One-to-two internal resonance in a micro-mechanical resonator with strong Duffing nonlinearity

This paper investigates the implementation of 1:2 internal resonance (InRes) in a clamped–clamped stepped beam resonator with a strong Duffing effect, focusing on its potential for frequency stabilization in micro-electro-mechanical systems (MEMS) resonators. InRes can arise in a nonlinear system of which mode frequencies are close to an integer ratio, facilitating the internal exchange of energy from an externally driven mode to an undriven mode. The presence of 1:2 InRes and Duffing hardening nonlinearity can result in frequency saturation phenomena, leading to a flat amplitude-frequency response range, which forms the basis for frequency stabilization. The stepped beam resonator design, combined with thermal frequency tuning, enables precise alteration of the frequency ratio between the second and third flexural modes required to achieve the desired 1:2 ratio for InRes. Experimental characterization and theoretical analysis revealed that frequency mismatch plays a significant role, with larger mismatch conditions leading to stronger energy exchange and a wider range of drive force for frequency saturation. The study highlights the frequency saturation mechanism utilizing 1:2 InRes and emphasizes the advantage of Duffing nonlinearity and larger intermodal frequency mismatch for broader frequency stabilization, providing valuable insights for the design and optimization of MEMS resonators.


Introduction
Micro-electro-mechanical systems (MEMS) are extensively utilized in sensing [1][2][3][4][5][6], timing [7][8][9], and energy harvesting [10][11][12][13] applications due to their miniaturized size, ultra-high frequency, and low power consumption.In these applications, MEMS resonators are typically designed to operate linearly Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
(figure 1(a)), aiming to minimize the adverse effects of non-linear noise mixing and simplify the associated control methods [14].However, due to the inherently low damping in MEMS resonators, nonlinearity often inevitably exists, leading to frequency fluctuations associated with the non-linear amplitude-frequency dependence (figure 1(b)).As a result, there has been significant interest in suppressing the nonlinearity of MEMS resonators to enhance their frequency stability [15][16][17][18][19]. Recently, researchers have proposed utilizing the non-linear phenomenon of internal resonance (InRes) as a new approach to stabilizing frequency fluctuations [20][21][22][23][24][25][26][27][28].In a non-linear system, InRes can arise in an undriven vibrational mode by internally transferring energy from another vibrational mode that is externally driven when the two modal frequencies are commensurable into an m:n ratio (where m and n are integers) [29,30].This non-linear intermodal interaction and internal energy exchange provide a foundation for stabilizing the mechanical fluctuation of the resonant frequency.
In recent years, advancements in InRes research have led to significant progress in understanding the underlying mechanisms of frequency stabilization.The pioneering experimental study by Antonio et al [22] demonstrated the use of 1:3 InRes to reduce frequency fluctuation, sparking further exploration in theoretical and experimental studies exploring 1:2 and 1:3 InRes for frequency stabilization [20][21][22][25][26][27][28].Even though the direct comparison between different MEMS designs is not always feasible, it appears that 1:3 InRes can achieve better stabilization than 1:2 InRes.Specifically, it has been observed that 1:3 InRes can reduce Allan deviation by 2-3 orders of magnitude, whereas 1:2 InRes can only achieve a reduction of one order of magnitude.The reason behind this performance difference can be explained by recent publications [26,27], which provide profound insight into the frequency stabilization mechanism for 1:3 InRes.Here, it was highlighted that the zero-dispersion (ZD) point, characterized by a vertical tangency in the frequency-amplitude relation on the Duffing backbone curve (refer to figure 1(c)), plays a crucial role in stabilizing frequency fluctuation attributed to the typical nonlinear frequency-amplitude dependence.At the ZD point, the Duffing hardening effect is counteracted by the softening effect attributed to the 1:3 InRes, mitigating the dispersion of frequency and the amplitudefrequency effect.Although previous studies on 1:3 InRes systems have explored the ZD effect, investigations of 1:2 InRes for frequency stabilization [20,21,25,28] did not involve a strong Duffing non-linear effect, leading to typical M-shaped resonance curves (figure 1(d)) without the ZD effect.
A diverse range of studies explores the non-linear dynamics of 1:2 InRes in micro/nano resonators.Most of these studies have demonstrated the M-shaped resonance curve with strong quadratic coupling compared to the Duffing nonlinearity.For instance, U-shaped resonators [31,32], disk resonators [14], arch beam resonators [33], and clamped-clamped beam resonators [20,21,28,34] have experimentally shown Mshaped resonances induced by 1:2 or 2:1 InRes.While these studies have contributed to understanding 1:2 and 2:1 InRes, the ZD point was not demonstrated due to the relatively small hardening effect.Limited research has been conducted on systems with 1:2 InRes and strong Duffing nonlinearity.For example, Kes ¸kekler et al [35] demonstrated the ZD point with 2:1 InRes in a graphene nano-mechanical resonator due to graphene's strong hardening effect and interpreted this as tailoring non-linear damping using 2:1 InRes.Ruzziconi et al [36] analyzed the 2:1 InRes dynamics of a microbeam resonator with cubic coupling and hardening nonlinearity in the presence of quadratic nonlinearity from electrostatic actuation.Although these studies on 2:1 InRes explored the Duffing hardening effect, a comprehensive analysis of the ZD effect in a 1:2 or 2:1 InRes systems for frequency stabilization has not been conducted.
In this current investigation, we focus on systems with 1:2 InRes and a strong Duffing effect, with particular emphasis given to the ZD point in relation to intermodal mismatch, aiming to deepen our understanding of frequency stabilization.Building upon our previous study [34], we designed and fabricated a clamped-clamped stepped beam resonator to demonstrate the 1:2 InRes with strong Duffing nonlinearity.Our prior study showed that a thin-layer stepped-beam resonator exhibits a 1:2 and 2:1 InRes with strong quadratic coupling and weak Duffing nonlinearity [28].However, its thinlayer structure with a 500 nm thickness limited its quality factor (Q-factor), and the design for out-of-plane vibration was not feasible for electrostatic transduction.Therefore, we modified the resonator design to feature in-plane vibration and increased the thickness to 20 µm, resulting in a higher Qfactor.The thicker structure also introduced stronger Duffing nonlinearity due to the increased axial stretching during vibrations.As a result, the dominant nonlinearities of the new resonator in this study comprise non-linear intermodal quadratic coupling and cubic Duffing effects.Increasing the thickness of the stepped beam reduced the effectiveness of electrostatic tuning, so we used the Joule heating method to compensate for fabrication variances and obtain the 1:2 ratio between flexural modes.We experimentally studied this system using precise thermal tuning to investigate the effects of intermodal mismatch on the InRes behavior and investigated the frequency saturation phenomena associated with the ZD point.We also developed a theoretical model to understand the non-linear dynamics of the resonator with 1:2 InRes, with a particular emphasis on the effect of intermodal mismatch on the ZD point.

Materials and methods
We propose implementing InRes in a stepped beam resonator with a clamped-clamped configuration.The clampedclamped beam resonator is favored in MEMS applications due to its simple modeling, ease of fabrication, and accessibility of vibrational modes compared to unconventional designs like Uand H-shaped resonators [31,37].Moreover, the stepped beam structure is suitable for satisfying the 1:2 commensurate condition between the second and third flexural modes, where the strongest coupling occurs via mid-plane stretching provided by the constrained clamped-clamped boundary condition [34].This design offers the flexibility to systematically adjust geometric parameters, such as relative length and width, to achieve the integer frequency ratio between the modes, unlike conventional prismatic beam designs with fixed non-integer frequency ratios.The dimensions of the stepped beam resonator were determined by finite element simulation with COMSOL, deliberately chosen to attain the desired 1:2 ratio between the second and third mode frequencies: the length of thick section (L 1 ) and thin section (L 2 ), depth of beam (b), width of thick section (h 1 ) and thin section (h 2 ) of the silicon microbeam are as follows: The fabrication of the stepped beam resonator followed conventional MEMS manufacturing techniques.The resonator structure was patterned on a silicon-on-insulator wafer using photolithography.The pattern was transferred onto the silicon device layer of the wafer using deep reactive ion etching to etch away the sacrificial areas selectively.The oxide layer was then etched using hydrofluoric acid wet etching to release the stepped beam structure.After the etching process, a thin film of gold was deposited onto the resonator's surface to enhance the conductivity of the Joule heating process.
Figure 2(a) depicts the optical layout image of the fabricated stepped beam resonator, while figure 2(b) shows a scanning electron micrograph (SEM) displaying the beam's crosssection suspended in the center adjacent to the driving electrodes.The resonator consists of eight electrodes, with six driving electrodes (three electrodes on the top and three electrodes on the bottom of the beam) and two DC-bias electrodes on the sides of the beam.To effectively drive the resonator's second and third flexural modes, two out of the six driving electrodes are utilized, while the remaining four of the driving electrodes remain floating.Specifically, the electrodes AC 0 and AC 2 were used to excite the second mode, while electrodes AC 0 and AC 3 were employed to excite the third mode.The variation in driving electrode position is due to the distinct mode shapes and nodal positions of the second and third flexural modes.The two electrodes on either side of the resonator anchor the stepped beam and apply a DC-bias voltage to facilitate Joule heating by supplying an electric current.
The experimental setup used to actuate, detect, and thermally tune the flexural modes of the resonator is illustrated in figure 2(c).The resonator was positioned within a vacuum probe station (Nextron MPS-PT) with an absolute pressure of 5 mTorr to minimize energy loss from air damping.The DC+ and DC− electrodes of the resonator were connected to the positive and negative terminals of the DC power supply, respectively.Additionally, a variable resistor was connected to the DC− electrode, allowing for control over the current passing through the beam, which enabled the regulation of the Joule heating temperature.An AC signal generated by the lock-in amplifier (Zurich Instruments H2FLI) was applied to the driving electrodes to actuate the resonator.It was swept across a range of frequencies encompassing the second flexural mode.A laser Doppler vibrometer (LDV) (Polytec OFV-534), of which laser spot was directed onto the beam through a 50x objective lens, captured the resonator's in-plane motion due to a subtle tilt angle of the beam [38].Finally, the LDV output signal was measured by the lock-in amplifier at each drive frequency.
The beam was thermally tuned using the Joule heating method to compensate for fabrication variances and obtain the desired 1:2 ratio between the second and third flexural modes.This method involved applying a voltage across the beam to induce a current, resulting in thermal expansion and the development of compressive stresses due to the constrained boundary conditions.These compressive stresses reduce the mode frequencies of the beam according to the strain-dependent correction for the Euler-Bernoulli beam theory.During the experiment, the DC voltage applied to the resonator was varied to precisely control the frequency ratio between the second and third flexural modes, aiming for a ratio varied around 1:2. Figures 3(a) and (b) show the decrease in the frequency of the second and third flexural modes as the DC voltage across the stepped beam increases.We also simulated the modal frequencies adjusted by thermal voltage using COMSOL, as shown by the red curves in figure 3(c), and the results align well with experimental results for the first three flexural modes.Figure 3(d) depicts third to second modal frequency ratios for each DC bias voltage.The frequency ratio smoothly increased from 1.985 to 2.025, and the exact 1:2 ratio was achieved when the DC voltage was at 16.3 V.

Results
We conducted in-depth investigations into the non-linear behavior of a resonator exhibiting a strong hardening effect while ensuring the commensurability between the resonant frequencies.Specifically, we examined the impact of intermodal mismatch on this nonlinear behavior by precisely tuning the frequency to be near the 1:2 ratio.For this purpose, we define the intermodal mismatch as σ (ω 2 = 2ω 1 − εσ) where ω 1 is the lower mode (second mode) frequency, ω 2 is the higher mode (third mode) frequency, and ε is a small scaling term.To obtain the forced response curve (FRC), we obtained the resonant amplitudes when the driving frequency was swept around the second mode.We repeated this characterization for different drive amplitudes from 100 mV to 600 mV with a step of 10 mV.The resulting color maps and line graphs in figure 4 show the FRC with varied drive amplitudes and compare the results at different intermodal mismatch values.
When the mode frequency ratio was slightly below 2.0000, specifically at 1:1.9995 (cf. Figure 4(a)), a typical hardening resonance was observed.Similarly, when the mode frequency ratio had an integer relationship, i.e. at 1:2.0000 (cf. Figure 4(b)), the same hardening Duffing backbone curve was observed, with no discernible differences between the 1:1.9995 and 1:2.0000 frequency ratio.In both of these cases, the InRes did not significantly influence the Duffing hardening behavior because the hardening effect resulted in relatively small resonant amplitudes at the frequencies where the exact 1:2 ratio is satisfied (i.e. when the drive frequency is half of the higher mode frequency, Ω ≈ ω 2 /2).To gain deeper insights into this phenomenon, we conducted further investigations using the theoretical model, which will be discussed in detail later.
To further study the effect of mistuning, we conducted sweeps with the mode frequency ratios set to 2.0004, 2.0005, and 2.0007.The characterization process involved increasing the driving amplitude from 100 mV to 700 mV in increments of 10 mV, as shown in figures 4(c)-(e).The results reveal the phenomenon of frequency saturation, where a drop-jump frequency remains pinned at a specific value within a certain range of driving amplitudes before transitioning to follow the typical Duffing backbone.Notably, increasing the frequency ratio (i.e., increasing intermodal frequency mismatch) led to a more pronounced saturation effect over a larger amplitude range.For instance, when the frequency ratio was 2.0004, the resonator's frequency remained saturated between 330 and 420 mV.At a higher frequency ratio of 2.0005, the frequency saturation occurred between 350 mV and 590 mV.Further increasing the frequency ratio to 2.0007 resulted in saturation within a driving voltage range between 370 mV and 610 mV.
Several studies have predicted and measured similar saturation behavior in jump-down frequency for the cases of 1:3 InRes [22,27,[39][40][41].Specifically, measurements presented by Antonio et al [22] demonstrated frequency saturation as the forcing amplitude increased.This measured behavior was attributed to the energy transfer to higher mode due to InRes, causing the upper branch solution to become unstable in the process yielding a jump-down.When externally pumped energy into the system is sufficiently large to overcome this InRes effect, the response passes through that range (Ω ≈ ω 2 /3 for 1:3 InRes, where Ω is the drive frequency) and follows the hardening Duffing curve.Detailed theoretical investigations presented in [27] further revealed that the frequency does not behave monotonically with vibration energy when approaching the InRes zone Ω → ω − 2 /3; instead, 1:3 InRes imposes a ZD of frequency in the FRC, effectively separating the stable upper branch of the FRC into two segments.As such, the amplitude-frequency effect is eliminated in the vicinity of ZD.Crossing through the InRes range results in Hopf bifurcations (HBs), in which amplitude modulations between two modes occur.However, beyond this threshold of forcing amplitude, the response follows a typical Duffing curve with a strong amplitude-frequency effect, thereby limiting the frequency stabilization effect of InRes.
Observations made in figure 4 suggested that mistuning plays a significant role in influencing the effective forcing amplitude range of frequency saturation (i.e.frequency stabilization) with 1:2 InRes.There also seems to be a threshold of the forcing value beyond which the response of the lower frequency mode follows the hardening backbone without jumping down to the lower branch of FRC.To examine the mechanism causing both the frequency saturation and the threshold of the forcing amplitude in the case of 1:2 InRes, we study the InRes problem through a two-degrees-of-freedom reduced order model with cubic nonlinearity and quadratic coupling terms that promote 1:2 InRes when the lower frequency mode (u 1 ) is harmonically driven with the forcing amplitude F and forcing frequency Ω.The equation of motion of two modes (u 1 , u 2 ) is written in the following form: Here ω i and ζ i are the linear natural frequency and damping ratio of mode-i, respectively, and γ 1 is the positive coefficient of the stiffening (i.e.hardening) term representing the midplane stretch of the lower frequency mode while α is the coupling term between two modes with coupling potential being It is important to note that the coupling of two vibrational modes also originates from the geometric effect through the midplane stretch [34].A steady-state solution to equation ( 1) for harmonic excitation F sin(Ωt) can be assumed in the single-harmonic form: where U 1 , U 2 , ϕ 1 , and ϕ 2 , are steady-state amplitude and phase of the first and second mode, respectively.Substituting equation ( 2) into equation ( 1) and balancing the corresponding harmonic terms yields the following set of non-linear algebraic equations: (3c,d) Combining equation (3b) utilizing sin ϕ 1 2 + cos ϕ 1 2 = 1, we obtain the frequency-amplitude relationship as The response amplitude of the first mode U 1 can be obtained by solving equations (3c,d) and (4) numerically.The stability of the steady-state solutions can be assessed by introducing perturbations v i (t) around the periodic solution u * i (t) such that u i = u * i (t) + v i (t) and evaluating the Floquet multipliers through monodromy matrix.In the presence of both cubic and modal coupling terms, two types of bifurcations are possible [27,[39][40][41]] (i) saddle-node (SN) bifurcations of typical Duffing curve when Floquet multipliers leave the unit circle at the real axis, (ii) HBs with two imaginary multipliers outside of the unit circle indicating quasi-periodic oscillations with amplitude modulations [29].
In figure 5, we present the FRCs at four intermodal mismatch conditions ω 2 = 1.9995, 2.0000, 2.0005, and 2.0012 and various forcing amplitudes F ⊂ [3,15] , and α = 0.02. Figure 5(a) corresponds to the case of negative mistuning (1:1.9995).In this scenario, the response follows the hardening curve with a small hump around the exact 1:2 frequency ratio (i.e.Ω ≈ ω 2 /2), and it exhibits strong amplitude-frequency dependence and SN bifurcations (red squares) due to the cubic nonlinearity term γ 1 .For the perfectly commensurable conditions of 1:2.0000, the response remains close to the hardening type Duffing curve except for HB taking place in the immediate vicinity of Ω ≈ ω 2 /2.Also noted here are the response amplitudes where the InRes is triggered.However, at these low energy levels at Ω ≈ ω 2 /2, the coupling between the modes is weak, which agrees with the measurements shown in figures 3(a) and (b) as there is no sign of frequency saturation in both predictions and measurements for ω 2 = 1.9995, 2.0000.
When the frequency mismatch is 1:2.0005 in figure 5(c), the intermodal coupling is seen to separate the FRC for F = 3 × 10 −6 at Ω ≈ ω 2 /2, yielding an isolated upper branch, which is referred to as an isola [42].The enlarged view of this curve in figure 5(d) illustrates the frequency gap imposed by the InRes around Ω ≈ ω 2 /2.This isolated solution branch would not be observed in sweep-up experiments as the system is likely to jump down to the lower branch at Ω ≈ ω 2 /2.In other words, until this isolated branch merges with the main curve, increasing the forcing amplitude will not significantly alter the jump-down frequency, resulting in the frequency saturation phenomena.For increased forcing amplitudes F ⩾ 6 × 10 −6 , the isola shown in figure 5(d) merges with the main curve exhibiting HB after passing through Ω ≈ ω 2 /2.Once this merge occurs, the upper branch of the FRC is very similar to those in figure 5(a), with strong amplitude-frequency dependence except for the small dip and HB.
For larger mismatching conditions, InRes is triggered with higher response amplitudes U 1 , resulting in a stronger energy exchange to the second mode, as shown in figures 5(c) and (d).Specifically, InRes separates FRCs F ⩽ 12 × 10 −6 when ω 2 = 2.0012 while forcing amplitude should be increased to 15 × 10 −6 for the isolated branch to merge with the main curve.This indicates that larger forcing amplitudes are required to merge two response curves at larger mismatching values.Additionally, it is noteworthy that the frequency gap for these isolated curves leads to the frequency saturation phenomena at Ω ≈ ω 2 /2, which is effective for frequency saturation.These theoretical findings qualitatively capture the measured behavior in figures 3(c)-(e).
We next consider backbone curves to better demonstrate this saturation behavior and the influence of frequency mismatch.The frequency Ω p at which the peak amplitude is reached can be implicitly obtained by setting ϕ 1 to π /2 in equation (3a): Here, cubic hardening nonlinearity γ 1 is seen to always increase the frequency Ω p .It is also noted here that the coupling term imposes a softening effect when Ω p < ω 2 /2.In the vicinity of InRes, this softening effect can balance the hardening term eliminating the amplitude-frequency dependency for a range of forcing amplitude F [27,43,44].To demonstrate this, figure 6 shows Ω p with respect to F at various mismatch conditions.Parallels can be drawn between the dashed lines of color plots shown in figure 4 and stable curves of figure 6.Here, a horizontal line corresponds to a linear response without any amplitude-frequency dependency denoted by a dasheddotted line.
The red line representing the perfect mismatch condition ω 2 = 2 displays Duffing amplitude-frequency dependency with no sign of saturation.This agrees well with the measured jump-down frequencies traced by a dashed line in the color plot of figure 4(a).When ω 2 = 2.0005 in figure 6(a), InRes yield frequency saturation at Ω p ≈ ω 2 /2, forming horizontal lines for a range of F ⊂ [2, 5.4] × 10 −6 .This range is important for frequency stabilization as in this flat frequency range (i.e.F ⊂ [0.6, 2.4] × 10 −5 for ω 2 = 2.0015), any fluctuation in forcing amplitude would not result in fluctuation of frequency Ω p .Further increase in F causes a jump to the hardening Duffing curve, where the frequency is highly dependent on the vibrational energy.A similar jump in Ω p was observed in experiments, as shown in figure 4(d), around a driving voltage of 600 mV.The range of F values covered in the stabilization region becomes larger with increasing σ 2 as demonstrated in figure 6(b), again qualitatively agreeing with the observation made in figures 4(c)-(e).This can be attributed to the stronger interactions occurring at higher energy levels.Hence, a larger mismatch proves advantageous for frequency stabilization, extending the stabilization range across a broader excitation span of F. Nevertheless, a noteworthy point is that the minimum required excitation force, F, for stabilization increases with ω 2 .This could pose a limitation, as the stabilization range must encompass the nominal operating excitation amplitude.If the forcing amplitude is not large enough for the upper branch to reach the InRes frequency zone (e.g.F < 0.5 × 10 −5 for ω 2 = 2.0015), the forced response adheres to the typical Duffing curve with no intermodal interactions or frequency stabilization.

Conclusion
In conclusion, our investigation focused on implementing InRes in a clamped-clamped stepped beam resonator with a strong Duffing effect, aiming to explore its potential for frequency stabilization.Through experimental characterization and theoretical analysis, several key findings were observed.The presence of Duffing hardening nonlinearity and quadratic coupling between the modes played a crucial role in determining the resonator's response.The softening effect of the coupling term near InRes balanced the hardening effect, resulting in a flat amplitude-frequency response range (i.e.ZD), contributing to frequency stabilization within a certain excitation range.This ZD range is shown as a frequency saturation phenomenon in the experiment, characterized by the pinned jump-down frequency within a certain forcing amplitude range, which was observed at specific intermodal mismatch conditions.The intermodal frequency mismatch played a significant role in determining the effective range of force for this frequency saturation.For the exact or negative intermodal mistuning, due to the low energy at Ω p ≈ ω 2 /2 attributed to the hardening effect, the resonator exhibited typical hardening behavior without frequency saturation.However, as the mismatch increased, the InRes effect became more prominent, resulting in frequency saturation at specific forcing amplitudes.Overall, the larger mismatch conditions led to stronger energy exchange and a broader range of effective force.
The combined experimental results and theoretical analysis provided valuable insights into the complex nonlinear behavior of the clamped-clamped stepped beam resonator with InRes.The study highlights the potential of 1:2 InRes as a mechanism for achieving frequency stabilization in micro/nano resonators, and it opens avenues for further research in the field of non-linear dynamics and MEMS applications.Overall, this work contributes to a deeper understanding of InRes dynamics and offers valuable guidance for designing and optimizing frequency-stabilized MEMS, holding immense promise for advancing the design and performance of future MEMS resonators and related applications.

Figure 1 .
Figure 1.Representative frequency response curves illustrating (a) a linear resonator, (b) a Duffing resonator without internal resonance (InRes), (c) a Duffing resonator with 1:3 InRes, and (d) a resonator with 1:2 InRes without the Duffing effect.The solid blue lines represent stable branches, dashed blue lines represents unstable branches, and black dash-dot lines depict backbone curves for cases without damping.

Figure 2 .
Figure 2. (a) Optical micrograph showing the system composed of a stepped beam, two DC electrodes, and six AC electrodes.(b) SEM image displaying the cross-section of the beam.The arrow in (a) and (b) indicates the direction of oscillation.(c) Experimental schematic utilized for actuating, sensing, and thermally tuning the mechanical motions of the resonator.

Figure 3 .
Figure 3. Experimentally obtained mode frequencies of the second (a) and third (c) flexural modes with varying DC voltages across the resonator.The experimentally measured mode frequencies are verified with the FEM simulation results in (c).(d) Shows the ratio of third to second flexural mode frequencies with respect to the DC voltages.