Investigations on Output Impedance Measurement of Digitally Controlled Power Converters with Wide Bandwidth Signals

Academic Editor: Frede Blaabjerg Received: 17 October 2025 Revised: 11 November 2025 Accepted: 20 November 2025 Published: 22 November 2025 Citation: Schwanninger, R.; Bosch, M.; Wunder, B.; März, M. Investigations on Output Impedance Measurement of Digitally Controlled Power Converters with Wide Bandwidth Signals. Energies 2025,18, 6121. https://doi.org/10.3390/ en18236121 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). Article Investigations on Output Impedance Measurement of Digitally Controlled Power Converters with Wide Bandwidth Signals † Raffael Schwanninger 1,* , Moritz Bosch 2, Bernd Wunder 2and Martin März 1 1Institute of Power Electronics, Friedrich-Alexander-University, 91054 Erlangen, Germany; [email protected] 2Fraunhofer Institute for Integrated Systems and Device Technology IISB, 91058 Erlangen, Germany; [email protected].de (M.B.); [email protected].de (B.W.) *Correspondence: raf[email protected] † This paper extends the research of our paper “Output Impedance Measurement of Digitally Controlled Power Converters in LVDC-Grids” previously published at the 2024 IEEE International Conference on DC Microgrids, Columbia, SC, USA, 5–8 August 2024. Abstract Impedance-based stability criteria can be a powerful tool in systems analysis of microgrids. Especially in terms of low-voltage direct current grids, the amount of available power converters continuously increases. Since commercial components are typically blackboxes for system analysis, reliable and fast measurement is required to allow for accurate stability estimations. Here, wide bandwidth signals offer reduced measurement time at high accuracy. Since higher power converters are often digitally controlled, this paper investigates requirements for proper wide bandwidth impedance measurement under the boundary conditions the digital control enforces on time and value quantization. Keywords: DC power systems; impedance measurement; control; system analysis and design; simulation; stability 1. Introduction The need for reliable and efficient energy distribution drives the development of low-voltage direct current (LVDC) grids. [ 1 – 3 ]. LVDC microgrids can either be connected to AC grids, with a certain level of autonomy, or operate completely autonomously, relying only on renewable energy sources. Through integration of battery storage systems and electric vehicles this autonomy can be even further increased [4–8]. However, maintaining stability in all operating conditions is required for ensuring the reliability of LVDC microgrid operation. This challenge becomes increasingly more difficult as the stability analysis needs to encompass an increasing number of components and converter types [ 9 – 12 ], making model-based predictions near impossible without knowledge about the internal system parameters. The application of measurement-based stability criteria can eliminate this issue. A promising group of stability criteria are based on the grid side impedance of converters [ 13 – 22 ], based on and adapted from [ 23 ]. Since impedance can be directly measured for a black-box system, stability information can be gained without further information about the converter itself. The only prerequisite is a measurement system capable of accurate impedance measurement of power converters. Since most higher power converters used for LVDC microgrids are digitally controlled, methods of impedance measurement need to be investigated for their effect on digital Energies 2025,18, 6121 https://doi.org/10.3390/en18236121 Energies 2025,18, 6121 2 of 26 control. Due to the added benefits in fast measurement, this paper will be focused on broad band measurements using pseudo random binary sequences (PRBSs). PRBS signals have been used for system identification in various power electronics applications. Often the PRBS is generated by the system itself and modulated onto an internal signal like the converters duty cycle [ 24 – 29 ], which can then be used to identify control loop transfer functions. Even using multiple orthogonal PRBSs simultaneously within a larger system have been investigated [ 30 – 32 ]. In AC systems these orthogonal signals can even be used to measure reactive and active current control loops as well as their interactions [ 33 , 34 ]. If the PRBS is used to excite a connected converter, or even the whole grid, impedance measurement of unknown converters is also possible [ 35 – 37 ]. PRBS signals have also been modified from being classical maximum length two state signals, to three state signals [ 34 , 38 , 39 ] or optimized two state signals [ 40 ], to enhance certain frequency component’s power density. While being widely used, the typical recommendation for the signal amplitude is somewhere in the low single digit percentage range [ 28 , 33 , 37 ], high single digit percentage range [ 35 , 36 ], or up to 25% [ 27 ] of the steady state values depending on application. These amplitudes all have been determined by trial and error, not based on fundamental principles. In [ 28 ] the effect of quantization on the measurement is discussed regarding the measurement system, but other power converters are not included. For a practical implementation of a measurement system, this diversity in recommended signal amplitude can lead to a high uncertainty in the measurement results. A more in-depth analysis on the required signal amplitude is therefore carried out in this paper. Additionally, as power converters are inherently nonlinear and time variant systems, the measurement needs to allow correct linearization for a given operating point. While this is easily possible with mono-frequency signals, wide bandwidth signals can lead to aliasing effects, thereby altering the measurement result. This requires a full analysis of the above Nyquist frequency behavior of power converters. In this paper, we will focus on the design of a measurement system, capable of measuring the output impedance of digitally controlled output converters with wide bandwidth signals. To ensure proper measurement, we first describe the consequences of digital control on measurement in Section 2, introduce wide bandwidth measurements in Section 3, and then verify the findings in simulation (Section 4) and actual measurement (Section 5). The paper is concluded with a discussion in Section 6and conclusion in Section 7. 2. Digitally Controlled Power Converters To allow for classical stability criteria to be used, a power converter needs to be described as a linear time invariant (LTI) system. Since power converters are generally nonlinear time variant systems, the measurement needs to accurately linearize the converters response. To correctly approximate an LTI system, two effects—value and time discretization— need to be considered when measuring a digitally controlled power converter. 2.1. Value Discretization To correctly linearize a system in measurement, a small signal perturbation is used for excitation. If the converter is analog in control, the smallness of the perturbation is theoretically unbound and only obstructed by the required signal to noise ratios of the measurement system. For digitally controlled systems, the analog to digital conversion leads to quantization in value. If a perturbation is too small, the system response of the converter will be adversely affected as shown in [8]. For the ADC to properly reconstruct the measured signal x with a digital signal y , the external excitation needs to be large enough to trigger a least-significant-bit ∆yLSB . The Energies 2025,18, 6121 3 of 26 effect is shown in Figure 1for a sufficient and insufficient excitation. If the test signal is insufficient, the converter cannot correctly react to the measurement and the output impedance will be altered. The effect of this under-excitation largely depends on the affected ADC. In practice, the most common form is the under-excitation of the voltage measurement ADC [ 8 ]. For this ADC to register the change ∆vGrid in the measured grid voltage vGrid , ∆vGrid needs to be larger than the lowest noticeable change ∆vLSB , that toggles one bit of the ADC. Different effects of under-excitation of mono-frequency signals are given in [8]. The effects for wide band measurement are discussed in Sections 3and 4. (a) (b) Figure 1. Reconstructed signal y of a measured signal x with sufficient (a) and insufficient (b) excitation. 2.2. Time Discretization While value discretization only affects digitally controlled converters, time discretization is inherent to any power converter due to its switching actions. To investigate the behavior of PWM on the converter’s transfer function, the simplified model of a 10 phase buck converter’s current control as shown in Figure 2is used. Each phase is controlled by a current controller GCC —acting on the difference in measured inductor current ILn of phase n to a reference current IL,set —with no interaction between the individual controllers. The input and output voltages are set as constant voltage sources to eliminate any interaction between the 10 phases. Therefore, if a small signal perturbation iL,set is added to IL,set, the perturbations iL,n measured in the individual phases should ideally be describable by the same closed loop transfer function GCL(s): iLn(s)=iL,set(s)·GCL(s)(1) modulator controller Figure 2. Simplified model of a 10 phase buck converter’s current control loop. The reason for the 10 phases is the noise cancelation in inductor currents. If all phases are perfectly interleaved, the switching noise in the sum of all iLn should cancel out up to five times the switching frequency. This should allow the investigation of GCL(s) Energies 2025,18, 6121 4 of 26 through simulation even close to and above switching frequency. We will therefore measure GCL(s)as GCL(s)=1 10·iL,set(s)· 10 ∑ n=1 iLn(s)(2) While GCC as well as L and internal resistances Ri are all linear time invariant, the influence of the PWM cannot as easily be transformed into the frequency domain. Instead, the effect of the modulator on d , can be explained by using a simplified time variant model shown in Figure 3. Similarly to the small signal perturbation in currents, we can first describe the duty cycle D as a constant operating point dependent on D0 and a small signal perturbation d: D(t)=d(t)+D0(3) Figure 3. Simplified time variant model of the switching behavior of a half bridge with a triangular carrier. (a) shows the carrier and the PWM-Modulators input signal, (b) depicts the resulting PWM, and (c) gives the small signal reaction to the perturbation. To generate the PWM, D is compared to a carrier signal vcar as shown in Figure 3a. vcar oscillates between 0 and 1 and is defined by its rising slope . vcar,r and falling slope . vcar,f . In this paper, we will assume a symmetric triangular carrier, where . vcar,r and . vcar,f normalized in amplitude and time are . vcar,r =−. vcar,f =. vcar =2 (4) If D is lower than the carrier, then the high side switch is turned on, if D is higher than the carrier, then the low side switch is turned on. This leads to a PWM signal with the amplitude equal to the input voltage vHV . D0 can be determined by fulfilling the following condition: D0·vHV =vLV +IL·Ri(5) By applying only D0 to the system, the PWM signal vPWM(D0) will be observed as shown in Figure 3b in blue. vPWM(D0) is not a DC signal and will already include the current ripple during unperturbed operation. If a small signal perturbation iL,set is applied, Energies 2025,18, 6121 5 of 26 the small signal perturbation d will change vPWM(D0) to vPWM(D) as indicated by the yellow PWM signal in Figure 3b. dtherefore leads to a perturbation PWM signal vd: vd(t)=vPWM(D,t)−vPWM(D0,t)(6) which is equal to the red PWM signal in Figure 3c. If like in Figure 3a triangular carrier signal is used, vd (red pulses in Figure 3c) has twice the frequency of vPWM(D0) and vPWM(D) , since there can be a change in D during turn on and turn off. If instead a sawtooth carrier were to be used, only the time of turn on t2n or turn off t2n−1can be perturbed. For each switching instant, we can define two points in time: tn where the half bridge would switch for D0 and tnd where the half bridge will switch due to the perturbation. To relate vd to d for a sufficiently low frequency perturbation—i.e., sufficiently lower period than the difference in time between tnand tnd—we can assume for d: d(tnd)≈d(tn)(7) For the high side switch turn off instant (e.g., t1 ) we can calculate the pulse width τd of vdby using the slope . vcar of vcar and d: τd=t(2n−1)d−t2n−1≈d . vcar =d 2(8) For the high side switch turn on the sequencing of tn and tnd is exchanged therefore the pulse width τdis τd=t2n −t2nd ≈d −. vcar,f =d . vcar =d 2(9) For a small signal perturbation, we can also assume tn1 ≈tn2 =tn(10) and approximate the pulses of vd as a series of δ -distributions as indicated by the green arrows in Figure 3c. For both to have the same effect on the inductor current, the only requirement is ZT 2(n+1) T 2n vHV·d . ucar ·δi(t−t1)dt =ZT 2(n+1) T 2nvddt (11) In the time domain, we can therefore describe the PWM signal as ∼ vd: ∼ vd(t)=d(t)·vHV 2· ∞ ∑ n=−∞ δ(t−tn)(12) A small signal model of the plant with the PWM can now be derived, using ∼ vd: iL(s)=∼ vd(s)·(Ri+sL)(13) Figure 4shows the effect of PWM in the frequency domain. By definition, the δ - comb of the PWM stays a δ -comb (red arrows) in the frequency domain. The distance between two δ -distributions is twice the switching frequency. If d is a mono-frequency perturbation, it will be represented as two δ -distributions as indicated by the blue arrows for a perturbation lower than switching frequency and as green arrows for a perturbation above switching frequency in Figure 4a. As ∼ vd results are from a multiplication in the time domain, it is a convoluted signal of the δ -comb and the mono-frequency d in the Energies 2025,18, 6121 6 of 26 frequency domain. ∼ vdf1 and ∼ vdf2 are caused by the excitations df1 and df2 and therefore gain additional frequency components as shown by the dashed arrows in Figure 4b. Figure 4. Effect of PWM on the open loop transfer function GOL and the feedback transfer function GOL,FB for two different frequency excitations df1 and df2 . The black arrow represents the origin axis at 0 Hz. (a) shows two sinusoidal perturbations (blue and green arrows) as well as the δ -comb representing the PWM (red arrows). The aliasing of the original perturbations are shown as dashed arrows in (b). In (c) these arrows are weighted with GOL . Due to aliasing, this affects GOL,FB as indicated in (d). Since d is a result of the linear controller, its frequency response can be directly calculated into an input signal iL,set via GCC: d(s)=GCC(s)·iL,set(s)(14) The linear open loop transfer function GOL depending on GCC and the converter is therefore given as GOL(s)=GCC(s)·(Ri+sL)(15) This linear open loop transfer function is shown as the black line—mirrored at the x-axis for negative excitations—in Figure 4c. Consequently, the measurable output current Energies 2025,18, 6121 7 of 26 should result from GOL as well as an equivalent transfer function of the PMW modulator GPWM acting on iL,set: iL(s)=GOL(s)·GPWM·iL,set(s)(16) Due to ∼ vdf1 and ∼ vdf2 not being mono-frequency signals, iL also includes other frequency components that are shaped by GOL as shown by the dashed arrows in Figure 4c. Still, if only the original excitation frequency (solid arrows) is considered, the PWM does not have any additional effect. Therefore, if the open loop transfer function is measured, and only the excitation frequency component is evaluated, the PWM transfer function GPWM must be GPWM(s)=1 (17) If the control loop is now closed, one would assume a linear closed loop transfer function GCL,Lin: GCL,Lin(s)=GOL(s) 1+GOL(s)(18) This is not the case, as shown in Figure 5for a simulation of the 10 phase closed loop current controller at a switching frequency fs of 100 kHz. While the simulated values GCL,sim show good accuracy with GCL,Lin at lower frequencies, they diverge greatly at twice the switching frequency. Figure 5. Simulation results and analytical models of a 10 phase buck converter’s current con trol loo p. To explain the difference, we need to include the natural aliasing of the PWM. Due to the two times switching frequency “sampling” of PWM, we can define an s0as s0=j·2π·2·fs(19) This s0 represents the complex angular frequency associated with the natural aliasing. Due to the aliasing of the higher frequency components, all high frequency components will lead to aliases at the excitation frequency. The measured iL at the switching actions will Energies 2025,18, 6121 8 of 26 be altered to ∼ iL as shown in Figure 4d. Therefore, the feedback transfer function GOL,FB needs to account for aliasing: GOL,FB(s)=GOL(s)+ ∞ ∑ n=0 GOL(sn+)−GOL(sn−) with sn+=n·s0+s sn−=n·s0−s (20) The true closed loop transfer function GCL , as plotted in Figure 5, therefore needs to be calculated as GCL(s)=GOL(s) 1+GOL,FB(s)(21) GCL(s) now accurately represents GCL,sim until the onset of switching noise at 500 kHz obscures the measurement. Interestingly, if GOL can be estimated as an integrator with a constant gain of K, GOL(s)=K s(22) which can be a good estimation for a buck converter with well-tuned PI-control, using the partial fraction decomposition of the cotangent, GCL can be simplified to GCL(s)= K s 1+Kcots s0(23) We can therefore interpret fs as the natural Nyquist frequency fNN of the converter, if a symmetrical carrier is used and D0 is close to 0.5. If a sawtooth carrier is used, there will be only one δ-distribution per switching frequency, leading to an fNN of half of fs: fNN =1 2fs(24) If a symmetrical carrier is used and D0 is close to 1 or 0, the δ -distributions will be so close that they can be interpreted as a single δ -distribution, again leading to an fNN of half of fs. If digital control is used, the ADC samplings will typically not coincide with the switching actions. This is shown for a digital sampling frequency fsamp of twice the switching frequency in Figure 6. The sampling happens at half the switching period T and multiples thereof. At these points, the digital duty cycle Dd is sampled from an analog signal consisting of an offset D0 and the perturbation d as indicated in Figure 6a. Since the switching actions are not at multiples of T/ 2, there is a time delay between sampling and the converter’s action as shown in Figure 6a–c. In this case, the transfer functions need to be adjusted for a delay of τ11 for turn off, or τ12 for turn on. Regarding aliasing, the effect of digital sampling on the actual Nyquist frequency fN depends on the switching frequency and the sampling frequency. For the digital sampling, a Nyquist frequency fNd exists, which is half of fsamp . As explained before, the PWM introduces its own natural Nyquist frequency fNN . Here, two cases can be distinguished. Either fNd is as high or higher than fNN , or fNd is lower than the converter’s fNN . In the first case, the digitally controlled converter’s Nyquist frequency fN will be equal to fNN . This can be intuitively seen, as an infinite sampling frequency would reproduce the analog control, with the delays τ11 and τ12 approaching zero, as fsamp is further increased. The effect will be visible in the transfer function but will not affect the measurement with wide Energies 2025,18, 6121 9 of 26 band width signals. In the latter case fN will be equal to fNd , since the sampling and therefore aliasing of the ADC will dominate. Figure 6. Simplified time variant model of the switching behavior of a half bridge with a triangular carrier and digital sampling. (a) shows the carrier and the PWM-Modulators input signal, (b) depicts the resulting PWM, and (c) gives the small signal reaction to the perturbation. 3. Wide Bandwidth Measurement System The most straight forward approach to impedance measurement is using monofrequent sinusoidal signals. To obtain a full frequency spectrum, selected frequency points are measured individually, with a fine-tuned bandpass filter acquiring only the relevant frequency component of the current and voltage waveforms. Requirements for monofrequency coupling are given in [8]. To probe a wide frequency range in small frequency steps, measurement of monofrequent signals takes a considerable amount of time, since the system should be at a steady state during measurement. The test signal must therefore be held at the same frequency for considerably longer, than the lowest time constant τmin of the to be investigated system [ 41 ]. The measurement time tmeas of n successive frequencies will correspondingly increase. Also, the total test signal energy Emeas as a product of measurement time and the average test signal power Ptest increases accordingly: Emeas >τmin·n·Ptest (25) If instead PRBS are used, the measurement time can be greatly reduced [ 8 , 24 – 41 ]. To be able to measure a converter, a system capable of injecting PRBS-shaped currents into a grid, as shown in Figure 7, is developed. The PRBS is coupled by turning a transistor on Energies 2025,18, 6121 16 of 26 trend in noise, but the deviations from the ideal measurement are also exactly identical for both, if the ratio of ∆QC to CDUT remains the same. Additionally, the signal to noise ratio not only improves at higher frequencies, but also at low frequencies, once Equation (38) is met. This indicates a frequency independence of SNR in terms of amplitude quantization. If a series of mono-frequency measurements were to be made, this would be different, since ∆QCwould be lower for higher frequencies. To observe the converter with change dynamics, the two simulations were repeated with changed droop resistances. For both the 200 µF and 400 µFCDUT , the droop resistances were doubled to 250 mΩand 125 mΩ, respectively. The results are shown in Figure 13. (a) (b) (c) (d) (e) (f) Figure 13. Simulation results of the output current of a converter with and without voltage discretization (ideal) for different excitation amplitudes. The results on the left are for a 200 µ F at 250 m Ω with (a) at 1.25 A, (c) at 2.5 A and (e) at 5 A, the ones on the right for a 400 µ F output capacitance at 125 mΩwith (b) at 2.5 A, (d)at5Aand(f) at 10 A. Energies 2025,18, 6121 17 of 26 Similarly to the lower droop resistance simulations, the requirement given in Equation (38) can be interpreted as a threshold for proper measurement. Compared to Figure 12, the simulation results in Figure 13 generally show lower noise. This can be explained by the reduced dynamics of the system. With the increased droop resistance, the impedance of the converters modeled exhibits a − 3 dB corner frequency of approximately 5 kHz, whereas the lower droop resistance leads to a corner frequency of 11 kHz. The effect of converter dynamics can be approached by taking the two most extreme cases. In the most dynamic case, the converter would have infinite dynamics leading to an SNR of 6.02 dB, as given by the dynamic range of a 1-bit analog to digital conversion, when Equation (38) is met. In the least dynamic case, the converter would have no dynamics and its impedance would be represented only by the analog output capacitor. In this case no quantization would occur. Equation (38) is therefore the limit, where an SNR of more than 6.02 dB can be obtained in any case, with SNR increasing with decreasing dynamics. In terms of required power to correctly measure the system, the PRBS current does not need to be drastically increased over a mono-frequency signal to properly excite the ADC. Using Equation (38) to estimate the RMS current required to measure at 30 kHz, a purely sinusoidal signal of 3.3 A would be needed. Using PRBS signals therefore greatly reduces the total signal energy for a full spectrum measurement. 4.2. Time Discretization Different than with value discretization, the time discretization of the converter itself impacts the output impedance inherently. As explained in Section 2, high frequency harmonics of the converter can directly impact the closed loop current transfer function. For the simulation model, a comparison of an averaged model with a switched model will therefore never lead to the same results. Instead, a single-phase model of a buck converter is implemented in simulation and tested at different values of fPRBS . The converter is still droop controlled, with a droop resistance of 125 m Ω converting from 1 kV to 500 V. The switching frequency is set to 100 kHz with a triangular carrier-based PWM modulator. For the analog controlled model, this leads to a Nyquist frequency fNN of 100 kHz. The system is then excited with an fPRBS = 100 kHz PRBS with an O of 12 and a fPRBS = 400 kHz PRBS with an O of 14. Both PRBSs show an fc,min of 24.2 Hz and are repeated 20 times to assure steady state conditions in simulation. The roll-off frequency fc is at roughly 30 kHz for the 100 kHz PRBS and 120 kHz for the 400 kHz PRBS. The simulation results for the pure analog control are shown in Figure 14. The lack of difference between the two measurements can be explained by the high damping of the to be aliased frequency parts. At 100.019 kHz, which would be the first frequency misinterpreted by the PWM modulation as 99.994 kHz, the output impedance is already dominated by the capacitor, which leads to the negligible effect of fPRBS . Higher misinterpreted frequencies, affecting frequencies where the control of the power converter dominates, are additionally filtered by the output capacitor, due to iPRBS being a current type signal. The frequency components at 190 kHz, which would affect the measurement at 10 kHz, are already damped by over 25 dB. To investigate the behavior at different duty cycles, the output voltage was varied to measure the system under a D0 of 0.5, 0.25, and 0.125. As shown in Figure 15, D0 has an almost negligible impact on the noise level of the impedance measurement. Due to the linearity of the buck converter, the change in output voltage also does not impact the impedance shape. For the following investigations, D0will therefore be kept to 0.5. Energies 2025,18, 6121 18 of 26 Figure 14. Impact of a 400 kHz and a 100 kHz PRBS on an analog controlled converter with fNN of 100 kHz. Figure 15. Impact of a 400 kHz PRBS on an analog controlled converter with fs of 100 kHz under different D0. If the converter is instead digitally controlled, the sampling frequency of the control can impact the aliasing. In simulation, the digital control is modeled through sampling and acts on the digital equivalent of the analog control. In this case, the vertical ADC resolution is quasi-infinite and—like with the simulated analog control—only limited by the floating-point resolution of the simulation. The results for time discrete sampling with 200 kHz are shown in Figure 16. Aside from a slight deviation in phase at 20 kHz, the result resembles that of the analog control. While the digitally introduced delay between sampling and switching changes the output impedance, the Nyquist frequency stays the same, resulting in the same negligible effect of fPRBS on the measurement. Energies 2025,18, 6121 19 of 26 Figure 16. Impact of a 400 kHz and a 100 kHz PRBS on a digitally controlled converter with an fNd of 100 kHz. If the sampling frequency is reduced to 50 kHz as shown in Figure 17, the introduced delay has a clearly visible effect on the output impedance. The formerly passive converter now clearly violates passivity at frequencies above 12.5 kHz. Additionally, application of a 400 kHz PRBS (blue dots) now leads to a clearly deteriorated measurement, with increased noise in gain and phase. Still, the effect is mild compared to under-excitation of the ADC for the examined converter. Figure 17. Impact of a 400 kHz and a 100 kHz PRBS on a digitally controlled converter with an fNd of 25 kHz. It is important to note that the effect of aliasing in wide bandwidth measurements is not unique to digital control, as shown for an analog controlled converter in Figure 18. Here, the control parameters were left unchanged but the switching frequency was reduced Energies 2025,18, 6121 20 of 26 to 25 kHz. While the effect on the shape of the output impedance—aside from artifacts at switching frequency—is not as prominent as with digital under-sampling, the deterioration of measurement of the 400 kHz PRBS (blue dots) compared to the 100 kHz PRBS measurement (red dots) is still clearly visible. Figure 18. Impact of a 400 kHz and a 100 kHz PRBS on an analog controlled converter with an fsof 25 kHz. When compared to value discretization, the effects of time discretization are comparatively small, since even the worst-case effects of time discretization (Figure 17) do not replicate only meeting the charge requirement in Equation (38) (Figures 12 and 13). For a synchronous buck converter, it can therefore be surmised that time discretization only has a very limited effect. It is important to note that this might not be true for any topology. Time discretization should therefore be at least kept in mind when selecting fPRBS. 5. Measurement Results To verify the simulated findings in practice, a digitally controlled synchronous buck converter is measured using the system described in Section 3. The converter is controlled by a cascaded voltage and current control using 12-bit ADCs to convert currents and voltages. The discretization in output voltage measurement is 195 mV, which—together with the output capacitance of 200 µF —leads to a required ∆QC of 39 µC . The converter has a switching frequency of 60 kHz, with the inductor current and input and output voltage sampled four times each switching period, leading to an effective fNd of 60 kHz. The measurement system excites the converter with 25 repetitions of a 100 kHz PRBS with an O of 12. To properly excite the system, a 4 A iPRBS is coupled with a 100 ΩRc at 400 V. The converter parameters are identical to the simulation model in Section 4.1. An exemplary current and voltage measurement, normalized to the maximum signal amplitude, is shown in Figure 19. Due to the 25 repetitions, the measurement includes 24 unusable frequency points for every PRBS constituent frequency. Since the measurement system does not induce any signal at these frequency points, they can be used as a measure of noise. As clearly visible in Figure 19, the noise has no noticeable impact in the frequency range up to 60 kHz. The system will be assumed unperturbed by noise up to 30 kHz with high confidence. Energies 2025,18, 6121 21 of 26 Figure 19. Comparison of signal and noise amplitudes of the measurement. To eliminate misinterpretation of effects due to insufficient excitation of the measurement system’s 16-bit ADC, the 4 A measurement raw data was retroactively reduced to 12-bit and 8-bit resolution as shown in Figure 20. While an under-excitation of the measurement system’s voltage ADC is highly unlikely for signals that can be picked up by the converter’s ADC, an under-excitation in the current measurement could still lead to a deteriorated measurement. As can be seen in Figure 20, only a reduction to 8-bit leads to a noticeable change in the measurement. We therefore conclude that a reduction in signal amplitude up to a factor of 16 would not negatively impact the measurement, when only considering the measurement system. Figure 20. Impact of the measurement system’s ADC resolution on the impedance measurement. The reduction to 12 bit has no impact on the measurement, with the red dots below the yellow dots (16 bit). Energies 2025,18, 6121 22 of 26 To investigate the influence of the digital control on the system, the measurement is carried out again with a 2 A and an 8 A iPRBS . The injected currents are shown in Figure 21. The results are shown Figure 22. While the increase in signal power to 8 A only marginally improves the measurement, the decrease in iPRBS to 4 A drastically limits the usability of the measurement, validating the stimulatory findings. Figure 21. First 100 bits of an injected PRBS-shaped current used to measure the output impedance of a power converter. Figure 22. Measurement results of the output impedance of a buck converter at different excitation currents. Regarding aliasing, no improvement was found if fPRBS was changed to 60 kHz, further indicating the low impact of aliasing on buck converters. It is important to note that this has to be expected of a well-designed digital control system, since perturbations in realistic environments will typically be wide band. A power converter not being able to handle multiple frequency components at once will therefore exhibit unpredictable behavior in a DC grid. Energies 2025,18, 6121 23 of 26 To further exemplify the validity of the wide bandwidth measurement, a set of 10 equal measurements with an iPRBS of 4 A were taken. Results with their standard deviation are given in Figure 23. As shown, the deviation between measurements is reasonably low, indicating high reproducibility. The results also show similar variance over a wide frequency range, with the exception of low frequencies. As expected from the simulation results, insufficient ADC triggering leads to wide bandwidth noise, visible in the whole frequency range. Figure 23. Measurement results of a set of 10 measurements at 4 A. The standard deviation is given as error bars for each frequency. 6. Discussion on Measurement Procedure When using external PRBS to measure a system, a few degrees of freedom can be selected which can affect measurement quality. While time discretization does not play a significant role in measurement quality for the converter investigated, we still advise to set fPRBS only as high as twice fN . Depending on whether digital or PWM sampling dominates, fPRBS can therefore be set to equal either fs or fsamp . If the power converter’s internal structure is unknown, we advise determining the switching frequency by means of external measurement and assume fsamp to be equal. If variable frequency topologies such as in [ 45 – 49 ] are used, fPRBS should be set close to the operating point frequency. With fPRBS set, Equation (38) can be used to determine the minimum excitation current. If the capacitance or value discretization are unknown, we advise a first measurement to judge the capacitance and assume an industry standard 12-bit ADC, adjusted for the maximum allowable voltage. The PRBS order O can be determined by the minimum frequency that is supposed to be measured, using Equation (27). This procedure should allow for accurate and fast impedance measurement using PRBS. If not, we suggest changing the excitation amplitude first, before changing fPRBS. 7. Conclusions Output impedance measurement of power converters can be a powerful tool in DC grid stability estimation. Here, excitation with wide bandwidth signals can speed up measurement at little cost to measurement accuracy. As shown in this paper, clear criteria for accurate wide bandwidth measurements can be deduced from theoretical considerations. Energies 2025,18, 6121 24 of 26 For parallel coupled current type PRBS signals, these criteria were verified in simulation as well as on a physical device under test. It is shown that especially the vertical resolution of the power converter’s digitalization can have a large impact on measurement. For the investigated type of converter and test signal type, horizontal resolution, i.e., sampling frequency, tends to only play a minor role relating to measurement. Further investigations of different types of signal coupling and their impact on aliasing as well as ADC excitation are still needed. Similarly, the investigations in this paper were limited to buck-type converters and might not be generalizable to any type of converter. While certain relations, like the capacitor–charge relation might still be applicable, especially aliasing might play a more pronounced role in different topologies. Additionally, means of softening the pulses, so as to only excite within a specified frequency range with less perturbation in the above Nyquist frequency band, might further enhance wide bandwidth excitation. Author Contributions: Conceptualization, R.S.; Methodology, R.S., B.W. and M.M.; Software, R.S.; Validation, R.S. and M.B.; Formal analysis, R.S.; Investigation, R.S. and M.B.; Writing—original draft, R.S.; Writing—review & editing, M.B., B.W. and M.M.; Visualization, R.S.; Supervision, M.M. All authors have read and agreed to the published version of the manuscript. Funding: This work was conducted within the project ECS4DRES. 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