High-intensity discharge lamp and Duffing oscillator—Similarities and differences

Baumann, Bernd; Schwieger, Joerg; Stein, Ulrich; Hallerberg, Sarah; Wolff, Marcus

doi:10.1063/1.4999979

The processes inside the arc tube of high-intensity discharge lamps are investigated using finite element simulations. The behavior of the gas mixture inside the arc tube is governed by differential equations describing mass, energy, and charge conservation, as well as the Helmholtz equation for the acoustic pressure and the Reynolds equations for the flow driven by buoyancy and Reynolds stresses. The model is highly nonlinear and requires a recursion procedure to account for the impact of acoustic streaming on the temperature and other fields. The investigations reveal the presence of a hysteresis and the corresponding jump phenomenon, quite similar to a Duffing oscillator. The similarities and, in particular, the differences of the nonlinear behavior of the high-intensity discharge lamp to that of a Duffing oscillator are discussed. For large amplitudes, the high-intensity discharge lamp exhibits a stiffening effect in contrast to the Duffing oscillator. It is speculated on how the stiffening might affect hysteresis suppression.

I. INTRODUCTION

High-intensity discharge (HID) lamps are encountered in many lighting applications. Metal halide (MH) lamps are state of the art lamps (Fig. 1) which possess a sun-like light characteristic. To avoid a demixing of the arc tube content and to reduce erosion of the electrodes, HID lamps are typically operated at 400 Hz alternating current. The size and the cost of the drivers could be reduced considerably if the lamps are operated at approximately 300 kHz.¹ Unfortunately, at these high frequencies, the periodic heating of the plasma induces acoustic resonances, which, in turn, result in light flicker and even severer problems.² In previous studies, we have presented the results of finite-element (FE) simulations, which have been carried out to obtain an understanding of the mechanisms responsible for light flicker.^3–5 Here, we elaborate on the interpretation of the findings in terms of a nonlinear oscillator.

FIG. 1.

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Design of a MH lamp (Philips 35 W 930 Elite). The arc tube is filled with argon, mercury, and metal halides. The distance between the electrodes is 4.8 mm. The upward bending of the plasma arc is due to buoyancy. The temperature in the plasma arc is about 6000 K and drops to ca. 1500 K at the ceramic walls.

II. MODELING

Dreeben pursued the most direct approach for the modeling of a HID lamp by setting up a transient model.⁶ In order to obtain results within a reasonable CPU-time, he had to restrict himself to a 2D-model corresponding to a lamp with an arc tube of infinite length. To attain a more realistic description, we use a 3D-model. We set up a time-independent model in the frequency domain to compensate for the additional degrees of freedom. The idea is to identify light flicker by the appearance of instable solutions. It is not possible to take advantage of the axisymmetric lamp design, since for a horizontally operated lamp, the upward bending of the plasma arc due to buoyancy violates the symmetry.

A relatively simple model describing the processes in the ceramic walls, the electrodes, and the interior of the arc tube comprises coupled partial differential equations (PDEs) representing the conservation of mass, energy, electric charge, and momentum, respectively (thermal plasma model²). The Reynolds equations (momentum conservation) have to be supplemented by the equation of state of an ideal gas. To model the generation and propagation of sound waves, three additional PDEs for the acoustic pressure, the acoustic temperature, and the sound particle velocity are needed (visco-thermal acoustic model⁷). It has been suspected that acoustic streaming (AS) plays an important role for the generation of light flicker.⁸ AS is a directed flow in a fluid, driven by high amplitude periodic sound waves.⁹ The Reynolds stress term appearing in the Reynolds equations describes the generation of AS.

The model is completed by appropriate boundary conditions and coefficients describing the physical properties of the materials. Some of these coefficients, such as viscosity and thermal conductivity, are temperature dependent and, therefore, field variables themselves. Their temperature dependencies are described by heuristic functions.⁴

Due to the complexity of this model, it is not suitable for the computer resources at our disposal. A considerable reduction of the required computer resources can be obtained by replacing the three PDEs for the modeling of sound waves by the inhomogeneous Helmholtz equation for the acoustic pressure. The inhomogeneity describing the generation of sound is identical to the resistive loss density minus the power density for the generation of electromagnetic radiation. The solution of the Helmholtz equation can be expressed in terms of an acoustic mode expansion, and the modes can readily be obtained numerically. This approach has the disadvantage of not taking acoustic loss into account. However, it is possible to incorporate loss by means of loss factors.^4,10,11 In our model, we have included viscous loss and loss due to thermal conductivity in the interior of the arc tube and at the inner surface of the arc tube as well. The simulations based on the mode expansion model consist of several consecutive steps (Fig. 2)

The thermal plasma model is solved. The main results of this step are the temperature field and the inhomogeneity term of the Helmholtz equation.
On the basis of the temperature field, the acoustic modes and natural frequencies are determined. The inhomogeneity term and the loss factors allow determining the amplitude of the sound pressure at a certain excitation frequency.
The sound particle velocity is calculated from the acoustic pressure. It has to be multiplied by a heuristical factor, which is equal to one in the interior of the arc tube and drops to zero near the wall. In this way, it is possible to account for the no-slip condition of the sound particle velocity.^12,13
Once the sound particle velocity is available, the Reynolds stresses driving AS can be determined and the complete procedure has to be repeated starting from step 1. Thus, a recursion loop is initiated. Once convergence has been reached, the recursion is terminated.
The procedure is repeated for a slightly shifted frequency until the frequency range of interest is covered (frequency scan). In the simulations, the converged solution of the previous frequency is used as an initial configuration for the simulation at the new frequency. This guarantees that the qualitative character of the solutions corresponding to neighboring frequencies is identical. Loss of convergence can then be interpreted as an indication that the respective solution does not represent a physically stable state anymore. (This interpretation is supported by the results obtained by the accompanying experiments.)⁵

FIG. 2.

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Schematic description of the recursion.

Using the mode expansion method, despite the necessity of the recursion, leads to results within manageable computing times. The advantage is due to the fact that the acoustic model, obtained as described above, is linear.

The differential equations used to obtain the results described below can be found in the Appendix. In order to keep the length of this presentation reasonably short, we refrain from discussing the differential equations, the boundary conditions, and the temperature-dependent transport coefficients, as well as the details of the implementation. This information can be found elsewhere.⁴

III. RESULTS

For the lamps supplied by Philips, the lowest light flicker frequency typically is near 42 kHz. The FE model identifies two acoustic modes in the vicinity of this frequency. The two modes are degenerate, if gravity and, therefore, the buoyancy force are switched off. In the presence of gravity, only the mode depicted in Fig. 3 is excited and generates sound waves. The depicted mode corresponds to the first iteration of the recursion loop, when the AS force is zero and the acoustic model (the Helmholtz equation) is not yet coupled to the plasma model. Close to resonance (see Sec. III A), a strong AS flow exists when the recursion has converged. The impact of the AS flow field is a shift of the natural frequency, without an alteration to the general pattern of the mode.

FIG. 3.

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Acoustic mode of natural frequency 47.8 kHz. Depicted is $| p |$ ⁠, the absolute value of the acoustic pressure in arbitrary units. Blue indicates the pressure node (⁠ $| p | = 0$ ⁠) and dark red an antinode (maximal values of $| p |$ ⁠). As a guide to the eye, the isobars are depicted along two gray planes, each plane cutting the arc tube into halves. The mode is normalized according to $\int_{V} p_{i}^{*} p_{j} d V = V δ_{i j}$ ⁠, and the correspondence of colors and numbers is of no importance here. The mode corresponds to the first iteration of the recursion loop (no AS force).

In principle, the interest is on the response of the acoustic pressure. The acoustic pressure data is available in the results of the FE simulations, but there is no simple way for its measurement. The quantity most easily accessible for measurement and also available from the data obtained in the FE simulations is the voltage drop between the electrodes. However, in contrast to the acoustic pressure, the voltage is not an amplitude of an oscillating quantity and there appears to be no reason that the two quantities show a similar behavior. Nonetheless, the results from the simulation show that the frequency response of acoustic pressure and voltage in fact are very much alike. We were even able to identify the reason for this similarity.⁵ Hence, we focus on the voltage instead of the acoustic pressure.

Figure 4 summarizes the corresponding simulation results. In the following, we interpret the figure by discussing the projections of the curve depicted in blue to the three coordinate planes. During the discussion, we will present evidence that the HID lamp behaves in some respects similar to a nonlinear oscillator. As a prerequisite, it seems helpful to recount some basic facts on oscillators.

FIG. 4.

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Connection of excitation frequency, natural frequency, and voltage drop resulting from the FE simulations (blue symbols). The black symbols result from projections of the blue curve to the respective plane. The meaning of the green line and of the distinction of data points represented by the circles and crosses are explained in Sec. III A.

An unforced, undamped oscillator, with a linear spring oscillates with its natural frequency, which is determined by the stiffness of the spring and the mass of the bob. For a nonlinear spring, the stiffness varies with the displacement and, since the stiffness determines the natural frequency, the natural frequency depends on the amplitude of the oscillation.

The bob of a forced oscillator moves with an amplitude that depends on the deviation of the forcing or excitation frequency and natural frequency. Therefore, for a nonlinear oscillator, a shifting of the excitation frequency results in a variation of the natural frequency. Since the physical behavior of the HID lamp is highly nonlinear, a corresponding behavior is expected and this assumption is confirmed in Sec. III A.

A. Natural frequencies

Figure 5 shows how the natural frequency varies with the excitation frequency. Assuming weak damping, natural frequency and resonance frequency practically coincide. Therefore, when the excitation frequency is tuned to the natural frequency, resonance occurs. Resonance corresponds to a point on the diagonal of Fig. 5.

FIG. 5.

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Natural frequency vs. excitation frequency resulting from the FE simulations. The inlay shows an enlargement of the downward scan data for f_exc ≤ 46.50 kHz.

When approaching resonance by an upward scan of the excitation frequency f_exc, the natural frequency has an almost constant value just below the natural frequency of the mode of the linear acoustic system depicted in Fig. 3. This is to be expected, since far off the resonance, the amplitude of the sound wave is small and the influence of the nonlinearity can be neglected. Only when the excitation frequency is near the resonance line, the nonlinearity shows in the data by a small drop of the natural frequency. The simulations cannot be extended to higher frequencies, since the simulations do not converge anymore. From the lack of convergence, we conclude that near the resonances, the system is unstable. We estimated the resonance frequency f_res by a fit of the data to the function $a log (f_{res} - f_{exc}) + b$ and found f_res = 47.15 kHz. The values of the parameters a and b are not important here. Figure 5 shows that the fit to the simulation results is excellent.

When approaching the resonance by a downward scan of the excitation frequency, the natural frequency drops by ca. 1 kHz. In this case, it is possible to get very close to the resonance line before convergence gets lost. As expected, the data reveals that a variation of the excitation frequency results in a sizable shift of the natural frequency. At a first glance, it appears as if the drop of the natural frequency in the downward scan is described by a straight line. A close inspection of the inlay in the figure reveals that the curve near the resonance systematically bends toward the resonance line. It is not clear to us which function should be used to describe the data. Therefore, we tried to estimate the corresponding resonance frequency by visual inspection. Our estimate is that it lies in the range between 46.25 kHz and 46.27 kHz.

There is a second reason, why the description of the downward scan data by a straight line cannot be correct: As mentioned earlier, far off the resonance, the influence of the nonlinearity can be neglected. Therefore, an extrapolation of the data to higher excitation frequencies should result in an asymptotic approach to the black horizontal line representing the natural frequency of the linear acoustic model. Figure 5 depicts a straight line fit to the downward scan data for the excitation frequency range 46.65 kHz to 47.20 kHz in the middle of the complete downward scan frequency range. The deviation of the fit line and the data for f_exc > 47.20 kHz indicate the beginning of the asymptotic approach.

B. Backbone curves

Next, we consider the plane spanned by the axis of the natural frequency and the axis of the voltage drop. As mentioned above, the natural frequency of a nonlinear oscillator varies with the amplitude of the unforced oscillation. A graph which depicts the connection of natural frequency and amplitude is called the backbone curve (in a previous publication, we used a different definition of the term backbone curve;⁵ in the leading order of a perturbation expansion, the two definitions lead to identical expressions).¹⁴ The backbone curve of a linear oscillator is trivial: since the natural frequency is constant, it is a straight line parallel to the amplitude axis.

Almost 100 years ago, Georg Duffing investigated a certain type of idealized nonlinear oscillator.¹⁵ The restoring force of the Duffing oscillator of angular frequency ω₀ = 1 is, after rescaling, described by

F (x) = x + γ x^{3},

(1)

where x is the dimensionless deflection of the bob from the position of equilibrium and γ is a constant. For γ > 0, the spring stiffens with increasing deflection and for γ < 0, the spring becomes softer when it is stretched or compressed. If in the softening case the amplitude exceeds a critical value x_crit, the force F(x) changes its character from attractive (F(x) < 0) to repulsive (F(x) > 0), i.e., experiences a zero-crossing. (A practical example for this is a ship floating in the water. If the ship is heeled by an external influence and then released, the righting moment will result in an oscillation of the ship. When the angle of heel increases to large values, the righting moment becomes weaker. At the critical angle of heel, the righting moment changes its sign and the ship will capsize.) Formally, F(x) ≤ 0 corresponds to a vanishing natural frequency.

The left part of Fig. 6 depicts the backbone curve of a Duffing oscillator with a softening spring. The curve has been obtained by solving the differential equation

\ddot{x} + F (x) = 0

(2)

numerically for γ = –0.8. The curve ends once the amplitude reaches the critical value x_crit. The right part of the figure depicts the backbone curve obtained from the simulations of the HID lamp. The backbone curves of the two systems show a similar general behavior. The most prominent difference is the steepening of the HID backbone curve, which is not present in the case of the Duffing oscillator. The steepening at the low frequency end of the curve, indicating the presence of a stiffening effect, continues until the slope of the backbone curve is vertical. An inspection of the flow field reveals that the occurrence of steepening is connected to the dominance by AS over buoyancy.

FIG. 6.

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Left: backbone curve of a Duffing oscillator with a softening spring. The vertical blue line is the (trivial) backbone curve of the corresponding linear oscillator (γ = 0). The dotted horizontal line corresponds to the dimensionless critical amplitude x_crit. Right: backbone curve resulting from the simulations of the HID lamp. The vertical blue line is the (trivial) backbone curve resulting for Reynolds stresses equal to zero.

In a still idealized but more realistic model of an oscillator build from a softening spring and a bob (modified Duffing oscillator), the steepening can be understood in the following way: At large amplitudes of the unforced oscillations, the spring is stretched so far that the spring degenerates to a wire, respectively, the spring is compressed until neighboring coils come into contact. In both cases, the system experiences a dramatic stiffening. The force exerted on the bob at such large deflections is much larger than at smaller deflections. For the sake of simplicity, it can be assumed that the stretching and the behavior for compression can be described in identical ways, i.e., that F(–x) = –F(x).

The forces acting on the bob at large deflections are practically identical to the forces acting on an elastic ball reflected between two walls. The bob/the ball is accelerated near the points of reversal and moves freely (ball) or almost freely (bob) during the rest of their trajectory. As long as the materials can be considered to be elastic, the bouncing frequency can be increased by applying appropriate initial conditions. In the case of the bob, this is linked to an additional stretching of the wire/a quenching of the contacting coils and, therefore, a larger amplitude of the oscillation. Concerning the backbone curve, this results in a steepening until the slope is vertical. It seems natural to assume that corresponding saturation effects emerge in the case of the lamp and account for the steepening in Fig. 6.

C. Forced oscillations

Finally, we consider the plane spanned by the axis of the excitation frequency and the axis of the voltage drop. The left part of Fig. 7 shows the response of a Duffing oscillator with a softening spring. The frequency scans have been calculated by numerically solving the differential equation

\ddot{x} + 2 ζ \dot{x} + F (x) = x_{0} cos (Ω τ),

(3)

with γ = –0.8, ζ = 0.028, and x₀ = 0.06, and by varying the angular frequency Ω in small steps. In the upward scan, the system jumps at Ω₁ = 0.875 to a considerably larger value of the amplitude and in the downward scan to a smaller amplitude value at Ω₂ = 0.602, a behavior well known from the Duffing oscillator. The two frequencies f₁: = Ω₁/2π and f_2:= Ω₂/2π are called the jump frequencies. The different behaviors of the system in the up- and downward frequency scans exhibit the characteristic of a hysteresis.

FIG. 7.

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Left: response of a Duffing oscillator with a softening spring. The dotted horizontal line corresponds to the critical amplitude. Outside the hysteresis region, the up- and downward data are practically identical and, therefore, not discernible. At the jump frequencies, the response shows overshoots, which are related to the dynamics of the jumps. Right: results from the simulations of the HID lamp for the response of the voltage drop. The dotted vertical lines indicate the location of the resonance frequencies (Sec. III A).

The right part of Fig. 7 shows the results for the response of the voltage drop obtained from the simulations of the HID lamp. Again, strong similarities in the behavior of the Duffing oscillator and the HID lamp can be detected. The jump phenomenon offers an explanation for the light flicker effect. Obviously, no data from the upward frequency scan exists above the resonance frequency resulting from the fit in Sec. III A. The estimated boundaries of the range of resonance frequencies for the downward scan are also shown in the figure. Similar to the backbone curves, the upper branches of the response curves for the Duffing oscillator and the voltage drop exhibit a pronounced distinction: Near the low frequency end, the voltage drop curve steepens strongly, while the response of the Duffing oscillator does not show such a behavior. It is obvious that this steepening is related to the stiffening effect described in Sec. III B. In the experimentally measured voltage data, a very similar behavior compared to the one depicted in the right part of Fig. 7 has been found.⁵

The top part of Fig. 8 qualitatively depicts the response curve of a Duffing oscillator with a softening spring, which is obtained by applying a perturbation expansion.¹⁶ All frequencies below f₂ and above f₁ correspond to a unique value of the amplitude. However, in the range from above f₂ to below f₁, the amplitude can attain three different values. The branch of the response ranging from point A to point B represents an unstable solution. Therefore, the frequency range from f₂ to f₁ is called the interval of bi-stability. It is in the nature of the process that the unstable branch of the solution does not show in the numerical results depicted in Fig. 7.

FIG. 8.

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Top: response of the Duffing oscillator with softening spring.¹⁶ The dashed part of the curve corresponds to an unstable solution. Middle and bottom: tentative suggestions of response curves for the modified Duffing oscillator, respectively, the HID lamp.

At the jump points A and B, the tangent to the response curve is vertical. Each of these two points constitutes a bifurcation point of a cyclic fold bifurcation.¹⁷ It would be interesting to identify the type of the bifurcation at the jump points of the response of the HID lamp. However, the determination of the system's bifurcation type would require to analyze the eigenvalues of the Jacobian and we are not able to do that.

In the following, we assume that the two stable branches of the response curve are connected by a branch representing an unstable solution. Finding a way to connect the stable branches of the response of the HID lamp, this is not possible without complications due to the steepening. If we rule out a scenario with crossing branches, we only have two choices:

The most obvious way is schematically depicted in the middle part of Fig. 8. This scenario possesses one additional point of the response curve with a vertical slope. For the HID lamp behavior we rule out this scenario, because it should show in the simulation.
The second possibility to connect the two stable branches is depicted in the bottom part of Fig. 8. In this case, the response curve is not smooth. Instead, it has a singularity at the point of bifurcation. This seems to be the most likely scenario.

To our knowledge, no qualitatively different scenarios offer a consistent description of the observed behavior so that no other consequences as the described ones are possible for the behavior of the corresponding physical system. Unless we drop the assumption that the stable branches of the response are connected by an unstable branch and the unstable branch must not cross a stable branch, we do not see a way to avoid such a scenario.

IV. HYSTERESIS SUPPRESSION

The main purpose of our work on HID lamps is to generate ideas of how to prevent light flicker at high frequency lamp operation. The discovery of a hysteresis effect and the related jump phenomenon offers a natural explanation for the appearance of light flicker. By establishing a connection of the HID lamp and the Duffing oscillator, we additionally obtain access to the methods, which have been invented to counteract the emergence of the hysteresis.^18–20

For the Duffing oscillator with a softening spring, it has been shown that applying an additional harmonic excitation of a frequency much higher than the operation frequency can lead to an additional apparent stiffness of the spring [fast harmonic excitation (FHE)].¹⁸ By varying the FHE frequency, it is possible to increase the jump frequency f₂ of Sec. III C towards the jump frequency f₁ and above, thus making the hysteresis vanish. The interesting question is, if this effect could be utilized for the HID lamp as well, thereby offering a way to solve the light flicker problem by electronic means.

As discussed in Sec. III B, the backbone curve of the voltage drop in the HID lamp shows, in contrast to the response of the Duffing oscillator, a pronounced steepening at the high amplitude end, indicating a stiffening effect. The question is, how the system would react to FHE, when it is in a state corresponding to the steepening part of the backbone curve. (Fahsi et al. consider the influence of FHE on the response curve. We decided to concentrate on the backbone curve instead. The relation between the two descriptions has been outlined in Sec. III.) Two scenarios seem possible:

The stiffening and the related steepening are due to some sort of saturation as outlined at the end of Sec. III B. In that case, FHE should not lead to an additional stiffening effect at the large amplitude end of the backbone curve. It is to be expected that it is possible to find a FHE frequency resulting in a straight vertical backbone curve corresponding to a linear oscillator (Fig. 6, left).
The stiffening observed for the HID lamp is not related to saturation. Then the stiffening effect of FHE would also affect the system, when it is in a state associated with the large amplitude end of the backbone curve. The effect of FHE would be to rotate the backbone curve clockwise and, therefore, change the system to one with the characteristic of a hardening spring (corresponding to γ > 0, see Sec. III B). In this case, the hysteresis would be replaced by a different kind of hysteresis and nothing is gained.

The processes involved in the development of the hysteresis behavior in the HID lamp comprise a sequence of steps.⁵ To summarize: AS causes a large temperature gradient near the upper wall of the arc tube, which leads to enhanced emission of electromagnetic radiation. The related loss of power results in a decrease of the average temperature and, therefore, to a drop in the static pressure and the module of compressibility (for an ideal gas, the module of compressibility is equal to the static pressure²¹). In this way, the softening effect (decrease of compressibility) is caused by AS. It is certainly difficult to predict in which way FHE will affect the various processes involved in the formation of a hysteresis, but it still seems plausible that at some point saturation will set in. In that case, FHE might be a solution to the flicker problem.

V. CONCLUSIONS

A stationary 3D FE model for the simulation of the processes in the arc tube of a certain type of HID lamp has been set up. The aim was to obtain a better understanding of the light flicker phenomenon, which has worried the lighting industry for decades. In order to restrict the necessary computer resources, the model includes a simplified description of the plasma and the acoustic streaming effect. The results of the simulations reveal that the response of the driven system exhibits properties which strongly resemble those of the forced Duffing oscillator with a softening spring. In particular, a hysteresis and the jump phenomenon have been observed. However, the HID lamp exhibits features which do not have a counterpart in the Duffing oscillator. It seems that it is necessary to use a new type of response curve to describe the behavior of the HID lamp properly.

ACKNOWLEDGMENTS

This research was supported by the German Federal Ministry of Education and Research (BMBF) under project reference 03FH025PX2 and Philips Lighting.

APPENDIX: DIFFERENTIAL EQUATIONS AND DEFINITIONS

Electric charge conservation

\vec{\nabla} \cdot (σ \vec{\nabla} ϕ) = 0

Power density of heat generation

H = σ | \vec{\nabla} ϕ |^{2} - q_{rad}

Elenbaas-Heller equation

\vec{\nabla} \cdot (- κ \vec{\nabla} T) + ρ c_{p} \vec{u} \cdot \vec{\nabla} T = H

Mass conservation

\vec{\nabla} \cdot (ρ \vec{u}) = 0

Reynolds equation

ρ (\vec{u} \cdot \vec{\nabla}) \vec{u} = \vec{f} + \vec{\nabla} \cdot [- P I + η (\vec{\nabla} \vec{u} + {(\vec{\nabla} \vec{u})}^{T}) - \frac{2}{3} η (\vec{\nabla} \cdot \vec{u}) I]

Force density (including Reynolds stresses and buoyancy)

f_{l} = - \frac{1}{2} \frac{\partial ρ {\hat{v}}_{k}^{*} {\hat{v}}_{l}}{\partial x_{k}} - δ_{l 3} ρ g

Sound particle velocity

\vec{\hat{v}} = \frac{1}{i ω_{j} ρ} \vec{\nabla} p

Helmholtz equation

\vec{\nabla} (\frac{1}{ρ} \vec{\nabla} p) + \frac{ω^{2}}{ρ c^{2}} p = i ω \frac{γ - 1}{ρ c^{2}} H

σ is the electrical conductivity, ϕ is the electric potential, q_rad is the power loss due to radiation, κ is the thermal conductivity, T is the temperature, ρ is the density, c_p is the specific heat capacity at constant pressure, $\vec{u}$ is the mean velocity field, P is the static pressure, η is the viscosity, δ_l₃ is the Kronecker delta, g is the acceleration of gravity, ω_j is the natural angular frequency, p is the acoustic pressure, ω is the angular frequency, c is the speed of sound, and γ is the specific heat ratio.

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2017

Author(s)

High-intensity discharge lamp and Duffing oscillator—Similarities and differences

I. INTRODUCTION

II. MODELING

III. RESULTS

A. Natural frequencies

B. Backbone curves

C. Forced oscillations

IV. HYSTERESIS SUPPRESSION

V. CONCLUSIONS

ACKNOWLEDGMENTS

APPENDIX: DIFFERENTIAL EQUATIONS AND DEFINITIONS

References

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Electric charge conservation	$\vec{\nabla} \cdot (σ \vec{\nabla} ϕ) = 0$
Power density of heat generation	$H = σ \| \vec{\nabla} ϕ \|^{2} - q_{rad}$
Elenbaas-Heller equation	$\vec{\nabla} \cdot (- κ \vec{\nabla} T) + ρ c_{p} \vec{u} \cdot \vec{\nabla} T = H$
Mass conservation	$\vec{\nabla} \cdot (ρ \vec{u}) = 0$
Reynolds equation	$ρ (\vec{u} \cdot \vec{\nabla}) \vec{u} = \vec{f} + \vec{\nabla} \cdot [- P I + η (\vec{\nabla} \vec{u} + {(\vec{\nabla} \vec{u})}^{T}) - \frac{2}{3} η (\vec{\nabla} \cdot \vec{u}) I]$
Force density (including Reynolds stresses and buoyancy)	$f_{l} = - \frac{1}{2} \frac{\partial ρ {\hat{v}}_{k}^{*} {\hat{v}}_{l}}{\partial x_{k}} - δ_{l 3} ρ g$
Sound particle velocity	$\vec{\hat{v}} = \frac{1}{i ω_{j} ρ} \vec{\nabla} p$
Helmholtz equation	$\vec{\nabla} (\frac{1}{ρ} \vec{\nabla} p) + \frac{ω^{2}}{ρ c^{2}} p = i ω \frac{γ - 1}{ρ c^{2}} H$

High-intensity discharge lamp and Duffing oscillator—Similarities and differences

I. INTRODUCTION

II. MODELING

III. RESULTS

A. Natural frequencies

B. Backbone curves

C. Forced oscillations

IV. HYSTERESIS SUPPRESSION

V. CONCLUSIONS

ACKNOWLEDGMENTS

APPENDIX: DIFFERENTIAL EQUATIONS AND DEFINITIONS

References

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

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