Control of the DC Link Voltage

12.3.4 Control of the DC Link Voltage

where V A PWM and V B PWM are the voltages measured between the middle point of the dc voltage, and the phases a and b Control of the dc link voltage requires a feedback control respectively. In a less straightforward fashion, the phase-to- loop. As already explained in Section 12.3.2, the dc voltage neutral voltage can be evaluated with the help of Eq. (12.53).

V D is compared with a reference V REF , and the error signal “e” obtained from this comparison is used to generate a tem- plate waveform. The template should be a sinusoidal waveform

V PWM (V AB CA = with the same frequency of the mains supply. This template PWM −V PWM ) (12.53)

AN

3 is used to produce the PWM pattern and allows controlling

12 Three-phase Controlled Rectifiers 229

V D C D dc load

Tn

i Dn

FIGURE 12.40 Current waveforms through the mains, the valves, and the dc link.

V D v dc link voltage V BRIDGE D must remain higher than the diode rectification voltage v BRIDGE (feedback control loop required).

FIGURE 12.41 The DC link voltage condition for the operation of the PWM rectifier.

the rectifier in two different ways: (1) as a voltage-source currents and forcing them to follow a sinusoidal current ref- current-controlled PWM rectifier or (2) as a voltage-source erence template, I_ref. The amplitude of the current reference voltage-controlled PWM rectifier. The first method controls template, I MAX is evaluated using the following equation the input current, and the second controls the magnitude and phase of the voltage V MOD . The current-controlled method is

I MAX =G C ·e=G C · (V REF −v D ) (12.54) simpler and more stable than the voltage-controlled method,

and for these reasons it will be explained first. where G C is shown in Fig. 12.43 and represents a controller

such as PI, P, Fuzzy, or other. The sinusoidal waveform of the

12.3.4.1 Voltage-source Current-controlled PWM

template is obtained by multiplying I MAX with a sine function,

with the same frequency of the mains, and with the desired This method of control is shown in the rectifier in Fig. 12.43. phase-shift angle ϕ, as shown in Fig. 12.43. Further, the tem- Control is achieved by measuring the instantaneous phase plate must be synchronized with the power supply. After that,

Rectifier

230 J. W. Dixon

FIGURE 12.42 PWM phase voltages: (a) PWM phase modulation; (b) PWM phase-to-phase voltage; and (c) PWM phase-to-neutral voltage.

v A =V M sin wt

v D LOAD

I_line(a)

I_line(b)

PWM generation

a,b,c

I_line(c)

C I_ref _ AB

Synchr.

sin( wt + j)

MAX

X G C V REF

sin( wt + j −120°)

sin( wt + j −240°)

FIGURE 12.43 Voltage-source current-controlled PWM rectifier.

the template has been created and is ready to produce the The blocks G 1 (s) and G 2 (s) in Fig. 12.44 represent the trans- PWM pattern.

fer function of the rectifier (around the operating point) and However, one problem arises with the rectifier because the the transfer function of the dc link capacitor C D respectively.

feedback control loop on the voltage V C can produce instabil-

= 3 · (V cos ϕ − 2RI S −L S I S s) (12.55) rectifier design. Upon introducing the voltage feedback and

ity. Then it becomes necessary to analyze this problem during

1 sented in a block diagram in Laplace dominion, as shown in

the G C controller, the control of the rectifier can be repre-

V D ( s)

(12.56) P 1 ( s) − P 2 ( = s) V D ·C D ·s Fig. 12.44. This block diagram represents a linearization of the

G 2 ( s) =

system around an operating point, given by the rms value of where P 1 (s) and P 2 (s) represent the input and output the input current, I S .

power of the rectifier in Laplace dominion, V the rms value of

12 Three-phase Controlled Rectifiers 231

FIGURE 12.44 Closed-loop rectifier transfer function.

the mains voltage supply (phase-to-neutral), I S the input cur-

I_line

D Q and R the resistance between the converter and power supply.

rent being controlled by the template, L S the input inductance, PWM

(a) I_ref

flip-flop According to stability criteria, and assuming a PI controller,

CLK the following relations are obtained

sampling clock

C D ·V D

3K hysteresis band adjust

I_line

2R ·K P +L S ·K I I_ref

These two relations are useful for the design of the current- controlled rectifier. They relate the values of dc link capacitor,

dc link voltage, rms voltage supply, input resistance and induc- I_line

I_err

tance, and input power factor, with the rms value of the input

current, I + S . With these relations the proportional and integral

+ PWM gains K P and K I can be calculated to ensure the stability of

(c)

kp + ki/s

– the rectifier. These relations only establish limitations for rec-

I_ref

tifier operation, because negative currents always satisfy the

V_tri

inequalities.

With these two stability limits satisfied, the rectifier will keep FIGURE 12.45 Modulation control methods: (a) periodical sampling; the dc capacitor voltage at the value of V REF (PI controller), (b) hysteresis band; and (c) triangular carrier. for all load conditions, by moving power from the ac to the

dc side. Under inverter operation, the power will move in the transitions is limited to the period of the sampling clock. This opposite direction.

characteristic determines the maximum switching frequency Once the stability problems have been solved and the sinu- of the converter. However, the average switching frequency is

soidal current template has been generated, a modulation not clearly defined. method will be required to produce the PWM pattern for

The HB method switches the transistors when the error the power valves. The PWM pattern will switch the power between I_ref and I_line exceeds a fixed magnitude: the hys-

valves to force the input currents I_line to follow the desired teresis band. As it can be seen in Fig. 12.45b, this type of current template I_ref. There are many modulation meth- control needs a single comparator with hysteresis per phase. ods in the literature, but three methods for voltage-source In this case the switching frequency is not determined, but current-controlled rectifiers are the most widely used ones: its maximum value can be evaluated through the following periodical sampling (PS), hysteresis band (HB), and triangular equation carrier (TC).

The PS method switches the power transistors of the rec-

max

tifier during the transitions of a square wave clock of fixed

(12.59) 4h ·L S frequency: the periodical sampling frequency. In each transi-

tion, a comparison between I_ref and I_line is made, and where h is the magnitude of the hysteresis band. corrections take place. As shown in Fig. 12.45a, this type of

The TC method, shown in Fig. 12.45c, compares the control is very simple to implement: only a comparator and error between I_ref and I_line with a triangular wave. This

a D-type flip-flop are needed per phase. The main advantage triangular wave has fixed amplitude and frequency and is of this method is that the minimum time between switching called the triangular carrier. The error is processed through a

232 J. W. Dixon proportional-integral (PI) gain stage before comparison with

the TC takes place. As can be seen, this control scheme is more complex than PS and HB. The values for k p and k i deter-

(a)

mine the transient response and steady-state error of the TC

(b)

method. It has been found empirically that the values for k p

(c)

and k i shown in Eqs. (12.60) and (12.61) give a good dynamic

(d)

performance under several operating conditions.

FIGURE 12.46 Waveforms obtained using 1.5 kHz switching frequency

k i =ω c ·K P

(12.61) and L S = 13 mH: (a) PS method; (b) HB method; (c) TC method (K P +

K I ); and (d) TC method (K P only).

where L S is the total series inductance seen by the rectifier,

ω c is the TC frequency, and V D is the dc link voltage of the

rectifier. In order to measure the level of distortion (or undesired har- with PI control are quite similar, and the TC with only propor- monic generation) introduced by these three control methods, tional control gives a current with a small phase shift. However, Eq. (12.62) is defined

Fig. 12.47 shows that the higher the switching frequency, the closer the results obtained with the different modulation meth-

2 − I_ref ) ods. Over 6 kHz of switching frequency, the distortion is very dt (12.62)

%Distortion =

(I _line

I rms T

small for all methods.

In Eq. (12.62), the term I rms is the effective value of the

12.3.4.2 Voltage-source Voltage-controlled PWM

desired current. The term inside the square root gives the rms

Rectifier

value of the error current, which is undesired. This formula Figure 12.48 shows a one-phase diagram from which the con- measures the percentage of error (or distortion) of the gener- trol system for a voltage-source voltage-controlled rectifier is ated waveform. This definition considers the ripple, amplitude, derived. This diagram represents an equivalent circuit of the and phase errors of the measured waveform, as opposed to the fundamentals, that is, pure sinusoidal at the mains side and THD, which does not take into account offsets, scalings, and pure dc at the dc link side. The control is achieved by creat- phase shifts.

ing a sinusoidal voltage template V MOD , which is modified in Figure 12.46 shows the current waveforms generated by the amplitude and angle to interact with the mains voltage V. In three aforementioned methods. The example uses an average this way the input currents are controlled without measuring switching frequency of 1.5 kHz. The PS is the worst, but its them. The template V MOD is generated using the differential digital implementation is simpler. The HB method and TC equations that govern the rectifier.

14 Periodical Sampling

Hysteresis Band

Triangular Carrier

(K P *+K I *)

Triangular Carrier

% Distortion

(only K P *)

Switching Frequency (Hz) FIGURE 12.47 Distortion comparison for a sinusoidal current reference.

12 Three-phase Controlled Rectifiers 233

i s (t) + V D C D LOAD

ac-dc conversion

FIGURE 12.48 One-phase fundamental diagram of the voltage source rectifier.

The following differential equation can be derived from the template V MOD to change in magnitude and phase so as to Fig. 12.48

have full unity power factor control of the rectifier. Compared with the control block of Fig. 12.43, in the

voltage-source voltage-controlled rectifier of Fig. 12.49, there v(t ) =L S

is no need to sense the input currents. However, to ensure

stability limits as good as the limits of the current-controlled Assuming that v(t ) =V

2 sin ωt , then the solution for rectifier, the blocks “–R–sL S ” and “–X S ” in Fig. 12.49, have to

i s (t ), to acquire a template V MOD able to make the rectifier emulate and reproduce exactly the real values of R, X S , and L S work at constant power factor, should be of the form

of the power circuit. However, these parameters do not remain constant, and this fact affects the stability of this system, mak-

i s (t ) =I max (t ) sin(ωt + ϕ) (12.64) ing it less stable than the system showed in Fig. 12.43. In theory, if the impedance parameters are reproduced exactly, the sta-

Equations (12.63), (12.64), and v(t) allows a function of bility limits of this rectifier are given by the same equations time able to modify V MOD in amplitude and phase that will as used for the current-controlled rectifier seen in Fig. 12.43 make the rectifier work at a fixed power factor. Combining (Eqs. (12.57) and (12.58)). these equations with v(t) yields

Under steady-state, I max is constant, and Eq. (12.66) can be written in terms of phasor diagram, resulting in Eq. (12.67).

√ v MOD (t ) = V 2 +X S I max sin ϕ − RI max

dI max

As shown in Fig. 12.50, different operating conditions for the

unity power factor rectifier can be displayed with this equation

V MOD V S − jX S S (12.67)

dt

With the sinusoidal template V MOD already created, a mod- ulation method to commutate the transistors will be required.

Equation (12.65) provides a template for V MOD , which is As in the case of current-controlled rectifier, there are many controlled through variations of the input current amplitude methods to modulate the template, with the most well known

I max . Substituting the derivatives of I max into Eq. (12.65) make the so-called sinusoidal pulse width modulation (SPWM), sense, because I max changes every time the dc load is modified. which uses a TC to generate the PWM as shown in Fig. 12.51. The term X S in Eq. (12.65) is ωL S . This equation can also Only this method will be described in this chapter.

be written for unity power factor operation. In such a case, In this method, there are two important parameters to cos ϕ = 1 and sin ϕ = 0. define: the amplitude modulation ratio or modulation index

√ m, and the frequency modulation ratio p. Definitions are

given by

With this last equation, a unity power factor, voltage source,

voltage-controlled PWM rectifier can be implemented as

shown in Fig. 12.49. It can be observed that Eqs. (12.65) and (12.66) have an in-phase term with the mains supply (sin ωt ) where V max

MOD and V TRIANG are the amplitudes of V MOD and and an in-quadrature term (cos ωt ). These two terms allow V TRIANG respectively. On the other hand, f S is the frequency of

max

234 J. W. Dixon

v A = VM sin ωt D

i dc I

B i v B s V D + LOAD

PWM generation

G C V REF Synchr.

sin( wt)

X –R–sL

cos( wt)

X –X s

FIGURE 12.49 Implementation of the voltage-controlled rectifier for unity power factor operation.

δ j wL S I S

V MOD

RI S

j wL S I S

V MOD

RI S

V MOD

I S =0

V MOD

j wL S I S

V RI S

V MOD

j wL S I S

V RI S

12 Three-phase Controlled Rectifiers 235

COMPARATOR V MOD A

V PWM A

V TRIANG

V TRIANG

FIGURE 12.51 Sinusoidal modulation method based on TC.

the mains supply and f T the frequency of the TC. In Fig. 12.51, an integer. If p is an odd number, even harmonics will be elim- m = 0.8 and p = 21. When m > 1 overmodulation is defined. inated. If p is a multiple of 3, then the PWM modulation of

The modulation method described in Fig. 12.51 has a har- the three phases will be identical. When m increases, the ampli- monic content that changes with p and m. When p < 21, it is tude of the fundamental voltage increases proportionally, but recommended that synchronous PWM be used, which means some harmonics decrease. Under overmodulation (m > 1), that the TC and the template should be synchronized. Fur- the fundamental voltage does not increase linearly, and more thermore, to avoid subharmonics, it is also desired that p be harmonics appear. Figure 12.52 shows the harmonic spectrum

1 p −4p−2p p+2p+4 2p − 5 2p − 1 2p 2p + 1 2p + 5 3p − 4 3p − 2 3p 3p + 2 3p + 4

236 J. W. Dixon switching frequency at 60 Hz or only 4.050 Hz in a rectifier

P =9

operating in a 50 Hz supply. This switching frequency can be managed by MOSFETs, IGBTs, and even Power Darlingtons.

P =15

Then a number p = 81, is feasible for today’s low and medium power rectifiers.

P = 21 P = 27

12.3.4.3 Voltage-source Load-controlled PWM

Rectifier

P = 39

A simple method of control for small PWM rectifiers (up

P = 81

to 10–20 kW) is based on direct control of the dc current. Figure 12.54 shows the schematic of this control system. The fundamental voltage V MOD modulated by the rectifier is pro- duced by a fixed and unique PWM pattern, which can be

FIGURE 12.53 Current waveforms for different values of p. carefully selected to eliminate most undesirable harmonics. As the PWM does not change, it can be stored in a permanent digital memory (ROM).

of the three-phase PWM voltage waveforms for different values The control is based on changing the power angle δ between of m, and p = 3k where k is an odd number.

the mains voltage V and fundamental PWM voltage V MOD . Due to the presence of the input inductance L S , the har- When δ changes, the amount of power flow transferred from monic currents that result are proportionally attenuated with the ac to the dc side also changes. When the power angle is the harmonic number. This characteristic is shown in the cur- negative (V MOD lags V), the power flow goes from the ac to rent waveforms of Fig. 12.53, where larger p numbers generate the dc side. When the power angle is positive, the power flows cleaner currents. The rectifier that originated the currents of in the opposite direction. Then, the power angle can be con-

= 450 V

Fig. 12.53 has the following characteristics: V D dc ,

trolled through the current I D . The voltage V D does not need

f −f = 220 V ac ,L S = 2 mH, and input current I S = 80 A rms . to be sensed, because this control establishes a stable dc volt- It can be observed that with p > 21 the current distortion age operation for each dc current and power angle. With these is quite small. The value of p = 81 in Fig. 12.53 produces characteristics, it is possible to find a relation between I D and an almost pure sinusoidal waveform, and it means 4860 Hz of δ so as to obtain constant dc voltage for all load conditions.

C D LOAD

Synchr.

d=f(Id)

DIGITAL CONTROL WITH

V FIXED PWM PATTERN

FIGURE 12.54 Voltage-source load-controlled PWM rectifier.

12 Three-phase Controlled Rectifiers 237 This relation is given by

The current sensors are located at the input terminals of the power source and these currents (instead of the rectifier cur- rents) are forced to be sinusoidal. As there are polluting loads

V cos δ ω L S − R sin δ −1 in the system, the rectifier is forced to deliver the harmonics

I D = f (δ) =

2 (12.70) that loads need, because the current sensors do not allow the

harmonics going to the mains. As a result, the rectifier cur- rents become distorted, but an adequate dc capacitor C D can From Eq. (12.70) a plot and a reciprocal function δ = f(I D ) is keep the dc link voltage in good shape. In this way the rectifier obtained to control the rectifier. The relation between I

D and δ

can do its duty, and also eliminate harmonics to the source. In

allows for leading power factor operation and null regulation. addition, it also can compensate power factor and unbalanced The leading power factor operation is shown in the phasor load problems. diagram of Fig. 12.54.

The control scheme of the voltage-source load-controlled

12.3.5.2 Frequency Link Systems

rectifier is characterized by the following: (i) there are neither Frequency link systems permit power to be transferred form input current sensors nor dc voltage sensor; (ii) it works with a one frequency to another one. They are also useful for linking fixed and predefined PWM pattern; (iii) it presents very good unsynchronized networks. Line-commutated converters are stability; (iv) its stability does not depend on the size of the widely used for this application, but they have some drawbacks

dc capacitor; (v) it can work at leading power factor for all that force-commutated converters can eliminate. For example, load conditions; and (vi) it can be adjusted with Eq. (12.70) the harmonic filters requirement, the poor power factor, and to work at zero regulation. The drawback appears when R the necessity to count with a synchronous compensator when in Eq. (12.70) becomes negligible, because in such a case the generating machines at the load side are absent. Figure 12.56 control system is unable to find an equilibrium point for the dc shows a typical line-commutated system in which a 60 Hz load link voltage. That is why this control method is not applicable is fed by a 50 Hz supply. As the 60 Hz side needs excitation to to large systems.

commutate the valves, a synchronous compensator has been required.

In contrast, an equivalent system with force-commutated

12.3.5 New Technologies and Applications of

converters is simpler, cleaner, and more reliable. It is imple-

mented with a dc voltage-controlled rectifier, and another The additional advantages of force-commutated rectifiers identical converter working in the inversion mode. The power

Force-commutated Rectifiers

with respect to line-commutated rectifiers, make them bet- factor can be adjusted independently at the two ac ter- ter candidates for industrial requirements. They permit new minals, and filters or synchronous compensators are not applications such as rectifiers with harmonic elimination capa- required. Figure 12.57 shows a frequency link system with bility (active filters), power factor compensators, machine force-commutated converters. drives with four-quadrant operation, frequency links to con- nect 50 Hz with 60 Hz systems, and regenerative converters for

12.3.5.3 Special Topologies for High

traction power supplies. Modulation with very fast valves such

Power Applications

as IGBTs permit almost sinusoidal currents to be obtained. High power applications require series- and/or parallel- The dynamics of these rectifiers is so fast that they can connected rectifiers. Series and parallel operation with force- reverse power almost instantaneously. In machine drives, cur- commutated rectifiers allow improving the power quality rent source PWM rectifiers, like the one shown in Fig. 12.35a, because harmonic cancellation can be applied to these can be used to drive dc machines from the three-phase sup- topologies. Figure 12.58 shows a series connection of force- ply. Four-quadrant applications using voltage-source PWM commutated rectifiers, where the modulating carriers of the rectifiers, are extended for induction machines, synchronous valves in each bridge are shifted to cancel harmonics. The machines with starting control, and special machines such as example uses sinusoidal PWM that are with TC shifted. brushless-dc motors. Back-to-back systems are being used in

The waveforms of the input currents for the series connec- Japan to link power systems with different frequencies.

tion system are shown in Fig. 12.59. The frequency modulation ratio shown in this figure is for p = 9. The carriers are shifted by 90 ◦ each, to obtain harmonics cancellation. Shifting of the

12.3.5.1 Active Power Filter

carriers δ T depends on the number of converters in series (or Force-commutated PWM rectifiers can work as active power in parallel), and is given by

filters. The voltage-source current-controlled rectifier has the capability to eliminate harmonics produced by other polluting

loads. It only needs to be connected as shown in Fig. 12.55.

238 J. W. Dixon

I_line(a)

I_line(b)

i B L i C L POLLUTING LOADS

C D V D LOAD

I_line(a)

I_line(b)

a,b,c

I_line(c)

PWM generation A B C I_ref _

Synchr.

I MAX

sin( wt + j)

X G C V REF

sin( wt + j −120°)

sin( wt + j −240°)

FIGURE 12.55 Voltage-source rectifier with harmonic elimination capability.

50 Hz

60 Hz

Excitation Voltage

Passive Filter

synchr.

Compensator

Master Control

Passive Filter

FIGURE 12.56 Frequency link systems with line-commutated converters.

12 Three-phase Controlled Rectifiers 239

50 Hz

60 Hz

50 Hz

V D 60 Hz

PWM +

PWM

DC LINK VOLTAGE

+ V D REF

CONTROL

POWER CONTROL

FIGURE 12.57 Frequency link systems with force-commutated converters. MAINS SUPPLY

V D(1)

SPWM generation

v MOD A B V v MOD C v T(1)

MOD

V D(2)

SPWM generation

C V T(2)

v MOD A v

v MOD B MOD

V D(3)

SPWM generation

v MOD A B v MOD C V v T(3) MOD

V D(M)

SPWM generation

v MOD A v MOD B v MOD C V T(4)

MONITOR

V M sin( ωt)

-R-sL s I MAX G C

REF

sin( ωt)

cos( ωt)

-X s

240 J. W. Dixon

Current in one of the four

converters in series