9.4. Filter capacitors

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9.4. Filter capacitors

9.4.1. Size modelling

Almost all conventional capacitors used in WECS are constructed based on plate capacitor structure. The volume of a plate capacitor (Vol_PC) is proportional to the area of the plates (A_PC) and the plate separation distance (d_PC). Also, the voltage applied between plates is limited by d_PC and the properties of the dielectric material (the breakdown electric strength, E_Bd). When an increase in the rated voltage (V_CN) of the capacitor is desired, the d_PC must be increased in order to avoid dielectric breakdown.

Figure 9.16 Example of inductor core losses and overall volume relationship for three different inductor technologies from Siemens. Three-phase reactor series 4EUXX with Cu and Al winding conductor, and DC iron core smoothing reactors series 4ETXX with Cu winding are considered. The lines show the calculated model based on Eq. [9.42].

Taking into account the capacitance definition and the previous mentioned relations, it can be shown that the Vol_PC is proportional to its capacitance (C) and the square of its V_CN, as follows:

$Vo l_{PC} \propto A_{PC} \cdot d_{PC} = (\frac{C \cdot d_{PC}}{ε}) \cdot d_{PC} \propto \frac{C}{ε} \cdot {(\frac{V_{CN}}{E_{Bd}})}^{2}$

[9.48]

Then, if a capacitor technology is kept (dielectric material, fabrication process and geometry) for different values of C and V_CN requirements, it is proposed to predict the overall capacitor volume (Vol_C) and capacitor total mass (Mass_C) by:

$Vo l_{C} = K_{VC 0} \cdot C^{K_{VC 1}} \cdot V_{CN}^{K_{VC 2}}$

[9.49]

$Mas s_{C} = K ρ_{C 0} \cdot {(Vo l_{C})}^{K ρ_{C 1}}$

[9.50]

where K_VC0, K_VC1, K_VC2, K_ρC0 and K_ρC1 are proportionality regression coefficients found by taking data from reference capacitor technology. Fig. 9.17 presents the relationship between Vol_C and C for different V_CN and different capacitor technologies. On the other hand, Fig. 9.18 displays Mass_C against Vol_C for the same families of capacitors considered in Fig. 9.17. The calculated parameters of the size and mass models for the capacitors considered in Figs 9.17 and 9.18 are presented in Table 9.4.

Figure 9.17 Film capacitor volume versus capacitance for different voltages ratings and capacitor technologies: (a) DC-link capacitors series MKP-B256xx from TDK; (b) DC-link capacitors series LNK-M3xx1 from ICAR; (c) AC filter capacitors series MKP B3236 from TDK; (d) AC filter capacitors series MKV-E1x from ICAR. The dash line shows the fitting curve as given in Eq. [9.49].

9.4.2. Capacitor dielectric losses

The capacitor losses (P_C) are modelled as the summation of the dielectric losses (P_εC) and the resistive losses (P_ΩC). The dielectric losses can be calculated by (EPCOS, 2012):

$P_{ε C} = π \cdot f_{c} \cdot C \cdot tan (δ_{0}) \cdot V_{Cac}^{2}$

[9.51]

where V_Cac is the maximum amplitude of the alternating voltage applied to the capacitor with effective frequency f_c, and tan(δ₀) is the dissipation factor of the dielectric. Normally, the dielectric dissipation factor depends on the dielectric material of the capacitor and it can be considered constant for all capacitors in their normal working frequency range. For example, a typical value of the dissipation factor for polypropylene is 2e-4 = 2 × 10⁻⁴ = 0.0002, which is the dielectric of the capacitors considered in Figs 9.17 and 9.18.

For DC capacitors, the voltage V_Cac is the peak value of the superimposed ripple voltage. When the ripple voltage is approximated to be a triangular waveform with frequency equal to f_sw of the converter and peak-to-peak amplitude equal to the maximum voltage ripple, the following approximation can be done based on Eq. [9.41]:

Figure 9.18 Film capacitor mass versus overall volume for the capacitor technologies considered in Fig. 9.17. The dashed line shows the fitting curve as given in Eq. [9.50].

Table 9.4

Parameters of capacitor model. DC and AC capacitor technologies from TDK and ICAR are considered. From TDK manufacturer: the DC-link capacitors series MKP-B256xx and the AC filter capacitors series MKP B3236. From ICAR: the DC-link capacitors series LNK-M3xx1and the AC filter capacitors series MKV-E1x

Parameter	DC capacitors		AC capacitors
Reference	TDK MKP-B256xx	ICAR LNK-M3xx1	TDK MKP B3236x	ICAR MKV-E1x
K_VC0	2.0734e-5	5.9622e-5	67.303e-5	13.406e-5
K_VC1	0.7290	0.7271	0.6770	0.5410
K_VC2	1.3796	1.2473	1.0706	1.2216
K_ρC0	1.3428e3	0.8821e3	1.8079e3	2.7496e3
K_ρC1	1.0543	0.9950	1.0923	1.2060
tan(δ₀)	2e-4	2e-4	2e-4	2e-4
K_ΩC0	4.069e-03	2.3056e-5	1.5711e-3	4.9369e-6
K_ΩC1	−0.3211	−0.0430	−0.3970	0.0783
K_ΩC2	−0.4661	0.4986	−0.4539	1.0316

$P_{ε C} = \frac{\sqrt{3}}{2} \cdot f_{sw} \cdot C \cdot tan (δ_{0}) \cdot δ_{Vdc}^{2} \cdot V_{DC}^{2}$

[9.52]

where δ_Vdc is the ratio of peak-to-peak voltage ripple to DC voltage (V_DC) of the converter.

For AC capacitors, the voltage V_Cac is the peak value of the fundamental component plus the peak voltage ripple, and the effective frequency can be approximated to the fundamental frequency, then dielectric losses can be expressed as:

$P_{ε C} = π \cdot f_{c 1} \cdot C \cdot tan (δ_{0}) \cdot {(1 + \frac{δ_{Vac}}{2})}^{2} \cdot V_{acp}^{2}$

[9.53]

where δ_Vac is the ratio of peak-to-peak voltage ripple to the peak fundamental voltage (V_acp) with fundamental frequency f_c1.

9.4.3. Capacitor resistive losses

The resistive losses occur in the electrodes, in the contacting and in the inner wiring. These losses can be calculated as follows (EPCOS, 2012):

$P_{Ω C} = R_{s C} \cdot I_{C}^{2}$

[9.54]

where I_C is the RMS value of the capacitor current, and R_sC is the series resistance at maximum hot-spot temperature, which is the sum of all resistance occurring inside the capacitor. The series resistance R_sC is a typical estimated value based on average film metallization parameters (EPCOS, 2012) and its value can be found from the datasheet of the reference capacitor technology. Due to the low thickness of the metalized layer, high-frequency effects, such as skin effects, are negligible (Mirjafari and Balog, 2011).

It is proposed to use a model based on C and V_CN to predict the value of series resistance, as follows:

$R_{s C} = K_{Ω C 0} \cdot C^{K_{Ω C 1}} \cdot V_{CN}^{K_{ΩC 2}}$

[9.55]

where K_ΩC0, K_ΩC1 and K_ΩC2 are the proportionality regression coefficients found by taking data from reference capacitor technology. The value of series resistance given in the datasheets is referenced to 20°C capacitor temperature, but a conversion factor of 1.25 can be used to estimate the resistance at nominal temperature (typically, 85°C for film capacitors) (EPCOS, 2012). Fig. 9.19 presents the relationship between the series resistance per energy storage and the capacitance for different V_CN and different capacitor technologies. The series resistance in Fig. 9.19 refers to the resistance at nominal temperature (85°C).

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Table of Contents for 9.4. Filter capacitors

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9.4. Filter capacitors

9.4.1. Size modelling

9.4.2. Capacitor dielectric losses

9.4.3. Capacitor resistive losses

Table of Contents for
9.4. Filter capacitors