Fast solver to get steady-state waveformsfor power converter design

Conference: PCIM Europe 2018; Published in: International Exhibition and Conference for Power Electronics, Intelligent Motion, Renewable Energy and Energy Management, Nuremberg, Germany, 2018, pp. 1-7

Guillaume Fontes, Power Design Technologies, France
Regis Ruelland, Power Design Technologies, France
Alvaro Morentin, Power Design Technologies, France
Thierry Meynard, Power Design Technologies, LAPLACE, Université de Toulouse, CNRS, INPT, UPS, France
Guillaume Delamare, Power Design Technologies, France
Nicolas Videau, Power Design Technologies, France
Adel Ziani, Power Design Technologies, France

Abstract

Designing a power converter requires a good prediction of electrical quantities. Steady-state waveforms are needed but using a time-domain simulator to determine accurately these steady-state waveforms is generally too slow, so the designer usually uses a simplified analytical formulation to estimate these electrical quantities. This is not always accurate, must be repeated for each new circuit, and can be tedious for complex topologies. In this paper we show that a frequency-domain solver based on a Modified Nodal Analysis of the circuit can be used to obtain accurate results in little time even for a complex circuit, which makes it an efficient building block for a design tool. Accuracy and speed is demonstrated using comparisons with the steady-state analysis of Plecs® – which relies on a time-domain solver.

1. Introduction

Power converter design is an important challenge in the industrial field, but few software tools can really assist designers in the “operational” design step. The need for new tools has been clearly stated several times[1]. In a power converter design process, determining the main electrical variables (average, maximum and RMS values) and steady-state waveforms of the circuit is usually made by the designer with analytical formulas.This must be done for each new circuit and may require assumptions to simplify the description of complex topologies, which reduces the accuracy. Using a time-domain solver could increase the accuracy, but in a design process it may be used iteratively to explore the design space and the computation time multiplied by the number of iteration might be unacceptable. To face the issue, frequency modelling of power converter has been studied these last years to get electrical quantities in little time: [2] introduced a frequency solver to get steady-state waveforms of powerconverters based on commutation cells with hard switchingand [3] studied the modeling of power converter topologies based on rectifier bridges.

Therefore,in this paper we detail the frequency solver adapted to commutation cells and based on the modified nodal analysis(MNA). It provides a generic method to get steady-state waveforms of power converters based on commutation cells with hard switching.

The first part of this paper deals with the principle of the proposed solver: a frequency solver adapted to the commutation cell. The second part deals with the principle of the modified nodal analysis and frequency resolution used in the proposed solver. The third part shows a performance evaluation of the proposed solver thanks to a comparison with the time-domain solver of the well-known simulation tool Plecs®[4]. At last, the fourth part deals with the useof the proposed solver in a computer-aided engineering platform dedicated to power converter design.

2. Principle of the frequency solver adapted to the commutation cell

The proposed frequency solver is based on the modified nodal analysis -described insection 3- and adapted to the commutation cell which builds most of power converters.

2.1. Commutation cell model

To be used with the frequency solver, the commutation cell is modelled by a current source 𝐼𝐻𝑉 at high voltage side and a voltage source 𝑉𝐿𝑉at low voltage side, forming two independent circuits as shown in Fig. 1.

Fig. 1 : Switching cell model

The values of current source 𝐼𝐻𝑉 and voltage source 𝑉𝐿𝑉 are analytically computed from:

𝑉𝐿𝑉(𝑡)=𝐷(𝑡)∙𝑉𝐻𝑉(𝑡) 𝐼𝐻𝑉(𝑡)=𝐷(𝑡)∙𝐼𝐿𝑉(𝑡)

Where 𝐷(𝑡) is the duty cycle of the commutation cell. Possible high frequency ripples in current 𝐼𝐿𝑉 and voltage 𝑉𝐻𝑉 are neglected to compute current 𝐼𝐻𝑉 and voltage 𝑉𝐿𝑉.

2.2. Main process

To illustrate the main process,the circuit of a simple dc-dc chopper is considered in Fig. 2, using the commutation cell model.

Fig. 2: Considered circuit with the commutation cell

The full process can be summarized with three steps shown in Fig. 3

  • Frequency spectrums of current 𝐼𝐻𝑉and voltage 𝑉𝐿𝑉are first obtained thanks to the Fast Fourier Transform (FFT) algorithm.
  • Then, the MNA resolution is applied on all frequencies of the spectrum and the different voltages or currents of the circuits (𝐼𝐿𝑉,𝑉𝐻𝑉,𝑉𝑐ℎ,𝐼𝑏𝑢𝑠…)are determined.
  • At last, the Inverse Fast Fourier Transform (IFFT) algorithm can be used to derive the time-domain quantities at both sides of commutation cell


Fig. 3: Frequency resolution adapted to the commutation cell

3 Principle of the Modified Nodal Analysis and frequency resolution

3.1. Principle of the modified nodal analysis

The Modified Nodal Analysis[5](MNA) isan equation formulation for circuits, which is based on Kirchoff’s voltage and current laws. Spice[6], [7] is a well-known circuit simulator based on this MNA. MNA applied to a circuit with passive elements, independent current and voltage sources and active elements results in a matrix equation of the form [𝐴]∙[𝑋]=[𝑍] such as:

Where [𝑋] is a vector which holds the unknown quantities (node voltages V and currents J). Any linear equation system solver can solve this system for [𝑋].

3.2. Frequency resolution

The principle of this circuit resolution is adapted to power converter circuit: it is applied to the spectrum of the different voltages and currents of the circuit. Thus, the matrix systemof(2) is builtfor each considered frequencyas illustrated in Fig. 4(a), gathered as a single matrix as shown in in Fig. 4(b), and then solved. Thereforethe [𝑥]vector holds the spectrums of the electrical quantities of the circuit.

Fig. 4: MNA Matrix system built for all considered frequency

4. Performance evaluation

4.1. Considered circuit

To evaluate the accuracy of this frequency resolution, a circuit composed of an inverter leg and a R-L-Vload is considered and shown in Fig. 5.

Fig. 5: Considered circuit composed of an inverter leg

The different parametervalues used are shown in Table 1.

Parameter nameValue
𝑉𝐻𝑉750 V
𝑉𝑅𝑀𝑆230 V
𝑓50 Hz
𝐼𝑅𝑀𝑆20 A
𝐿5 mH
𝑅0.01 Ω

Table 1: Parameter values used

4.2. Results and comparison

The results obtained with the proposed frequency solver are compared with the steady-state analysis ofPlecs®.The frequency solver bandwidth is 1024 kHzand thetime-domain solverusesa variable time-step.For this example, the Plecs®analysisran in 6.1645 sand the frequency resolution in 0.404s,more than 15times faster.

Fig. 6 and Fig. 7 present waveform and spectrum comparisons of 𝑖𝐻𝑉 and 𝑖𝐿𝑉 currentsfor both solvers showing good matching. As explained above in 2.1, the ripple of 𝑖𝐿𝑉 is neglected when determining the waveforms atthe high-voltage side and we have a significant difference on the time domain waveform of current 𝑖𝐻𝑉 for example, as shown in the zoom of Fig. 6(a). However, differences of the spectrum of 𝑖𝐻𝑉 current in the frequency domain are very small as shown in Fig. 6(b). Table 2 shows the harmonic magnitudes of 𝑖𝐻𝑉 current obtained with Plecs® and the proposed frequency solver with a maximum relative error of 1 %.

Yet, when it comes to filter design, accuracy in the frequency domain is certainly what matters, so this formulation offers a good accuracy/computation time ratio, i.e. a very good performance/cost ratio in the context of converter design.

(a) Waveform of current 𝑖𝐻𝑉
(b) Frequeny spectrum of current
Fig 6: Waveform and spectrum of current

(a)Waveform of current 𝑖𝐿𝑉
(b)Spectrum of current 𝑖𝐿𝑉
Fig. 7: Waveform and spectrum of current 𝑖𝐿𝑉

Frequency [Hz]Plecs®Frequency solverRelative error
10011.6011.590.001
40008.548.500.005
80003.053.040.005
120001.131.120.01

Table 2: Harmonic magnitudes of 𝑖𝐻𝑉 current: PLECS® and frequency solver

5. Application to a power converter design tool

The proposed solver is implemented in a computer-aided engineering platform dedicated to power converter design: Powerforge®[8]. This platform proposes to help users design power converters including multilevel topologies as a freedom degree. Main steps or the design process are given in 5.1. Two conversion stage examples are then considered: a non-isolated dc-dc converter (Fig. 8) with specifications in Table 1 Table 3 and a three-phase inverter(Fig. 9) with specifications in Table 4. For each conversion stage, the CPU time (on an Intel Core I5 1.6 GHz with 8 Go of RAM) needed to perform the design and the semiconductor losses will be highlighted.

Fig. 8: Circuit of a multilevel non-isolated dc-dc converter
𝑈𝐻𝑉540 V
𝑈𝐿𝑉200 V
𝑃𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟10 kW
𝑓10 kHz

Table 3: Specifications of the non-isolated dc-dc converter

Fig. 9: Circuit of a multilevel three-phase inverter
𝑈𝐷𝐶750 V
𝑈𝐴𝐶(𝑅𝑀𝑆𝑝ℎ𝑎𝑠𝑒−𝑝ℎ𝑎𝑠𝑒)380 V
𝑃𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟10 kW
𝑓050 Hz
𝑓𝑠𝑤10 kHz

Table 4: Specifications of the three-phase inverter

For each conversion stage, three designs were performed with respectively three semiconductor references leading to three different topologies. These semiconductor references and topologies are presented in Table 5 and Table 6 respectively for the non-isolated dc-dc converter and the three-phase inverter.

Semiconductor referenceTopology
SCT30N1203-level, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 2
SCT2160KE5-level, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 4
IKB20N60H37-level, 𝑛𝑐𝑒𝑙𝑙(𝑓𝑐) = 2, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 3

Table 5: Semiconductor references vs topology for the non-isolated dc-dc converter

Semiconductor referenceTopology
C2M0045170D2-level
C2M0040120D3-level, 𝑛𝑐𝑒𝑙𝑙(𝑓𝑐) = 2
SCT3080KL3-level, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 2

Table 6: Semiconductor references vs topology for the three-phase inverter

5.1. Main steps of the design process

From specifications(voltage, power levels, ambient temperature, filtering requirements…) and user choices (semiconductor reference, switching frequency, materials), the main steps arepresented in Fig. 10

  • Design of the multilevel switching cell: determination of the number of cells in parallel 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) and of the number of cells in series 𝑛𝑐𝑒𝑙𝑙(𝑓𝑐)[9]
  • Design of the filters
  • Design of the cooling device
  • Performance evaluation
  • Constraintcheck(maximum junction temperature, filtering requirements…)
Fig. 9: Circuit of a multilevel three-phase inverter

5.2. CPU time

Table 7 shows the different CPU time spent to design several multilevel topologies in the case of a non-isolated dc-dc converter. As these times are very short, it leaves room to explore different options to find the most relevant design regarding user requirements.

Multi-level topologyCPU Time
Non-isolated DC-DC3-level, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 20.15 – 0.2s
5-level, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 40.25 – 0.3s
7-level, 𝑛𝑐𝑒𝑙𝑙(𝑓𝑐) = 2, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 30.3s
Three-phase inverter2-level2.7 – 4.4s
3-level, 𝑛𝑐𝑒𝑙𝑙(𝑓𝑐) = 23.3 – 4.4s
3-level, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 25.2 – 6s

Table 7: CPU time todesign different multilevel topologies

5.3. Loss comparison

Table shows the comparison of semiconductor losses computed by Powerforge® and Plecs®. Differences are small enough to be considered negligible.

Multi-Level TopologyLosses in PowerForgeLosses in Plecs
Non-isolated DC-DC3-level, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 2137.2 W136.16 W
5-level, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 4181.5 W181 W
7-level, 𝑛𝑐𝑒𝑙𝑙(𝑓𝑐) = 2, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 3219.3 W219.25 W
Three-phase inverter2-level66.5 W67.4 W
3-level, 𝑛𝑐𝑒𝑙𝑙(𝑓𝑐) = 2109 W107.6 W
3-level, 𝑛𝑐𝑒𝑙𝑙(𝑝𝑎𝑟) = 249.8 W53.4 W

Table 8: Comparison of semiconductor losses between Powerforge® and Plecs®

6. Conclusion

The principle of a frequency solver adapted to the commutation cell has been introduced. It allows power electronics designer to get steady-state waveforms much faster than a time-domain solver and harmonic spectrum is obtained with a very good accuracy. At last, an application of this frequency solver to a computer-aided engineering platform dedicated to the design of power converters is presented. Some design examples have been proposed and show a very good compromise between the computation time and the accuracy of the losses which are computed in the semiconductors. References :

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