Analytical evaluation of DC capacitor RMS current and voltage ripple in neutral-point clamped inverters

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1 Sādhanā Vol. 4, No. 6, June 7, DOI.7/s y Ó Indian Academy of Sciences Analytical evaluation of DC caacitor RMS current and voltage rile in neutral-oint clamed inverters K S GOPALAKRISHNAN, *, SANTOSH JANAKIRAMAN, SOUMITRA DAS and G NARAYANAN Deartment of Electrical Engineering, Indian Institute of Science, Bangalore 56, India Deartment of Electrical and Electronics Engineering, National Institute of Technology Goa, Farmagudi, Ponda 44, India ashwinkrishnan@gmail.com MS received Aril 6; acceted November 6 Abstract. The sizing of the DC-link caacitor in a three-level inverter is based on the RMS current flowing through it. This aer analyses the DC-link caacitor RMS current in a neutral-oint clamed (NPC) inverter and exresses the same as a function of modulation index, line-side current amlitude and ower factor. Analytical closed-form exressions are derived for the caacitor RMS current for single-hase half-bridge, single-hase full-bridge and three-hase three-leg toologies of a three-level inverter. The worst-case caacitor current stress is determined for each toology based on the analytical exressions. Further, analytical exressions are derived for the RMS values of low-frequency and high-frequency caacitor currents. These exressions are then used to estimate voltage rile across the DC caacitor for sinusoidally modulated three-hase NPC inverter. The analytical exressions for the RMS current and voltage rile are validated exerimentally over a wide range of oerating oints. Keywords. Current control; current stress; diode-clamed inverter; full-bridge inverter; half-bridge inverter; multi-level inverter; roortional-resonant controller; sinusoidal modulation; vector control; voltage rile.. Introduction Extensive research has been carried out on two-level and three-level inverters for DC AC ower conversion [, ]. In a two-level inverter, the mid-oint of each leg can be connected to either the ositive DC terminal or the negative DC terminal. In a three-level neutral-oint-clamed (NPC) inverter, the mid-oint of each leg could also be connected to the DC-bus mid-oint, otherwise known as the DC-bus neutral [, 4]. The NPC inverter offers several advantages over the traditional two-level inverter. With devices of the same voltage rating, this inverter can handle higher DC-bus voltage. Hence, this finds alication in medium-voltage alication [5]. At the same DC-bus voltage, each device blocks only half the DC-bus voltage in the off state. Therefore, the voltage stress on each device is reduced. Hence, the switching loss is reduced []. The NPC inverter is advantageous at high switching frequencies in low-voltage alications []. The NPC inverter has the ability of getting connected to the DC-bus midoint as stated earlier. This caability leads to better waveform quality than that of a two-level inverter [6]. However, the connection to the mid-oint also causes *For corresondence DC voltage imbalance due to injection of the load current into the DC bus during certain intervals. A number of methods have been devised to reduce this roblem of voltage imbalance [7, 8]. The caacitor RMS current is imortant in both two-level and three-level inverters from the ersective of sizing and cost of the DC caacitor. The worst-case caacitor RMS current in either case can be obtained through reeated digital simulations of the converter under various oerating conditions. However, it would be elegant to determine the worstcase caacitor RMS current through an analytical exression, relating the caacitor current to the oerating conditions such as modulation index, line current amlitude and ower factor. Analytical exressions for the caacitor RMS current in a two-level inverter have been derived in [9, ]. For a threelevel inverter, this exression is more involved as there are three ossible connections to the converter mid-oint and more devices are involved. More recently, an exression has also been reorted for the caacitor RMS current in a sinusoidally modulated three-hase NPC inverter [, ]. This aer derives such analytical exressions for singlehase half-bridge figure a, single-hase full-bridge figure b and three-hase three-leg figure c toologies of an NPC inverter. These analytical exressions are validated 87

2 88 K S Goalakrishnan et al (c) Figure. Toologies of neutral-oint clamed inverters. through extensive simulation and exerimental studies, both under oen-loo and closed-loo conditions. Another stress on the DC caacitor is the voltage rile, caused by the rile current flowing through it [, ]. The caacitor voltage rile is more deendent on the low-frequency current (e.g., third harmonic) than on high-frequency currents (i.e., switching frequency comonents) of the caacitor current [, 4]. Closedform exressions are derived here for the RMS values of low-frequency and high-frequency caacitor currents. An analytical exression is then derived to rovide an estimate of the DC voltage rile in a sinusoidally modulated three-hase NPC inverter. This is then validated exerimentally at different modulation indices and ower factor angles. A reliminary version of this work was ublished [5], which reorts only the analytical exression for caacitor RMS current ertaining to the three-leg toology and its exerimental validation under oen-loo conditions. This extended version reorts such analytical exressions corresonding to all three toologies in figure, and also validates these exressions under oen-loo as well as closed-loo conditions, as mentioned earlier. Further, the aer also includes evaluation of DC caacitor voltage rile as indicated earlier.. Derivation of analytical exression for caacitor RMS current Analytical closed-form exressions are derived for the caacitor RMS current ertaining to a single-hase halfbridge NPC inverter figure a, single-hase full-bridge NPC inverter figure b, and three-hase NPC inverter figure c in this section. Sine-triangle PWM is considered.. Duty ratio In each leg of an NPC inverter, switches SR and SR are comlementary, and switches SR and SR4 are also comlementary [4]. The duty ratio d R of the to switch SR in the half-bridge inverter is given by d R ¼ MsinðxtÞ; d R ¼ ; ðþ where M is the modulation index, and is also the eak duty ratio of the to switch; x ¼ f and f is the fundamental frequency. The duty ratio d Y of the to switch SY in the single-hase full-bridge inverter is hase shifted by 8 from d R in (), as indicated in figure a.

3 Analytical evaluation of DC caacitor RMS current and voltage rile 89 d R.8 d R.8 d Y.8 Modulation index (M).6.4. Modulation index (M).6.4. Modulation index (M).6.4. d R d Y d B Fundamental angle (in degrees) Fundamental angle (in degrees) Fundamental angle (in degrees) (c) Figure. Plot of duty ratios of switches SR, SY and SB for single-hase, half-bridge, single-hase, full-bridge and (c) threehase NPC inverters. Similarly, d Y and d B are the duty ratios of the to switches in the Y-hase and B-hase legs, resectively, in the three-hase inverter. These duty ratios d Y and d B are hase shifted by and 4, resectively, from the d R in (). They are illustrated in figure, considering M =.8.. Single-hase half-bridge inverter In a single-hase half-bridge inverter, the load current flows through the ositive DC link only when the switch SR is on. Therefore the instantaneous DC-link current (i HB ) can be written as the roduct of switching function (S R ) and load current (i R ) as follows: i HB ¼ S R i R : ðþ Ignoring the harmonic comonents, the load current can be exressed as shown in (), where / is the fundamental ower factor angle: i R ¼ I m sinðxt /Þ: ðþ Now, based on () and (), the fundamental-cycle-average of the DC-link current (I avg HB ) can be evaluated as follows: I avg HB ¼ Z d R i R dxt ¼ MI m cosð/þ: ð4þ 4 Similarly, the exression for DC-link RMS current (I rms HB ) can be found as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I rms HB ¼ d R i R dxt MIm ¼ þ cosð/þ : ð5þ The exressions for the DC-link RMS and average currents obtained in (5) and (4), resectively, are used to derive an exression for the DC-link caacitor RMS current (I c HB ) as shown below: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I c HB ¼ Irms HB I avg HB ð6þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I c HB ¼ MIm ð cosð/þ ð þ Þ M 6 cos ð/þþ: ð7þ The normalized caacitor current is lotted as a function of modulation index at different ower factors of,.,.4,.6,.8 and in figure a. As seen, the caacitor RMS current increases with ower factor for a fixed modulation index. Also, for a given ower factor, the caacitor RMS current increases with modulation index. Thus the maximum RMS current flows through the caacitor when the ower factor is unity and the modulation index is. The maximum caacitor RMS current in half-bridge inverter is found to be.8 times the eak load current.. Single-hase full-bridge inverter The instantaneous DC-link current (i FB ) and the fundamental-cycle-average DC-link current (I avg FB ) of a singlehase full-bridge inverter are exressed as shown here. S R and S Y are the switching functions of the switches SR and SY, resectively. i FB ¼ S R i R þ S Y i Y ; Z ð8þ I avg FB ¼ ðd R i R þ d Y i Y Þ dxt: ð9þ Since the oeration of the full-bridge inverter is symmetrical, it is sufficient to consider the interval xt for analysis. Based on symmetry, Eq. (9) can be simlified, and I avg FB can be evaluated as shown in ():

4 8 K S Goalakrishnan et al I c-hb /I m cos(φ)=.6 cos(φ)= cos(φ)= cos(φ)=.8 cos(φ)= Modulation index (M) I c- φ /I m cos(φ)=.4 cos(φ)=.8 I c-fb /I m cos(φ)= cos(φ)= cos(φ)=. cos(φ)=.6 cos(φ)= cos(φ)=.8 cos(φ)= cos(φ)=. cos(φ)= cos(φ)=.4 Modulation index (M) Modulation index (M) cos(φ)=.6 Figure. Variation of caacitor RMS current with modulation index and ower factor single-hase, half-bridge NPC inverter single-hase, full-bridge NPC inverter and (c) three-hase NPC inverters. (c) Z I avg FB ¼ d R i R dxt ¼ MI m cosð/þ: ðþ Further, an exression for DC-link RMS current (I rms FB ) can be derived as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I rms FB ¼ d R i R dxt MIm ¼ þ cosð/þ : ðþ An exression for the DC-link caacitor RMS current (I c FB ) can be obtained using the exressions for the DClink RMS and average currents obtained in () and (), resectively, as follows: I c FB ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Irms FB I avg FB ðþ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I c FB ¼ MIm ð cosð/þ ð þ Þ M 4 cos ð/þþ ðþ From figure b, it can be seen that the caacitor RMS current in a full-bridge inverter increases with ower factor u to a modulation index of.8, and decreases thereafter with increase in ower factor. The maximum caacitor

5 Analytical evaluation of DC caacitor RMS current and voltage rile 8 RMS current is.46 times the eak load current. This haens when the ower factor is zero, and the modulation index is unity..4 Three-hase NPC inverter Ignoring the switching frequency harmonics, the ole currents i R, i Y and i B for the three-hase toology are defined as follows: i R ¼ I m sinðxt /Þ; i Y ¼ I m sinðxt i B ¼ I m sinðxt þ /Þ; /Þ: ð4þ.4a DC bus current: The instantaneous DC-link current, which is the sum of all to switch currents, is given as follows: i / ¼ i R S R þ i Y S Y þ i B S B : ð5þ In this exression, S R, S Y and S B are the switching functions of the switches SR, SY and SB, resectively, in figure c..4b Average DC bus current: Since the three-hase duty ratios and three-hase currents are symmetric, an exression for the fundamental-cycle-average DC-link current (I avg / ) can be derived as shown by the following equations: I avg / ¼ I avg / ¼ Z Z ði R S R þ i Y S Y þ i B S B Þ dxt; d R i R dxt þ Z ð6þ! d B i B dxt ; ð7þ I avg / ¼ MI m cosð/þ: ð8þ 4.4c RMS DC-link current: The RMS DC-link current (I rms / ) for a three-hase inverter can be exressed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z I rms / ¼ ði R S R þ i Y S Y þ i B S B Þ dxt: ð9þ By virtue of three-hase symmetry, Eq. (9) reduces to the following equation: Z Irms / ¼ ðs R i R þ S RS B i R i B Þ dxt: ðþ Square of a switching function is the same as the switching function. Further, the switching function S R is always zero between and (see figure ). Similarly, S B is always zero between = and 4=. Hence Irms / ¼ Z Z S R i R dxt þ S R S B i R i B dxt : ðþ From table, the roduct of the switching function S R S B reduces to either S R or S B in different intervals as follows: S R S B ¼ S R ; S R S B ¼ S B ; Then, based on () and (), Irms / ¼ Z Z S R i R dxt þ 6 S R i R i B dxt Z! þ S B i R i B dxt : 6 ðþ ðþ Now, the DC-link RMS current can be rewritten in terms of the three-hase duty ratios as follows: Irms / ¼ Z Z d R i R dxt þ 6 d R i R i B dxt Z! ð4þ þ d B i R i B dxt : 6 By evaluating the integrals in (4), an exression for the DC-link RMS current is obtained as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Im I rms / ¼ Mð þ cosð/þ ffiffi Þ ð5þ 4.4d DC-link caacitor RMS current: Using the exressions for the DC-link RMS and average currents obtained in (5) and(8), resectively, an exression for the DC-link caacitor RMS current (I c / ) can be derived as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Im I c / ¼ M ffiffiffi þ ffiffiffi cosð/þ ði mmþ cos ð/þ: ð6þ In the case of three-hase three-level inverter, as can be seen from figure c, the caacitor RMS current increases with ower factor in the modulation index range M\:95; the caacitor RMS current decreases with increase in ower factor in the range :95\M\. The maximum caacitor RMS current is.459 times the maximum load current; this Table. Switching functions of the switches in the three legs. Angular duration S R S Y S B S R S Y S Y S B S B S R xt\ 6,, S R 6 xt\,, S B xt\, xt\,

6 8 K S Goalakrishnan et al haens at unity ower factor and a modulation index of.6. Thus, the caacitor RMS current is close to 46 er cent of the eak load current in case of single-hase fullbridge as well as three-hase NPC inverters. It is noteworthy that the caacitor RMS current rating is comarable for the single-hase and three-hase cases for the same current rating, desite the higher ower handling caability of the three-hase inverter. The analysis here is alied to three-level converters with one, two and three legs. This can certainly be extended to threelevel converters with any number of legs (e.g., multihase motor drives) or multile converters connected to a common DC bus. Also, the above analysis ignores dead-time effect as this is not exected to be significant at the oerating conditions considered here. At high switching frequencies, when the ratio of dead time to switching eriod is considerable, the dead-time effect could be incororated into the analysis, extending the aroach in [6] for two-level converters.. Simulated and exerimental results under oenloo conditions The analytical exressions derived in the revious section are first validated through simulations and measurements under oen-loo oeration of the NPC inverter.. Simulated and exerimental waveform A single-hase half-bridge NPC inverter, realized using one leg of an NPC inverter, is simulated using MATLAB Simulink. The devices are assumed to be ideal. The inverter is switched using sine-triangle PWM (see section.) at a carrier frequency of.5 khz. The ac load is considered as a sinusoidal current sink. The simulated waveforms of the gating ulses to the to device (i.e., S R ), the DC caacitor current and the load current are shown in figure 4a for a modulation index M of.8, eak load current I m of 5 A, and load ower factor of.8 (lag). The exerimental waveforms, corresonding to the simulated ones in figure 4, are resented in figure 4b. The exerimental NPC inverter is built with SEMIKRON SK5MLI66T (5 A/ V) IGBTs. Each leg consists of two 47 uf, 45 V, A electrolytic caacitors in series. The DC source is an Agilent 65A ( 5V, 5 A, W) rogrammable ower suly. The caacitor current is measured using a Fluke 4S current robe. A TMSF47- DSP-rocessor-based digital controller is used for control imlementation and generation of gating ulses. The simulated and exerimental sets of waveforms, corresonding to a single-hase full-bridge inverter, are shown in figure 5a and b, resectively, for M =.8, cosð/þ =.8 (lag) and I m = 5 A. This full-bridge inverter is realized using two NPC inverter legs, described earlier..5 (i) SR (ii) Caacitor (iii) Load Time (s) Figure 4. Simulated waveforms of gating ulses to to device SR, DC caacitor current and load current in a single-hase, halfbridge NPC inverter at M =.8 and cos / =.8 (lag). Exerimental gating ulses to to device SR (trace ), DC caacitor current (trace ) load current (trace ) in a single-hase, half-bridge NPC inverter at M =.8 and cos / =.8 (lag), [scale: trace, V/division; Trace, 5 A/division; Trace, 5 A/division].

7 Analytical evaluation of DC caacitor RMS current and voltage rile 8.5 (i) SR.5 (ii) Caacitor (iii) Load s Time Figure 5. Simulated waveforms of gating ulses to to device SR, DC caacitor current and load Current in a single-hase, fullbridge NPC inverter at M =.8 and cos / =.8 (lag). Exerimental gating ulses to to device SR (trace ), DC caacitor current (trace ) and load current (trace ) in a single-hase, full-bridge NPC inverter at M =.8 and cos / =.8 (lag), [scale: trace, V/division; trace, 5 A/division; Trace, 5 A/division]. Similarly, the simulated waveforms ertaining to the three-hase NPC inverter at M =.6 and cosð/þ =.4 (lag) are resented in figure 6a. Similar results at M =.6, but a different ower factor of.8 (lag), are shown in figure 6b. Figure 7a and b resents the exerimental results corresonding to figure 6a and b, resectively. As one can see, the simulated and exerimental waveforms agree reasonably well though certain differences are evident. While the simulations consider the DC voltage to be constant, there is a rile in the DC voltage in the actual case. Aart from such inevitable differences between the simulation model and the actual system, the bandwidth of the current robe is also limited to accurately cature the DC caacitor current ulses with fast rising/falling edges and rather short widths. Hence the simulated and measured DC caacitor current waveforms aear different. However, the distortions caused by the measurement system are mostly limited to the high frequency comonents (such as related to the edges of the ulses). The low frequency comonents, which are dominant, are measured with reasonable recision. This is borne out by the fact that the caacitor RMS currents based on analysis, simulation and measurements are in good agreements as shown in the following sections.. Validation of analytical exressions for caacitor current For the single-hase half-bridge, single-hase full-bridge and three-hase full-bridge toologies, caacitor RMS current is evaluated through simulation at different oerating conditions as indicated in table. The various oerating conditions include different modulation indices, different ower factors and different values of eak load current as seen from the table. The caacitor RMS current is also measured at each of these oerating conditions. The simulated and exerimental values are comared with analytically obtained ones (based on the exressions in section ) in table. As seen, the values redicted by the analytical exressions are in good agreement with the simulated and measured values. 4. Simulation and exerimental results under closed-loo conditions Closed-loo control is imlemented for the three toologies. The caacitor RMS current is measured at different oerating conditions under closed-loo control. The simulated and measured values of caacitor RMS current under closed-loo control are comared with the analytically redicted values in this section. 4. Closed-loo current control A roortional resonant (PR)-controller-based current control [7] is imlemented on a single-hase half-bridge

8 84 K S Goalakrishnan et al SR SR Caacitor Caacitor Load Time (s) Load Time (s) Figure 6. Simulated waveforms of gating ulses to to device SR, DC caacitor current and load current in a three-hase, NPC inverter at M =.6, cos / =.4 (lag) and M =.6, cos / =.8 (lag). Figure 7. Exerimental gating ulses to to device SR (trace ), DC caacitor current (trace ) and load current in a three-hase NPC inverter (trace ) at M =.6, cos / =.4 (lag) and M =.6, cos / =.8 (lag), [scale: trace, V/division; trace, 5 A/division; Trace, 5 A/division]. inverter. The PR controller has a transfer function as shown in (7), where x o is the resonant frequency: G s ¼ K þ K is s þ x ð7þ o The controller arameters K and K i are selected as discussed in [7], considering x o = 4 rad/s and carrier frequency f c ¼ khz. The measured dynamic resonse of the current controller to a ste change in current reference is shown by the oscillogram in figure 8. As seen from the exerimental result, the dynamic resonse of the controller is satisfactory. The PR current controller for the single-hase, fullbridge inverter is also designed along similar lines. Figure 9 shows the dynamic resonse of the exerimental system to a ste change in the current reference. As seen. the dynamic resonse is satisfactory.

9 Analytical evaluation of DC caacitor RMS current and voltage rile 85 Table. Analytical, simulated and exerimental values of caacitor RMS currents for three toologies with oen-loo control. Toology -hase halfbridge -hase fullbridge -hase fullbridge M / (lag) I m (A) Caacitor RMS Analytical Simulation Ext 8: : : : : : : : : : : : : : : : : : : : : : : : : : : : : For the three-hase NPC inverter, closed-loo control is imlemented in the synchronously revolving d q reference frame [8]. The three-hase currents (i R, i Y and i B ) are transformed into the synchronous reference frame (SRF) as d-axis and q-axis currents, namely i d and i q. Closed-loo control is imlemented for both i d and i q [8]. The PI controller arameters for the two current controllers are chosen based on [8]. Figure a and b shows the dynamic resonse of the current controller for the three-hase NPC inverter. The d- axis current reference is set to zero. The q-axis current reference is used to obtain the desired eak value of load current. When the q-axis current reference is changed, the actual q-axis current and the actual load current change at a large rate and settle down at a new value. Figure 8. Half bridge inverter current controller resonse: (i) Reference current and (ii) load current; scale: trace : 5 A/division; trace : 5 A/division. Figure 9. Full bridge inverter current controller resonse: (i) Reference current and (ii) load current, scale: trace : 5 A/division; trace : 5 A/division. 4. Validation of analytical exression for caacitor current The three toologies are simulated under closed loo conditions using MATLAB Simulink. The simulated values of caacitor RMS current at different oerating oints are tabulated in table. The caacitor RMS current is also measured in case of all three toologies at different oerating conditions. These results are also tabulated in table. As seen from table, the analytical, simulated and measured values of the DC caacitor current are in close agreement at the various oerating oints. Thus, the closed-form analytical exressions for the caacitor RMS current, derived in section, are validated both through simulations and actual measurements. The

10 86 K S Goalakrishnan et al Figure. Three-hase inverter-current controller resonse, scale: trace : 5 A/division and Trace : 5 A/division. Exerimental waveforms: (i) q-axis current reference and (ii) q-axis current. Exerimental waveforms: (i) q-axis current and (ii) load current. Table. Analytical, simulated and exerimental values of caacitor RMS currents for three toologies with closed-loo control. Toology -hase halfbridge -hase fullbridge -hase fullbridge M / (lag) I m (A) Caacitor RMS Analytical Simulation Ext 6: : : : : : : : : : : : : : : : : : : : : : : : : : : : : caacitor RMS current contains both low-frequency and high-frequency comonents. The RMS values of the lowfrequency and high-frequency comonents are analyzed in the following section. 5. Analysis of low-frequency and high-frequency caacitor current To evaluate the low-frequency comonents in the DC-caacitor current, one can consider the switching-cycle-average i C;avg defined as follows: i C;avg ¼ i ;avg I avg / ð8þ where I avg / is the average DC-link current over a fundamental cycle as derived in (8). Since the caacitor current has a eriodicity of, considering the interval \xt\, the switching-cycle-average (i ;avg ) can be exressed as follows: i ;avg ¼ d R i R þ d B i B ; \xt\6 ; ¼ d R i R ; 6 \xt\ : ð9þ Based on Eqs. (8), (9), (8), (4) and (), the switchingcycle-average caacitor-current i C;avg is exressed as shown in (): i C;avg ¼ M 4 I m½cosð/þþcosðxt 4 /ÞŠ; \xt\6 ; i C;avg ¼ M 4 I m½cosð/þþcosðxt /ÞŠ; 6 \xt\ : ðþ Further, the RMS value of the low-frequency caacitor current comonents can be evaluated as follows:

11 Analytical evaluation of DC caacitor RMS current and voltage rile 87.8 Analytical Result Exerimental Result.8 Analytical Result Exerimental Result.6.6 I C,LF.4 I C,LF Modulation index.8 Analytical Result Exerimental Result Modulation index I C,LF Modulation index Figure. Analytical and exerimental values of low-order harmonic RMS current for SPWM scheme at lagging ower factors of.49,.669 and (c).866. (c) I C;LF ¼ Z = i C;avg dxt ¼ 9M Im 6 cos ð/þð ffiffiffi Þþ ffiffi : ðþ The RMS value of the high-frequency caacitor current is obtained as shown in (), where I c / is the total RMS current through the DC-link caacitor, the exression for which is derived earlier in (6): I C;HF ¼ I c / I C;LF : ðþ Substituting the exressions for I c / and I C;LF from (6) and (), resectively, the following exression for the high-frequency RMS caacitor current can be derived: IC;HF ¼ I m M 4 ð ffiffi þ ffiffiffi 9M I m 6 cos ð/þð cosð/þþ 9 6 I m M cos ð/þ ffiffi ffiffi Þþ : ðþ The voltage rile contributed by the low-frequency and high-frequency comonents of the caacitor current is discussed in the next section. 6. Estimation of caacitor voltage rile The total RMS value of all the low-frequency comonents in the DC caacitor current in a sinusoidally modulated three-hase NPC inverter can be exressed as shown in (). As has been established reviously [4], the rincial low-frequency comonent in the DC caacitor current of a sinusoidally modulated NPC inverter is the third harmonic (i.e., f) comonent. Hence, assuming I C;LF to be the RMS value of the f comonent, the voltage rile V rms LF caused by the same can be evaluated as shown in (4), where C dc is the DC link caacitance and ESR is the equivalent series resistance seen by the third harmonic: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I C;LF V rms LF ¼ ð Þ þði C;LF ESR Þ : ð4þ ðf ÞC dc

12 88 K S Goalakrishnan et al Analytical Result Exerimental Result.6.5 Analytical Result Exerimental Result V rms-c.4. V rms-c Modulation index Modulation index Analytical Result Exerimental Result V rms-c.4... The high-frequency caacitor current consists of comonents around integral multiles of switching frequency f sw. The first side-band comonents (i.e., frequencies around f sw ) are more dominant than the other side bands [4]. Also, the caacitive reactances seen by the higher side-bands are much lower than that seen by the first side band. Further, the variation of ESR at high frequencies is not very significant [9]. Hence, the entire high-frequency caacitor current of RMS value I C;HF is assumed to be of switching frequency f sw for the urose of evaluating the RMS highfrequency voltage rile as shown by (5), where ESR SF is the ESR value at switching frequency f sw : V rms HF sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ I C;HF ð Þ þði C;HF ESR SF Þ : ðf sw ÞC dc ð5þ The total RMS voltage rile across the DC caacitor is evaluated further as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V rms C ¼ Vrms LF þ V rms HF: ð6þ Modulation index Figure. Analytical and exerimental values of RMS voltage rile for SPWM scheme at lagging ower factors of.49,.669 and (c).866. (c) 7. Validation of analysis of voltage rile The exerimental set-u consists of a MOSFET-based -kva inverter. The caacitor bank consists of two sets of three arallel caacitors of value 47 lf each. The load consists of a three-hase R L load. The load inductance er hase is fixed at mh. The rheostat is varied to obtain the desired ower factor angle. The controller latform is a TMSLF47A DSP rocessor. Sine-triangle PWM is considered as mentioned earlier. The switching frequency is.5 khz. The DC-bus voltage is varied to maintain the load current at a constant amlitude of A. The caacitor current is measured using a Fluke 4S robe. The FFT of the measured caacitor current is carried out in the MATLAB latform. Considering the frequency comonents less than half of the switching frequency, the lowfrequency caacitor RMS current is calculated. The analytically and exerimentally obtained values of the lowfrequency caacitor RMS current I C;LF are lotted in figure. As seen, the exerimental and analytical results tally with each other reasonably well.

13 Analytical evaluation of DC caacitor RMS current and voltage rile 89 The voltage rile across the caacitor is measured using a fluke multimeter. The analytical and exerimental values of voltage rile for SPWM scheme are shown lotted in figure. As seen from the lots, the exerimental and analytical values agree reasonably well. Also, it is seen that the voltage rile increases with both modulation index and ower factor angle. Therefore, the voltage rile is maximum when the ower factor is zero, and the modulation index is. 8. Conclusions Analytical exressions for the DC-link caacitor RMS current ertaining to single-hase half-bridge, single-hase full-bridge and three-hase three-level inverter toologies are derived, considering a sinusoidal PWM scheme. The analytical exressions for the three toologies are validated extensively, using simulation studies and exerimental measurements, under both oen-loo and closed-loo conditions. The analytical exressions yield the worst-case caacitor RMS current and the oerating condition at which this occurs for all three toologies. The caacitor RMS current is found to be the highest at the maximum modulation index and unity ower factor for the half-bridge toology, whereas for the full-bridge toology at the maximum modulation index and zero ower factor. For the three-hase NPC, on the other hand, the caacitor RMS current is maximum at a modulation index of.6 at unity ower factor. The worst-case caacitor RMS current is always a ercentage of the eak load current. This is 8 er cent of the eak load current for the half-bridge inverter, while it is 46 er cent for the other two toologies. Analytical exressions are also derived for the RMS values of low-frequency and high-frequency caacitor currents for the sinusoidally modulated three-hase NPC inverter. These exressions are used to evaluate the voltage rile across the DC caacitor at various oerating conditions. 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