The Physics of Oscillations and Waves: With Applications in Electricity and Mechanics

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Results show that the n i and n n oscillate in an opposed-phase form, which indicates that the ions and neutrals have the relationship of predator and prey. Although this model can reflect the complex processes of low-frequency oscillations, its boundary conditions are unreasonable. In the original predator-prey model, it means without the existence of the predator, the growth rate of the prey is proportional to its own numbers.

In the ionization model, it implies that the rate of neutrals entering the ionization zone is proportional to the number density of neutrals in the ionization zone, when in fact, neutrals are introduced into the channel at a constant rate. If the rate of neutrals arriving is fixed, the system would be stable unconditionally. Detailed calculation results are shown in [ 42 ]. It shows that the boundary conditions of the predator-prey model have a significant effect on system stability, which is unreasonable. In order to give reasonable boundary conditions, we can use the simplified equation set to reflect the characteristics of low-frequency oscillations.

The predator-prey model is derived from the ion continuity equation and the neutral continuity equation. We begin with the following equation set:. With operating conditions of a neutral gas flow rate of 3. Simulation results show that the system is stable. The results show that even with equation set 2 — 4 , the mechanism of ionization oscillations is incomplete.

Further, we introduce the dynamic electric field as follows:. H 0 is the magnetic field at the cathode, and H x is the profile of the transverse magnetic field. The details of the parameter values and model can be found in [ 43 ]. The results show that low-frequency oscillations are related with the changes of electric field. Changes in mean neutral number density and mean ion number density in the ionization zone with a constant electric field.

Changes in mean neutral number density and mean ion number density in the ionization zone with a dynamic electric field. For a fixed electric field, the ions acceleration characteristic is fixed. If a disturbance occurs in the ionization process and the ionization intensity increases, the n n would decrease and the rate of ions leaving would be increased; the effects of these two processes cause the system balance again. From Eq. The numerical results show that the feedback of the electric field, together with the replenishment and ionization avalanche of neutral, brings the oscillation.

Analyzing particle movement process, the ionization zone is upstream of the acceleration zone. Electrons go into the ionization zone from the acceleration zone. The ionization zone and acceleration zone interact with each other due to this movement. The electric field equation is essentially the electron momentum equation which reflects the interact between the acceleration zone and the ionization zone.

Therefore, the import of a dynamic electric field makes the system into whole. The dynamic electric field is a highly important influence factor for low-frequency oscillation. The dynamic electric field is also applied and studied in many research areas, and these research results shown in the electric field have a plentiful scientific connotation [ 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 ].

The stabilizing method of low-frequency oscillation is an essential challenge for the space applications of thrusters. Researchers have tried many different ways to mitigate low-frequency oscillations. Previous studies indicated that the low-frequency oscillations are sensitive to filter parameters and can be mitigated to an acceptable level with proper filter values [ 59 ].

A filter unit is always involved between the thrusters and the power supply. The traditional filter consists of an inductor and a capacitor, and sometimes a resistor is also applied. This type of filter is a low-pass filter designed to isolate the interfering signal from the thrusters to the power supply. Recently, the role of the filter in the oscillation control was introduced by Yu et al.

It is noted that the filter regulates the voltage across itself according to the variation of discharge current so as to decrease its fluctuation in the discharge circuit, which is the function of a controller. Therefore, the matching network between the thrusters and power supply has two functions, which are those of a filter and controller. However, it is highly difficult to mix the function of a filter and controller in one network, because the aim and function of these two parts are quite different. This may be the reason that there have been no design methods of the matching network until now.


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The parameters are in practice always obtained through a trial-and-error method. Therefore, it follows that we should separate the matching network into two stages. The first stage is the filter, which aims to isolate the interfering signal from the thrusters to the power supply. The second stage is the controller, which provides a regulated voltage to decrease the low-frequency discharge current oscillation.

As a matching network between the thruster and power supply, the function of a filter is to protect the power supply, that is, to obtain a stable load for the power supply and, therefore, enable the power supply to operate in its normal electrical state. Another requirement is that the filter has low insert impedance. In a manner of speaking, the design goal is to ensure stable current or voltage on the power supply side, even though the current fluctuates on the thruster side.

Thus, from the point of view of energy conservation, the filter needs a capacitor large enough to store energy and release it to a fluctuating load. Connecting the capacitor between the positive and negative terminals of the power supply directly may be unreasonable, because there will be a large current peak in the circuit when the power supply is turned on. Therefore, designers must choose a component to connect between the capacitor and the power supply, with the aim of charging the capacitor. This chosen component should have low DC impedance because it is connected in the discharge circuit.

Moreover, the component should have high AC impedance. The reason can be understood as follows: The thruster consumes power with a pulsating mode, which would cause fluctuation in the capacitor voltage. The voltage difference between the power supply port and the thruster port would draw a fluctuating current from the power supply, and we do not expect that situation to occur. Thus, the component should have high AC impedance to suppress this kind of current fluctuation.

Obviously, the component meeting these criteria is an inductor. Therefore, the filter stage consists of an inductor and a capacitor. On the thruster side, we use specific properties of charge storage to stabilize the discharge voltage, and on the power supply side, we use the inductor to achieve constant current.

In order to avoid the LC circuit resonance and electromagnetic interference after extinguishing the thrusters, a resistor is connected parallel with the inductor. According to the preceding analysis, the inductor in the filter stage should be sufficiently large to achieve a small power supply current fluctuation. On the other hand, the capacitor should be sufficiently large to ensure a small thruster voltage fluctuation.

According to Fourier analysis, a fluctuating signal can be expressed as summation of different frequency components. Therefore, the supply current can be expressed as.

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If we temporarily neglect to consider the existence of the resistor, the inductor voltage can be expressed as. Additionally, we introduce the Fourier analysis method, and therefore, Eq. Thus, we obtain. Substituting Eq. Therefore, the phase relationship between u L and u C is opposite.

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The phase relationship between i s and i C is also opposite. However, it is highly difficult to simplify Eq.

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From the angle of decrease for the low-frequency oscillation, it is also unreasonable to consider waves of all frequencies to set the parameters of the inductor and capacitor. Thus, we simply take into account the main frequency component as low-frequency oscillation f 0. The inductor and capacitor of the filter stage must satisfy.

Under these assumptions, the product of the inductance and capacitance should be greater than or equal to 5. If we choose the inductance to be 0. The inductance and the capacitance values could be different combinations as long as they satisfy Eq. However, the value of the inductor and capacitor should also take certain factors into consideration, such as the DC power dissipation, volume, and weight.

After the determination of the inductance and capacitance, it can be seen that an LC network would have a resonant peak in its characteristic frequency. This would induce undesired electromagnetic interference. Thus, a resistor is usually connected in parallel with the inductor. Excessive resistance will cause the damping coefficient to be too small, and the electromagnetic interference will increase.

Undervalued resistance will weaken the effects of the inductor and cause an increase in power supply current fluctuation. Thus, the resistance of the resistor is often chosen on the basis of experiments. It can be seen that the current attenuation ratio increases with the increase in inductance and capacitance in the filter stage. Though the larger inductance and capacitance yields a larger current attenuation ratio, it is also accompanied by an increase in the capacitor volume and inductor weight. Therefore, the reasonable range of current attenuation ratio is approximately 5— In this system, the Hall thruster is the controlled object, the discharge power supply voltage is the reference signal, and the filter is the controller.

Therefore, we designate the filter stage as the controller stage. The controller stage filter regulates the voltage across itself according to the variation of discharge current, so as to affect the electric field distribution in the discharge channel and therefore decrease the discharge current fluctuation in the discharge circuit [ 8 , 10 , 12 , 26 , 33 ]. The simplest component that can provide a varying voltage with a change in current is an inductor.

However, the magnetized plasma of a Hall thruster exhibits varied oscillations ranging from kilohertz to gigahertz. High-frequency current oscillation will cause an undesirable high-amplitude oscillation of the inductor voltage. Thus, we require the controller stage to have sufficient gain in the low band but to decay the signal in the high-frequency band. If we consider the oscillation current as the input signal and the voltage of the control filter stage the voltage at the thruster as the output signal, the transfer function can be expressed as. Obviously, the parameters of the controller stage relate to the required gain of the system.

If we suppose the minimum gain is g 0 with the low-frequency oscillation frequency f , it must satisfy. Rewriting Eq.

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The resistor affects only the gain near the turnover frequency, and thus, we temporarily ignore the effects of the resistor. We then obtain.


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  8. Considering Eq. A proper filter parameter can provide a regulated voltage with a suitable amplitude and phase angle to control the oscillation of plasma density in the channel so as to decrease the current oscillation in the discharge circuit. Though it is difficult to determine the phase angle of plasma density in the ionization region, the phase relationship between the discharge current and the plasma density in the ionization region can be obtained by analyzing the propellant ionization process in the discharge channel.

    When the propellant is ionized in the ionization region, the plasma density in the ionization region reaches its highest point. Subsequently, the produced ions are accelerated to their high velocity in the accelerating region, and the discharge current reaches its maximum. From the analysis of the propellant ionization process, it can be seen that there is a lag of time between the discharge current and the plasma density of the ionization region in the time scale of low-frequency oscillation.

    Thus, when the plasma density in the ionization region increases, the control stage filter should provide a voltage to balance the large ion production.

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    Thus, the regulated voltage of the filter has a phase-angle difference from the discharge current. This phase angle can be provided by the resistor paralleled between the inductors. The performer must be singing a note that corresponds to the natural frequency of the glass. As the sound wave is directed at the glass, the glass responds by resonating at the same frequency as the sound wave.

    With enough energy introduced into the system, the glass begins to vibrate and eventually shatters. How much energy must the shock absorbers of a kg car dissipate in order to damp a bounce that initially has a velocity of 0. Assume the car returns to its original vertical position. Calculate the energy stored in the spring by this stretch, and compare it with the gravitational potential energy. Explain where the rest of the energy might go.

    The rest of the energy may go into heat caused by friction and other damping forces. Suppose you have a 0. There is simple friction between the object and surface with a static coefficient of friction. Assume it starts at the maximum amplitude. Engineering Application: A suspension bridge oscillates with an effective force constant of. Skip to content Increase Font Size. Oscillatory Motion and Waves. Learning Objectives Observe resonance of a paddle ball on a string.

    Observe amplitude of a damped harmonic oscillator. You can cause the strings in a piano to vibrate simply by producing sound waves from your voice. The paddle ball on its rubber band moves in response to the finger supporting it.

    At higher and lower driving frequencies, energy is transferred to the ball less efficiently, and it responds with lower-amplitude oscillations. Amplitude of a harmonic oscillator as a function of the frequency of the driving force. The curves represent the same oscillator with the same natural frequency but with different amounts of damping.

    Resonance occurs when the driving frequency equals the natural frequency, and the greatest response is for the least amount of damping. The narrowest response is also for the least damping. In , the Tacoma Narrows Bridge in Washington state collapsed. Heavy cross winds drove the bridge into oscillations at its resonant frequency. Check Your Understanding. A famous magic trick involves a performer singing a note toward a crystal glass until the glass shatters. A periodic force driving a harmonic oscillator at its natural frequency produces resonance.

    The system is said to resonate. The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies. Previous: Damped Harmonic Motion.



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