Abstract—Radio frequency (RF) noise can severely degrade the performance of electronic circuits and wireless systems. This article proposes an analytical method to evaluate the spectrum of RF noise caused by multicarrier multipactordischarge.First, a simplified model is developed to analyze the electric current characteristics of the resonant multicarrier multipactor discharge. Then, an analytical formula for the spectrum of the resonant multicarrier multipactor discharge is derived. Finally, the theoretical method is evaluated for dual-carrier operation inside a silver plated rectangular waveguide. Simulation results coincided well with theoretical findings show that the proposed method is encouraging.
Index Terms—Multicarrier, multipactor discharge, parallel plate model, rectangular waveguide, RF noise
Multipactor is a nonlinear phenomenon which is caused by the interaction between the charged particles and the applied RF fields , . Multipactors are generally classified into two types on the basis of the carriers of the applied RF fields, namely, single-carrier and multicarrier multipactor discharge. The main effort of this work is focused on the multicarrier mutipactor cases; the theoretical analysis and the resulting equations developed in this article are valid only for monoenergetic secondary emission.
As is well known, when there is an electric field between two small parallel plates, the initial electrons of the parallel plate region will be accelerated by the electric field and then impact against the surface of the plate. One or more electrons are released after impact. If some of the electrons synchronize with the electric field, such process as aforementioned is repeated until a steady state is reached (the growth of multipactor electrons ceases to continue) . Multipactor discharge causes a number of negative influences such as the RF noise, passive intermodulation, impedance mismatch, and signal distortion, which can severely degrade the performance of microwave circuits –.
In the past decades, various models and analytical methods were proposed to suppress or predict multicarrier multipactor breakdown in microwave devices –. For example, in , a new quasi-stationary (QS) prediction method for multipactor breakdown determination in multicarrier signals has been presented; the experimental results show that the QS prediction method offers better predictions than that the popular 20-gap-crossing rule. In , the performance of the most popular multipactor breakdown prediction method, i.e., the 20-gap-crossing rule, for multicarrier signals, has been checked by experiments. In , the Monte Carlo method has been proposed to find the thresholds and the global “worst case” waveforms of both single-event and long-term multipactors. However, in practical situations, it is almost impossible to completely suppress multipactor discharge for all cases. Therefore, it is necessary to understand the spectrum characteristics of multipactor discharge, as it is useful for designing the noise compression components and filters to eliminate the RF noise generated by multipactor discharge.
The spectrum characteristics of the single-carrier multipactor discharge have been well investigated by some works –. For example, Sorolla et al.  presented a model to evaluate the power spectrum of a single-carrier multipactor discharge. Semenov et al.  analyzed the amplitude spectrum of the multipacting electrons in rectangular waveguide with single carrier signal. Jimenez et al.  measured the power spectrum of the multipactor event excited in a single-carrier microwave circuit based on rectangular waveguides. However, a specific theory, which is used to analyze the spectrum characteristics of multicarrier multipactor discharge, attracts little attentions to the best of the authors’ knowledge .
In this article, a theoretical analysis method is presented to analyze the spectrum characteristics of the multicarrier multipactor discharge. It should be noted that the multipactor electrons are considered to resonate with the fundamental RF carrier. This article is organized as follows: Section II analyzes the frequency characteristics of the multicarrier RF field briefly; it will be used to find the period, harmonics, and subharmonics of the resonant multicarrier multipactor discharge in later Sections. In Section III, a simplified model of the multipactor electrons is stated first, and then a numerical expression for evaluating the spectrum of multicarrier multipactor discharge is derived. In Section IV, simulations are conducted to demonstrate the effectiveness of the proposed theories. Section V draws a conclusion of this article.
II. FREQUENCY OF MULTICARRIER RF FIELD
In this section, the frequency characteristics of multicarrier signals are proposed. A multicarrier signal u(t) (RF field) is composed of k carriers with frequencies ωi , angular phases ϕi , and amplitudes Ui , mathematically
where ωi = 2π fi , i are integers, and I = 1, 2…k. Furthermore, as detailed in , if fi are integers, the frequency f of the multicarrier signal u(t) is the greatest common divisor of the set carrier frequencies fi , mathematically
In practical microwave circuits, the precision of fi is usually limited to decimals; thus, the integer frequencies can be obtained by normalizing fi . For example, when f1 is equal to 2.33 GHz, the normalized integer frequency of f1 will be written as 2330 MHz.
Three cases (A, B, and C), when ϕi = 0, Ui = 1, and the carrier numbers are 2, 2, and 3, are shown in Table I to explain the analytical results in (2). As illustrate in Table I, the frequencies of cases A, B, and C are 0.5, 0.6, and 0.05 GHz, respectively. These demonstrated results will be used to analyze the period, harmonics, and subharmonics of the multicarrier multipactor discharge.
III. NUMERICAL ANALYSIS METHOD
In this section, the analytic model and mathematical expression are developed to analyze the spectrum characteristics of the resonant multicarrier multipactor discharge.
A. Analytical Model
As shown in Fig. 1, a parallel plate model is adopted to investigate the spectrum characteristics of the resonant multicarrier multipactor discharge. The length, height, and width for the parallel plate model are defined as d, h, and w, respectively. As shown in Fig. 1(a), the electrons resonate with the applied electric field and grow exponentially. The parallel plate model will be filled with the resonant electrons after a few nanoseconds, and the current of those resonant electrons is directly related to the multipactor noise . Therefore, the cruces of eliminating the multipactor noise are the analysis on the spectrum characteristics of the resonant electron current.
B. Mathematical Expressions of the Spectrum Characteristic
As shown in Fig. 1(b), we assume that all resonant electrons are localized in a thin sheet L, and L is located at zL . Here, the bottom and top plates of parallel plate model are located at z = 0 and z = h, respectively. The distance between the thin sheet L and the bottom plate is |zL |. Because the resonant electrons are driven by the RF field u(t), the motion of the thin sheet L is repeated with the period of the RF field u(t). Furthermore, the direction of this thin sheet L orients only in the z-axis, when d L and w L are assumed; the theoretical model of this article is developed in Fig. 1(c). Therefore, the current density of the thin sheet L can be calculated as 
where v(t) is the instantaneous velocity of the thin sheet L, in the units of meter per second (m·s−1);
z is the motion direction of the thin sheet L (does not carry units); ρ is the volume charge density, measured in coulombs per cubic meter (C·m−3), and ρ can be further measured with putting a positive probe in the multipacting components . Since the thin sheet L is directly derived by the RF field u(t), according to the Newton–Lorentz force law, the relationship between the instantaneous velocity v(t) and the RF field u(t) can be written as
where m and e are the electronic mass and the electronic charge, respectively. As Ui = U0 and ϕi = 0 are first assumed in (1), and then integrating (4), the instantaneous velocity v(t) can be expressed as
where t0 and v0 are the initial time and the initial velocity of the thin sheet L, respectively, and v(t0) = v0. T is the period of the RF field u(t). N is the multipactor discharge resonance order, and N = 1, 3, 5, 7, … odd.
Equation (5) consists of two parts, which are the oscillation velocity vosc(t) and the constant velocity vcon(t). As shown in (6), vosc(t) changes continuously with time t and thus contributes only to the basic carrier frequencies of the current density J . vcon(t) represents an antisymmetric square-wave function with period NT, which changes in a step-like way after each impact against the surface of the parallel plate; the instantaneous jump of vcon(t) will lead to distortion of the carrier signals. Therefore, the harmonics and subharmonics of the current density J are completely determined by vcon(t).
As also shown in (6), vcon(t) is an antisymmetric step function with the period of NT. Therefore, vcon(t) can be expanded by Fourier series; the Fourier series is an expansion of a periodic function in terms of an infinite sum of harmonically related sinusoids . Specially, the Fourier series of vcon(t) can be rewritten as
Furthermore, the Fourier coefficients a0, an, and bn of vcon(t) can be calculated as
Substituting (8) into (7), vcon_FS(t) can be rewritten as
where n is the orders of harmonics and subharmonics, and s belongs to positive integers. Since vcon_FS(t) is an expansion of vcon(t), vcon_FS(t) is equal to vcon(t), then v(t) = vosc(t) + vcon_FS(t).
When taking both vcon(t) and vosc(t) into considerations and ignoring the motion direction
z, the current density J can be rewritten as
Equation (10) shows an analytical expression to evaluate the spectrum of the current density of multicarrier multipactor discharge. The following conclusions can be drawn from (10): 1) in the first-order multipacting resonance N = 1, the odd harmonics ( f , 3 f , 5 f,…, and f = 1/T ) of the RF field u(t) are generated. For example, the harmonics are determined by the last term in (10) and expressed in
∞ n=2s−1 sin[2πn/T (t − t0)], where s belongs to positive integers, and n = 2s − 1; therefore, the harmonic components are 1/T , 3/T , 5/T,··· , n/T (i.e., f , 3 f , 5 f , ··· , nf, f = 1/T ). 2) in the higher order multipacting resonance N ≥ 3, the subharmonics are generated and can be described by 2πnf/N. It means that the frequencies of the subharmonics are directly related to the order of multipacting resonance N and the frequency f of the RF field u(t). 3) the magnitude of the multicarrier multipactor noise, including harmonics and subharmonics, decreases with the order n. For example, in (10), the amplitudes of harmonics and subharmonics are expressed in
∞ n=2s−1[v0 − (U0e)/(hmωi)
k i=1 sin(ωi t0)]4/(nπ ), and thus for a particular case of multicarrier multipactor discharge (i.e., the coefficients of v0, U0, k, h, and ωi are fixed), the amplitudes are only affected by n and decrease with it.
Two cases, which are used to further illustrate the frequency components calculated by (10), are shown in Fig. 2. The data employed to analyze the frequency components in the above two cases are set as follows: k = 2, t0 = 0, U0 = 30 V, v0 = 6.68 eV. Form the example, we see that the third, fifth, harmonics (3 f , 5 f ) and the seventh, eleventh, thirteenth subharmonics (7 f /3, 11 f /3, and 13 f /3) are very close to carrier frequencies and need to be addressed carefully.
In this section, a dual-carrier multipactor discharge, which occurs in a rectangular waveguide, is designed to demonstrate the proposed theories in Section III. The length and width of the rectangular waveguide are 10 and 8 mm, respectively. The height of the waveguide covers two different sizes, 0.6 and 2.4 mm, in order to provide the multipactor orders of 1 and 3. Silver-plated waveguide surfaces have been assumed; thus, the standard silver parameters in ECSS  are used as follows. δmax = 2.22, Emax = 165 eV, E1 = 30 eV, and taking v0 = 6.68 eV, SEY at low energies of 0.5.
The multipactor noises are investigated in the time domain and the frequency domain, respectively. A conformal time domain finite integration theorem (TDFIT) and particle in cell (PIC) hybrid method are used to simulate the dual-carrier multipactor discharge in the time domain –. Furthermore, two separate simulations, electromagnetic (EM) simulation and dual-carrier multipactor, are conducted in this work. Except for the difference of seed electrons, other simulation conditions such as RF fields, SEY, and material parameters are the same in both cases. In the dual-carrier multipactor simulation, the number of seed electrons is set as 600, which is employed to analyze the multipactor noise. In the EM simulation, the number of seed electrons is set as 0, which is employed as a controlled experiment to confirm that the RF noise only relates to the multipactor discharge.
The simulation results are shown in Figs. 3 and 4, where Fig. 3 illustrates the changes of the total electrons in the silverplated rectangular waveguide. It is easy to see that there is a significant increase in the number of the resonant electrons after 6 ns, which means the multipacting resonance occurs at t > 6 ns
Fig. 4 shows the results of the EM simulation (no multipactor) and dual-carrier multipactor discharge simulation in a silver-plated rectangular waveguide. The red dashed line represents the simulation result of the EM simulation, where the multipactor discharge is not presented due to the absence of seed electrons. The blue dashed dotted line represents the simulation result of the dual-carrier multipactor discharge. It is important to notice that a time-varying convection current is generated in the dual-carrier multipactor discharge simulation. This convection current is caused by the resonant electrons and always treated as multipactor noise because they can affect the integrity of the propagating signals , . Comparing the results between Figs. 3 and 4, it is easy to see that the intensity of multipactor noise increases with the total number of the multipactor electrons.
In order to study the spectrum characteristics of the dualcarrier multipactor discharge, the simulation results in Fig. 4 are further analyzed in the frequency domain using a standard Fourier transform algorithm, and the corresponding results are shown in Fig. 5. The blue dashed dotted line and the red dashed line represent the power spectrum of the dual-carrier multipactor simulation and EM simulation, respectively. We can clearly see that the odd harmonics ( f , 3 f , 5 f , …) and subharmonics ( f /3, f , 5 f /3, …) emerge and decrease with the frequency ft, and ft is the frequencydomain variable of time t. As also shown in Fig. 5, a good agreement between the simulation results and the theoretical analysis results (i.e., the cyan open circles) is observed.
Although it is well known that the RF noise could be caused by the multicarrier multipactor, the frequency components of the multicarrier multipactor noise have not been sufficiently investigated. In this article, an analytical method of evaluating the spectrum characteristics of the muticarrier multipactor has been proposed. According to our investigation, the frequencies of the multipactor noise can be described by 2πnf/N, which are directly related to the multipactor discharge resonance order N and the frequency of the RF field. This conclusion is useful for designing the noise compression components or filters to eliminate the RF noise caused by multicarrier multipactor discharge.
The authors would like to thank prof. V. E. Semenov, Institute of Applied Physics, Nizhny Novgorod, Russia, for his helpful discussions in the theoretical analysis. The authors would also like to thank Dr. Yiming Zhang, Maritime Institute, Nanyang Technological University, Singapore, for his helps in the simulations and in the proofreading of this article.