Methodology for Designing Broadband DC Link Filters for Voltage Source Converters

Abstract

This paper presents a new methodology for the design process of DC ripple filters for voltage source converters. It focuses on fast-switching, wide-bandgap-material-based converters. Therefore, a wide frequency range of up to 100 MHz is taken into consideration during the whole process. Different tools like analytic calculations, time-domain modelling, and the finite element method are used for different tasks in order to generate a realistic model in terms of filter effect and reliability. The models are validated by small-signal measurements using a vector network analyser as well as realistic high-power tests. The contribution of this paper is to provide a tool for DC link filter design to estimate the filter efficiency and the current stress on the filter elements with a special focus on WBG hardware.

1. Introduction

For the design of a DC link differential mode (DM) disturbance filter, often called a DC ripple filter, of a voltage source converter (VSC), the main criteria are the current load on the filter and the remaining voltage ripple at the filter’s quiet port. Classically, the design process is based on analytic calculations to determine a filter that fulfils these criteria while also depending on the modulation scheme that is used [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17].

Not considered in the named calculations is the limitation of switching overvoltage for protection of the switching devices. Low-inductance capacitors are necessary for this task. In the past, the matching filter was mostly determined empirically [18]. Especially in wide-bandgap (WBG) devices, recent papers also analyse the voltage oscillation caused by the switching of the power device via mathematical models [19,20,21,22].

WBG devices allow for higher power densities and reduced filter component sizes. This enables and, with regard to the fast switching of these devices, at the same time requires compact system design [23,24,25,26,27]. According to [28], capacitors are the most common cause of errors in VSCs, along with various other components such as printed circuit boards, semiconductors, solder joints, and connectors. There are many different types of capacitors. For use as a DC-side filter capacitor, three different types are used [29]:

• Aluminum electrolytic capacitors:
  Electrolyte capacitors are mostly used in industrial applications. Their dielectric strength is limited to approx. 450V (although there are also special versions with a maximum applicable voltage of 600 V), which is why typical applications in the range 600–900 V usually use two capacitors connected in series. To increase the total capacity and/or current-carrying capability, several capacitors are usually connected in parallel. Compared to other types of capacitors, they achieve the highest energy density and the lowest cost per joule. However, the large space requirement due to the low current-carrying capability, wear and tear caused by the evaporation of the electrolyte solution, is disadvantageous. Furthermore, compared to the film capacitor, it has a much higher, strongly frequency-dependent ESR and thus a significantly lower efficiency and a lower current-carrying capability. Electrolyte capacitors are polarised and can only be used for DC voltage applications. The main areas of application are industrial applications.
• Multilayer ceramic capacitors:
  Multilayer ceramic capacitors (MLCCs) have smaller construction volumes in comparison to electrolyte capacitors, larger frequency ranges, and higher operating temperatures of up to 200 °C. Disadvantages, however, are higher costs and mechanical sensitivity. Parasitic effects like DC and AC bias dependence have to be taken into account [30,31,32,33,34,35,36]. They are often manufactured in surface mounted device (SMD) designs and are used for complex applications on printed circuit boards.
• Metallised film capacitors:
  Metallised Polypropylene film capacitors (MPPFCs) offer a balanced design for higher-voltage applications (e.g., over 500 V) in terms of cost and ESR, capacitance, ripple, current, and reliability. However, they have the disadvantages of a large volume and a moderate upper operating temperature. Parasitic effects like eddy currents and skin and proximity effects influence their filtering effect, especially for higher frequencies, and have to be taken into account [37,38,39,40,41,42,43]. Related to the current-carrying capability, the costs of MPPFCs are about one third of the costs for electrolyte capacitors. Because of this, they are preferred for applications with high distortion currents, such as in the drive train of electric vehicles [44].

For Si-based VSCs, where electrolyte capacitors are applied for DC ripple filtering, the switching overvoltage filter can be designed separately. Electrolyte capacitors feature comparably high ohmic damping [29], which reduces the risk of resonances with high resonant circuit quality. This statement cannot be applied to modern VSCs based on fast-switching wide-bandgap (WBG) devices. Especially when high switching frequencies are introduced, the tasks of switching overvoltage protection and DC ripple filtering overlap. Therefore, a holistic design approach has to be taken for filtering DC link DM disturbances.

This paper reflects the task of redesigning the DC link DM disturbance filter for a WBG-based VSC in order to improve its HF-filter efficiency. In the scope of this work, it is used as an example for explaining the model-based holistic design methodology, which is the key finding of this paper. The considered two-level VSC provides three phase legs that are connected to a common DC link, as can be seen in Figure 1.

Table 1. Parameters of the VSC used as an example.

Parameter	Value
Switching frequency range	10 kHz … 200 kHz
Current-carrying capability (RMS)	500 A
Maximum DC-terminal voltage	1.2 kV
Modulation scheme	Space vector modulation (SVM)

states the most important parameters for the design of the DC link DM disturbance filter.

2. Basic Filter Design

As mentioned before, the designated filter performance should be very broadband to fulfil different tasks:

• Low -frequency (LF) filter: Reduce the DC voltage ripple generated from the modulation scheme.
• High-frequency (HF) filter: Limit the switching overvoltage to a fixed level.

• Course of action:

The methodology presented is structured as a basic course of action, which is described in the following:

1.

First estimations based on analytic calculations.

2.

Component selection for LF filter.

3.

Time-based simulation model of VSC and filter.

4.

Component selection for HF filter.

5.

Determine impedance of mechanical components.

6.

Integration in time-based simulation model.

7.

Validation on experimental setup.

3. Exemplary Filter Design

3.1. First Estimations Based on Analytic Calculations

By using the analytic calculation described in [1], the necessary current-carrying capability of the capacitors for the LF filter can be determined. By using the analytic calculations from [2], the necessary capacity for the LF filter for a defined remaining voltage ripple $U_{\mathrm{C}1\sim }$ can be determined. Here, $U_{\mathrm{C}1\sim }=8\mathrm{V}$ is defined.

• This results in

• Current-carrying capability: $I_{\mathrm{C}1}=325\mathrm{A}$;
• Necessary capacity: $C_1=450\mathsf{\mu }\mathrm{F}$.

3.2. Component Selection for LF Filter

Depending on the application, electrolytic capacitors, film capacitors, or a combination of both may be preferable. Capacitors should have sufficient low impedance at the dominant amplitude of the VSC’s DC-side current amplitude spectrum. Depending on the modulation scheme and the topology, the current amplitude spectrum contains different dominant harmonic amplitudes [3,45,46,47,48]. They can be calculated analytically as described in [49] or be determined by simulation.

Here, simulation is used. Figure 2 shows the VSC’s DC-side current amplitude spectrum for switching frequencies $f_{\mathrm{s}}=10\mathrm{kHz}$ (a) and $f_{\mathrm{s}}=200\mathrm{kHz}$ (b), respectively. For the SVM modulation scheme, the dominant amplitude appears for $2·f_{\mathrm{S}}$. Different capacitors are compared regarding their impedance. The necessary data can be taken from manufacturers’ datasheets or be determined via impedance measurement.

Figure 3 shows the measured impedances of different capacitors, all with identical capacity. It becomes apparent that the impedance and so their filter efficiency strongly depend on parasitic attributes, namely, the parasitic inductance $L_{\mathrm{C}1}$. Results (A) and (B) are for capacitors with a cylindrical design. For result (C), the filter is built from several smaller capacitors with comparable low $L_{\mathrm{C}1}$. Here, 22 capacitors with a capacity of $22\mathrm{\mu }\mathrm{F}$ each are used. It can be seen that this filter has significantly lower impedance for $f>40\mathrm{kHz}$ and therefore improved filter efficiency. Because of these advantages, variant (C) is chosen to be used as the LF filter.

3.3. Time-Based Simulation Model Step I

Using a discrete-time model of the modulator, the DC-side current ripple $i_{\mathrm{E}\sim }$ of the VSC is defined according to (1). Equation (2) states that the DC-side current ripple for the top switch, $i_{\mathrm{E}\sim ,\mathrm{T}n}$ (with n representing the phases U, V, and W, respectively) is divided into the top switches’ transistor currents $i_{\mathrm{E},\mathrm{tT}n}$ and their diode currents $i_{\mathrm{E},\mathrm{dT}n}$. Equations (3) and (4) calculate the DC-side current ripple for the top switch $i_{\mathrm{E}\sim ,\mathrm{T}n}$ and for the bottom switch $i_{\mathrm{E}\sim ,\mathrm{B}n}$, respectively. $s_n$ represents the switching commandos for all three phase legs:

• $s_n=+1$: Top switch of phase leg n on, corresponding bottom switch off.
• $-1<s_n<+1$: Commutation between top and bottom switches.
• $s_n=-1$: Top switch of phase leg n off, corresponding bottom switch on.

As shown in Figure 4, the current sources for $i_{\mathrm{E}\sim ,\mathrm{T}n}$ as well as for $i_{\mathrm{E}\sim ,\mathrm{B}n}$ are ideally interconnected, respectively. This means that for model step I, the DC-side ripple current could be described by one single current source. The division into six different current sources is performed here in view of the more accurate simulation model step II, where this becomes necessary.

The switching behaviour of the power modules is approximated using a PT2s function with damping of 1 (aperiodic borderline case) and natural angular frequency of $5/t_{\mathrm{r}}$, with $t_{\mathrm{r}}$ being the rise-time of the power modules as stated in the datasheet.

$$\begin{array}{cc}\hfill i_{\mathrm{E}\sim }& =i_{\mathrm{E},\mathrm{TU}}+i_{\mathrm{E},\mathrm{TV}}+i_{\mathrm{E},\mathrm{TW}}-i_{\mathrm{E}\left(0\right)}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill i_{\mathrm{E}\sim ,\mathrm{T}n}& =i_{\mathrm{E},\mathrm{tT}n}+i_{\mathrm{E},\mathrm{dT}n}-{\displaystyle \frac{1}{3}}{i}_{\mathrm{E},n\left(0\right)}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill i_{\mathrm{E}\sim ,\mathrm{T}n}& =\lfloor {\displaystyle \frac{s_n+1}{2}}\rceil ·i_n+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\mathrm{d}{s}_n/\mathrm{d}t}{|\mathrm{d}{s}_n/\mathrm{d}t|}}+1\right)·\left|i_n\right|-{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{3}{4}}\sqrt{2}·I_{\mathrm{N}}·m_{\mathrm{a}}·cos\left(\phi \right)\right)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill i_{\mathrm{E}\sim ,\mathrm{B}n}& =\lfloor {\displaystyle \frac{s_n-1}{2}}\rceil ·i_n+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\mathrm{d}{s}_n/\mathrm{d}t}{|\mathrm{d}{s}_n/\mathrm{d}t|}}-1\right)·\left|i_n\right|-{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{3}{4}}\sqrt{2}·I_{\mathrm{N}}·m_{\mathrm{a}}·cos\left(\phi \right)\right)\hfill \end{array}$$

(4)

Figure 4 shows the diagram of the model for the three-phase system. The signal $s_n$ relates to the switching signals of the half-bridges. Capacitors are simulated by an RLC series resonant circuit. This is sufficient as long as the influence of skin and proximity effects as well as eddy current effects is negligible. Otherwise, more complex models need to be applied [42].

Figure 5 shows simulation results for the time-based simulation model step I. The pulsed DC-side current $i_{\mathrm{E}\sim }$ is filtered by the LF filter capacitor, which results in the remaining voltage ripple $u_{\mathrm{RC}1\sim }$. As there is no HF filter, the voltage ripple $u_{\mathrm{C}1\sim }$ shows high-frequency peaks with high amplitude which are neither calculated exactly nor considered in this design step. The simulation was carried out for the operating point with maximum current load at the LF filter capacitor when the analytic calculations are $m_{\mathrm{a}}=0,6$, $\phi =0°$ and $f_{\mathrm{S}}=10\mathrm{kHz}$. The RMS value of $i_{\mathrm{C}1}$ over one fundamental period is $325\mathrm{A}$, which exactly fits the analytic calculation for this operating point. The RMS value of $u_{\mathrm{RC}1\sim }$ is 5.6 V here. Although this operating point creates the maximum current load on the filter capacitor, the maximum voltage ripple according to the analytic calculations is reached for $m_{\mathrm{a}}=1.15$, $\phi =90°$, and $f_{\mathrm{S}}=10\mathrm{kHz}$.

3.4. Component Selection for HF Filter

For HF filtering, switching speed $(\mathrm{d}{i}_{\mathrm{E}}/\mathrm{d}t)$ is more relevant as it strongly influences the HF part of $i_{\mathrm{E}\sim }$. Comparing the amplitude spectrum of $i_{\mathrm{E}\sim }$ from Figure 2 with the chosen capacitor impedance (C) shown in Figure 3 indicates that filter performance needs to be improved for $f>1\mathrm{MHz}$. Therefore, ceramic capacitors with a series resonance frequency of $15\mathrm{MHz}$ are installed as close to the power modules as possible. In Figure 6, the impedance of one of these HF filter capacitors is compared to the chosen variant (C) for the LF filter. As can be seen, to significantly improve the filter performance for $f>1\mathrm{MHz}$, multiple of these HF filter capacitors are necessary.

3.5. Determine Impedance of Mechanical Components

Mechanical components like PCBs and busbars are necessary for electric connection and mounting of the filter components. This primarily introduces additional inductance in the DC link. Therefore, for a more reliable simulation, the parasitics introduced by the mechanical parts need to be considered. A field solver calculates the electrical parameters of the mechanical parts. Available commercial tools offer front-ends which simplify familiarisation with this topic. Here, a field solver using the boundary method is used. Using the boundary method, radiated emissions are neglected, and so no boundary conditions have to be set. On the other hand, this means the results become inaccurate when the radiated emission reaches relevant levels. As a rule of thumb, it can be stated that the wavelength of the maximum representable frequency should be greater than ten times the dimension of the simulated geometry. Care must be taken when placing the sinks and sources in the field solver tool to ensure that the current loops are considered correctly, as this has influence on the mutual inductance calculated. The results are implemented in the time-based simulation model step II.

3.6. Integration in Time-Based Simulation Model

Applying all the information gathered in Section 3.4 and Section 3.5 to the time-based simulation model results in the diagram shown in Figure 7. It allows determination of the current load on all components and switching voltage overshoot at any places relevant outside the power module. It takes into account the additional stray inductance due to the mechanical structure and the HF filter components. As the impedances of mechanical parts are determined by field solvers, this results in a grey-box model. This approach is also used successfully in the literature, as on the one hand the load on individual components can be determined and on the other hand it offers an efficient method for taking into account parasitic effects of mechanical components [50].

• Results for different HF filter configurations:

As the model is built from linear components, a linear analysis of the model is possible. Figure 8 shows the input impedance for the old, existing filter concept as well as the design result using the presented approach. Additionally, two different configurations of the HF filter capacitors are compared:

(a)

Existing DC link: HF filter consists of six film capacitors close to the half-bridge power modules. LF filter is built from electrolyte capacitors.

(b)

HF Filter Variant 1: 78 ceramic capacitors are close to the half-bridge power modules.

(c)

HF Filter Variant 2: 60 ceramic capacitors are set in series with low-inductance resistors for damping resonances.

To cope with requirements of insulation coordination, two ceramic capacitors have to be put in series, respectively. With the PCB as a carrier for all the HF filter capacitors, a low-inductive connection of the half-bridges of $4\mathrm{nH}$ could be realised. Comparing (a) and (b) in Figure 8 determines the improvement in high-frequency filter efficiency introduced by the new filter concept. The new filter concept, variant (b), shows a series resonance at approximately $7\mathrm{MHz}$. The time-based model shows that this resonance leads to relatively high current load on the ceramic capacitors. For variant (c), this current load is reduced as described before. It can also be seen by looking at Figure 8.

For quantification, time-based simulations are carried out. The simulation was carried out for an AC-side phase current of 50 A at a switching frequency of $10\mathrm{kHz}$ and a modulation level of 0.6. The results are stated in

Table 2. Simulation results to quantify different configurations.

	(a)	(b)	(c)
LF filter: Single capacitor peak current	-	5.5 A	5.1 A
HF filter: Single ceramic capacitor RMS current	-	0.7 A	0.6 A
HF filter: Single ceramic capacitor peak current	-	5.1 A	6.4 A
Voltage overshoot at power module	29 V	18 V	20 V

. As the existing DC link uses different types and numbers of capacitors compared to the presented new approach, a comparison of current load on the single capacitors is not applicable for variant (a).

For the maximum AC RMS current of 500 A, the values stated in Table 2 are about 10 times higher. The results quantify the improved HF filter performance of the new approach compared to the existing DC link. As can be seen, configuration (c) makes a compromise between voltage overshoot and current load on the filter components.

4. Validation of Simulation Model

For validation of the model, measurements were made on an experimental setup. It is presented in Figure 9.

Air coils are connected between the mid-points of the half-bridges for phases U and W, respectively, as AC-side load and the VSC is operated like a full-bridge converter. The VSC is powered via a DC power supply. As the newly designed filter variants (b) and (c) do not exist yet, the existing filter variant (a) is used here. To measure the current load of one of the film capacitors, a rogowski coil was used. It has an HF bandwidth of 30 MHz and can measure a maximum current rise of $2\mathrm{kA}/\mathsf{\mu }\mathrm{s}$. The current rise measured in the setup was $0.5\mathrm{kA}/\mathsf{\mu }\mathrm{s}$.

Figure 10 shows the results for both measurement and simulation at an identical operating point of the VSC. It can be seen that there is good agreement.

Further validation is performed for different operating points.

Table 3. Validation of simulation model for $F_{\mathrm{s}}=10$ kHz and different AC-side currents.

AC-Side RMS Current	Modulation Level	HF Capacitor Current Stress		Relative
$I_{\mathrm{N}}$	$m_{\mathrm{a}}$	Measurement	Simulation	Error
$20\mathrm{A}$	0.0176	1.1$\mathrm{A}$	0.7$\mathrm{A}$	+36%
$40\mathrm{A}$	0.0250	2.0$\mathrm{A}$	1.6$\mathrm{A}$	+20%
$60\mathrm{A}$	0.0308	3.0$\mathrm{A}$	2.5$\mathrm{A}$	+17%
$80\mathrm{A}$	0.0358	3.8$\mathrm{A}$	3.4$\mathrm{A}$	+11%
$100\mathrm{A}$	0.0400	4.7$\mathrm{A}$	4.4$\mathrm{A}$	+6%
$120\mathrm{A}$	0.0442	5.6$\mathrm{A}$	5.6$\mathrm{A}$	+0%
$140\mathrm{A}$	0.0475	6.5$\mathrm{A}$	6.7$\mathrm{A}$	−3%
$160\mathrm{A}$	0.0508	7.5$\mathrm{A}$	7.9$\mathrm{A}$	−5%
$180\mathrm{A}$	0.0542	8.4$\mathrm{A}$	9.2$\mathrm{A}$	−10%
$200\mathrm{A}$	0.0567	9.3$\mathrm{A}$	10.4$\mathrm{A}$	−12%
$220\mathrm{A}$	0.0600	10.0$\mathrm{A}$	11.7$\mathrm{A}$	−17%

shows the simulation as well as the measurement results for different AC-side RMS currents. The VSC is operated at a fixed switching frequency of 10 kHz. The results show good agreement for $60\mathrm{A}\le I_{\mathrm{N}}\le 200\mathrm{A}$. For $I_{\mathrm{N}}\le 60\mathrm{A}$, accuracy of the measurement equipment comes into consideration with $0.41\mathrm{A}$ for $I_{\mathrm{N}}=20\mathrm{A}$ ($2\%$ of reading for conductor position and $0.05\%$ of reading for linearity error). For $I_{\mathrm{N}}=220\mathrm{A}$, the maximum capable current of the rogowski coil used is reached.

Additionally, Figure 11 shows the comparison for different switching frequencies $F_{\mathrm{s}}$. It can be seen that the method presented shows valid results also for different switching frequencies.

Table 4. Validation of simulation model for different switching frequencies and different AC-side currents.

AC-Side RMS Current	Modulation Level	Relative Error
$I_{\mathrm{N}}$	$m_{\mathrm{a}}$	$F_{\mathrm{s}}=\mathbf{20}$ kHz	$F_{\mathrm{s}}=\mathbf{80}$ kHz	$F_{\mathrm{s}}=\mathbf{120}$ kHz
$20\mathrm{A}$	0.0176	+36%	+29%	+38%
$40\mathrm{A}$	0.0250	+40%	+10%	+30%
$60\mathrm{A}$	0.0308	+25%	+8%	+18%
$80\mathrm{A}$	0.0358	+18%	+6%	+9%
$100\mathrm{A}$	0.0400	+12%	+9%	+7%
$120\mathrm{A}$	0.0442	+6%	+8%	+5%
$140\mathrm{A}$	0.0475	+5%	+11%	−4%
$160\mathrm{A}$	0.0508	−2%	+8%	−3%
$180\mathrm{A}$	0.0542	−5%	+5%	+2%
$200\mathrm{A}$	0.0567	−6%	+6%	+1%
$220\mathrm{A}$	0.0600	−11%	+5%	0%

shows the relative error for the different operating points and switching frequencies from Figure 11. The deviations between simulation and measurement are in the same range for the different switching frequencies.

5. Non-Linear Loss Effects in Film Capacitors

In Figure 12a, a simple capacitor model is shown. It takes into account the voltage drop at the ESR of the capacitor and parasitic, voltage-dependent losses. The skin and proximity losses of the capacitor are taken into account by the inductance $L_{\mathrm{C}1}$ representing the ESL of the capacitor. The series connection of $L_{\mathrm{C}1}$ and $C_1$ forms a series resonant circuit whose resonance condition is specified in Equation (5). The frequency at which the series resonance occurs is called the series resonance frequency $f_{\mathrm{SR}}$.

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\omega C_1}}& =\omega L_{\mathrm{C}1}\mathrm{for}\omega ={\omega }_{\mathrm{SR}}=2\pi f_{\mathrm{SR}}|R_{\mathrm{C}1,\mathrm{P}}>>{\displaystyle \frac{1}{{\omega }_{\mathrm{SR}}{C}_1}}\hfill \end{array}$$

(5)

The limits of the influence of $L_{\mathrm{C}1}$ on the HF behaviour of the capacitor impedance and the need to implement further elements in the component model are discussed below. For this purpose, the result of an impedance measurement is presented. The measurement was carried out using a vector network analyser. Capacitor (A) from Figure 3 is measured here. To illustrate the effects of the observed phenomenon, real and imaginary parts of the measurement result are determined. These are presented in Figure 13. From the diagram it can be seen that the real part of the measured impedance increases for $100\mathrm{kHz}\le f\le 200\mathrm{kHz}$. The result for the simple model (Figure 12a) does not show this behaviour. Instead, the complex model (Figure 12b) addresses this effect.

The measured behaviour of the impedance is adequately described in the literature and can be attributed to the following phenomena, which are not sufficiently described by means of ESL [37]:

• Skin effect;
• Proximity effect;
• Eddy currents.

The cause of these phenomena is attributed to the magnetic fields generated at high frequencies in the windings of the capacitor [39,51]. The result of these magnetic fields are eddy currents which influence the current density distribution within the capacitor. This leads to impedance fluctuations and resonance effects at certain frequencies, whose position also depends on the geometry of the capacitor. Especially in the range of the capacitors’ resonance frequencies, a considerable increase in the ohmic internal resistance is expected. The visible consequence of this is the occurrence of local heating [41]. To take into account the named phenomena, according to [42], the model can be enhanced as follows:

1.

To map skin and proximity losses within the capacitor, the inductance at $L_{\mathrm{C}1}$ mapping the ESL is substituted. The skin losses are represented by a first-order high-pass filter consisting of the resistor $R_{\mathrm{C}1,\mathrm{Skin}}$ and the inductance $L_{\mathrm{C}1,\mathrm{Skin}}$. To account for the proximity losses, the inductance $L_{\mathrm{C}1,\mathrm{Prox}}$ is also implemented in the capacitor model.

2.

To represent the eddy currents and the additional losses caused by them, a second-order high-pass filter (parallel dipole) consisting of the inductance $L_{\mathrm{C}1,\mathrm{AR}}$, the resistance $R_{\mathrm{C}1,\mathrm{AR}}$, and the capacitance $C_{1,\mathrm{AR}}$ is implemented in the capacitor model.

The resulting model, addressing all these effects, is shown in Figure 12b. In the following, the equivalent circuit data of this model are determined. The parametrisation of the model is carried out in an exemplary fashion on the basis of the measurement results of the experimental setup generated by means of the network analyser.

The parameters of $R_{\mathrm{C}1,\mathrm{Skin}}$, $L_{\mathrm{C}1,\mathrm{Skin}}$, and $L_{\mathrm{C}1,\mathrm{Prox}}$ are determined based on the measurement results. In (7), the real and imaginary parts of ${\underline{Z}}_{\mathrm{C}1}$ are given. The values for the real and imaginary parts are taken directly from the measurement. They are chosen for $f>500\mathrm{kHz}$ to avoid the influence of the rapid increase in the real part introduced by eddy currents. In addition, the context given in (6) is applied.

$$\begin{array}{cc}\hfill L_{\mathrm{C}1}& =L_{\mathrm{C}1,\mathrm{Skin}}+L_{\mathrm{C}1,\mathrm{Prox}}\hfill \end{array}$$

(6)

The component values can be calculated by inserting the measured values for the real and imaginary parts as well as (6) into (7). If the resulting equations have more than one solution, the physically most sensible solution is chosen.

$$\begin{array}{cc}\hfill \Re \left({\underline{Z}}_{\mathrm{C}1}\right)& =R_{\mathrm{C}1,\mathrm{S}}+{\displaystyle \frac{R_{\mathrm{C}1,\mathrm{P}}}{C_1^2{R}_{\mathrm{C}1,\mathrm{P}}^2{\omega }^2+1}}+{\displaystyle \frac{L_{\mathrm{C}1,\mathrm{Skin}}^2{R}_{\mathrm{C}1,\mathrm{Skin}}{\omega }^2}{L_{\mathrm{C}1,\mathrm{Skin}}^2{\omega }^2+R_{\mathrm{C}1,\mathrm{Skin}}^2}}\hfill \\ \hfill \Im \left({\underline{Z}}_{\mathrm{C}1}\right)& =L_{\mathrm{C}1,\mathrm{Prox}}·\omega -{\displaystyle \frac{C_1{R}_{\mathrm{C}1,\mathrm{P}}^2\omega }{C_1^2{R}_{\mathrm{C}1,\mathrm{P}}^2{\omega }^2+1}}+{\displaystyle \frac{L_{\mathrm{C}1,\mathrm{Skin}}{R}_{\mathrm{C}1,\mathrm{Skin}}^2\omega }{L_{\mathrm{C}1,\mathrm{Skin}}^2{\omega }^2+R_{\mathrm{C}1,\mathrm{Skin}}^2}}\hfill \end{array}$$

(7)

Also shown in Figure 12 is the high pass formed by $C_{1,\mathrm{AR}}$, $L_{\mathrm{C}1,\mathrm{AR}}$, and $R_{\mathrm{C}1,\mathrm{AR}}$. This element is mainly relevant to address the rapid increase in the real part. The equivalent circuit diagram data of the high-pass filter are determined from the frequency response of the real part of the measurement result. The criteria taken from the measurement are the quality $Q_{\mathrm{AR}}$ and and the resonant angular frequency ${\omega }_{\mathrm{R},\mathrm{AR}}$. Here, $Q_{\mathrm{AR}}=1,1$ and ${\omega }_{\mathrm{R},\mathrm{AR}}=150·{10}^3\mathrm{rad}/\mathrm{s}$ was chosen. For $R_{\mathrm{C}1,\mathrm{AR}}$ the following applies:

$$\begin{array}{cc}\hfill R_{\mathrm{C}1,\mathrm{AR}}& =\Re \left({\underline{Z}}_{\mathrm{C}1}\left({\omega }_{\mathrm{R},\mathrm{AR}}\right)\right)\hfill \end{array}$$

(8)

For the $Q_{\mathrm{AR}}$ and ${\omega }_{\mathrm{R},\mathrm{AR}}$ results,

$$\begin{array}{cc}\hfill Q_{\mathrm{AR}}& ={\displaystyle \frac{1}{R_{\mathrm{C}1,\mathrm{AR}}}}\sqrt{{\displaystyle \frac{L_{\mathrm{C}1,\mathrm{AR}}}{C_{\mathrm{AR}}}}}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\omega }_{\mathrm{R},\mathrm{AR}}& ={\displaystyle \frac{1}{\sqrt{L_{\mathrm{C}1,\mathrm{AR}}{C}_{\mathrm{AR}}}}}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill L_{\mathrm{C}1}& =L_{\mathrm{C}1,\mathrm{AR}}+L_{\mathrm{C}1,\mathrm{SP}}\hfill \end{array}$$

(11)

By converting (9) and (10) to $C_{\mathrm{AR}}$, $C_{\mathrm{AR}}$ can first be substituted and $L_{\mathrm{AR}}$ can be calculated. Then $C_{\mathrm{AR}}$ can also be determined using (9) or (10).

The qualitative comparison between the simple model, the complex model, and the measurement results shows the better fitting of the complex model to the measured results. Especially when looking at the real parts, clear differences are noticeable between the simple model and the measurement results, which mean inaccurate representation of the capacitors’ high-frequency losses. As the complex model consists only of linear components, calculation effort during the simulation is not much higher compared to the simple model.

6. Discussion

This paper describes a simulation-based design methodology for the DC link filter of WBG-based VSCs. The methodology is described using an example of a three-phase VSC. Validation shows that moderate model precision can be reached with only a few iterative steps. The model enables us to determine the current load on the individual components, which is often not measurable in the real application. This allows faster and more precise design of the DC link especially for fast-switching WBG devices where HF filtering becomes crucial. Nevertheless, a model validation on an experimental setup is obligatory, as not all effects are considered in the model approach, including the following:

• Temperature and current dependent behaviour;
• Production-related component tolerances;
• Resonances occurring in the composite system.

Implementation of these effects should be considered only if necessary, as it may significantly increase the computing time of the models and influence the convergence of the solver. The necessity can be determined by experimental validation.

Furthermore, this paper describes briefly what parasitic effects influence a DC link capacitor’s filter efficiency and how they can be taken into account in the model.

• Model restrictions:

Usually, VSCs are not controlled via setting a defined set-point sine wave for the modulator. Instead, a superimposed control structure, for example, a PI current controller, is used to adjust the modulator’s set point in such way that the set-point current is applied to the load. To focus on the objective here, the influence of this feedback loop is neglected in the structure shown in Figure 7. Figure 14 now shows the measured and the simulated amplitude spectrum $|{\underline{I}}_{\mathrm{C}}|$ of the capacitor load current $i_{\mathrm{C}}$ for $I_{\mathrm{N}}=220$ A, $f_{\mathrm{s}}=80$ kHz and $f_1=100$ Hz. Although on the test-bench, a superimposed PI current controller is used to adjust the set-point, the simulation is only controlled in a feedforward manner. It can be seen from the results that in the case under consideration, the odd harmonics (mainly first, third, and fifth) of the switching frequency are especially influenced by the superimposed controller, which is the main reason for the deviations between simulation and measurement. This means that simulation results could be improved by including the superimposed controllers. But on the other hand, this would result in a significantly more complex simulation model, as the AC-side load circuit, measurement data processing, and digital controllers need to be added to the model.

Not implementing the superimposed control structure also means that dynamic processes like load changes cannot be simulated exactly as the systems control dynamic is neglected. The model is intended to determine the load current for all quasi-stationary operating points that occur in real-life operation. Other modelling approaches briefly describe the HF behaviour [19]. In real hardware, various hardware-specific effects like the non-linear behaviour of the transistor output capacitance cause high-frequency oscillations [52,53,54,55]. These effects influence switching losses, especially when high switching frequencies (in the MHz range) are introduced [56]. The goal of the approach mentioned in this paper is mainly to have a design tool for engineers to estimate their DC link capacitor current load and the remaining voltage overshoot, which are two crucial factors for DC link filter design. Therefore, a trade-off was made between the model accuracy and the practical usability for design engineers, who often do not have access to such specific information like the non-linear behaviour of the transistor output capacitance.

• Higher switching frequencies

Figure 15 shows the measured amplitude spectrum $|{\underline{I}}_{\mathrm{C}}|$ of the capacitor load current $i_{\mathrm{C}}$ for $I_{\mathrm{N}}=40$ A, $f_{\mathrm{s}}=200$ kHz and $f_1=100$ Hz. The deviation especially of the first harmonic amplitude is higher than in Figure 14, although it is for the same reason. It is related to the superimposed PI controller not taken into account in the simulation model. It is observed that the deviation becomes more for higher switching frequencies, as the controller reacts to switching noise and forces unnecessary switching actions that generate additional DC-side ripple current. This could be improved by using a more sophisticated superimposed controller.

7. Future Development

Deviations between simulation and measurement can also be seen in Figure 14, looking at the frequency range from $2$ MHz to $10$ MHz. The amplitude spectrum generated by simulation starts decreasing for frequencies higher than $3$ MHz, while in the measurement it starts decreasing at about $5.6$ MHz. This deviation comes from an inaccuracy in the simulation of the switching behaviour of the semiconductor switches. It is approximated using a PT2s function with a damping of 1, as described in Section 3.3. This is a very simple method that saves computational effort and that does not need too much specific information about the semiconductor. But as these high-frequency amplitudes resulting from the switching behaviour can stimulate resonances in the DC filter, a more precise simulation of the switching behaviour is desirable. This will be part of future developments.