0 Introduction
In the 21st century, with the rapid development of power electronic devices such as IGBTs, frequency converters have been widely used in various industries. Traditional topologies mainly include two-level converters, diode-clamped multilevel converters, flying capacitor-clamped multilevel converters, and H-bridge cascaded converters. In recent years, based on these, many manufacturers and suppliers have derived numerous improved topologies to optimize product performance, driving the improvement of product performance and the development of the drive industry.
The two-level AC-DC-AC main circuit topology of a traditional power electronic device is shown in Figure 1. The three-phase AC power at power frequency is connected to the input side via a rectifier transformer, then rectified by a rectifier bridge to obtain DC voltage, which is then converted to the desired voltage by a controllable three-phase inverter circuit. The filter capacitor's function is to stabilize the DC voltage and filter it. However, the voltage across the capacitor fluctuates during actual operation; this fluctuation is called ripple voltage. Ripple voltage causes current to flow through the capacitor; this current is called ripple current.
Figure 1. Topology of AC-DC-AC converter circuit
Fig . 1TopologyofAC-DC-ACconvertercircuit
Excessive ripple voltage and ripple current can cause capacitors to overheat and reduce their lifespan. Therefore, in engineering applications, different capacitor models have specific specifications for ripple voltage and ripple current in their parameter manuals. This article focuses on the B43310 series capacitors manufactured by EPCOS, identifying several key factors affecting ripple voltage and ripple current. Simulation software is used to calculate the characteristics of these factors influencing ripple voltage and ripple current, providing a reference for capacitor selection in engineering projects.
Introduction to Filter Capacitors
1.1 Introduction to Equivalent Circuit
The equivalent circuit of the capacitor is shown in Figure 2, consisting of an equivalent resistance RSR, a capacitance Cd, and an equivalent inductance LESL connected in series. The equivalent resistance RSR is formed by connecting the equivalent resistances of the lead resistance, polarization loss, and ionization loss in series, while the equivalent inductance LESL is formed by connecting the lead inductance and the equivalent inductances on the two plates of the capacitor in series.
Figure 2. Equivalent circuit of capacitor
Fig . 2Equivalentcircuitofcapacitor
RESR increases with increasing temperature and decreases with increasing operating frequency. LESL has little effect at low frequencies but a significant effect at high frequencies.
During the simulation, the effect of temperature fluctuations on RSR is not considered, and RSR is taken as a constant value. At the same time, since the operating frequency is very small, the effect of LESR is ignored, thereby simplifying the simulation model.
1.2 Introduction to Capacitor Selection Methods
The size of the filter capacitor determines the stability of the bus voltage and output voltage, and also affects the ripple current, which in turn affects the temperature rise and service life. Too small a capacitance will lead to unstable DC bus voltage and output voltage; too large a capacitance will greatly increase the cost and size of the device.
The capacitance value is usually calculated based on the magnitude of the intermediate circuit current of the frequency converter, and the specific calculation formula is shown in Equation 1.1 .
Where Id is the rated load current; Udc is the DC bus voltage; Q is the stored charge; and t is the capacitor charging and discharging time. It is generally assumed that the system needs to remain powered off for one cycle without tripping; therefore, for power frequency voltage, t is taken as 20ms.
Simulation circuit design
A simulation model was built using Psim software, and the simulation circuit is shown in Figure 3. The three-phase power frequency supply is rectified to obtain a DC voltage Vdc, which is then converted into a PWM waveform by a controllable inverter and applied across the three-phase symmetrical RL load. The 380V power frequency three-phase voltage is rectified to obtain a 540V DC voltage, which is then converted by SVPWM and applied to the three-phase symmetrical resistive-inductive load.
First, the capacitor capacitance is calculated. We specify the system power P = 55kW. In Equation 1.1 , Udc = 540V, t = 20ms, and Id = P/Udc ≈ 102A. Therefore, C = 3778μF. Referring to the B43310 capacitor model, the capacitance should be 4700μF.
The power factor cosφ = 0.75 , therefore the apparent power S = P/cosφ, and the apparent power S = 3 * Uphase * Iphase. The phase voltage Uphase = 220V, so the phase current Iphase = (1000/9) ≈ 111.11A . Therefore, based on P = 3 * I² * R and tanφ = (ωL/R), the parameters selected under power frequency conditions are R = 1.485Ω and L = 4.165mH . To ensure that the active power and power factor meet the requirements, the resistance and inductance parameters need to be adjusted under different transformer leakage inductance conditions. The goal of the adjustment is to ensure that the load active power and power factor meet the requirements when the modulation ratio is 1 under these leakage inductance conditions.
Figure 3 Simulation circuit topology diagram
Fig . 3Topological structure of simulation circuit
3 Simulation Calculation Results and Analysis
The simulation results are divided into four parts . Sections 3.1-3.3 do not consider ESR and perform simulations by treating the capacitor as a pure capacitor, obtaining the effects of leakage inductance, sampling frequency, and modulation ratio on ripple voltage and equivalent ripple current. Section 3.4 considers ESR and obtains the effects of the above factors under this condition, comparing them with the case where ESR is not considered.
3.1 The Influence of Leakage Inductance on Ripple Voltage and Ripple Current
The carrier frequency was set to 4000Hz, and the modulation ratio was set to 1. The leakage inductance was set to 7%, 5%, 3%, 1%, and no leakage inductance, respectively. Taking 7% as an example, the actual leakage inductance of the transformer was 0.6238mH . The ripple voltage waveform across the filter capacitor at this time is shown in Figure 4. Its maximum value is 492.34V , and its minimum value is 486.29V , with a difference of 6.06V between the maximum and minimum values. The leakage inductance was adjusted to 5%, i.e., 0.4456mH . The ripple voltage waveform at this time is shown in Figure 5.
Figure 4 shows the ripple voltage on the filter capacitor when the transformer leakage inductance is 7%.
Fig . 4Ripplevoltageoffiltercapacitorwhenleakageinductanceoftransformeris7%
Its maximum value is 500.10V , and its minimum value is 492.1V , with a difference of 8.0V between the maximum and minimum values. Taking this as an example, if the transformer leakage inductance is set to 3% and 1% respectively, the difference between the maximum and minimum values will be 12.03V and 29.3V respectively.
Figure 5 shows the ripple voltage on the filter capacitor when the transformer leakage inductance is 5%.
Fig . 5Ripplevoltageoffiltercapacitorwhenleakageinductanceoftransformeris5%
Figure 6 shows the trend of ripple voltage variation with leakage inductance. It can be seen that as the leakage inductance increases, the ripple voltage fluctuation across the capacitor decreases.
Figure 6. Curve of ripple voltage fluctuation as a function of leakage inductance
Fig . 6Variation of ripple voltage with leakage inductance
Taking a 7% leakage inductance as an example, the ripple current flowing through the filter capacitor is shown in Figure 7. To further analyze whether the ripple current meets the requirements, it is necessary to perform FFT analysis on the ripple current at the power frequency of 50Hz, and then convert the ripple current components at each frequency to 120Hz.
Figure 7 shows the ripple current on the filter capacitor when the transformer leakage inductance is 7%.
Fig . 7RipplecurrentofthefiltercapacitorWhentheleakageinductanceis7%
The FFT transformation result of the ripple current is shown in Figure 8. It can be seen that the main current components are concentrated in the range of 0~50000Hz. In order to convert the current, it is necessary to know the frequency conversion factor corresponding to different frequency components. Then, the current component needs to be divided by the conversion factor to obtain the current component at 120Hz, and then the squares are summed for calculation.
Figure 8. Fast Fourier Transform (FFT) of ripple current waveform.
Fig . 8FFTofripplecurrentwaveform
According to the B43310 parameter manual, the frequency conversion factor IAC , f/IAC , and the variation curves at 120Hz for different frequency components are shown in Figure 9.
Figure 9 shows the frequency conversion factor variation curve of the B43310 series capacitors.
Fig . 9FrequencyconversionfactorofB43310seriescapacitor
The FFT results can be approximated using the curve characteristics. For example, the conversion factor is 0.2 when the frequency is between 0 and 10 Hz, and 0.3 when the frequency is between 10 and 15 Hz. After processing the data from the FFT results, the equivalent ripple current converted to 120 Hz is 32.1 A.
Similarly, the leakage inductance is set to 5%, 3%, and 1% respectively, and the equivalent ripple current waveform is shown in Figure 10.
Figure 10. FFT results corresponding to different leakage inductances.
Fig . 10FFTresultscorrespondingtodifferentleakageinductance
As shown in Figure 10, with the increase of leakage inductance, the low-frequency component of the current gradually decreases, while the current frequency distribution range becomes wider. After conversion using FFT, the equivalent ripple currents are 34.51A , 41.84A , and 79.77A , respectively, and the trends are shown in Figure 11.
Figure 11. Equivalent ripple current variation under different leakage inductances.
Fig . 11Equivalentripplecurrentatdifferentleakageinductance
The above analysis shows that when the load consumes 55kW of active power, the power factor is 0.75 , and the three-phase output PWM modulation ratio is 1, the larger the transformer leakage inductance, the smaller the ripple voltage fluctuation, and the smaller the equivalent ripple current at 120Hz. In the extreme case where the leakage inductance is 0, although the ripple voltage fluctuation range is still within 10% of the average bus voltage, the equivalent ripple current at 120Hz can reach over 110A.
3.2 The Influence of Carrier Frequency on Ripple Voltage and Ripple Current
With the transformer leakage inductance set to 5% and the modulation ratio set to 1, the carrier frequencies were set to 2000Hz, 4000Hz, 6000Hz, and 8000Hz, respectively, and the analysis method was the same as above. The ripple voltage fluctuations at different sampling frequencies can be obtained, as shown in Figure 12. Then, an FFT transformation was performed on the ripple current to convert the components at different frequencies, and the equivalent ripple current at 120Hz was calculated, as shown in Figure 13.
Figure 12. Ripple voltage fluctuation at different carrier frequencies
Fig . 12Ripple voltage at different carrier frequencies
Figure 13. Equivalent ripple current variation at different carrier frequencies
Fig . 13Equivalentripplecurrentatdifferentcarrierfrequencies
Therefore, it can be concluded that the carrier frequency has little effect on the ripple voltage fluctuation and equivalent ripple current on the filter capacitor.
3.3 Effect of Modulation Ratio on Ripple Voltage and Ripple Current
With the transformer leakage inductance set to 5%, the carrier frequency set to 4000Hz, and the modulation ratio set to the range of 0.1 to 1, the analysis method is the same as above. The ripple voltage fluctuation and equivalent ripple current changes under different modulation ratios can be obtained, and the variation characteristics of the curves under different leakage inductance conditions can be compared, as shown in Figure 14.
Figure 14. Ripple voltage and equivalent ripple current as a function of modulation ratio under different leakage inductance conditions.
Fig . 14Ripplecurrentandripplevoltageatdifferentmodulation
As can be seen from the figure, under the same leakage inductance conditions, the ripple voltage fluctuation of the filter capacitor increases with the increase of the modulation ratio M. However, in a longitudinal comparison, the increase of the transformer leakage inductance can reduce the voltage fluctuation of the filter capacitor.
For ripple current, for the same modulation ratio output, the larger the leakage inductance, the smaller the equivalent ripple current. However, for the same leakage inductance condition, when the modulation ratio increases, the equivalent ripple current reaches its maximum value at around 0.9 . Beyond this point, increasing the modulation ratio will lead to a decrease in the equivalent ripple current.
3.4 The Influence of ESR on Ripple Voltage and Ripple Current
The simulation results above are all based on the premise that the filter capacitor is equivalent to a pure capacitor. Under normal temperature conditions, the equivalent series inductance (ESL) of the capacitor is on the order of μH and can be ignored. Therefore, only the effect of the equivalent series resistance (ESR) is considered.
According to the parameter manual, the ESR of the B43310 series 400V/4700μF capacitor is about 40mΩ under normal operating temperature. We added it to the simulation circuit to observe its ripple voltage and equivalent ripple current variation characteristics, and compared it with that without considering ESR.
Figure 15 shows the characteristics of ripple voltage fluctuation and ripple current variation. It can be seen that the trend is similar to that without considering ESR, that is, under the same leakage inductance condition, as the modulation ratio M increases, the ripple voltage fluctuation of the filter capacitor increases. However, in a longitudinal comparison, increasing the transformer leakage inductance can reduce the voltage fluctuation of the filter capacitor.
For ripple current, for the same modulation ratio output, the larger the leakage inductance, the smaller the equivalent ripple current. However, for the same leakage inductance condition, when the modulation ratio increases, the equivalent ripple current reaches its maximum value at around 0.9 . Beyond this point, increasing the modulation ratio will lead to a decrease in the equivalent ripple current.
Figure 15 shows the variation of ripple voltage and equivalent ripple current with modulation ratio at ESR=40mΩ and different leakage inductance.
Fig . 15RipplecurrentandripplevoltageatdifferentmodulationwhenESR=40mΩ
Taking a leakage inductance of 7% as an example, the results of comparing the cases with and without considering ESR are shown in Figure 16.
Figure 16 Comparison of ESR ripple voltage and equivalent ripple current with and without ESR
Fig . 16RipplevoltageandripplecurrentwithorwithoutESR
As can be seen from the figure, the presence of ESR can increase the fluctuation of the ripple voltage across the filter capacitor, but has little effect on the equivalent ripple current.
4. Conclusions and Outlook
1) The larger the leakage inductance of the transformer, the smaller the ripple voltage fluctuation and the equivalent ripple current. The sampling frequency has little effect on the ripple voltage and the equivalent ripple current. However, the increase of the modulation ratio can increase the ripple voltage fluctuation. When the leakage inductance is small, the equivalent ripple current increases. When the leakage inductance is large, it first increases to the peak value and then decreases. However, during this process of first rising and then falling, the equivalent ripple current does not change much.
2) According to the parameter manual, when ESR is considered, the trends of ripple voltage and equivalent ripple current affected by the three factors are similar to those when ESR is not considered. However, when ESR is present, under the same external factors, the ripple voltage fluctuation is smaller, while the presence of ESR has little impact on the equivalent ripple current.
3) In summary, when selecting a capacitor, it is necessary to consider whether the ripple voltage fluctuation and equivalent ripple current under extreme conditions meet the requirements of the capacitor model. These extreme conditions refer to the ripple voltage fluctuation and equivalent ripple current when the actual transformer leakage inductance is at a modulation index of 1.
Simulation calculations reveal that even in the most extreme cases, the ripple voltage fluctuation remains below 50V, meeting the requirement that the fluctuation be within 10% of the average DC bus voltage. However, when the transformer leakage inductance is less than 3%, the maximum equivalent ripple current exceeds 80A, while when it exceeds 3%, the value is no greater than 40A – a significant difference that directly impacts the selection result. Therefore, transformer parameters should be rationally designed in practical applications. Furthermore, factors such as ambient temperature and capacitor aging affect the ESR and R value of the capacitor, but since their impact is minor, this model can serve as a theoretical basis for practical capacitor selection.
