verfahrens kann die Integration elektrischer Systeme in multifunktionale Produkte realisiert werden. In vielen Bereichen, auch in Hochfrequenz (HF)-Anwendungen, bietet das LDS®-Verfahren hohes Nutzenpotenzial hinsichtlich Funktionalität und Integrationsdichte, wodurch ein kompaktes Kommunikationssystem mit genauen Abmessungen sowie mit reduzierter Anzahl an Verbindungsstellen geschaffen werden kann. Hierfür müssen die mittels LDS®-Verfahren hergestellten Bauteile zahlreiche Anforderungen bezüglich ihrer HF-Eigenschaften erfüllen. Passive Intermodulation (PIM) als eine der Leistungsanforderungen ist zu einem wachsenden Anliegen hinsichtlich des Designs und der Fertigung von HF-Bauteilen geworden. Die Evaluierung von PIM an den mittels LDS®-Verfahren hergestellten Mikrostreifenleitungen steht in dieser Arbeit im Fokus. Eine im Rahmen dieser Dissertation durchgeführte Untersuchung kommt zu dem Ergebnis, dass sowohl der PIM-Pegel als auch die Qualitätsmerkmale von den Laserprozessparametern abhängig sind. Durch die Einstellung der Laserprozessparameter kann der PIM-Pegel beeinflusst werden. Aus den präsentierten Forschungsergebnissen ergibt sich, dass das LDS®-Verfahren für HF-Anwendungen unter Berücksichtigung von PIM einsetzbar ist.
... A distinct aspect of the published works is the use of specific physical models of intrinsic nonlinearity. While being phenomenological and subject to experimental validation, such models were based on the predefined physical mechanisms, e.g., an assumed dependence of the superfluid density on the electric current density in superconductors, [20], or electrothermal coupling in imperfect conductors, [21], [22]. In contrast, characterization of the intrinsic nonlinearities in planar circuits fabricated on commercial RF laminates and operated in a broad temperature range is complicated by that the dominant mechanisms and location of nonlinearities are typically unknown a priori. ...
... Indeed, a typical two-tone PIM analyzer permits measurement only of a lowersideband PIM product magnitude with minimum 200 kHz (few megahertz in case of the third-order PIM) offset of the CW carrier frequencies. This proved to be insufficient for detection of the low-frequency dynamics, such as electrothermal, [21], [22], and memory effects, whereas the instruments capable of the extended nonlinear characterization have the residual PIM level higher than that in the low-PIM PCB laminates. While the dynamic nonlinearity of passive devices may affect practical signal waveforms, the available data suggest that its effect is hardly discernible, as compared with the static nonlinearity. ...
Products of passive intermodulation (PIM) generated by weak nonlinearities of passive circuits subjected to relatively high transmit power of multicarrier signals may cause strong interference in emerging broadband and multiradio communication systems. This paper presents a new approach to a characterization of distributed nonlinearities in printed circuits fabricated on commercial grade microwave laminate materials. An efficient procedure for PIM characterization has been devised using the commercial RF-CAD software. The phenomenological model has been developed to take into account concurrent distributed nonlinearities of printed transmission lines and to evaluate PIM products of arbitrary order. It has been observed for the first time that the sources of nonlinearity in typical microstrip circuits may have highly uneven distributions which require a different means for PIM characterization and modeling. The proposed methodology has been validated by accurate predictions of the PIM response of complex circuit layouts. The results of this paper pave the way to a holistic approach to the design of planar microwave circuits and devices under given linearity constraints.
... This is even further critical in high-throughput satellite (HTS) payloads operating at higher frequency bands to meet throughput demands [8], [9]. Indeed, higher operational frequencies lead to a wavelength reduction, with a consequent increase in the surface current density, a typical PIM ignition factor [10], [11], [12], [13]. ...
The effect of the phase of the input carriers has been traditionally neglected in the characterization of passive intermodulation (PIM) since standard two-tone PIM tests seem to be unaffected by phase variations of the excitation signals. However, the phase of the input carriers can be of relevance in many practical applications. This article is aimed at filling this gap in the technical literature. First, the existing theory explaining why the phases do not affect the measured PIM contribution for a two-carrier excitation but can be of relevance for generic multicarrier scenarios is summarized. PIM measurements for complex signals composed of several carriers with varying phases are then reported, enabling the practical characterization of this effect for the first time. Experimental results of the test campaigns are fully aligned with theoretical predictions, thus allowing us to identify those PIM contributions that can be affected by the carrier phases and assess the expected variation range in their amplitude level.
... Oxidized and contaminated metallic joints also contribute to this type of PIM. Examples in this group include connections between waveguides, cable connectors, duplexers, attenuators, terminations, and antennas [24], [43]- [52]. ...
Passive Intermodulation (PIM) is a well-known phenomenon that poses a significant challenge to modern wireless communication systems. Although efforts have been made to reduce or mitigate the PIM through hardware design, frequency planning, and/or band separation techniques, the impact of PIM on radio performance remains significant. This is especially true for radio transceivers operating in the frequency division duplex mode, which are known to be susceptible to PIM interference. Moreover, with the adoption of carrier aggregation and advanced multi-antenna technology, PIM is increasingly becoming a major problem in multi-band multi-standard radio systems. Therefore, innovative approaches that exploit PIM detection, avoidance, and cancellation techniques are required to effectively mitigate or reduce its impact. In response to these challenges, this study highlights various PIM interference mechanisms and their interactions which cause potential receiver sensitivity degradation. The discussion further explores PIM mitigation strategies, and underscores the potential of digital PIM cancellation solutions to offer a hardware-independent resolution for reducing PIM impact in multi-band radios, in the 5G era and beyond in radio systems.
... The basic mechanisms of PIM generation have been studied in contact junctions [10]- [13], printed transmission lines [14]- [17], cable assemblies [6], [7], [15], coaxial connectors [18], [19] and antennas [9]. The main sources of PIM in conductor joints include charge tunnelling in Metal-Insulator-Metal (MIM) junctions [10]- [13], electro-thermal effects [19], [20], and mechanical deformations of contact surfaces [21]- [23]. ...
Contacts of conductors with rough surfaces cause losses and nonlinear distortion of the high-power signals at high frequencies. The contact nonlinearities are analysed in a pair and array of the contact asperities in metal-insulator-metal (MIM) junctions. An improved model of the tunnelling resistivity is proposed and its accuracy is demonstrated for the harmonic signals. The model is also applied to the analysis of the thermal effect on the contact resistivity of rough conductors.
MIM Junction, Metal-Insulator-Metal (MIM), MIM Diode, Passive Intermodulation (PIM), Metal-to-Metal Contact, Simmons Formula, Nonlinear Devices, RF and Microwave Applications, Metal-Insulator Transition, Schottky Barrier, Tunneling Current, Nonlinear Response, Intermodulation, Contact, Rough Surfaces, Insulator Layer, Quantum Tunneling, Junction, WKB, WKBJ, Tunnel Junctions.
... The physical mechanisms that cause nonlinear responses are complex and diverse [6]. Some examples are provided below: electron tunneling semiconductor effect [7][8][9]; micro-discharge between the voids in metal-metal interface [10][11][12]; electro-thermal effect [13][14][15]; contact resistance effect [16][17][18]; and ferromagnetic effect [19][20][21]. ...
In modern wireless communication systems, the signal-to-noise ratio (SNR) is one of the most important performance indicators. When the other radio frequency (RF) performance of the components is well designed, passive intermodulation (PIM) interference may become an important factor limiting the system’s SNR. Whether it is a base station, an indoor distributed antenna system, or a satellite system, there are stringent PIM level requirements to minimize interference and enhance network capacity in multicarrier networks. Especially for systems of high power and wide bandwidth such as 5G wireless communication, PIM interference is even more serious. Due to the complexity and uncertainty of PIM, measurement is the most important means to study and evaluate the PIM performance of wireless communication systems. In this review, the current main PIM measurement methods recommended by International Electrotechnical Commission (IEC) and other standard organizations are introduced, and several key challenges in PIM measurement and their solutions (including the design of PIM tester, the location of the PIM sources, the design of compact PIM anechoic chambers, and the evaluation methods of PIM anechoic chambers) are highlighted. These challenges are of great significance to solve PIM problems that may arise during device characterization and verification in real wireless communication systems.
... The proposed method is implemented in a SolidWorks simulation environ Then, the maximum stress point under the resonance frequency response [24][25][26] is found. After that, according to the analysis of structural harmonic response deformation and stress distribution, structural optimization design is carried out. ...
Passive intermodulation (PIM) generated from antennas is a nonlinear distortion phenomenon and causes serious problems to communication quality. Traditional radio frequency (RF) solutions focus on testing the final product to find the PIM source. However, it cannot solve the stability of PIM after the antenna is vibrated. This paper introduces a new method to improve the stability of PIM in the design phase. By studying the mechanism of PIM generation, a simulation method is proposed in this paper by applying mechanical finite element simulation and simulating the structural design of the device under test. Then, the stress at the PIM source is reduced, thereby the PIM stability of the product is improved. This paper adopts this method by studying a typical product, finding the root cause that affects the product PIM magnitude and stability, and optimizing its design. The PIM value of the new scheme is stable by making a prototype and testing. The method provided in this article can effectively improve product development efficiency and assist designers in avoiding the risks of PIM before the product’s manufacturing.
... At present, the research on the generation mechanism of PIM mainly focuses on contact nonlinearity [4][5][6][7] and electrothermal coupling phenomena [8]. Actually, there are few studies on the frequency dependence of PIM, such as the influence of power and frequency on PIM [9], and the influence of electric-thermal and beat frequency on PIM [10]. However, due to the skin effect, the carrier frequency will affect the resistance, then affect the PIM product. ...
Passive intermodulation (PIM), which is one of the inevitable interferences in radio frequency circuits and systems, has an important impact on communication quality. It exhibits dynamic characteristics with changes in carrier frequency and beat frequency. Aiming at the coaxial structure, this paper proposes a calculation model for studying the influence of carrier frequency and beat frequency on PIM characteristics, in which the influence of skin effect on resistance is analyzed. This work is helpful to understand the frequency-dependent characteristics of PIM.
... In high power applications, the contact junctions can work as the nonlinear sources and produce serious mixing components to disturb its normal performances [1]. The passive intermodulation (PIM) is always considered as a thorny issue for most microwave devices [2] [3]. It is recognized that most contact nonlinearity induced PIM distortion effect can never be attributed to a single factor. ...
This work presents a numerical modeling of passive intermodulation (PIM) product on contact junction under the multiphysics condition. The carrier power, temperature, pressure impacts are combined into the model using design of experiment method. Based on several sets of optimized PIM test conditions, an empirical PIM model versus multiphysics can be established. The experiment demonstrates the proposed PIM modeling method can predict PIM product with the max error of 7 dB. Index Terms-contact junction, design of experiment, passive intermodulation, response surface methodology.
... Passive intermodulation (PIM) is an intractable problem in high power and multiple input multiple output frequency division communication systems. 1-3 Up to now, as the PIM modeling to different nonlinear source is complicated, the PIM modeling draws great attentions, 4 however, most of these studies are about the PIM generation, few is about the PIM combination effect on a practical device, 5 proves a model to study the PIM combination along a transmission line. But this is far from solving a practical microwave device with plenty nonlinear sources located in a small area. ...
A compact passive intermodulation calculation method for the nonlinear source with noticeable size is proposed and demonstrated. By approximating the transmission factors of linear carrier and nonlinear passive intermodulation to be the non‐dimensional ratios between the current and power based on the reciprocal calculation principle, the PIM synthetize on a nonlinear source with notable size can ignore the complicated microwave characteristic analysis, while just need the accumulation of each PIM product of each nonlinear cell, which significantly simplifies the calculation process for the passive intermodulation product on a practical microwave device. The study proposed a compact solution to solve PIM product from the nonlinear source with big scale, which can be extended to the PIM analysis of practical microwave device with unevenly distributed PIM product.
... The curve exhibits a low-pass characteristic with a slope of 10 dB per decade, which is the defining characteristic of electro-thermal distortion. 26 The measured PIM power vs the two-tone frequency separation Δf for antennas #1 and #2 are shown in Figures 14 and 15. The range of the frequency separation is 0.05 to 0.15 GHz, which is limited by the bandwidth and accuracy of the measurement system. ...
A theoretical treatment of electro‐thermally induced passive intermodulation (PIM) is developed for printed dipole antennas, yielding an expression of third‐order intermodulation distortion based on the surface current distribution. The simulation procedure of third‐order PIM products with the full‐wave frequency‐domain method was given to evaluate the PIM level. In particular, the PIM dependencies on input power, two‐tone frequency separation, and substrate parameters are analyzed leading to design guidelines for low distortion antennas. It is shown in this paper that the thermal factors have a noticeable impact on the PIM power generated by printed dipole antennas. Finally, two antenna samples are fabricated on different substrates, Rogers 5880 and FR4, and a two‐tone test at 2‐GHz band using a reflective PIM test system is reported. The PIM evaluation method and the design guidelines to reduce the PIM on printed antennas proposed in this paper are of great significance for telecommunication systems.
... The electro-thermal (ET) phenomenon belongs to the latter category [1], [2]. Namely, the high frequency heating due to conductor and/or dielectric losses is essentially nonlinear process, which causes self-modulation of the media conductivity and permittivity [3]- [5]. This results in generation of harmonics, frequency mixing and PIM, which distort the transmitted and received EM signals. ...
Surface roughness and texture of printed conductors increase dissipative losses and enhance the effect of electro-thermal nonlinearity. Although this nonlinearity is weak it can significantly affect and distort wave packets and modulated signals. In this work, the effects of conductor roughness on integrity of multifrequency wave packets is explored and elucidated.
... The contact nonlinearities are reviewed before, including metal-metal contacts, metal-oxide-metal contacts [9], and tunneling [10]. The material nonlinearities are also studied, such as material electrically nonlinear characteristics of ferromagnetic materials [4], [11], piezoelectric materials [12], and material electrothermal nonlinearity [13]. These nonlinearities can occur anywhere in the RF path. ...
With emerging wireless communication technology, undesired interference caused by passive intermodulation (PIM) becomes a major obstacle to the network efficiency and the electromagnetic compatibility (EMC) of the devices. Accurate measurement techniques for PIM are fundamental to identify and characterize the electromagnetic interferences (EMIs). Anechoic chambers are commonly used in the antenna PIM test. The noise of PIM test chamber usually determines the accuracy of PIM measurement. The additional PIM sources caused by the chamber can be classified into three classes: those due to the absorbers, those due to the design of the shielded enclosure, and those due to the extraneous signals caused by an ineffective shielding. In this paper, a measurement method is proposed for the performance verification of PIM test chamber. The proposed method has two key points. One is adjusting the measurement distance between antenna and absorbers for gain compensation, which can evaluate the maximum measurable antenna gain of the PIM test chamber. The other is choosing the multiple measurement points according to the beamwidth of antenna, which can define the PIM level in worse case of the chamber. Theoretical analysis of the proposed method and measured results for two PIM chambers by the method are given in this paper.
When the cross section of a transmission line (TL) with a complex shape affects the internal electromagnetic field distribution of the conductor, it is necessary to combine the numerical method and the cross-sectional shape of the conductor to calculate the electrothermal passive intermodulation (ET PIM) current density caused by the uneven current distribution. Therefore, to analyze the influence of TL cross-sectional geometry on ET PIM, an ET PIM model of TL considering conductor cross-sectional geometry and transmission loss is derived. In the model, the modified gradient model is defined to characterize the nonlinear conductivity and conductor loss. Meanwhile, the PIM measurement experiments of the microstrip line, meshed microstrip line and parallel plate waveguide are designed to compare numerical solutions of the ET PIM3 model. Finally, numerical analyses of cross-sectional geometry effects on ET PIM3 generation are made to explain the phenomenon of ET PIM3 measurement. Combined with the measurement results, the numerical solution shows that the ET PIM produced in the high current density region is affected by the cross-sectional geometry, frequency offset, and thermal flux.
We explore the relationship between rough surface conductors and the phenomenon of passive intermodulation. The underlying surface is taken to be the boundary of a Lipschitz domain, and a characteristic angle of the domain is used to track boundary dependence on the various fields. To model electro‐thermal passive intermodulation in particular, we consider a specific type of temperature‐dependent conductivity and determine conditions on the conductivity under which one can use fixed point arguments to solve an induction heating and Joule heating problem on a Lipschitz domain. In the latter problem, we also consider a time‐dependent permittivity function . Finally, weak solutions to a magneto‐quasi‐static problem are obtained when the permeability µ is temperature dependent and is allowed to degenerate in a certain way. An interesting effect of the rough surface is the inherently limited Sobolev regularity of the electric field, which can be improved if one assumes additional constraints on the boundary.
Considering the distributed electrothermal coupling effect, passive intermodulation (PIM) products in very long microstrip lines (MLs) have been deeply studied theoretically and experimentally; however, the complex PIM behaviors of short MLs (SMLs) arising from edge defects are still not well presented, and the nonlinearity mechanism has not been revealed until now. In this article, we propose the microscale single-and double-triangle structures to represent intended random defect patterns on the edge of SMLs. Then, we fabricate multiple 20-mm-long SML samples on sapphire crystal wafers employing the metal lift-off technology (MLOT). By PIM measurements and scanning electron microscope (SEM) image observations, we find statistically that the PIM levels of the SMLs with nanoscale depressions and holes on the top edge are over 10 dB higher than that of the SMLs with dense and smooth edges. Based on the statistical approach, we confirm that the discontinuous nanoscale defects on the top edge are the nonlinearity origin. And the microscale structure will enhance the localized electrical field concentration according to the electrical simulations. Considering the electrical field enhancement effect of microscale structure and the nanoscale defects as nonlinearity sources, an analytical PIM model for SMLs is established.
In this paper, a concept of non-contact PIM evaluation method using balanced transmission lines is proposed for impedance-matched PIM measurement systems. In order to evaluate the PIM characteristics of a MSL by using its image model, measurement system using balanced transmission line is introduced. In non-contact PIM measurement, to reduce undesirable PIM generation by metallic contact and the PIM-degradation in repeated measurements, a non-contact connector which is applicable without any design changes in DUT is introduce. The three-dimensional balun composed of U-balun and balanced transmission line is also proposed so that it can be applicable to conventional unbalanced PIM measurement systems. In order to validate the concept of the proposed system, a sample using nickel producing high PIM is introduced. In order to avoid the effect of the non-contact connection part on observed PIM, a sample-configuration that PIM-source exists outside of the non-contact connection part is introduced. It is also shown using a sample using copper that, nickel-sample can be clearly differentiated in PIM characteristics while it is equivalent to low-PIM sample in scattering-parameter characteristics. Finally, by introducing the TRL-calibration and by extracting inherent DUT-characteristics from whole-system characteristics, a method to estimate the PIM characteristics of DUT which cannot be taken directly in measurement is proposed.
This work presents the passive intermodulation (PIM) induced by vias in microwave circuits. An analytical model combining the equivalent nonlinear source theory and the vector superposition principle is established to evaluate the PIM product of the vias array, which is influenced by the combination effect of the vias location, diameter, and number. The proposed model is verified and evaluated at 2.6 GHz. Good agreement between theory and experiment proves the logic and practicability of modeling. This work provides guidance for the low PIM optimization design of vias.
The occurrence of passive intermodulation (PIM) is inevitable in high-power microwave devices. It interferes with wireless communication and radar systems, resulting in severe deterioration of receiving performance. So far, narrowband PIM analyzers have been realized for different business bands. However, they are incapable of testing wideband systems. To solve this problem, rack-mounted analyzers piled with multiple narrowband analyzers were implemented. While such solutions significantly increase the volume, cost, and power consumption, they cannot provide a wide, continuous bandwidth for PIM localizations. Aiming to solve this problem, we propose a novel solution for realizing ultrawideband PIM analyzers. Based on the principle of cooperative radio frequency and baseband (RF-BB) nulling, we demonstrate an experimental prototype working within a continuous frequency band covering three octaves, exhibiting promising potential in PIM testing applications.
In this article, two wideband choke flanges (Flange 1 and Flange 2) are designed to reduce the passive intermodulation (PIM) level of the orthmode transducer (OMT). Since flange connections are important sources of PIM, circular Flange 1 introduces a choke groove and achieves a power suppression level of 41.95 dB. To meet the requirement of miniaturization and lightweight, Flange 2 with rectangular structure and rectangular choke groove is proposed for reducing height waveguide. Compared with conventional circular choke flange for reduced height waveguide, it has a smaller cross area and better power suppression level (39.92 dB). To evaluate the PIM performance of the proposed OMT with two flanges, temperature circulation test is executed. It is proved that the third-order PIM level is about −135 dBm from measurement results. Therefore, the proposed flanges can be utilized in large-scale multibeam array for low PIM satellite communication applications.
This paper 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 electro-thermal coupling PIM.
A novel nonlinear model describing the passive intermodulation (PIM) behavior in coaxial connectors is developed based on sine function. Two types of coaxial connectors are selected as test samples, and a series of two-tone PIM tests are conducted. The coefficients of the sine function model for these connectors are obtained by fitting the test results using ordinary least squares (OLS) method. The predicted results of the sine function model agree well with the test results, which proves that the sine function model can be used to describe the third-order and the fifth-order PIM performance of coaxial connectors accurately. The differences between the sine function model and other nonlinear models are compared.
Coaxial connectors play an important role in communication systems. However, passive intermodulation (PIM) and cross modulation resulting from slight nonlinear characteristics in coaxial connectors may negatively impact signal transmission quality. In addition, degradation of coaxial connectors tends to exacerbate such effects. In the current work, the nonlinear characteristics of pristine coaxial connectors are analyzed and modeled. From these results, a new polynomial behavior model is developed to describe the nonlinear characteristics for connectors with various levels of degradation for the first time. A series of two-tone PIM tests on pristine connectors and degraded connectors are conducted with the input power from 35 dBm to 44 dBm. The test results validate the model for pristine connectors and the model for connectors with various levels of degradation. The error vector magnitude (EVM) is selected as the index to quantify the level of signal distortion. The power of PIM and cross modulation interference, and the impacts of PIM and cross modulation on EVM are theoretically calculated and simulated. This work serves to provide a behavior model that describes the nonlinear characteristics of pristine and degraded connectors, and a theoretical basis for evaluating the impacts of PIM and cross modulation on signal distortion.
Environmental temperatures may affect the high frequency electrical performance of passive components, including passive intermodulation (PIM) characteristics. This work proposes a theoretical model of the third-order intermodulation distortion (IM3) signal as a function of the material properties and contact characteristics in a connector. A cross-sectional model of the inner conductor supported by finite-element analysis (FEA) simulations was used to investigate the effect of temperature on the current density distribution in different material regions and its impact on nonlinear performance. Based on electrical contact theory and temperature-dependent modeling, mathematical expressions were developed to calculate the impact of the various contact parameters, such as contact pressure, contact area and contact resistance. Combining the effects of both material nonlinearity and contact nonlinearity, the IM3 product powers resulting from coaxial connectors associated with temperature change were predicted theoretically. A series of experiments were designed and conducted to measure the IM3 product powers of connectors for a variety of environmental temperatures. The model predictions show good correlation with these experimental results.
This work demonstrated the directional passive intermodula-tion (PIM) product distribution effect in a microwave two-port network using a distributed nonlinear model. The analysis is demonstrated through the comparison of PIM products at the double ends of a general two-port network by a microstrip line. The reverse and forward PIM products are compared with the nonlinear source distribution and carrier frequency, the model and experiment demonstrates that the directional PIM effect (the difference by reverse and forward PIM) is caused by the PIM accumulation effect from multiple nonlinear sources, what is more, the frequency dependent phase relation can be one reason for the frequency dependent PIM product on wideband transmission line. K E Y W O R D S accumulated PIM signal, directional PIM product, distributed nonlinearity, frequency dependent PIM
This work demonstrated the directional passive intermodula-tion (PIM) product distribution effect in a microwave two-port network using a distributed nonlinear model. The analysis is demonstrated through the comparison of PIM products at the double ends of a general two-port network by a microstrip line. The reverse and forward PIM products are compared with the nonlinear source distribution and carrier frequency, the model and experiment demonstrates that the directional PIM effect (the difference by reverse and forward PIM) is caused by the PIM accumulation effect from multiple nonlinear sources, what is more, the frequency dependent phase relation can be one reason for the frequency dependent PIM product on wideband transmission line. K E Y W O R D S accumulated PIM signal, directional PIM product, distributed nonlinearity, frequency dependent PIM
An empirical modeling of contact nonlinearity induced intermodulation effect on coaxial connector is presented in this work. The intermodulation weights on inner and outer conductors are respectively clarified using measurement method. The contact degeneration induced intermodulation evolution is quantized with considering the contact coupling effect between the inner and outer conductors. This work demonstrated a set of test methods to quantify the oxide induced nonlinearity with contact degeneration effects, these methods can evaluate the contact intermodulation products and further predict the low intermodulation lifetime of passive devices.
Multipactor is a much studied resonant AC discharge that is harmful to microwave components and circuits. There is substantial current interest in this topic because of its threat to satellite communications. In this paper, an analytical transmission line model is presented to assess the effects of multipactor, should it happen, on the distortion of a signal. The model is applicable to any input signal (analog or digital) once the electron multipactor current it causes is specified. Examples using planar and coaxial transmission lines are given on a simple dynamical model of the multipactor current. Single and multitone signals are treated, with some results presented in the In-phase and Quadrature plots.
A design method is proposed in this paper for on-line (online measurement: measurement when producing devices) and on-site (on-site measurement: measurement when using devices) passive intermodulation (PIM) measurement with small size anechoic chamber. In a PIM measurement, the role of an anechoic chamber is to shield extraneous electromagnetic signal and reduce reflection, so that an accurate measurement result can be obtained. However, the chamber itself may introduce new PIM sources due to the nonlinearity in absorbing material and shielding enclosure. Traditional method to lower the influence of chamber to PIM measurement is to design a large size PIM test chamber in order to reduce the PIM level by path loss. In this paper, an innovative design concept of PIM test chamber is presented by controlling PIM source level of anechoic chamber instead of increasing the size of chamber. It can significantly reduce the chamber size using this method, particularly suitable for on-line and on-site PIM measurement. To implement the concept, absorbing material and shielding enclosure structure with extremely low PIM are developed. The design concept includes defining the worst-case reflection level, estimating the PIM level due to reflected signals, and determining the smallest size of chamber that meets the design specifications. Finally, two PIM test chambers are built to demonstrate the design concept. PIM measurement results are given to verify the theoretical analysis.
This paper proposes a new method to comprehensively predict the passive intermodulation (PIM) distortion in loose contact coaxial connectors by the time-domain reflectometry (TDR) test. The proposed approach treats the insulating film current and the electrothermal (ET) coupling as PIM sources. An ET coupling model is introduced to analyze the generating process of the PIM caused by the ET coupling in the loose contact coaxial connectors. The simulation and measurement results show good agreement. The proposed method is a cost-effective approach for the evaluation of the PIM distortion, because it uses a common TDR method to replace professional PIM test instruments. The proposed method can also be extended to all the loose metallic contact cases.
This paper presents an analysis for coaxial connector PIM using statistical method. A novel PIM prediction method for coaxial connector considering random contact behavior is proposed and demonstrated. In this work, based on Monte-Carlo approximation and micro-measurement to solve the micro contact force on contact surface, the PIM level from each single contact unit can be calculated based on the height distribution of contact tip. In experiment, PIM of coaxial connector was tested and compared with predication. A good agreement between theory and test proves the proposed PIM prediction method using contact statistic method is efficient. It gives a PIM confidence interval for the potential PIM values rather than a single value.
In this paper, emphasis is put on passive intermodulation (PIM) caused by multipaction. The results are calculated by particle-in-cell simulation under the conditions of 2-tone and 4-tone carriers. A temporal evolution of the number of particles with different carrier powers is presented. The input and output signals in the time domain and the power spectrum densities of these signals in the frequency domain under different conditions are compared. The relationship between multipaction and PIM interference is discussed. It shows that the appearance of the PIM phenomenon follows the occurrence of saturation of multipactor discharge and the powers of PIM products basically increase as the carrier powers grow. In addition, to suppress the occurrence of PIM interference, an external magnetic field is added in the direction perpendicular to the electric field, which is a practical way to inhibit multipaction, and the effectiveness and the reliability of this method are also verified by simulation.
This paper presents a novel passive intermodulation (PIM) prediction method considering random contact behavior using a Monte Carlo method for a coaxial connector. A smart contact model for a contact unit at a microcosmic level is proposed. Using Monte Carlo approximation and micromeasurements, different random distributed contact samples for different contact components inside the coaxial connector are reconstructed. In the experiment, PIM on inner and outer conductor was tested and compared with predication. A good agreement proves the proposed PIM prediction method is efficient. Rather than generating a single PIM prediction value, this method will give a PIM confidence interval for all the potential PIM values considering the contact force statistical behavior. The work will help analyze fluctuated PIM on coaxial connectors and inspire a new method to predict PIM risk.
The intermodulation distortion produced by passive and tunable liquid metal antennas is investigated. Four kinds of monopoles, including a passive copper monopole, a varactor-tuned copper monopole, a passive liquid metal (LM) monopole, and a tunable LM monopole using electrochemically controlled capillarity (ECC), are compared for their linearity and power handling capabilities. Linearity is assessed using a two-tone distortion test in which two fundamental tones close in frequency result in third-order intermodulation (IM3) tones at frequencies above and below the original tones separated by the frequency difference. The passive LM monopole has comparable linearity to the passive copper monopole while the linearity of the ECC-tuned LM monopole is at least 40 dB better than that of the active varactor monopole. The reconfigurable liquid metal antenna also handles higher power (31 dBm) before failure than does the active varactor-tuned antenna (24 dBm). IEEE
A novel field-circuit hybrid numerical method is proposed to analyze the passive intermodulation (PIM) caused by nonlinear contacts. In this hybrid numerical method, the electromagnetic field part is simulated by a self-developed time-domain finite integration theory. Meanwhile, the circuit part is modeled by a number of nonlinear lumped elements, which can introduce the nonlinear feature of nonlinear contacts. After solving a multiscale problem, the field part and the circuit part can couple with each other self-consistently via the generalized Ampere’s equation. To verify the accuracy of the proposed method, a practical example is discussed. The results show that the frequencies of each PIM order can match analytical results very well. Moreover, the PIM levels calculated by the proposed numerical method have a very good agreement with the measured results.
Applying properties of the Laplace transform, the transient heat diffusion equation can be transformed into a fractional (extraordinary) differential equation. This equation can then be modified, using the Fourier Law, into a unique expression relating the local value of the time-varying temperature (or heat flux) and the corresponding transient heat flux (or temperature). We demonstrate that the transformation into a fractional equation requires the assumption of unidirectional heat transport through a semi-infinite domain. Even considering this limitation, the transformed equation leads to a very simple relation between local time-varying temperature and heat flux. When applied along the boundary of the domain, the analytical expression determines the local time-variation of surface temperature (or heat flux) without having to solve the diffusion equation within the entire domain. The simplicity of the solution procedure, together with some introductory concepts of fractional derivatives, is highlighted considering some transient heat transfer problems with known analytical solutions.
The temperature of microwave FETs is found to vary significan tly at extremely high frequencies even though the fun- damental thermal time constant is only a few kHz. This affects third-order intermodulation through a process involving temperature rise due to power dissipation at the fundamental and second harmonic frequencies of signal, which is mixed with the fundamental and second-order products of the device's inherent nonlinearity. The process is strongly influenc ed by the spacing between the frequencies of signal components, which is exploited in a proposed method for characteriz- ing the frequency dependence of self-heating. The impact on circuit performance is that distortion and intermodulatio n, which vary with bias and load conditions, additionally vary with self-heating. This additional dependence is overlooked in present circuit models and analysis techniques.
An ac technique for measuring the thermal conductivity of dielectric solids between 30 and 750 K is described. This technique, the 3ω method, can be applied to bulk amorphous solids and crystals as well as amorphous films tens of microns thick. Errors from black‐body radiation are calculated to be less than 2% even at 1000 K. Data for a‐SiO 2 , Pyrex 7740, and Pyroceram 9606 are compared to results obtained by conventional techniques.
The temperature response of field-effect transistors to instantaneous power dissipation has been shown to be significant at high frequencies, even though the self-heating process has a very low time constant. This affects intermodulation at high frequencies, which is examined with the aid of a signal-flow description of the self-heating process. The impact on broad-band intermodulation is confirmed with measurements over a range of biases. Intermodulation measurements are then used to obtain parameters that describe the heating response in the frequency domain. This description is then implemented in a time-domain model suitable for transient analysis and compared with measured heating and cooling step responses.
The intermodulation performance of an FET in the common-source configuration is dependent on the impedance presented to its gate and drain terminals, not only at fundamental, but also at harmonic and baseband frequencies. At baseband frequencies, these terminating impedances are usually determined by the bias networks, which may have varying impedance over the frequencies involved. This can give rise to asymmetry in two-tone intermodulation levels, and changing intermodulation levels with tone spacing, as previous studies have shown. In this paper, an FET is analyzed to gain an understanding, useful to the circuit designer, of the contributing mechanisms, and to enable the prediction of bias points and the design of networks that can minimize or maximize these effects. Compact formulas are given to facilitate this. An amplifier was tested, showing good agreement between the theoretical and measured results.
From a study of the experimental results of the previous chapter, the following are the main features that emerge regarding the electrical resistance of the metallic elements:
1.
Metals have low resistivities lying in the range 1.5 to 150 πΩ-cm at room temperature, whereas semiconductors have resistivities that are 106 to 1012 times larger than these and insulators 106 to 1012 times larger still.
2.
To a first approximation the resistivity is linear in temperature for most metals above 0.5θ.
3.
Below 0.25θ the ideal or thermal component of the resistivity decreases faster than linearly—roughly as T
3 in many metals and as T
5 in several others, particularly the monovalent ones.
4.
Some dependence of the magnitude of the resistivity on position in the periodic table is evident from Table I, both as regards valency and atomic number.
5a.
The total resistivity is composed of thermal and impurity contributions; in magnetic metals there is a term of magnetic origin as well.
5b.
The separation of the total resistivity into the above independent components, commonly known as Matthiessen’s rule, may be said to be generally valid [Equation (1.5)].
6.
In some supposedly pure metals, notably Cu, Ag, Au, and Mg, a minimum occurs in the resistance close to the 5 to 15°K temperature region.
7.
Nearly half the metals and a good many alloys and compounds become superconducting at low-enough temperatures.
This text describes the statistcal behavior of complex systems and shows how the fractional calculus can be used to model the behavior. The discussion emphasizes physical phenomena whose evolution is best described using the fractional calculus, such as systems with long-range spatial interactions or long-time memory. The book gives general strategies for understanding wave propagation through random media, the nonlinear response of complex materials, and the fluctuations of heat transport in heterogeneous materials.
This article proposes a geometric interpretation of discrete fractional order controllers based on sampling time scaling property. Due to its clear interpretation, satisfactory accuracy and easy programming, the property could be used as a reliable simulation and realization method for fractional order control systems. The experiments of one-inertia speed control with fractional order integral controllers realized by the proposed sampling time property are also carried out to verify the theoretical robustness of fractional 1/sα systems. The experimental results show the superior robustness performances of fractional 1/sα systems against saturation non-linearity and inertia variation, which highlights the promising aspects of fractional order control.
Electrical Resistance of Eighteen Elements.—The paper contains a brief summary of an extensive series of measurements which are to be published in detail elsewhere made to determine the effect of pressures up to 12000 kg. per sq. cm. and of temperatures from 0° to 275° C. on the resistance of lithium, sodium, potassium, gallium, bismuth, mercury, calcium, strontium, magnesium, titanium, zirconium, arsenic, tungsten, lanthannum, neodymium, carbon (amorphous and graphitic), silicon, and black phosphorus. The data for tungsten and magnesium are improvements on data previously published; the data for the other substances are new. The first six of these elements were studied in both the liquid and the solid states. The pressure coefficients of solid calcium, solid strontium, and both solid and liquid lithium are positive; the coefficient of bismuth is positive in the solid state, but negative in the liquid.
It is well known that dielectric absorption plays a critical role in determining the accuracy of analog sampled-data systems that are based on charge storage, such as sample-and-holds and switched-capacitor ADCs. Less appreciated, but no less important, is the role it plays in determining the quality factor, or Q, of the capacitor. Dielectric absorption has both storage and loss components that act, and are significant, over the entire usable frequency range of the capacitor. Along with background information on the origins of dielectric absorption, this paper presents two models for a capacitor that exhibits dielectric absorption. The first is the relatively well known model proposed by Dow. The second is a model with relatively little exposure that is based on the dielectric permittivity model of Cole and Cole. This model has fewer parameters than the Dow model and is predictive over a very broad range of frequencies.
Here, we should mention the most important function used in fractional calculus — Euler’s Gamma function, which is defined as (2.1) This function is generalization of a factorial in the following form: (2.2)
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An overview of electro-thermal modeling of microsystems is presented. We consider the most important coupling between thermal and electrical phenomena, and then focus on the industry's central concern, that of Joule heating. A description of different solution approaches for the heat transfer partial differential equation, which constitutes the central part of electro-thermal simulation, is given. We briefly review the analytical solutions and consider further the numerical approaches, which are based on spatial discretization of the thermal domain. Lastly, we describe the final level of approximation, the dynamic compact thermal modeling. We emphasize the formal model order reduction methods, because they directly follow the spatial discretization, and thus preserve the investment into the finite element modeling.
An original method for modelling and simulation of fractional systems is presented in this article. The basic idea is to model the fractional system by a state-space representation, where conventional integration is replaced by fractional one with the help of non-integer integrator. This operator is itself approximated by a N dimensional system composed of an integrator and of a phase-lead ÿlter. This method is compared to other techniques like direct discretization of the fractional derivator and diiusive representation. Numerical simulations exhibit the general applicability and exibility of this new approach to diierent types of fractional models and to non-conventional non-integer derivation with limited spectral range. ? 2003 Elsevier B.V. All rights reserved.
Application of the methods of fractional integro-differential analysis to an inhomogeneous canonical heat-conduction (diffusion) equation with inhomogeneous boundary conditions has enabled us for the first time to reduce the canonical heat-conduction equation to three equations of lower order that contain fractional-derivative operators. Examples and an analysis of those fundamental new possibilities that are opened up by such an approach to a wide class of problems of heat and mass exchange, combustion, self-propagating high-temperature synthesis, etc., have been given.
This paper presents a Fractional Derivative Approach for thermal analysis of disk brakes. In this research, the problem is idealized as one-dimensional. The formulation developed contains fractional semi integral and derivative expressions, which provide an easy approach to compute friction surface temperature and heat flux as functions of time. Given the heat flux, the formulation provides a means to compute the surface temperature, and given the surface temperature, it provides a means to compute surface heat flux. A least square method is presented to smooth out the temperature curve and eliminate/reduce the effect of statistical variations in temperature due to measurement errors. It is shown that the integer power series approach to consider simple polynomials for least square purposes can lead to significant error. In contrast, the polynomials considered here contain fractional power terms. The formulation is extended to account for convective heat loss from the side surfaces. Using a simulated experiment, it is also shown that the present formulation predicts accurate values for the surface heat flux. Results of this study compare well with analytical and experimental results.
In this paper, the time-scaled trapezoidal integration rule for discretizing fractional order controllers is discussed. This interesting proposal is used to interpret discrete fractional order control (FOC) systems as control with scaled sampling time. Based on this time-scaled version of trapezoidal integration rule, discrete FOC can be regarded as some kind of control strategy, in which strong control action is applied to the latest sampled inputs by using scaled sampling time. Namely, there are two time scalers for FOC systems: a normal time scale for ordinary feedback and a scaled one for fractional order controllers. A new realization method is also proposed for discrete fractional order controllers, which is based on the time-scaled trapezoidal integration rule. Finally, a one mass position 1/sk control system, realized by the proposed method, is introduced to verify discrete FOC systems and their robustness against saturation non-linearity.
This paper explores the generation of passive intermodulation distortion products through the self heating of resistive elements in a microwave attenuator. A compact electrothermal resistor model requiring only two physical parameters is developed for platinum resistors and applied to a resistive attenuator pi network to predict electrothermal PIM. The electrothermal model is verified by comparing measured and predicted results when the attenuator is excited by a two tone signal at 400MHz with tone spacing from 1 to 100 Hz
The solution of the time-dependent diffusion equation in a semiinfinite planar, cylindrical, or spherical geometry with common initial and asymptotic boundary conditions is considered. It is shown that this boundary value problem may be described by a single equation which involves only a first order spatial derivative and a half order time derivative. The replacement is exact in the planar and spherical geometry cases but approximate in the cylindrical case. This replacement permits the solution of the original boundary value problem to be written for any boundary condition at the origin. It also leads to a simple relationship between the boundary flux and the boundary intensive variable, which does not require a calculation of the intensive variable at all positions and times.
This paper presents an intermodulation distortion measurement system based on automated feedfoward cancellation that achieves 95 dB of broadband dynamic range. A single tone cancellation formula is developed requiring only the power of the probing signal and the power of the combined probe and cancellation signal to predict the required phase shift for cancellation. This formula is applied to a two path feedfoward cancellation system, combined with the DUT probe path in a bridge configuration. The dynamic range and cancellation capabilities of this system are confirmed by measuring the passive intermodulation distortion generated by a low PIM microwave chip termination. The cancellation method extends the intermodulation distortion measurement dynamic range by at least 20 dB.
The appearance of PIM products bring a lot of harm into modern satellite communication systems. In this paper, the fundamental concept, mechanism and prediction of PIM are studied. And one mathematical model is given, which is able to predict high order PIM level by using low order PIM level.
The relationship between low-frequency dispersion and the
intermodulation performance of AlGaAs/GaAs HBTs has been demonstrated
for the first time. The theoretical analysis and experimental results
indicate that IM3 will depend strongly on the frequency
spacing (Δf=f2-f1) in the two-tone
measurement
Dual-gate gallium-arsenide field-effect transistors have been fabricated which give power gains greater than those which can be obtained from single-gate devices of similar gate length. Unconditionally stable gains in excess of 12 dB at 5 GHz have been measured for these devices.
A new model is presented for the calculation of passive intermodulation (PIM) in waveguide connections. The model considers the roughness of interconnecting waveguide surfaces and the presence of an insulator layer (oxide and contaminants) on these metal surfaces. This results in the generation of a contact resistance, which can excite the PIM level. In particular, the case in which metal–insulator–metal regions are the PIM source is especially investigated. The intermodulation level response is calculated for different waveguide junction parameters like applied mechanical load, surface finish, or metal properties showing qualitative agreement with the measured data published by previous authors.
Wire-wound type resistors are often used as reference elements in high-precision electronic instruments of industrial frequencies (50-60 Hz). One of the main factors in the measurement uncertainty of these instruments is the "self-heating" affected drift of the built-in wire-wound resistors. The materials, geometry, construction and the thermal environment of the resistors determine the dynamics of the load-affected drift. This paper investigates a thermal model that can be used for on-line estimation of the load-affected thermal dynamic drift of commercially available encapsulated precision wire-wound resistors. Using the suggested model, the achievable estimation error of the relative change in resistance is below 10%. The model and the model-based on-line error estimation allow reducing the measurement uncertainty of industrial high-precision calibrators and extending the power range of standard resistors in laboratories.
Drain current dispersion effects are investigated in AlGaN-GaN HEMTs by means of pulsed, transient, and small-signal measurements. Gate- and drain-lag effects characterized by time constants in the order of 10-5-10-4 s cause dispersion between dc and pulsed output characteristics when the gate or the drain voltage are pulsed. An activation energy of 0.3 eV is extracted from temperature-dependent gate-lag measurements. We show that two-dimensional numerical device simulations accounting only for polarization charges and donor-like traps at the ungated AlGaN surface can quantitatively reproduce all dispersion effects observed experimentally in the different pulsing modes, provided that the measured activation energy is adopted as the energetic distance of surface traps from the valence-band edge. Within this hypothesis, simulations show that surface traps behave as hole traps during transients, interacting with holes attracted at the AlGaN surface by the negative polarization charge.
A physics-based compact analytical expression for the thermal
impedance of SOI MOSFET's is presented. This new model extends the
steady-state thermal model of Goodson and Flik (1992) to allow for
transient and ac analyses, while improving self-consistency for large
devices. The modified steady-state model compares favorably to
measurements. Using the software package Thermal Impedance Pre-Processor
(TIPP), a multiple-pole circuit can be fitted to the thermal-impedance
model. The new model is compared to three-dimensional (3-D) ANSYS
transient simulations with good results. The thermal-equivalent circuit
is used in conjunction with a modified version of SOISPICE to give
efficient electrothermal simulations in the dc and transient regimes
Available: http://www.ips.gov
- Ed Wilkinson
- Hobart
- Australia
Wilkinson, Ed., Hobart, Australia, 2004, pp. 1–8. [Online]. Available:
http://www.ips.gov.au/IPSHosted/NCRS/wars/wars-2004/index.htm
- Amer
- Soc
Amer. Soc. Mech. Eng. Tech. Design Eng. Conf., Dec. 2007, pp. 1–6.
Lowfrequency dispersion and its influence on the intermodulation performance of AlGaAs/GaAs HBTs
- K Lu
- P M Mcintosh
- C M Snowden
- R D Pollard
K. Lu, P. M. McIntosh, C. M. Snowden, and R. D. Pollard, "Lowfrequency dispersion and its influence on the intermodulation performance of AlGaAs/GaAs HBTs," in IEEE MTT-S Int. Microw. Symp.
Dig., 1996, pp. 1373-1376.
- H S Carslaw
- J C Jaeger
H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics. New York: Dover, 1963.