Chapter 21
Harmonic and Interharmonic Components Generated by Adjustable Speed Drives
R. Langella, A. Testa
21.1 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 distortions are 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 Line Commutated Inverter (LCI) drives and, then, for synchronous sinusoidal Pulse Width Modulation (PWM) drives. Moreover, some considerations about harmonics and interharmonics amplitude variability are given. Afterwards, a proper symbolism is proposed to make it possible to recognize the interharmonic origins. 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. Finally, some probabilistic modelling aspects are discussed.
21.2 LCI Drives
LCI drives are 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. 21.1.
Figure 21.1. LCI drive scheme.
21.2.1 Harmonic and Interharmonic Frequencies
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.
A. DC-Link
The harmonics due to both the rectifier and the inverter are present in idc(t). Their frequencies are, respectively:
v=1,2,3,…. (21.1)
j=1,2,3,…. (21.2)
being hssdc and hosdc the order of the DC-link harmonic due to the supply side rectifier and to the output side inverter, fss (fos) the supply system (output) fundamental frequency and qss (qos) the rectifier (inverter) number of pulses.
The interharmonics due to the intermodulation operated by the two converters are also present [8]. Their frequencies are:
v=1,2,3, j= 1,2,3,…. (21.3)
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 and inverter (j) harmonics produces a corresponding interharmonic frequency. In general, two or more couples may give even the same values of interharmonic frequencies.
B. Supply System Side
The harmonic frequencies of iss(t) derive from the modulation of the DC component and of the components (21.1) of idc(t) operated by the rectifier and are:
n=1,2,3… (21.4)
being hss the order of the supply side harmonic.
The sign “+” in (21.4) determines positive sequences while the sign “-” negative sequences.
The interharmonic frequencies of iss(t) derive from the modulation of the inverter harmonic components (21.2) of idc(t) operated by the rectifier and are:
n=1,2,3,… j= 1,2,3,… (21.5)
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 (21.5);
- there is the sign “-” in (21.5) and ;
- the opposite, when there is the sign “-” in (21.5) and .
- not definable, when , which means that the interharmonics become dc components.
C. 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 (21.4) and (21.5) respectively, in order to obtain the harmonic and interharmonic frequencies of the output side current, ios(t):
j=1,2,3… (21.6)
j=1,2,3,… n=1,2,3… (21.7)
being hos the order of the output side harmonic.
As for the sequences, the same considerations about (21.4) and (21.5) still apply.
21.2.2 Harmonic and Iterhamronic Amplitudes
Harmonic amplitudes can be predicted by means of analytical formulas. On the other side, it seems not possible to predict by means of analytical formulas the amplitude of interharmonics. It depends on their frequency values and from the system impedance behaviour, in particular resonances. To give an idea of the entity of the variability, reference is made to the case-study specified in [19] where the results obtained for all the output frequencies from 17.5 to 47.5 Hz of a case-study based on an LCI like that of Fig. 21.1, for constant flux operation, have been considered. The results are reported in Fig. 21.2, where the frequency component amplitudes are referred to the 50Hz fundamental component.
It is possible to observe that the amplitude of the main interharmonic component, generated by the intermodulation between the fundamental rectifier ac harmonic and the first inverter dc harmonic, is of some percentages (<3%) for output frequencies around the nominal value, while in the output frequency range from 21 to 37.5Hz it reaches very high amplitudes (up to 35%). This great variability is a consequence of the resonance phenomenon due to the interaction between the capacitance present on the motor side and the series of the DC link inductance together with the supply side inductance modulated by the rectifier operation (see. Fig. 21.1).
Figure 21.2. LCI harmonic and interharmonic amplitudes referred to the 50Hz component versus output frequency: 5th and 7th harmonics: (); main interharmonic (…).
21.3 PWM Drives
Nowadays, from low to medium-high power applications, voltage source inverters are 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 (21.2) and (21.6) 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. 21.3.
Figure 21.3. PWM drive scheme.
21.3.1. Harmonic and Interharmonic Frequencies
A. DC-Link
The harmonics produced by the rectifier, those produced by the inverter and the interharmonics due to the interaction between the two converters are present in both DC-link currents, idcr(t) and idci(t). The harmonic frequencies generated by the rectifier can be calculated using formula (21.1) because the rectifier operation does not change.
The harmonic frequencies generated by the inverter (see Appendix B) are evaluated as:
(21.8)
with mf the modulation ratio, j and r integers depending on mf as reported in Table 21.1; the dependency from mf is related to the switching strategy adopted as shown in the following Section VI. In particular, Table 21.1 shows that:
ü both even and odd harmonics are present for even mf;
ü only even harmonics are present for odd mf;
ü only triple harmonics are present for triple mf.
Table 21.1: Values of parameters j and r for different mf choices
mf / Odd / evennon triple / j / r / j / r
Even integers / Þ / even integers / even integers / Þ / integers
Odd integers / Þ / odd integers / odd integers / Þ
triple / j / r / j / r
even integers / Þ / even triple integer / even integers / Þ / triple integers
odd integers / Þ / odd triple integer / odd integers / Þ
The interharmonics due to the intermodulation operated by the two converters are also present [9]. Their frequencies are evaluated according to the following relationship:
(21.9)
with=1,2,3,…, j and r as in Table 21.1.
B. Supply System Side
For the harmonic frequencies of iss(t) the same considerations developed in Section III apply and formula (21.4) is still valid.
The interharmonic frequencies of iss(t) derive from the modulation of the inverter harmonic components (21.8) operated by the rectifier and can be evaluated according to the following relationship:
(21.10)
with=1,2,3,… j and r as in Table 21.1.
As for the sequences, the same considerations about (21.4) and (21.5) still apply.
C. Output Side
The harmonic frequencies of ios(t) are evaluated as shown in Appendix B; it is:
(21.11)
with j and k as in Table 21.2.
Table 21.2: Values of parameters j and k for different mf choices
mf / odd / EvenJ / k / j / k
even integers / Þ / odd integers / even integers / Þ / integers
odd integers / Þ / even integers / odd integers / Þ
The interharmonic frequencies of ios(t) derive from the modulation of the rectifier harmonic components (21.1) operated by the inverter and can be evaluated according to the following relationship:
(21.12)
with j and k as in Table 21.1, =1,2,3,….
Also in this case, the sequences of the harmonics in (21.11) determine the sequences of the interharmonics in (21.12) according to the same rules shown in Section III.
21.3.2 Harmonic and Interharmoic Amplitudes
As for the case of LCI drives, also in the case of PWM drives it seems not possible to predict by means of analytical formulas the amplitude of interharmonics. To have an idea of the entity of the variability, reference is made to the case-study specified in [9] where the results obtained for all the output frequencies from 5 to 50 Hz of a case-study based on a PWM like that of Fig. 21.3, have been considered. The results are reported in Fig. 21.4, where the frequency component amplitudes are referred to the 50 Hz fundamental component.
Fig. 21.4 confirm that, as it is well known, the relative harmonic distortion is greater when the load consumption is lower. On the contrary, the relative interharmonic distortion is not so regular: it may increase with the load and locally be amplified due to resonances.
Fig. 21.4. PWM harmonic and interharmonic amplitudes referred to the 50Hz component versus output frequency: 5th, 7th, 11th and 13th harmonics (___) __; main interharmonics (---).
21.4 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.