The primary current and resonant inductance current of transformer begin to decrease. At this point, the voltage of the switch junction capacitor is. At , the resonance ends, and at this time , the resonance process time is obtained by substitution The larger and the smaller , the shorter the dead time. Therefore, the dead-time can be adjusted according to the load current , so that the switch can achieve a wide range of soft-switching, which is called automatic dead-time technology.
Figure 10 is the automatic dead-zone task flow chart. The output current sampling is connected with ADC channel 9, and result register for ADC channel 9 is read to obtain the current load. The generation mechanism of the phase-shifted PWM signal is shown in Figure The on-chip analog comparator compares the primary current of the transformer with the peak current reference of slope compensation. The comparator output is connected to the PWM generator.
The ePWM1 module is set up to run in the incremental and subtractive counting mode, while the other PWM modules run in the incremental counting mode. The driver chip is selected as UCC, which can provide high peak current for capacitive load. The synchronous rectifier driver circuit is shown in Figure The full-bridge switch tube on the original side of the transformer is also driven by UCC, but transformer isolation must be provided.
The primary current ip of the transformer passes through the current transformer and is sampled by the resistor to obtain a voltage signal, which is sent to the AD sampling port. Figure 13 is the primary current sampling circuit of the transformer.
The output voltage sampling circuit is shown in Figure The poststage circuit is connected with a voltage tracker whose amplification factor is 1, and the operational amplifier chip adopts OP27GS.
The main experimental parameters are shown in Table 1. The driving circuit, the detecting circuit, and the main circuit are as shown in Figure Figure 17 shows the drain-source voltage and driving voltage waveforms of the lagging arm before and after dead-time adjustment under light load. It shows that the soft switch with wide range can be realized by adding automatic dead-zone technology. The relationship between load and efficiency under different input voltages are shown in Figure Dynamic dead-time control technology is proposed, which realizes a wide range of soft-switching, reduces switching losses and improves power supply efficiency.
The data used to support the findings of this study are available from the corresponding author upon request. Authors acknowledge the assistance and encouragement from colleagues, financial support by Henan mine power electronics device and control innovative technology team Science and Technology Planning Project of Henan Province of China, Grant No.
This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Article of the Year Award: Outstanding research contributions of , as selected by our Chief Editors. Read the winning articles. Journal overview. Special Issues. Guest Editor: Morris Brenna. Received 21 Jul Accepted 24 Oct Published 24 Dec Abstract Vehicle charging power supply is widely used because of its small size and portability. Introduction The electric vehicle has many advantages, such as high energy utilization rate, no pollution, and so on [ 1 ]. Figure 1. Figure 2. Waveform of primary current of the transformer before and after disturbance.
Figure 3. Figure 4. Duty cycle change caused by output voltage error disturbance. Figure 5. Table 1. Figure 6. Figure 7. Figure 8. However, the on-line calculation of explicit trigonometric functions implies a greater consump- tion of resources. Therefore, it is interesting to improve the accuracy of the implementations based on two integrators by increasing the order of the approximation, as proposed in the fol- lowing.
The second-order Taylor series that appear in 3. It should be noted that only even values of n T are possible. If the correction 3. In the same manner, the application of correction 3. It can be appreciated that the deviation increases with Ts and h f1 , and it is kept constant when the product of both terms is the same i. Each surface represents a different order n T for the Taylor series approximation in 3.
A great improvement can be appreciated with respect to the original second-order expression, when higher values of n T are employed. TABLE 3. Table 3. From Table 3. That would be also the case if low resonant frequencies were required in combination with lower fs for example, in order to reduce losses in high-power converters [—]. It should be remarked that these results can be also applied to other implementations, apart from those based on two integrators, in which 3.
Note that this problem does not only apply to implementations with two integrators. This difference explains why GVPIh z becomes unstable at high frequencies, as reported in [56, 57], if delay compen- sation is not included.
However, this approach is not developed in this study due to the fact that it would add a significant complexity to the original schemes based on two integrators. However, as illustrated in Fig. Therefore, an alternative expression should be sought to provide a more accurate compensation of the plant delay for PR controllers. It can be observed in Fig. Nevertheless, 3. The diagrams of negative frequencies are symmetrical of these.
As depicted in Fig. In any case, it should be remarked that delay compensation is more critical for high frequencies [56—58, 70, , ]. The closed-loop Bode diagram depicted in Fig. The same parameters of Fig.
This implies that individual phase mar- gins can be defined for each resonant controller. Note the multiple 0 dB crossings multi- ple phase margins. The second 0 dB crossing is the one considered because it is more critical. Figure 3. On the other hand, 3. It is also interesting to note that, from 3.
Therefore, in order to avoid uncertainties that could lead to an unstable system or a poor performance, it is preferable to seek for an alternative method to perform the discrete-time delay compensation.
Equation 3. In fact, it Fig. This fact explains the leading angle inaccuracy of 3. The numerator in 3. Furthermore, it can be observed that both Fig. Nevertheless, the variation of the reference signal frequency is less critical for the delay com- pensation techniques than for the resonant peak locations.
A small deviation in the resonant frequency leads to an important steady-state error, but slight errors in the actual phase margin with respect to its expected value are not so relevant. Large frequency deviations should occur in order to cause a substantial phase margin degradation. Therefore, the trigonometric expressions of Fig. It is possible to obtain a much more accurate simplification than 3. Finally, if very low frequency deviations are expected, it may be convenient to implement Fig.
In this manner, zeros are calculated off-line and implemented as fixed coefficients. It can be noted that Fig. By comparison of Figs. Actually, from Fig. It should be also remarked that, due to the explicit trigonometric terms, the error curves shown in Figs.
On the other hand, if Figs. Furthermore, it can be appreciated in Fig. Therefore, these advantages of Figs. Only the tuning values of the resonant controllers are different. Identical gains have been chosen for each harmonic order to achieve similar bandwidth for all values of h.
In this manner, as the selectivity becomes independent of the harmonic order, it will be possible to assess in the experiments the dependence of the resonance deviation on h, which is one of the main objectives of the experiments. The delay compensation technique proposed in Fig. Note that u, y, x, w1 and T represent the input of the controller, the output, the internal variables, the fundamental frequency and the inputs for the Ch variable c in the code correction as shown in Fig.
The multiplications by the sampling period have been relocated to different points with respect to those shown in Fig. The accuracy provided by n T greater than 6 may become more significant for higher res- onant frequencies or lower fsw. However, it should be taken into account that compensating beyond such high frequency values or such frequencies with lower fsw may be problematic due to appearance of side-band harmonics [70].
The average execution times of the current controller GC z , with different values of n T , are shown in Table 3. Bearing in mind that there are 23 resonant controllers involved, it can be appreciated that the calculation time increase with small variations of n T is not significant specially if it is compared with the great improvement achieved regarding steady-state error.
This includes the proposed target leading angle expression 3. Consequently, the results shown in Fig. The same experiment has been performed with Figs. On the other hand, the original delay compensation technique based on two integrators shown in Fig.
This fact proves that the stability margins provided by the proposed approaches are indeed larger, allowing for stable operation in more demanding conditions. However, Fig. Thus, the effectiveness of Fig. The ac voltage source performs a f1 transient from 25 Hz to 90 Hz in ms. Obviously, so great deviations from nominal frequency are not usual in grid-connected con- verters, and so many resonant controllers are usually not required either.
The main reason to test resonant controllers tuned at such high frequencies and so large frequency variations is to serve as an useful test for applications in which both requirements are common, such as aero- nautic APFs, ac drives or torque ripple minimization in high speed permanent magnet drives.
The objective is to confirm that, in those cases, the proposed Fig. It can be ob- served that Fig. It can be appreciated that at these fre- quencies the resonant controllers are also able to achieve a very high rejection of the harmonic components.
The former provided a similar behavior to 3. These proposals provide a significant improvement over the previously existing schemes based on two discrete integrators, in terms of lower steady-state error and larger stability margins, while their low computational burden and good frequency adaptation are maintained. The resonant frequency error is reduced by means of a correction of the input frequency.
In this manner, the order of the Taylor series approximation is increased. It is proved that a fourth-order expansion is a very adequate solution for tracking frequencies such that the rela- tion between resonant frequency and sampling frequency is not really large until the former becomes ten times smaller than the latter, approximately.
These improvements lead to wider stability margins and less sensitivity to frequency deviations around the resonant frequencies.
Finally, experimental results obtained with a laboratory prototype have validated the main outcomes of the theoretical approach and the improvement provided by the proposed digital implementations of resonant controllers. Contributions of this chapter have been published in the journal IEEE Transactions on Power Electronics [5] and presented at an international conference [16].
The analysis and design of PR controllers is usually performed by Bode diagrams and phase margin criterion. However, this approach presents some limitations when resonant frequencies are higher than the crossover frequency defined by the proportional gain. This occurs in selective harmonic control and applications with high reference frequency with respect to switching frequency, such as high-power converters with low switching frequency.
In such cases, ad- ditional 0 dB crossings phase margins appear, so the usual methods for simple systems are no longer valid.
Additionally, VPI controllers always present multiple 0 dB crossings in their frequency response. In this chapter, the proximity to instability of PR and VPI controllers is evaluated and optimized by means of Nyquist diagrams. A systematic method is proposed to obtain the highest stability and avoidance of closed-loop anomalous peaks, as well as an improved transient response: it is achieved by minimization of the inverse of the Nyquist trajectory distance to the critical point, that is, the sensitivity function.
Fi- nally, several experimental tests, including an active power filter operating at low switching frequency and compensating harmonics up to the Nyquist frequency, validate the theoretical approach. In many practical situations, this is enough to achieve satisfying results. However, this method has some limita- tions in more demanding scenarios. When resonant controllers are needed to track frequencies higher than fc , two additional 0 dB crossings phase margins appear around each of these fre- quencies.
These extra stability margins may be much lower than the phase margin at fc , and hence lead to instability of the system. Consequently, the value of this phase margin is no longer representative of the whole system stability. This happens in high-power converters tracking har- monics while operating at low switching frequency, which is actually often required in order to reduce switching losses of semiconductor devices in high-power applications [—].
Furthermore, even in cases in which resonant frequencies are not that high with respect to the switching frequency, it may be also desirable to set a low fc in order to have very selective control; in that manner, the inverter rating can be reduced while tracking the most relevant harmonics, and greater robustness to parameter uncertainties is achieved [56, 57, 64, ]. Therefore, it is of paramount interest to establish an effective and simple method to perform the analysis and design of PR controllers in these situations.
Concerning vector proportional-integral VPI controllers, indications to tune them in order to obtain certain closed-loop bandwidth around each resonant peak have been exposed in [56, 57].
However, a method to measure and optimize the proximity to instability of VPI controllers has not been proposed. Whereas PR controllers only present more than one 0 dB crossing when higher resonant frequencies than fc are employed, the VPI controllers always exhibit this kind of complex frequency response.
Thus, the tuning of VPI resonant controllers also requires to seek for a more adequate method than the usual approaches involving Bode plots and the phase margin criterion. The maximum of the sensitivity function is known as sensitivity peak, which is a more compact and reliable indicator of stability than gain and phase margins []. In fact, it is proved in this chapter that the minimization of the sensitivity peak permits to achieve better results in resonant controllers rather than by maximizing the gain or phase margins, the latter of which is the most common approach [45, 51, 65, 68, 70, , , , , , , ].
Nyquist diagrams have been employed for resonant current controllers with delay compen- sation in [51, ]. In those works, the Nyquist plots are used to check the stability the critical point is not encircled when delay compensation is performed.
However, as shown in this chap- ter, the Nyquist diagrams are also a powerful tool to measure and optimize the actual proximity to instability not only to check if the system is stable or not. Consequently, it is also interest- ing to analyze the influence of each freedom degree on the plots, studying how they affect the sensitivity function, and to establish the optimum combinations.
A problem of resonant controllers regarding their closed-loop frequency response has been reported in [57]: due to non-compensated terms of the plant, undesired gain peaks may appear at frequencies close to the resonant frequencies. This fact may cause magnification of inter- harmonics found in the vicinity of integer harmonics e.
Moreover, it would also cause a significant amplification of integer harmonics, instead of unity gain perfect tracking , when small frequency deviations occur. In this manner, by minimization of the sensitivity function, not only stability is improved, but these anomalous peaks are also avoided.
It is proved that minimization of the sensitivity peak permits to achieve a greater performance and stability than maximizing the gain or phase margins. The proposed method is valid even when there are multiple 0 dB crossings. Finally, several experimental tests, including an APF operating at 2 kHz and compensating harmonics up to the Nyquist frequency, have been performed to validate the theoretical approach. It can be also obtained from 2.
In this manner, the analysis carried out in this chapter acquires a generic value. Actually, E z is the Z transform of the signal e shown in Fig. It can be concluded from 4. In simple systems where the open-loop frequency response is monotonically decreasing and it does not present abrupt changes, either gain of phase margins provide enough information about the stability.
However, in complex systems with more complicated trajectories these indicators do not give a reliable stability measurement. Resonant controllers define the behavior of the system within a very narrow band of fre- quencies [51, 58, 65, , , ]. Consequently, the stability of the whole system can be studied by separated analysis of the Nyquist diagrams that correspond to each individual reso- nant controller included in GC z. The well-known fact that KIh is directly related to this bandwidth around the resonance [64, ] is also corroborated.
In this manner, the following conclusions concerning stability of PR controllers can be extracted from inspection of the Nyquist diagrams. Figure 4. This is an important fact, since PMP is often considered to be the most important indicator of stability, even in PR implementations [65, 68, , , , , , ]. Thus, KP T should be tuned to achieve a tradeoff between fast transient response, selective filtering and large upper bounds of stability margins.
Nevertheless, it can be inferred from Fig. For instance, in Fig. On the other hand, the trajectories in Fig. Additionally, for each resonant controller, there is always a positive and a negative phase margin. For instance, Fig. This can be appreciated, for instance, in the situation depicted in Fig. In this manner, the optimum values for the freedom degrees of the whole GC z con- troller will result from the combination of the best parameters for the Nyquist plot of each resonant controller GPR d z.
This permits to achieve a significant improvement over the two samples phase lead. However, as proved in this chapter, there are other possibilities that allow for even better performance and stability than trying to compensate the delay of the plant, specially when KPT is large. This equation is quite relevant from a practical point of view. In this manner, the laborious trial and error process is avoided. This can be avoided by assuring that the crossover frequency is lower than a decade below the switching frequency i.
If this is not true, the Ts variable in 4. It should be also noted that the maximum KP T should satisfy the initial hypothesis that the plant can be modeled as an L filter in the range of frequencies to be controlled.
In this manner, 4. The most restrictive of the two values should be the one implemented in the final control. Note that G Figure 4. On the contrary, it can be seen in Fig.
These aspects permit to improve the stability margins with respect to those of PR controllers, and they also facilitate GC z design.
In a similar manner to the case of PR controllers, Fig. This explains why GVPIh z becomes unstable at high frequencies, as reported in [56, 57], if delay compensation is not included. As the gain becomes greater, two effects appear simultaneously: the range of frequencies affected by the resonance widens as stated in [56, 57] and the asymptotes move slightly away from the origin.
In any case, it can be appreciated in Fig. Note that the effect of Kh on the stability margins is negligible. These Kh values are high enough to obtain large bandwidths.
It can be seen that these Kh values lead to unnecessary large bandwidths, which are not required in most applications. Therefore, it is corroborated that in normal conditions the asymptotes pass very close to the origin.
Concerning transient response, its time is defined by each Kh. This can cause a significant magnifi- cation of inter-harmonics found in the vicinity of integer harmonics, or amplification of integer harmonics when small frequency deviations occur.
Furthermore, these undesired closed-loop resonances are also related to low damping in the resonant poles at those frequencies; transient response becomes more oscillatory as the anomalous peaks become greater [].
In the fol- lowing, a direct method to quantify the presence of closed-loop anomalous peaks by means of the sensitivity function is exposed. Thus, in order to establish a systematic approach to analyze the presence of anomalous gain peaks, the relation between the closed-loop response and S z should be studied.
A high S z does not necessarily mean a large CL z. It can be deduced from 4. This situation can be observed in the examples depicted in Figs. In Figs. It can be observed in Figs. This fact is quite compromising, because of the proximity between A and B. Finally, Figs. These diagrams are related to those of S z according to 4. Furthermore, the proximity to instability would be also reduced greater stability margin and transient response would be improved higher damping.
This is approached in the following section. This issue is approached in the following. From 4. This implies that the best stability margins and avoidance of anomalous closed-loop peaks [lowest S z around h f1 ] are achieved. However, for greater bandwidths, the effect of KP T GPL z should be also taken into account in order to obtain the largest distance to the critical point. On the contrary, if large fluctuations of h f1 may occur, other options such as low order approximations or look-up tables seem to be good alternatives.
According to 4. Because of the linearity of 4. TABLE 4. Its equivalent series resistance is RF. The dc-link voltage v dc is kept constant by an HP A power supply, so no fundamental current is required to maintain v dc. Table 4. This signal is converted to an analog voltage output and compared in an oscilloscope appropiately scaled with the actual current value i, which is sensed in the power circuit by means of a current sensor.
The digital control GC z includes two resonant controllers: one tuned at the fundamental frequency f1 of the ac source voltage v ac , and another one tuned at the resonant frequency h f1. The resonant controller used in Figs. When the reference frequency coincides with the resonant frequency h f1 , it can be appre- ciated in Figs. It can be also seen in Fig. The reference frequency has been modified around the nominal value h f1 to seek for the presence of anomalous gain peaks.
This situation is shown in Fig. Nevertheless, no appreciable increase over unity gain h h could be noticed in CL z. In the following experiment, shown in Fig. It should be noted that, besides the undesired high gain at Hz, this fact also reveals that the system may become easily unstable due to uncertainties or variations in GPL z parameters.
On the other hand, it has been checked that the proposed expression for the leading an- gle provides an effective avoidance of closed-loop anomalous gain peaks around the resonant frequency.
Therefore, it can be concluded that the proposal based on S z minimization provides a more satisfying performance and stability margins than the approaches aimed at maximizing the phase margin i. Note that the latter avoids the anomalous gain peak.
As expected, Figs. Particularly, Fig. In this manner, the high stability margins and avoidance of closed-loop anomalous peaks provided by 4.
An APF application at low switching frequency is se- lected because it is a very demanding situation, in which most frequencies to be tracked are higher than the first crossover frequency fc set by KP T. The constraint of setting a low switching frequency is actually a requirement in high-power converters, in which a small fsw is needed in order to reduce switching losses of semiconductor devices [—]. There are only some particular differences, which are exposed in the following.
As shown in Table 4. The tuning values of the resonant controllers are also different in this case: the proposed approach based on Nyquist diagrams and sensitivity function is employed. The nonlinear load employed in this experiment instead of the programmable load shown in Fig.
Its total harmonic distortion THD with respect to the fundamental component has been computed up to the harmonic 50, which resulted in The leading angles are adapted by means of a look-up table in the case of PR controllers and with a simple linear relation [according to 4. Note that the proposed design method effectively maximizes D z around each resonant frequency. In this manner, fc is more than a decade below the switching frequency, so there is no need to further reduce the proportional gain in order to achieve a satisfying filtering of the commutation harmonics.
It can be observed that the proposed design method effectively maximizes D z around each resonant frequency. The steady-state currents obtained with the prototype are shown in Fig. This fact is corroborated by the i S spectrum shown in Fig.
The APF transient response when the harmonic compensation is enabled is shown in Fig. The most significant harmonic during this transient is the fifth order component. This finds its explanation in the fact that the Nyquist trajectory is very close to the critical point for frequencies around Hz see Fig.
This may be interesting, for instance, in case of fluctuating loads. An effective maximization of D z is achieved at all frequencies by means of the proposed design method. Its harmonic spectrum, shown in Fig. It is also interesting to note that, while in the PR controller some harmonics were more dominant during transients than others, in VPI controllers the amplitudes of each harmonic decay with a similar speed.
This is due to the fact that, as shown in Fig. In [25] it was concluded that in the former the transient response was very dependent on the value of the resonant frequency, but in the latter it was quite independent. The effect of each freedom degree on the trajectories is studied, and their relation with the sensitivity function and its peak value is assessed.
It is proved that the minimization of the sensitivity peak permits to achieve a better performance and stability in resonant controllers rather than by maximizing the gain or phase margins.
A systematic method, supported by closed-form analytical expressions, is proposed to obtain the highest stability and avoidance of closed-loop anomalous peaks, as well as a significant improvement in transient response greater damping : it is achieved by maximization of the Nyquist trajectory distance to the critical point minimization of the sensitivity function.
Finally, several experimental tests, including an active power filter operating at low switching frequency and compensating current harmonics up to the Nyquist frequency, validate the theoretical approach.
Chapter 5 Conclusions and Future Research 5. Its main contributions and conclusions are summarized below. The optimum discrete-time implementation alternatives are assessed, in terms of their influence on the resonant peak location capability of achieving zero steady-state error and phase versus frequency response stability.
It is proved that some popular implementations, such as the ones based on two interconnected integrators, cause a large steady-state error unless the quotient between resonant frequency and sampling frequency is kept low enough. Discretization methods such as Tustin with prewarping and impulse invariant are demon- strated to be more advantageous due to their resonant frequency accuracy and positive influence on stability.
These enhanced schemes achieve higher performance by means of more accurate resonant peak locations and delay compensation, while maintaining the advantage on low computational burden and good frequency adaptation of the original ones. The effect of each freedom degree on the Nyquist trajectories is studied, and their relation with the distance to the critical point inverse of the sensitivity function is established.
A systematic method, supported by closed-form analytical expressions, is proposed to obtain the highest stability and avoidance of closed-loop anomalous peaks, as well as an improved transient response greater damping : it is achieved by minimization of the sensitivity function and its peak value.
The contributions of each paper are summarized in the following. Particularizations of the contributions of this thesis to this ap- plication can be studied, with special emphasis on the issues associated to the implemen- tation of resonant controllers in a fundamental positive-sequence synchronous reference frame.
Due to the need for tracking several frequencies and sequences simultaneously, it seems to be a promising field for application of resonant controllers.
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Choi, and W. Blasko and V. Dannehl, F. Fuchs, and P. Twining and D. Shen, X. Zhu, J. Zhang, and D. Dannehl, M. Liserre, and F. Teodorescu, and F. You're going to remove this assignment. Are you sure? Yes No. Additional information Data set: ieee. Publisher IEEE. Fields of science No field of science has been suggested yet.
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