Section 6 - Chapter 10 – Chapter 23
Interharmonic Components Generated by Adjustable Speed Drives
Francesco De Rosa, Roberto Langella, Adolfo Sollazzo and Alfredo Testa
1023.1 Introduction
Adjustable Speed Drives (ASDs) based on double stage conversion systems generate interharmonic current components in the supply system side, the DC link and output side, in addition to harmonics typical for single stage converters[1]-[4]. Under ideal supply conditions, interharmonics are generated by the interaction between the two conversion systems through the intermodulation of their harmonics [5]. When unbalances, background harmonic and interharmonic distortionsare present in supply voltages more complex intermodulation phenomenons take place [6].
To characterize the aforementioned process, the interharmonic amplitudes (and phases), frequencies and origins have to be considered. The interharmonic amplitude importance for compatibility problems is evident: a comprehensive simulation or analysis for each working point of the ASD is needed and proper deterministic or probabilistic models must be used in accordance with the aim of the analysis. Various models are available: experimental analogue, time domain and frequency domain models [7]- [11]. Each of them is characterized by a different degree of complexity in representing the AC supply-system, the converters, the DC link and the AC supplied system. Whichever is the model, particular attention must be devoted to the problem of the frequency resolution and of the computational burden [12].
As for the interharmonic frequencies, whichever is the kind of analysis to be performed, it is very important to forecast them, for a given working point of the ASD. This allows evaluating in advance the Fourier fundamental frequency that is the maximum common divisor of all the component frequencies that are present and gives the exact periodicity of the distorted waveform. Other reasons are related to the effects of interharmonics such as light flicker, asynchronous motor aging, dormant resonance excitation, etc. The frequency forecast requires: i) in ideal supply conditions, only the study of the interactions between rectifier and inverter, which are internal to the ASD; ii) when unbalances and/or background harmonic and interharmonic distortion are present in supply voltages, also the study of the further interactions between the supply system and the whole ASD.
As for the origins of a given interharmonic, the availability of a proper symbolism for interharmonic turns out to be very useful. In fact, while the frequency is a sufficient information to find out the origin of harmonic components (in a defined scenario or conversion system), the same is not true for interharmonic components. These components have frequencies which vary within a wide range, according to the output frequency, and can assume harmonic frequencies or even become DC components. Moreover, they overlap in situations in which two or more components of different origin assume the same frequency value. In general, the presence of non ideal supply conditions makes impossible to recognize the origins without complex analyses. Anyway, the knowledge of the normal ASD behaviour in ideal conditions may help in recognizing what derives from the ASD internal behaviour and what from interactions with non ideal supply system.
In this chapter, reference is made to ideal supply conditions which allow recognising the frequencies and the origins of those interharmonics which are generated by the interaction between the rectifier and the inverter inside the ASD. Formulas to forecast the interharmonic component frequencies are developed firstly for LCI drives and, then, for synchronous sinusoidal PWM drives. Afterwards, a proper symbolism is proposed to make it possible to recognize the interharmonic origins. Finally, numerical analyses, performed for both ASDs considered in a wide range of output frequencies, give a comprehensive insight in the complex behaviour of interharmonic component frequencies; also some characteristic aspects, such as the degeneration in harmonics or the overlapping of an interharmonic couple of different origins, are described.
1023.2 Harmonics and Interharmonics in LCI Drives
LCI drivesare still largely used in high power applications. In this type of ASDs the DC-link is made up of an inductor according to the typical scheme shown in Fig. 23.1.
Figure.23.1. LCI drive scheme.
In the following, formulas for the evaluation of harmonic and interharmonic frequencies of the DC-link (dc), supply system (ss) and output side (os) currents are reported. The formulas are obtained applying the principle of modulation theory fully developed in[1],[2]and[6] and summarised in Appendix A. The majority of the symbols in the following sections are defined in the Nomenclature.
23.3 DC-Link
The harmonics due to both the rectifier and the inverterare present in idc(t). Their frequencies are, respectively:
v=1,2,3,….(23.1)
j=1,2,3,….(23.2)
being hssdc and hosdc the order of the DC-link harmonic due to the supply side rectifier and to the output side inverter.
The interharmonics due to the intermodulation operated by the two convertersare also present [8]. Their frequencies are:
v=1,2,3, j= 1,2,3,….(23.3)
where the absolute value, here and in the following formulas, is justified by the symmetry properties of the Fourier Transform [16]. Each couple of rectifier ()[MS1] and inverter (j) harmonics produces a corresponding interharmonic frequency. In general, two or more couples (j) [MS2]may give even the same values of interharmonic frequencies.
1023.43 Supply System Side
The harmonic frequencies of iss(t) derive from the modulation of the DC component and of the components (23.1) of idc(t) operated by the rectifier and are:
[MS3]=1,2,3… (23.4)
being hss the order of the supply side harmonic.
The sign “+” in (23.4) determines positive sequences while the sign “-” negative sequences.
The interharmonic frequencies of iss(t) derive from the modulation of the inverter harmonic components (23.2) of idc(t) operated by the rectifier and are:
[MS4]=1,2,3,… j= 1,2,3,…(23.5)
The sequence of the interharmonics compared to the hss-th harmonic sequence, is:
- the same, when one of the following conditions apply:
- there is the sign “+” in (23.5);
- there is the sign “-” in (23.5) and ;
- the opposite, when there is the sign “-” in (23.5) and .
- not definable, when , which means that the interharmonics become dc components.
1023.54 Output Side
Due to the structural symmetry between the rectifier and the inverter, it is simply necessary to change fss with fos and qss with qos in (23.4) and (23.5) respectively, in order to obtain the harmonic and interharmonic frequencies of ios(t):
j=1,2,3… (23.6)
j=1,2,3,… [MS5]=1,2,3…(23.7)
being hos the order of the output side harmonic.
As for the sequences, the same considerations about (23.4) and (23.5) still apply.
23.6 Harmonics and Interharmonics in PWM Drives
Nowadays, from low to medium-high power applications, voltage source invertersare more and more used in ASDs. The DC-link is made up of a capacitor while, in high power applications, an inductor is added on the rectifier output side to smooth the current waveform.
Formulas (23.2) and (23.6) become useless because of both the different structure and inverter operation, and a different analysis has to be considered. The harmonics generated by the inverter depend on the control strategy of the inverter switches, in particular, from the modulation ratio, mf.
For the sake of brevity, here reference is made to the synchronous sinusoidal PWM, which is the most used for high power ASDs, and a method to forecast the produced harmonic frequencies is reported in the Appendix B. Other modulation techniques, such as harmonic elimination and random modulations, require different formulas for harmonic (and interharmonic) frequencies evaluation but the phenomenon of intermodulation between rectifier and inverter still takes place.
Being the inverter operated by a PWM technique, the rectifier is an uncontrolled diode bridge, as reported in the typical scheme shown in Fig. 23.2.
Figure.23.2. PWM drive scheme.
- DC-Link
The harmonics produced by the rectifier, those produced by the inverter and the interharmonics due to the interaction between the two convertersare present in both DC-link currents, idcr(t) and idci(t). The harmonic frequencies generated by the rectifier can be calculated using formula (23.1) because the rectifier operation does not change.
The harmonic frequencies generated by the inverter (see Appendix B) are evaluated as:
(23.8)
with j and r integers depending on the modulation ratio as reported in Table 23.1I; the dependency from mf is related to the switching strategy adopted as shown in the following Section VI[MS6]. In particular, Table 23.1Ishows that:
both even and odd harmonics are present for even mf;
only even harmonics are present for odd mf;
only triple harmonics are present for triple mf.
Table 23.1I: Values of parameters j and r for different mf choices
mf / Odd / evennon triple / j / r / j / r
Even integers / / even integers / even integers / / integers
Odd integers / / odd integers / odd integers /
triple / j / r / j / r
even integers / / even triple integer / even integers / / triple integers
odd integers / / odd triple integer / odd integers /
The interharmonics due to the intermodulation operated by the two convertersare also present [9]. Their frequencies are evaluated according to the following relationship:
[MS7](23.9)
with [MS8]=1,2,3,…, j and r as in Table 23.1I.
- Supply System Side
For the harmonic frequencies of iss(t) the same considerations developed in Section III[MS9]apply and formula (23.4) is still valid.
The interharmonic frequencies of iss(t) derive from the modulation of the inverter harmonic components (23.8) operated by the rectifier and can be evaluated according to the following relationship:
(23.10)
with [MS10]=1,2,3,… j and r as in Table 23.1I.
As for the sequences, the same considerations about (23.4) and (23.5) still apply.
- Output Side
The harmonic frequencies of ios(t) are evaluated as shown in Appendix B; it is:
(23.11)
with j and k as in Table.23.2II.
Table 23.2II: Values of parameters j and k for different mf choices
mf / odd / EvenJ / k / j / k
even integers / / odd integers / even integers / / integers
odd integers / / even integers / odd integers /
The interharmonic frequencies of ios(t) derive from the modulation of the rectifier harmonic components (23.1) operated by the inverter and can be evaluated according to the following relationship:
(23.12)
with j and k as in Table 23.1II, [MS11]=1,2,3,….
Also in this case, the sequences of the harmonics in (23.11) determine the sequences of the interharmonics in (23.12) according to the same rules shown in Section III[MS12].
23.710.6 Symbolism Proposal
In order to find out the origins of a given interharmonic component produced by the interaction between rectifier and inverter inside the ASD and under ideal supply conditions, it seems useful to introduce a proper symbolism. Reference is made to the interharmonic components in both the supply system and output sides. Though harmonic order calculation is different in LCI and PWM drives it is, anyway, possible to find a unified and, at the same time, easy to understand symbolism.
Concerning the supply system side interharmonics, it seems suitable to introduce the symbols:
, (23.13)
,(23.14)
where:
-the subscripts evidence the common origin of the couple of components (23.13) and (23.14) related to the rectifier harmonic of order hss (see (23.4)) and to the inverter DC-link harmonic of order hosdc (see (23.2) for LCI and (23.8) for PWM) involved in the intermodulation process;
-the superscript distinguishes the relative position of the components, being “u” for the component (23.13), which stands “up” in frequency due to sign “+” in (23.5), for LCI, and (23.10), for PWM, and “d” for the other component (23.14), which stands “down” in frequency due to the sign “-” in (23.5), for LCI, and (23.10), for PWM.
An example of the application of the aforementioned symbols is reported in the following Section[MS13] (Fig. 23.4 for LCI and Fig. 23.8 for PWM ).
Concerning the output side interharmonics, it seems suitable to introduce the symbols:
,(23.15)
,(23.16)
where:
-the subscripts evidence again the common origin of the couple of components (23.15) and (23.16) related to the inverter harmonic of order hos (see (23.6) for LCI and (23.11) for PWM) and to the rectifier DC-link harmonic of order hssdc (see (23.1) for both LCI and PWM) involved in the intermodulation process;
-the use of the superscript is the same as for the supply side components (23.15) and (23.16) but, this time, it is related to the signs (“+” and “-“) in (23.7), for LCI, and (23.12), for PWM.
An example of the application of the aforementioned symbols is reported in the following Section (Fig. 23.5 for LCI and Fig. 23.9 for PWM).
23.810.7 Numerical Analyses
Several numerical analyses were carried out on both LCI and PWM typical drives, both characterized by qss and qos equal to six; their other parameters are fully referenced in [8] and [9] respectively. The models proposed in [8] and [9], validated comparing their results with those obtained by means of the well known software EMTP and Power System Blockset under Matlab environment, were used.
Some experimental verifications of the formulas developed in the chapterare reported in [3] and [13] for LCI drives and in [14] and [15] for PWM drives. In all the cases the frequency components expected according to the formulas (23.4)-(23.7) and to (23.10)-(23.12) respectively, were always actually present and of prevalent amplitude. Some additional frequency components were measured and it was demonstrated that their origin was the interaction between the non ideal supply system and ASDs, not covered by the formulas of Section III and IV[MS14].
- LCI Drive
Figure 23.3 shows the LCI drive interharmonic frequency values versusoutput frequency values, for both the supply system and output sides, obtained using (23.5) and (23.7). For the sake of clarity, only some specified interharmonic components are reported.
Figures 23.4a) and 23.5a) show the amplitudes of current interharmonics versus their frequencies for fos=40Hz, and refer to the supply system and output sides, respectively. Figures 23.4b) and 23.5b) show how to use the plots of Figs. 23.3a) and 23.3b), respectively, to forecast the interharmonic component frequencies. The couple of the main interharmonics in Fig. 23.4a), Iu1,6 and Id1,6, is generated by the intermodulation between the 1st supply system harmonic and the 6th DC-link harmonic produced by the inverter. The main interharmonic in Fig. 23.5a) is Id11,6, while Id7,6, of smaller but still remarkable amplitude, seems particularly notable due to its low frequency (20Hz), which can create problems to asynchronous motor life[17].
Figure.23.3. LCI drive interharmonic frequencies versusoutput frequency:
a) supply system side, all interharmonics due to ([MS15]j) of (5) = (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3), (4,1), (4,2), (4,3);
b) output side, all interharmonics due to (j,[MS16]) of (7) = (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3), (4,1), (4,2), (4,3).
Figure.23.4. LCI drive supply system current interharmonics for fos= 40Hz:
a) amplitudes in percentage of 230A versus interharmonic frequency;
b) frequencies (abscissa) versusoutput frequency (ordinate).
Fig. ure 23.5. LCI driveoutput current interharmonic for fos= 40Hz:
a) amplitudes in percentage of 165A versus interharmonic frequency;
b) frequencies (abscissa) versusoutput frequency (ordinate).
- PWM drive
The inverter modulation strategy selected is represented in Fig. 23.6 in terms of actual, fsw, and mean, f*sw, switching frequency (see Appendix B); a constant value of f*sw for all the output frequencies is assumed equal to 350Hz. Fig. 23.6 reports also the modulation ratio, mf. It is possible to observe that the modulation ratio varies during the ASD operation so changing the order of the harmonic components injected both into the DC-link and into the output side.
The plots in Fig. 23.7, obtained by means of (23.10) and (23.12), show the interharmonic frequency values versus the output frequency both in the supply system and output sides, according to the switching frequency trend of Fig. 23.6. Only some specified interharmonic components are reported to make it easier to understand the plots: f.i. in Fig.23.7a) the interharmonic components due to the intermodulation of the first harmonic of the rectifier ([MS17]=1) with the harmonic components generated by different couples of j and r, that is to say the intermodulation produced by specified side bands of the DC spectrum produced by the inverterare represented.
Concerning the supply system side, it is evident that the overlapping of different interharmonic components occurs for some output frequency ranges. The most remarkable overlappings are highlighted by the grey shadows in Fig. 23.7a). Of course, such overlapping leads to synergic combinations of interharmonic components originated from different intermodulations. As for the output side, the aforementioned overlapping occurs only for discrete frequency values.
Figures 23.8a) and 23.9a) show the amplitudes of the current interharmonics versus their frequencies for fos=40Hz and mf=9, with reference to the supply system and output sides, respectively.
Figures 23.8b) and 23.9b) show how to use the plots of Figs. 23.7a) and 23.7b) to forecast the interharmonic component frequencies. It is worthwhile noting that overlapping causes that the interharmonic components due to the values of the parameters (,[MS18]mf,j,r) in (10) equal to (1,9,0,6) and to (1,9,1,3) have the same frequencies producing Iu1,6 and Id1,6 (see Fig. 23.8a)).
In Fig. 23.10 the behaviours of the two supply side interharmonic components due to (1,mf,0,6) and (1,mf,1,3) in (23.10) in terms of amplitude (Fig. 23.10a) and Fig. 23.10b)) and frequency (Fig. 23.10c)) variations versus the output frequency are shown. The overlapping occurs in the output frequency range (see Fig. 23.10c) in the interval 3741Hz where mf=9). Finally, it can be observed that the interharmonic due to (1,mf,1,3) of (23.10) degenerates in the 5-th harmonic twice (fos=15Hz and fos=20Hz).
Figure.23.6. PWM drive: switching frequency (fsw) and modulation ratio (mf.)
Figure.23.7. PWM drive interharmonic frequencies versusoutput frequency:
a) supply system side: all interharmonics due to [MS19]=1, mf as in Fig. 23.6 and
(j,r) of (23.10) = (0,6), (1,3), (1,9), (2,0), (2,6), (3,3), (3,9), (4,0), (4,6);
b) output side: all interharmonics due to mf as in Fig. 23.6,[MS20]=1 and
(j,k) of (23.12) = (0,1), (0,3), (1,0), (1,2), (2,1), (2,3), (3,0), (3,2), (4,1), (4,3).
Figure.23.8. PWM drive supply system current interharmonics: a) amplitudes in percentage of 248A versus interharmonic frequency; b) frequencies (abscissa) versusoutput frequency (ordinate) with switching frequency of Fig. 23.6.
Figure.23.9. PWM driveoutput current interharmonics: a) amplitudes in percentage of 423A versus interharmonic frequency; b) frequencies (abscissa) versusoutput frequency (ordinate) with switching frequency of Fig. 23.6.