Multilevel Voltage Source Modulation

6.2 Multilevel Voltage Source Modulation

Before proceeding with the discussion of multilevel modulation, a general multilevel power converter structure will be introduced and notation will be defined for later use. Although the primary focus of this chapter is on power conversion from DC to an AC voltages (inverter operation), the material presented herein is also applicable to rectifier operation. The term multilevel converter is used to refer to

a power electronic converter that may operate in an inverter or rectifier mode. Figure 6.1 shows the general structure of the multilevel converter system. In this case, a three-phase

motor load is shown on the AC side of the converter. However, the converter may interface to an electric utility or drive another type of load. The goal of the multilevel pulse-width modulation (PWM) block is to switch the converter transistors in such a way that the phase voltages v , v

bs , and v cs are equal to commanded voltages v as , v bs , and v cs . The commanded voltages are generated from an overall supervisory

FIGURE 6.1 Multilevel converter structure.

control [14] and may be expressed in a general form as

as = 2v s cos () θ c (6.1)

v bs = 2v s cos 

 θ c – ------  3 

cs =

2v ∗

s cos  + ------ 2 θ c π  

(6.3) where is a voltage amplitude and ∗ v s θ c is an electrical angle. To describe how the modulation is accom-

plished, the converter AC voltages must be defined. For convenience, a line-to-ground voltage is defined as the voltage from one of the AC points in Fig. 6.1 ( a , or b , or c ) to the negative pole of the DC voltage (labeled g in Fig. 6.1 ). For example, the voltage from a to g is denoted v ag . It is important to note that the converter has direct control of the voltages v ag , v bg , and v cg . The next step in defining the control of the line-to-ground voltages is expressing a relationship between these voltages and the motor phase voltages. Assuming a balanced wye-connected load, it can be shown that [15]

v as

2 – 1 – 1 v 1 ag

v bs = -- 3 – 1 2 – 1 v bg (6.4)

v cs

– 1 – 1 2 v cs

Because an inverse of the matrix in Eq. (6.4) does not exist, there is no direct relationship between commanded phase voltages and line-to-ground voltages. In fact, there are an infinite number of voltage

sets { v ag v bg v cg } that will yield a particular set of commanded phase voltages because any zero sequence components of the line-to-ground voltages will not affect the phase voltages according to Eq. (6.4). In a three-phase system, zero sequence components of { v ag v bg v cg } include DC offsets and triplen harmonics of θ c . To maximize the utilization of the DC bus voltage, the following set of line-to-ground voltages may be commanded [16]

ag =

v dc

------ 1 +

m cos () θ c – ---- cos ( 3 θ c )

v dc

v bg = ------ 1 + m cos  θ c – 2 ------ π  – m  ----  cos ( 3 θ c )

v dc 2

cg = ------ 1 + m cos  θ c + ------  –  ----  cos ( 3 θ c )

where m is a modulation index. It should be noted that the power converter switching will yield line-to- ground voltages with a high-frequency component and, for this reason, the commanded voltages in Eqs. (6.5) to (6.7) cannot be obtained instantaneously. However, if the high-frequency component is neglected, then the commanded line-to-ground voltages may be obtained on a fast-average basis. By substitution of Eqs. (6.5) to (6.7) into Eq. (6.4), it can be seen that commanding this particular set of line-to-ground voltages will result in phase voltages of

vˆ as = ----------- cos ()

mv dc

2 θ c (6.8)

dc = 2 -----------

mv

cos 

c – ------  3 

vˆ bs

vˆ cs = ----------- cos 

mv dc

c + ------ π  

22v ∗ s

m = ----------------

v dc

It should be noted that in H-bridge-based converters, the range of line-to-ground voltage is twice that of converters where one DC voltage supplies all three phases (as in Fig. 6.1 ). The modulation method here can accommodate these converters if the modulation index is related to the commanded voltage magnitude by

2v ∗ m s

H = ------------

v dc

The modulation process described here may be applied to H-bridge converters by substituting m H for m in the equations that follow. The benefit of including the third harmonic terms in Eqs. (6.5) to (6.7) is an extended range of modulation index [16]. In particular, the range of the modulation index is

0 ≤ m ≤ ------- 2 (6.13)

It is sometimes convenient to define a modulation index that has an upper limit of 100% or

m = ------- m 3 (6.14)

The next step in the modulation process is to define normalized commanded line-to-ground voltages, which will be referred to as duty cycles. In terms of the modulation index and electrical angle, the duty cycles may be written:

d a = -- 1 + m cos () θ c – m ---- cos ( 3 θ c )

d b = -- 1 + m cos  – ------ 2 π  – ---- m

cos ( 3

d c = -- 1 + m cos  θ c + ------ π  – ---- cos ( 3 )

To relate the duty cycles to the inverter switching operation, switching states must be defined that are valid for any number of voltage levels. Here, the switching states for the a-, b-, and c-phase will be denoted s a ,s b , and s c , respectively. Although the specific topology of the multilevel converter is covered in the next section, it may be stated in general for an n-level converter that the AC output consists of a number of To relate the duty cycles to the inverter switching operation, switching states must be defined that are valid for any number of voltage levels. Here, the switching states for the a-, b-, and c-phase will be denoted s a ,s b , and s c , respectively. Although the specific topology of the multilevel converter is covered in the next section, it may be stated in general for an n-level converter that the AC output consists of a number of

v ag = ----------------

s a v dc

( (6.18)

0, 1, … n ( – 1 )

v bg = ---------------- s b v dc s b =

( (6.19)

0, 1, … n ( – 1 )

n – 1 ) ---------------- s c v v dc cg =

0, 1, … n ( – 1 – ) ( (6.20)

As can be seen, a higher number of levels n leads to a larger number of switching state possibilities and smaller voltage steps. An overall switching state can be defined by using the base n mathematical expression

a + ns b + s c (6.21) Figure 6.2 shows the a-phase commanded line-to-ground voltage according to Eq. (6.5) as well as line-

sw 2 = n s

to-ground voltages for two-level, three-level, and four-level converters. In each case, the fast-average of v ∗

ag will equal the commanded value v ag . However, it can be seen that as the number of voltage levels increases, the converter voltage yields a closer approximation to the commanded value, resulting in lower harmonic distortion.

The next step in multilevel modulation is to relate the switching states s a ,s b , and s c to the duty cycles defined in Eqs. (6.14) through (6.16). Here, the multilevel sine-triangle technique will be used for this purpose [17, 18, 19]. The first step involves scaling the duty cycles for the n-level case as

d am = ( n – 1 )d a (6.22)

d bm = ( n – 1 )d b (6.23)

d cm = ( n – 1 )d c (6.24) The switching state may then be directly determined from the scaled duty cycles by comparing them to

a set of high-frequency triangle waveforms with a frequency of f sw . For an n-level converter, n − 1 triangle waveforms of unity amplitude are defined. As an example, consider the four-level case. Figure 6.3a shows the a-phase duty cycle and the three triangle waveforms offset so that their peaks correspond to the nearest switching states. In general, the highest triangle waveform has a minimum value of (n − 2) and

a peak value of (n − 1). The switching rules for the four-level case are fairly straightforward and may be specifically stated as

 0 d am < v tr1

 tr2

tr1

am

(6.25)

 2 v tr2 ≤ d am < v tr3

 3v tr3 ≤ d am

Figure 6.3b shows the resulting switching state based on the switching rules. As can be seen, the form is similar to that of Fig. 6.2d and, therefore, the resulting line-to-ground voltage according to Eq. (6.17) will have a fast-average value equal to its commanded value. These switching rules may be extended to

FIGURE 6.2 Power converter line-to-ground output voltages.

any number of levels by incorporating the appropriate number of triangle waveforms and defining switching rules similar to Eq. (6.25). It should be pointed out that the sine-triangle method is shown here since it is depicts a fairly straightforward method of accomplishing multilevel switching. In practice, the modulation is typically implemented on a digital signal processor (DSP) or erasable programmable logic device (EPLD) without using triangle waveforms. One common method for implementation is space-vector modulation [20–22], which is a method where the switching states are viewed in the voltage reference frame. Another method that may be used is duty-cycle modulation [23], which is a direct calculation method that uses duty cycles instead of triangle waveforms and is more readily implementable on a DSP. It is also possible to perform modulation based on a current-regulated approach [22, 24], which is fundamentally different than voltage-source modulation and results in a higher bandwidth control of load currents.

FIGURE 6.3 Four-level sine-triangle modulation technique.