Nr 59
Prace Naukowe Instytutu Maszyn, Napędów i Pomiarów Elektrycznych Politechniki Wrocławskiej Nr 59
Studia i Materiały
Nr 26
2006
AC/DC line-side converter, PWM rectifier, Hysteresis-Band Modulation, Carrier-Based Sinusoidal Modulation, Space-Vector Modulation, overmodulation, fixed switching frequency
Michał KNAPCZYK * , Krzysztof PIEŃKOWSKI * F
F
F
ANALYSIS OF PULSE WIDTH MODULATION TECHNIQUES FOR AC/DC LINE-SIDE CONVERTERS
The paper presents the analysis of the modulation strategies for the AC/DC line-side converters. The application of the different modulation methods to the control system of the AC/DC converter has been presented. The operation of the AC/DC converter in different dynamic states strongly depends on the modulation method applied. The theoretical background of Hysteresis-Band Modulation, Carrier-Based Sinusoidal Modulation and Space-Vector Modulation has been presented. The issue of overmodulation has been discussed. Voltage Oriented Control of the AC/DC line-side converter has been chosen to examine the presented modulation methods. The influence of the discussed modulation methods on the line current distortion and the switching frequency has been examined. The simulation results of the presented techniques have been demonstrated and concluded.
1. INTRODUCTION The dynamic development of the power- and microelectronics devices sustains continual progress in design and realization of modern adjustable speed drives. The interest of researchers in the elaboration of advanced control techniques for voltage source inverters was in last two decades aroused by AC/DC line-side converters called also PWM rectifiers (synchronous rectifiers). These front-end rectifiers due to their properties systematically displace the diode bridges becoming an important part of the modern frequency converters for the intelligent motion control applications [1,5]. The three-phase two-level AC/DC line-side converters provide sinusoidal line currents and bidirectional power flow at the unity power factor (UPF). These properties have decided of the use of the PWM rectifiers in the applications improving the electrical power quality [7]. __________ *
Politechnika Wrocławska, Instytut Maszyn, Napędów i Pomiarów Elektrycznych, 50-370 Wrocław, ul. Smoluchowskiego 19,
[email protected],
[email protected].
The AC/DC converters consist of power electronics devices like Insulated Gate Bipolar Transistors (IGBT) or Gate Turn-Off thyristors (GTO) that are characterized by switch mode operation. The capability of forming sinusoidal currents is provided by the introduction of the sophisticated technique called Pulse-Width Modulation (PWM). This technique provides the sequences of width-modulated pulses to control power switches. Many PWM techniques have been developed according to special requirements and optimization criteria. The choice of the particular PWM technique arises from the desired performance of the synchronous rectifiers [2,3]. Generally pulse-width modulation techniques for frequency converters may be classified as follows: Carrier-Based Sinusoidal PWM, Hysteresis-Band PWM, Space Vector PWM, Selected Harmonic Elimination PWM, Minimum Current Ripple PWM, Sinusoidal PWM with Instantaneous Current Control and Random PWM. This paper presents basic assumptions and applications of selected, most frequently used modulation techniques applied to PWM rectifiers. For the comparative analysis Voltage Oriented Control of the AC/DC line-side converter has been chosen to examine the proposed modulation methods. 2. VOLTAGE SOURCE AC/DC LINE-SIDE CONVERTER The topology of the voltage source AC/DC converter connected to the grid is presented in Fig.1. The power circuit of the synchronous rectifier stems from the topology of the three-phase PWM voltage inverter.
Fig.1. Voltage source AC/DC line-side converter
The PWM rectifier’s bridge consists of six fully-controlled IGBT transistors connected to the supply line throughout the three symmetrical line inductors. The voltage drop over line chokes has to be controlled to provide sinusoidal line currents. Equations (1) describe the dynamic model of the PWM rectifier in natural (A,B,C) coordinates. Sa, Sb, Sc represent states [1,0] of power switches in respective converter legs.
1 d i gA = dt Lg
U dc ⎡ ⎤ ⎢e gA − R g i gA − 3 (2 S a − S b − S c )⎥ ⎣ ⎦
1 d i gB = dt Lg
U dc ⎡ ⎤ ⎢e gB − Rg i gB − 3 (− S a + 2 S b − S c )⎥ ⎣ ⎦
1 d i gC = dt Lg
U dc ⎡ ⎤ ⎢e gC − Rg i gC − 3 (− S a − S b + 2 S c )⎥ ⎣ ⎦
(1)
1 d (S a ⋅ igA + S b ⋅ i gB + S c ⋅ i gC − iload ) U dc = dt Cd
3. PULSE-WIDTH MODULATION IN APPLICATION TO PWM RECTIFIERS The major aim of the pulse width modulation in the AC/DC line-side converters is the control of the amplitudes of the main harmonics of the converter input PWM three-phase voltages. Besides the appropriate forming of the line current harmonic spectrum should be provided. In general the modulation techniques should be characterized by the wide range of the linear operation, fixed switching frequency and lower influence on producing of the higher harmonics in the line currents [6]. The modulation process during the operation of the PWM rectifier should provide the maximal use of the DC-link voltage. The modulation index m expresses the capability of the DC-link voltage utilization in generating the PWM voltages at the input of the AC/DC converter during the control process.
Fig.2. Range of Pulse-Width Modulation for the AC/DC line-side converters
Fig. 2 presents the dependence of the converter input voltage on the modulation index with respect to the DC-link voltage. The modulation bandwidth is generally divided into the linear and the nonlinear range. The limitation of the modulation region to the linear range is sufficient for the proper operation of the PWM rectifier. Yet for the excellent dynamic performance of the synchronous rectifier during the transients the operation in the overmodulation range must be provided [1,8]. However this technique introduces the line current distortions due its nonlinearity and may be inadvisable in the applications of the AC/DC converters improving the electrical power quality. The issue of the modulation index and the modulation range for the basic PWM techniques will be presented in detail in the next sections of this paper. 3.1. HYSTERESIS-BAND PULSE-WIDTH MODULATION
The Hysteresis Pulse-Width Modulation, named also bang-bang current control consists in direct forcing of the line current flow according to the current reference signals i*gA, i*gB, i*gC. This kind of the modulation is performed in the nonlinear control circuit with the hysteresis relays [7]. The Voltage Oriented Control of the PWM rectifier with the hysteresis pulse-width modulation is presented in Fig.3.
Fig.3. Voltage Oriented Control with Hysteresis-Band PWM (HB-PWM)
When the instantaneous value of the line phase current exceeds its reference value than the respective grid phase is instantly connected to the negative node of the DClink voltage. Otherwise the grid phase is switched to the positive node in the DC-link. This process is carried out simultaneously and independently for two other phases.
Equations (2) describe bang-bang current control:
H ⇒ Sa = 0 2 H ≤ − ⇒ Sb = 0 2 H ≤ − ⇒ Sc = 0 2
H ⇒ Sa = 1 2 H ≥ ⇒ Sb = 1 2 H ≥ ⇒ Sc = 1 2
Δi gA ≤ −
Δi gA ≥
Δi gB
Δi gB
Δi gC
Δi gC
(2)
where Δi g = i g − i * g is the line current error and H is the hysteresis bandwidth in [A]. In result the voltage source AC/DC line-side converter with the hysteresis modulation operates as the source of directly formed current. The idea of the hysteresis pulsewidth modulation is demonstrated in Fig.4.
Fig.4. The idea and control signals of Hysteresis-Band PWM
The basic problem in the bang-bang modulation is the alternating switching frequency that depends on the following significant factors: hysteresis bandwidth, electromagnetic time constant of the grid and chokes circuit, IGBT dead-time and the difference between DC-link voltage and instantaneous value of the respective grid phase voltage. In order to obtain the privileged switching frequency the additional carrier signal may be added to the current error at the input of the relays. The other solution is
the synchronization of the switching process i.e. generation of the firing pulses in every fixed sample step. This type of bang-bang modulation is usually applied in the microprocessor-based digital control systems and is called Δ-modulation. The line currents oscillate around the current reference signals within the boundaries defined in the hysteresis relays. Fig.5 presents the line phase voltage and the respective line phase current for the control system of the AC/DC converter with Hysteresis-Band PWM depicted in fig.3.
Fig.5. Line voltage (1) and respective line current (2) for Hysteresis-Band PWM
The narrower hysteresis bandwidth is the better reconstruction of the sine-shaped currents is obtained. However the absolute reduction of the hysteresis bandwidth is unacceptable due to the restricted values of the switching frequency of the power transistors. Hence the minimal bandwidth of the hysteresis relays should provide no higher switching frequency as results from the value of dead-time for IGBT devices.
Fig.6. Line current harmonic spectrum for Hysteresis-Band PWM
In case of non-synchronized Hysteresis-Band PWM the line current harmonic spectrum contains all higher harmonics including sub-harmonics (fig.6). The Total Harmonic Distortion ratio of the line current from fig.5 is equal THDigA=9.3%. The Hysteresis-Band Pulse-Width Modulation technique is mostly used in the analog control systems of the PWM rectifiers. The hardware implementation of the hysteresis relays can be realized using simple circuit applications with the operational amplifiers [1]. The dynamic performance of the PWM rectifier’s control system with the bang-bang modulation technique applied is excellent since the hysteresis relays do not require tuning and provide robustness to line disturbances or parameter mismatch. 3.2. CARRIER-BASED SINUSOIDAL PULSE-WIDTH MODULATION
Carrier-Based Sinusoidal Pulse-Width Modulation is based on the comparison of the converter voltage reference signals with the carrier signal of the triangular shape [4]. Unlike Hysteresis-Band PWM with the reference current signals, this modulation technique provides the firing pulses upon the converter voltage reference signals. The Voltage Oriented Control of the PWM rectifier with Carrier-Based Sinusoidal PulseWidth Modulation is presented in Fig.7.
Fig.7. Voltage Oriented Control with Carrier-Based Sinusoidal PWM (CB-SPWM)
The reference voltage signals U*A, U*B, U*C are usually sinusoidal of the grid frequency. Their amplitudes are proportional to the expected amplitudes of the main harmonic of the converter input PWM three-phase voltages. The frequency of the carrier signal is usually hundredfold higher then the frequency of reference signals. The modulation index m is defined by the mutual rate of the reference signals and the carrier signal amplitudes (3):
m=
U * A, B ,C U carrier
(3)
The converter voltage reference signals U*d, U*q are provided by two linear line current controllers operating in the (d-q) rotating frame. The idea of the Carrier-Based Sinusoidal Pulse-Width Modulation is demonstrated in Fig.8.
Fig.8. The idea and control signals of Carrier-Based Sinusoidal PWM
The range of the Carrier-Based Sinusoidal Pulse-Width Modulation is constrained due to the fact that the modulation index can reach the maximal value of m=1. Then the amplitudes of the sinusoidal converter voltage reference signals and the carrier signal are equal. The equation (4) describes the amplitude of the main harmonic of the converter input PWM phase voltage by the unity modulation index m=1. U U convA = U *max = dc (4) 2
According to the equation (4) the RMS value of the line-to-line converter input PWM voltage is described by the following expression: U convAB ( RMS ) =
3 3 U convA( RMS ) = 2 U dc = 0.612 ⋅ U dc 2 2
(5)
The maximal utilization of the DC-link voltage in case of Carrier-Based SPWM is about 83%. Due to the fact that the DC-link is not connected with the neutral node of the grid it is admissible to modify the sinusoidal converter voltage reference signals by adding so called Zero Sequence Signals of the third harmonic frequency [9].
Fig.9. Line voltage (1) and respective line current (2) for Carrier-Based Sinusoidal PWM
This solution extends the linear range of the modulation up to m=2/√3=1.15. Then the utilization of the DC-link voltage increases to 91%. Fig.9 presents the line phase voltage and the respective line phase current for the control system of the AC/DC converter with the Carrier-Based Sinusoidal PWM depicted in Fig.7.
Fig.10. Line current harmonic spectrum for Carrier-Based Sinusoidal PWM
Fig.10 demonstrates the line current harmonic spectrum for the Carrier-Based Sinusoidal Pulse-Width Modulation. The higher harmonics are located in the vicinity of the carrier frequency fs=5 kHz and its multiple. The Total Harmonic Distortion ratio of the line current presented in Fig.9 is equal THDigA=6.1%. The Carrier-Based Sinusoidal Pulse-Width Modulation is acknowledged as the classical type of the modulation for the power converters. The analog realization is relatively simple and is based on the application of the operational amplifiers and the monolithic carrier signal generators. For the digital implementation a slightly different type of the Carrier-Based Sinusoidal PWM technique has been elaborated. It is called Regular Modulation and consists in the fixed sampling of the sinusoidal converter voltage reference signals. In result the sequences of the pulses of the precisely defined widths are obtained. The modern evaluation boards with the Digital Signal Processors are equipped with the hardware implementation of Carrier-Based Sinusoidal Modulation. These digital devices are purpose-dedicated to control the power converters. 3.3. SPACE VECTOR PULSE-WIDTH MODULATION
The power transistors of the three-phase two-level synchronous rectifier during its operation provide eight different states of the conduction (Fig.11).
Fig.11. Conduction states of the AC/DC line-side converter
The converter input voltage can be represented by the space vector U [8,9]. The space vector U once can take one of eight different positions resulting from the permissible combinations of the conduction states. Fig.12a presents the diagram of the possible locations of the space vector U decomposed over (α-β) orthogonal coordinates oriented with the line phase A. Vectors U1 to U6 have fixed modulus of (2/3)⋅Udc and are phase-shifted by π/3. They are called active vectors and refer to the conduction states of the power switches during which the respective phases are supplying the DC-link load. While the three upper or three lower transistors are conducting simultaneously, the supply line is short-circuited. These states are described by two zero vectors U0 and U7. The zero vectors are located in the origin of (α-β) coordinates and they are represented by the two concentric points.
Fig.12. Space Vector representation of the converter input voltage (a) and control signal pattern (b)
If six active conduction states were successively forced in the rectifier, the converter input PWM voltage space vector would change its position every π/3 inside the hexagon. Hence active vectors allocate six equal sectors. During the SV-PWM the converter input voltage vector U should be appropriately formed in order to map the reference vector U*. The (d-q) components of the reference vector U* are generated by two PI current controllers in the control system depicted in Fig.13. The optional position of the reference vector U* inside the sector can be reached by providing the symmetrical control pulses represented by the following switching sequence: U0-U1-U2-U7-U2-U1-U0
(6)
in case when the reference vector U* is moving throughout the first sector. In the other sectors two next adjacent vectors should be considered. The period TS of each symmetrical control sequence is fixed and corresponds to the frequency fS as follows:
fS =
1 TS
(7)
Fig.12b presents the control sequence for the first sector. During the TS cycle the adjacent and zero vectors are applied for unambiguously determined times t1, t2, t0. These durations are calculated with the help of the trigonometrical relationships (8). t1 =
3 ⎛π ⎞ ⋅ m ⋅ TS ⋅ sin ⎜ − ω g t ⎟ 3 ⎝3 ⎠
3 ⋅ m ⋅ TS ⋅ sin (ω g t ) 3 T t 0 = S − (t1 + t 2 ) 2 t2 =
(8)
where 0 ≤ωgt ≤ π/3 (in the first sector). The argument of the sin functions should be appropriately modified according to the range of the other sectors every π/3.
Fig.13. Voltage Oriented Control with Space-Vector PWM (SV-PWM)
The reference vector U* rotates with mains pulsation ωg. To achieve the smooth rotation of the reference vector U* a relatively small value of period TS is necessary. In practice the switching frequency of fS=5 kHz is sufficient to obtain the sinusoidal line
currents with the minimum ripple. Fig.14 presents the line phase voltage and the respective line phase current for the control system of the AC/DC converter with the Space Vector Pulse-Width Modulation depicted in Fig.13.
Fig.14. Line current and respective line voltage for Space Vector PWM
The Total Harmonic Distortion ratio of the line current presented in Fig.14 is equal THDigA=5.7%. For the linear modulation the maximal length of the reference vector U* is restricted to the radius of the circle inscribed into the hexagon (Fig.12a) and is equal U*max = Udc/√3. The modulation index can reach the maximal value m=1.15 and the utilization of the DC-link voltage equals 91%. The Space Vector Modulation corresponds to the sinusoidal modulation with the additional Zero Sequence signals.
Fig.15. Line current harmonic spectrum for Space Vector PWM
Fig.15 presents the line current harmonic spectrum for the Space Vector PulseWidth Modulation. The higher harmonics are located in the vicinity of the switching frequency fs=5 kHz thus have no practical influence on the grid.
3.4. OVERMODULATION
The nonlinear Pulse-Width Modulation occurs when the modulus of the reference vector U* exceeds the radius of the hexagon’s incircle. Then the hodograph of the converter input voltage vector U is not circular any more since the top of the vector remains at the hexagon being unable to cross its boundaries.
Fig.16. Regions beyond linear modulation: a) overmodulation; b) six-step operation
Fig.16 presents the regions of the nonlinear modulation. The overmodulation proceeds when the modulation index is located in the range of 1.15