Phase Response Error Analysis in Dynamic Testing of Electric Drivetrains: Effects of Measurement Parameters

Abstract

The development of NVH (Noise, Vibration, and Harshness) characteristics in vehicles is facing new challenges with the widespread utilization of electric drivetrains. This shift introduces new requirements in several areas, such as reduced noise and vibration levels, the need for advanced nonlinear characterization methods, and tuning/masking the typically more prominent tonal noise components. More accurate simulation and measurement techniques are essential to meet these demands. This study focuses on the experimental frequency response function (FRF) testing of electric drivetrain components, specifically on potential phase errors caused by inappropriate measurement settings. The influencing parameters and their quantitative effects are analyzed theoretically and demonstrated using real measurement data. A novel numerical approach, termed Maximum Phase Error Analysis (MPEA), is introduced to systematically quantify the largest potential phase errors due to arbitrary alignment between resonance frequencies and discrete spectral lines. MPEA enhances the robustness of phase accuracy assessment, especially critical for lightly damped systems and closely spaced resonance peaks. Based on the findings, optimal testing parameters are proposed to ensure phase errors remain within a predefined limit. The results can be applied in various dynamic testing scenarios, including durability testing and rattling analysis.

1. Introduction

Electric drivetrains in automotive engineering have fundamentally changed the field of Noise, Vibration, and Harshness (NVH). Unlike conventional internal combustion engines—generating noise that is often less prominent and low-frequency and can be effectively managed by engine and muffler acoustics—electric motors often produce a tonal whining sound at discrete harmonics of their rotational speed, as well as high-frequency vibration from inverter switching and gearbox interactions [1]. Consequently, NVH engineers face new challenges: not only must overall vibration and noise levels be reduced [2], but the characterization and mitigation of narrow-band tonal components become critical for passenger comfort and perceived quality [3]. At the core of many NVH analyses lies the Frequency Response Function (FRF), which describes how a structure or component responds in amplitude and phase across a range of harmonic excitation frequencies. Accurate FRF measurement is essential for modal parameter identification, component tuning, and subsequent numerical model updating. However, the accuracy of the measured FRF—particularly its phase content—depends sensitively on experimental settings such as sampling frequency, record length, windowing, and spectral averaging. A well-known source of error in FRF measurement is inadequate frequency resolution of the discrete Fourier transform (DFT). When the frequency grid is too coarse relative to the bandwidth of resonance peaks, both amplitude and phase estimates at or near resonance can become distorted [4]. Amplitude errors manifest as under- or overestimation of the peak magnitude, leading to biased estimates of modal damping and modal mass. These phenomena have been discussed in our earlier works [5,6]. The present work focuses on phase response artifacts, which arise from “bin-centering” errors, where the true resonant frequency—or any other frequency of interest—lies between discrete spectral lines, causing a calculated phase that deviates from the ideal.

Even though vibration amplitude is generally considered the main reason for NVH issues, phase relations play a significant role in numerous phenomena, e.g., in rattling or fatigue loading. Consequently, accurate phase information is inevitable for reliable predictions in these fields [7,8].

2. Theoretical Background

In Chapter A5.1 of [9], Lalanne discusses the errors introduced during the digitization of transfer functions, specifically emphasizing measurements near resonance frequencies. Lalanne shows that precise frequency resolution (∆ω) near resonance is crucial to accurately determine the resonance parameters. Insufficient sampling density near resonance peaks (illustrated in Figure 1) will result in substantial errors in determined frequency and damping values.

Figure 1. Amplitude response around resonance with fine (a) and coarse frequency resolution (b). Change in apparent peak frequency and bandwidth is demonstrated.

Studying a 2-DOF (degree of freedom) system allows one to demonstrate digitalization inaccuracies in the determination of resonance parameters: f_n = ω_n/2π natural frequency and Q = 1/2ξ quality factor, ξ being the damping ratio. Such low-DOF analytical models help reveal the error mechanisms and identify the role of dynamic parameters in them, whereas numerical methods (e.g., the MPEA presented in Maximum Phase Error Analysis (MPEA) Section) are effective tools for quantifying the errors in real, multi-DOF systems.

Inadequate frequency resolution affects both amplitude and phase estimations, as shown in Figure 2. When inserting the determined, inaccurate damping value into simulations as an empirical system parameter, erroneous amplitude and phase prediction will be the consequence—see Figure 3.

Figure 2. Effect of frequency resolution on the amplitude and phase accuracy of Frequency Response Functions (FRFs) of a 2-DOF system: (a) amplitude comparison; (b) amplitude error relative to the finest resolution; (c) phase comparison; (d) phase error relative to the finest resolution.

Figure 3. Effect of damping ratio (ξ) on the amplitude and phase of Frequency Response Functions (FRFs) of a 2-DOF system: (a) amplitude comparison; (b) amplitude deviation relative to ξ = 0.005; (c) phase comparison; (d) phase deviation relative to ξ = 0.005.

The present work collects and evaluates multiple methods to handle and limit phase errors. An analysis of potential amplitude errors has already been conducted in our previous works [5,6].

2.1. Calculation of Phase Error at Resonance Frequency

In the following, the theoretical derivation of phase error due to inappropriate frequency sampling will be presented. The results are then applied and evaluated on real measurement data. Based on the findings, a method is proposed for determining frequency resolution to keep phase error below a limit value.

Let us focus first on a 1-DOF system’s phase response near resonance (shown graphically in Figure 4):

$$\varphi \left(\omega \right)=arctan\left(-{\displaystyle \frac{2\xi {\omega }_n\omega }{{\omega }_n^2-{\omega }^2}}\right)$$

(1)

where ξ is the damping ratio, ω is the actual (angular) frequency, and ω_n is the natural (angular) frequency. Note: numerically, this function gives values between −π/2 … +π/2. To obtain the real, physical phase values, in case of frequencies above resonance, the result should be shifted by π [10]. Furthermore, the negative sign depends on the interpretation of phase response (as a “phase delay”, it should be positive), and on which physical value we consider (displacement, velocity, acceleration, etc.).

Figure 4. (a) Phase response of a 1-DOF system as a function of frequency ratio for different damping values [11]; (b) the visualization of phase error due to discrete frequency resolution.

Considering the resonance case:

$$\varphi \left(\omega ={\omega }_n\right)=\mathit{arctan}\left(\infty \right)=\pi /2$$

(2)

Mathematically, Equation (2) is true when we approximate resonance from above, i.e., ω > ω_n. Approximation from below results in ϕ(ω = ω_n) = −π/2.

For small deviations, let ω = ω_n + ε_ω, the last term being the frequency error (or deviation) coming from the difference between the discretized and real resonance frequency (see Figure 4). In that case, the phase response is as follows:

$$\varphi \left({\omega }_n+{\epsilon }_{\omega }\right)=arctan\left(-{\displaystyle \frac{2\xi {\omega }_n\left({\omega }_n+{\epsilon }_{\omega }\right)}{{\omega }_n^2-{({\omega }_n+{\epsilon }_{\omega })}^2}}\right)=arctan\left({\displaystyle \frac{2\xi {\omega }_n\left({\omega }_n+{\epsilon }_{\omega }\right)}{2{\omega }_n{\epsilon }_{\omega }{+\left({\epsilon }_{\omega }\right)}^2}}\right)$$

(3)

Equation (3) precisely describes the phase of a single-degree-of-freedom (1-DOF) system when the excitation frequency is slightly offset from the natural frequency. It accounts for both the damping ratio ξ and the frequency deviation ε_ω.

Finally, the (discrete) phase error (ε_ϕ,d)—meaning the deviation between the phase value at natural frequency (ω = ω_n) and the phase value at the next discrete frequency value (upper neighbor: ω = ω_n + ε_ω—see Figure 4)—can be expressed as follows:

$${\epsilon }_{\varphi ,d}=\varphi \left({\omega }_n+{\epsilon }_{\omega }\right)-\varphi \left({\omega }_n\right)$$

(4)

Inserting Equations (2) and (3) into (4), the following can be obtained:

$${\epsilon }_{\varphi ,d}=arctan\left({\displaystyle \frac{2\xi {\omega }_n\left({\omega }_n+{\epsilon }_{\omega }\right)}{2{\omega }_n{\epsilon }_{\omega }{+\left({\epsilon }_{\omega }\right)}^2}}\right)-{\displaystyle \frac{\pi }{2}}$$

(5)

Using a dimensionless parameter named specific frequency error of r = ε_ω/ω_n, one can obtain the relative phase error near resonance based on (5) as follows:

$${\epsilon }_{\varphi ,d}\left(r\right)=arctan\left({\displaystyle \frac{2\xi \left(1+r\right)}{2r+r^2}}\right)-{\displaystyle \frac{\pi }{2}}$$

(6)

Equation (6) gives the phase result if the actual frequency is slightly above the natural frequency. The derivation can be executed analogously for frequencies below the natural frequency, i.e., ω = ω_n − ε_ω. Without giving the details, the phase error for that case will be as follows:

$${\epsilon }_{\varphi ,d}\left(r\right)=arctan\left({\displaystyle \frac{2\xi \left(1-r\right)}{r^2-2r}}\right)+{\displaystyle \frac{\pi }{2}}$$

(7)

Absolute values of Equations (6) and (7) are plotted in Figure 5. One can conclude that—due to the asymmetrical characteristic of phase response (see Figure 4)—the phase error will be slightly different for frequencies above and below the natural frequency, even with the same frequency error (ε_ω). Phase error is more sensitive to negative frequency error; however, this difference is considerable only with high damping and large frequency error.

Figure 5. Phase error of a 1-DOF system as a function of normalized frequency error for different damping values—considering discrete frequency lines.

Using Figure 5, one can define the limiting frequency resolution so as to keep the discrete phase error below a pre-defined threshold value for a single resonance peak with given damping.

2.2. Phase Error via Interpolation

The above derivation results in the phase error between the theoretical natural frequency and the nearest frequency line of the discrete frequency sampling. There are, however, two reasons for further analysis:

• On the one hand, most evaluation software is capable of calculating interpolated results between two neighboring frequency samples—to get a more accurate phase response even at the exact resonance frequency or at any other frequency.
• On the other hand, we can assume that phase error will not reach its maximum at resonance, but slightly away from it—considering the curvature characteristics in Figure 4 (i.e., at resonance, the inflexion point will result in a smaller chord distance than at highly curved regions below and above). This assumption will be proven in this chapter.

Based on these considerations, the estimation of the interpolated phase error (ε_ϕ,int—see Figure 4) of a 1-DOF system will be given in the following, using the phase values available at the two nearest frequ––ency bins, ω₁ and ω₂ (ω₁ ≤ ω < ω₂, ω₂ − ω₁ = Δω, the latter being the frequency resolution). This value can be calculated not only at the natural frequency, but at any other frequency—as it is shown later.

The exact phase response at ω frequency (ϕ(ω)) can be calculated by Equation (1). The linearly interpolated phase value at ω frequency—calculated from ϕ₁ = ϕ(ω₁) and ϕ₂ = ϕ(ω₂) discrete phase responses—can be given as follows:

$${\varphi }_{int}(\omega )={\varphi }_1+{\displaystyle \frac{\omega -{\omega }_1}{{\omega }_2-{\omega }_1}}\left({{\varphi }_2-\varphi }_1\right)$$

(8)

Finally, the interpolated phase error is given as the difference in the exact and the interpolated phase responses:

$${\epsilon }_{\varphi ,int}\left(\omega \right)=\varphi \left(\omega \right)-{\varphi }_{int}\left(\omega \right)$$

(9)

Aiming for higher accuracy, from this point on, the given/calculated phase error always refers to the interpolated value, and not the discrete one (unless otherwise stated).

To be able to give an insight into the characteristics of the interpolated phase error around a resonance, a simplification is applied in the following: we assume that the phase error reaches its maximum at exactly the middle point of two neighboring frequency bins, that is, when ω = ω₁ + Δω/2 = ω₂ − Δω/2. Furthermore, the frequency ratio R = ω/ω_n and specific frequency resolution ΔR = Δω/2ω_n are introduced. Using these conventions, Equation (1) can be reformulated as follows:

$$\varphi \left(R\right)=arctan\left(-{\displaystyle \frac{2\xi R}{1-R^2}}\right)$$

(10)

while ϕ₁ = ϕ(ω − Δω/2) and ϕ₂ = ϕ(ω + Δω/2) can be expressed as follows:

$${\varphi }_1\left(R\right)=arctan\left(-{\displaystyle \frac{2\xi (R-∆R)}{1-{(R-∆R)}^2}}\right)$$

(11)

$${\varphi }_2\left(R\right)=arctan\left(-{\displaystyle \frac{2\xi (R+∆R)}{1-{(R+∆R)}^2}}\right)$$

(12)

and the interpolated phase response (Equation (8)) simplifies to the following:

$${\varphi }_{int}(R)={\displaystyle \frac{{{\varphi }_2\left(R\right)-\varphi }_1\left(R\right)}{2}}$$

(13)

Lastly, the interpolated phase error as a function of R is calculated from Equations (10) and (13) as follows:

$${\epsilon }_{\varphi ,int}\left(R\right)=\varphi \left(R\right)-{\varphi }_{int}\left(R\right)$$

(14)

This phase error can be visualized as a function of R, for different damping ratio (ξ) and specific resolution (ΔR) values—see Figure 6.

Figure 6. Interpolated phase error as a function of frequency ratio (R = ω/ω_n) for different damping ratios. (a) Medium frequency resolution; (b) coarse frequency resolution.

Figure 6 demonstrates well the character of phase error: if damping is small (ξ = 0.001), a rather large phase error (up to 55.7°) might be obtained even with moderate frequency resolution (ΔR = 0.004 equals, e.g., Δω = 2 Hz @ ω_n = 250 Hz). With a coarser resolution (ΔR = 0.016, e.g., Δω = 8 Hz @ ω_n = 250 Hz), phase error exceeds 80° with ξ = 0.001. On the other hand, increasing the damping ratio significantly decreases the phase error: for ξ < 0.05, the phase error is below 5° for the examined cases.

One should note that the phase error at ω = ω_n is close to, but not exactly 0—due to the slightly asymmetric characteristic of phase response curves. For the same reason, the amplitude of the left-hand side peak is marginally larger than that of the right-hand side one.

From an engineering perspective, the phase error near resonance is of primary interest, as this region is critical for accurate modal parameter identification and is most susceptible to practical misinterpretation. As the vibration amplitude of a damped system reaches its maximum slightly below ω_n—where the phase error is marginally larger too, compared to the peak value above ω_n—the most significant phase error issues are expected somewhat below the natural frequency.

3. Measurement Evaluation Method

Now, let us put the findings into practice. First, let us consider the discrete phase error: Assuming a coarse frequency resolution, the peak amplitude will be identified above or below the exact peak frequency (see Figure 1). Depending on the coincidence of frequency lines and the natural frequency, the frequency error is ${\epsilon }_{\omega }$. In the worst case, the peak frequency lies exactly between two frequency lines, meaning that the frequency error is half of the frequency resolution (${\epsilon }_{\omega }= ∆\omega /2$). The resulting phase error can be calculated according to Equation (6) or (7)—or can be determined from Figure 5.

As discussed above, the discrete frequency error (calculated at resonance) is not always the best indicator of potential phase error issues. Interpolated frequency error not only gives a better approximation at resonance but can be calculated in a wider range around the resonance frequency. The given method for this calculation works well if there are well-separated resonance peaks, but for FRFs with multiple interfering peaks, the applicability of the theory-based methods is strongly limited. For such cases, another solution is more practical: One can measure the system with a very fine frequency resolution (considered as giving back the accurate phase response) and compare it to a coarse one (resulting in some phase response error). The difference in these two functions will give back the actual phase error as a function of frequency:

$${\epsilon }_{\varphi }\left(f\right)=\left|{\varphi }_{coarse}\left(f\right)-{\varphi }_{fine}\left(f\right)\right|$$

(15)

This is a simple, generally applicable method; however, it needs more and longer measurement runs and will not yield the theoretically maximum phase error, just the one resulting from the actual coincidence of frequency lines and the resonance frequencies. (That is: if a frequency line hits exactly the frequency of interest, even if a very coarse frequency resolution was used, the phase error will be zero at that point. Furthermore, a significantly finer frequency resolution may result in higher error, if it does not hit the frequency exactly.) Consequently, the results of this method can be used only for the given system with the exact resonances. If there is any slight change in the FRF, the phase error value is not valid anymore.

The manifestation of this limitation is demonstrated in Figure 7, where the (interpolated) phase error is given for ∆ω = 11.7 Hz frequency resolution (with fixed frequency lines) for a damped, 1-DOF system with slightly shifted resonance frequencies. It is now obvious that the phase error is dependent on the coincidence of the natural frequency and the frequency lines of the used, discrete resolution. Moreover, the maximum phase error value is near the natural frequency only if the damping is low. An increased damping—e.g., 5% in the presented case—will result in phase error maxima further away from ω_n. This finding is in accordance with the interpretation of Figure 6: the greater the damping, the further the phase error maximum is from the resonance frequency. The difference between Figure 6 and Figure 7 is that the former evaluates a continuous frequency scale (like a sliding window), while the latter calculates the error from the measured, discrete frequency lines (and linear interpolation between them).

Figure 7. Phase error with the same frequency resolution (Δω = 11.7 Hz) but slightly shifted resonance frequency: (a) medium damping; (b) small damping.

Turning it into practice: Measuring a new component of the same type, the resonance frequency can easily be shifted by 1–2 Hz due to slight material, manufacturing, and assembly differences. Consequently, the measured phase error might alter from component to component—according to the results in Figure 7.

Now it is clear that the phase error is highly sensitive to the changes in resonance frequencies, and to keep the phase error below a threshold value, surely, a more detailed analysis should be executed, which is described in the following. In contrast to the previous analytical considerations, which focused on the vicinity of individual resonance peaks, the subsequently proposed method is a post-processing framework for real measured data that can be applied to any FRF (sweep or impulse response) of a complex multi-DOF system.

Maximum Phase Error Analysis (MPEA)

The idea of the method is to consider all possible relative positions of the discretized frequency values (lines) and the resonance frequencies—as demonstrated in Figure 8. It is similar to what was discussed for interpolated phase error calculation based on theoretical considerations, with two major differences: Firstly, the following numerical method works with measured data, and thus can be applied to any FRF, not just to 1-DOF systems. Secondly, the numerical calculation capabilities make it possible to consider the phase error at not just the middle point of the “chord” (i.e., the linearly interpolated phase response), but to calculate the error at all points and take the biggest error.

Figure 8. Demonstration of multiple frequency-shifted coarse sampling from a fine spectral signal.

For this goal, we should have an FRF of the examined system (amplitude and phase response, the latter being in our interest now) with a very fine frequency resolution ($∆{\omega }_f$). The coarser frequency resolution data ($∆{\omega }_c$) are generated by skipping some of the frequency lines—e.g., keeping just the 1st, 11th, 21st … frequency lines will result in a ten times coarser resolution. This rough resolution is then shifted in small steps ($∆{\omega }_f$) by starting with not the 1st, but the 2nd, 3rd … frequency line, and keeping the same $∆{\omega }_c$ resolution (every 10th line is kept in this example). The last starting frequency should be the 10th one, before we reach the same case as the very first one. In this way, all possible relative positions of the coarse frequency lines and the FRF peaks are generated—numerically, 10 cases in our example.

The next step is to determine the maximum phase error, i.e., calculating the phase differences between the fine resolution FRF and the 1st, 2nd … 10th rough resolution case at the examined f_i frequency (ε_ϕ1, ε_ϕ2, … ε_ϕ10)—as illustrated in Figure 8—and take the maximum of the phase error values.

By scanning the whole frequency range with f_i and calculating the phase error maxima, one can obtain the maximum possible phase error for the whole frequency range with the given, rough resolution—independently of the exact positions of the resonance peaks.

The numerical implementation of this method is explained in detail in Appendix A; here, only a condensed explanation is provided as follows. Assuming the original, finely sampled frequency domain signal exists between ω_min and ω_max with the number of frequency lines N, the frequency resolution of this fine dataset is equal to $∆{\omega }_f={\displaystyle \frac{{\omega }_{max}-{\omega }_{min}}{N-1}}.$

Coarse datasets are generated with a frequency resolution of Δω_c = M · Δω_f using the following indexing method:

$${i_c}^k=k+M·j$$

(16)

where i_c^k denotes the indices of the kth coarse dataset (referring to the values of the original fine dataset, i_c^k ϵ [1, 2, … N]), k = 1, 2, … M and j runs from 0 to $⌊{\displaystyle \frac{N}{M}}⌋-1$. In such a way, M pieces of coarse datasets are generated from the values of the original fine dataset, each one shifted by Δω_f compared to the previous one.

The next step is to calculate the phase deviations (ε_ϕ^k,i) between the original and the kth coarse datasets for all k = 1 … M coarse datasets, for all f_i (i = 1 … N) frequency values (f₁ being ω_min and f_N being ω_max).

Finally, the maximum phase error can be calculated as a function of frequency (ε_ϕ^max(f)), i.e., considering all coarse datasets for all f_i (i = 1 … N) frequency values.

This method can be executed for any M ϵ [2, 3, 4 …] value (any multiplication of the original, fine resolution), making it possible to generate the phase error function for a high variety of coarse resolutions and find the optimal one.

4. Experimental Test

A measurement campaign was conducted to validate the theory and methods introduced in the previous chapters. The tested system was an electric powertrain unit (e-Axle), as illustrated in Figure 9. The objective of this measurement was to assess the dynamic characteristics of the powertrain and to establish a correlation between the theoretical/model predictions and the empirical data regarding phase error.

Figure 9. Impact hammer excitation and acceleration response measurement on an e-Axle.

For the measurement, a PCB 086C03 impact hammer (PCB Piezotronics, Inc., Depew, NY, USA) was used with a white plastic tip—this setup proved reliable even at higher frequencies up to 450 Hz. The system’s response was captured using a PCB HTJ356B01/NC triaxial accelerometer (PCB Piezotronics, Inc.), positioned close to the driving point to ensure we capture the local response to the Dirac-type impact. The accelerometer was fixed using a dedicated superglue (Loctite 454, Henkel, Düsseldorf, Germany). Data was sampled at 3000 Hz, which ensured appropriate temporal resolution for our purposes. Throughout the tests, the FFT window size was varied to directly study how frequency resolution influences the clarity of modal peaks and phase response. Table 1 summarizes the FFT settings and resulting frequency resolutions applied during the evaluation.

Table 1. Parameters of the measurement.

FFT Window Size	Frequency Resolution
256	11.72
512	5.86
1024	2.93
2048	1.46
4096	0.73
8192	0.37
16,384	0.18
32,768	0.09
65,536	0.05

To minimize random errors and to check repeatability, each measurement was repeated three times. This allowed us to estimate the maximum coherence more reliably—a key step for validating modal results in practice.

In the results section, the processed measurement data of this system are presented and evaluated using the methods introduced above.

5. Results and Discussion

In Figure 10, the system’s amplitude and phase response are presented around a dominant resonance, with different frequency resolutions. Amplitude response demonstrates clearly the distortion of the resonance peak in the case of inadequate resolution—resulting in a shifted peak value and frequency, and altered (apparent) damping. In case of phase response, however, the largest deviations between the fine and coarse resolution data arise not at the resonance frequency (ca. 358 Hz), but before and after it—in accordance with the findings described before and illustrated in Figure 6 and Figure 7.

Figure 10. (a) Amplitude response functions and (b) phase response functions processed with different frequency resolutions.

It has to be mentioned that in the case of real, multi-DOF systems, the resonances are characterized by sudden phase changes: the phase may increase or decrease as well, and the change is generally less than 180°—in contrast to an ideal 1DOF system, as shown in Figure 4.

According to the evaluation of the amplitude data, the properties of the resonance shown in Figure 10 are the following: ω_max = 358.1 Hz, ξ = 0.032, from which the natural frequency can be calculated as ${\omega }_n={\displaystyle \frac{{\omega }_{max}}{\sqrt{1-2{\xi }^2}}}=358.5 \mathrm{H}\mathrm{z}$.

Considering the frequency error ε_ω = Δω/2, the resulting discrete phase error values according to Equations (6) and (7) are collected in Table 2. Please note that with the defined ε_ω value, the specific frequency error (r) equals the specific frequency resolution (ΔR).

Table 2. Discrete phase error values for the measured system.

Δω (Hz)	r (=ΔR) (-)	εϕ,dhigh (°)	εϕ,dlow (°)
0.18	0.0003	−0.459	0.459
1.4	0.0021	−3.650	3.657
2.9	0.0041	−7.263	7.292
5.8	0.0082	−14.271	14.384
11.7	0.0165	−26.871	27.251

Results show that considering the nearest discrete frequency instead of the accurate resonance frequency may result in a phase error up to 27.25° using a Δω = 11.7 Hz resolution. To keep the discrete phase error below 5°, less than 2 Hz resolution should be used.

The interpolated phase error was calculated too around resonance with Equation (14), and the resulting curves are shown in Figure 11 for the medium (Δω = 2.9 Hz) and coarse (Δω = 11.7 Hz) resolutions. The former results in maximum ε_ϕ,int = 0.3° at R = 0.981 (ω = 351.7 Hz), while the coarse resolution gives ε_ϕ,int = 4.5° at R = 0.979 (ω = 351.0 Hz).

Figure 11. Interpolated phase error for medium (Δω = 2.9 Hz → ΔR = 0.004) and coarse (Δω = 11.7 Hz → ΔR = 0.016) frequency resolutions.

One can conclude that using interpolation for phase calculation decreases the potential phase error significantly: ε_ϕ,int < 5° limit can be kept even with Δω = 11.7 Hz.

The above-discussed methods are based on the characteristics of a 1-DOF system. In the examined test case, even if a rather clear resonance is evaluated, the transfer functions are influenced by many other resonances and dynamic phenomena—this can be seen clearly in the amplitude and phase response functions (Figure 10). To handle this, the empirical/numerical methods for phase error analysis (introduced above) are executed and evaluated, too: the determination of phase error based on the difference in fine and coarse resolution data (Equation (15)), and the MPEA method.

The deviation of the coarse resolution phase response values from the finest one (Δω = 0.05 Hz) is shown in Figure 12 with solid lines. The lower phase error in the narrow vicinity of the resonance frequency (358 Hz) is clearly visible—in accordance with Figure 6 and Figure 7. The maximum actual phase error for Δω = 11.7 Hz resolution is around 5.45° at 368.3 Hz. This value is somewhat larger than the error estimation from the interpolated error analysis, which is due to the deviation of the real system from a pure 1-DOF response (it can be seen evidently from the amplitude response in Figure 10 at around 370 Hz).

Figure 12. Phase error of coarse resolution measurements—the actual, measured phase error (solid lines) and the calculated maximum values (dashed lines).

Maximum Phase Error Analysis (MPEA)—described in Maximum Phase Error Analysis (MPEA) Section—is applied to the measured data, using the finest resolution measurement as a basis. Coarse resolution response curves are generated—in accordance with the real measurements—by taking every 4th (M = 4, Δω_c = 0.18 Hz), 32nd (M = 32, Δω_c = 1.4 Hz), 64th (M = 64, Δω_c = 2.9 Hz), 128th (M = 128, Δω_c = 5.8 Hz), and 256th (M = 256, Δω_c = 11.7 Hz) frequency sample. The starting frequency is then increased step-by-step until it reaches the 2nd sample of the coarse resolution. The resulting phase error values are determined for each frequency, for each step (bin), for the selected resolutions—as shown in Figure 13. Finally, the maximum error value is calculated for each frequency line (of the fine resolution input data)—i.e., along the “Bin” axis. This result is plotted in Figure 12 with dashed lines, meaning the theoretical maximum of phase error. As expected, the real phase error is always below the theoretical maximum value. An exception may occur due to measurement inaccuracies only, as the “theoretical maximum” is calculated based on the finest resolution measurement, while the “actual phase error” data came from different measurements with coarser frequency resolutions.

Figure 13. Phase error waterfall plot based on multiple frequency-shifted coarse sampling referenced to fine spectral measurement data. The maximum phase error for 0.18 Hz frequency resolution is negligible: for 1.4 Hz, it is only a few tenths of a degree, and for 2.9 Hz, it is around 1°. For 5.8 Hz resolution, the maximum phase error is around 4°, while for 11.7 Hz resolution, it is more than 8°.

The results of MPEA show that with 11.7 Hz frequency resolution, the phase error may reach 8.25° at 330.5 Hz—more than 50% higher than the actually determined maximum phase error (5.45° at 368.3 Hz).

As a final conclusion, based on the MPEA results, we can state that using 11.7 Hz frequency resolution for this type of e-Axle is enough to limit the phase error at 8.25°. If this is too much, the phase error limit can be reduced to 3.8° by using 5.8 Hz frequency resolution, or even kept below 1.15°, for which 2.9 Hz resolution must be used. The actually measured phase error is generally somewhat smaller than these maximum values, while the 1DOF approximation results in even smaller errors—especially for fine resolution cases. A detailed list of results is shown in Table 3.

Table 3. Maximum phase error for the measured e-drive.

Δω (Hz)	Interpolated Phase Error from the 1DOF Approximation for fr = 358 Hz	Measured Real Phase Error (Relative to Δωf = 0.04 Hz Response)	Determined Maximum Phase Error Based on MPEA Method
0.18	0.001°	0.018°	0.02°
1.4	0.069°	0.27°	0.29°
2.9	0.303°	1.12°	1.15°
5.8	1.161°	3.55°	3.84°
11.7	4.505°	5.45°	8.25°

6. Conclusions

The introduced Maximum Phase Error Analysis (MPEA) method offers a robust numerical approach to quantify the highest possible phase error arising from the arbitrary alignment of discrete frequency lines with resonance peaks. This technique is particularly beneficial for multi-DOF systems with small damping (ξ < 0.05), where slight deviations in resonance frequencies can significantly influence phase accuracy. MPEA thus provides a practical guideline to determine the frequency resolution required for reliably minimizing phase response errors in dynamic NVH testing scenarios.

The executed analyses and their most important findings can be summarized as follows:

• Phase error is analyzed theoretically around resonance for 1DOF systems. Both the error at the nearest frequency line and the error in the case of phase interpolation were determined. The proposed method can be used to maximize the phase error around a dominant, single resonance peak. However, this theoretical approach was proven to be inaccurate in real, multi-DOF cases, especially for high-frequency resolutions.
• For a more complex, real, multi-DOF system, the phase error should be determined more accurately by comparing the coarse resolution data with an extremely fine one. In the case of the tested system (an electric drive assembly—e-Axle), this phase error was around 5.45°—slightly higher than the value obtained for the interpolated phase error analysis of the corresponding 1-DOF system (4.5°).
• The detailed analysis confirmed the hypothesis that—in contrast with the amplitude error—the largest phase error does not occur at the resonance frequency, but somewhat below and above. It was proven analytically for 1-DOF systems, and empirically on the measured e-Axle too.
• Finally, a numerical method (Maximum Phase Error Analysis—MPEA) was proposed and demonstrated to determine the theoretically maximum phase error—considering any possible coincidence of the coarse frequency lines and the resonance frequency. It is especially important when the damping is small (ξ < 0.05) and slight changes in resonance frequency may occur. In such cases, setting the frequency resolution limit based on one measured piece is not appropriate; a significantly higher phase error may occur due to a small shift in resonance frequencies (more than 50% higher in our case).

The presented results show that frequency resolution is a key factor in the vibration analysis of electric drivetrains. Both the calculations and the measurements make it clear: if the frequency step is too large, important details—especially sharp phase changes near resonance—might be missed or distorted. This leads to less reliable identification of the system’s dynamic properties. For accurate NVH work, it is not enough to just “collect data”—the settings of the FFT, especially the frequency resolution, must match the targeted amplitude and phase error limit. The novelty of the present study lies in providing a concrete workflow to limit phase error, and its application to a real, complex system (an e-drive assembly) is demonstrated. As electric vehicles become more common, getting these details right will be even more important for engineering work to solve NVH issues.

Outlook—Future Work

The proposed methodology was successfully applied to an electric drive. However, to demonstrate the general applicability of the presented solution, repeatability and comparative tests—including assemblies with different damping characteristics and assembly tolerances, as well as further statistical analyses—should be conducted next. Another step could involve the use of simulation results for a detailed sensitivity and robustness analysis of the proposed method.

From a broader perspective, the application of the MPEA method to a wider range of NVH testing scenarios should be pursued (e.g., speed order tracking analysis). Investigating the feasibility of integrating the method into commercial testing software—for instance, by exploring adaptive frequency resolution testing strategies based on MPEA results—could represent a promising practical direction.