state space-vector modulation fundamentals

State Space-Vector Modulation Fundamentals

Brendan Graham, March 2015http://usysinc.com

Table of Contents

3-ϕ peak interphase voltage ratio..........................................................................................................1PWM waveform fundamental (first harmonic) amplitude....................................................................1Half-bridge fundamental (first harmonic) amplitude............................................................................2Six-step modulation characteristic voltages..........................................................................................2State Space-Vector Modulation.............................................................................................................3

3-ϕ peak interphase voltage ratio

Let two unit vectors v⃗a and v⃗b connected at the origin be out of phase by 120 degrees. A resultant third vector v⃗ r

connecting the ends of these two unit vectors is drawn. A line is drawn through the origin, normal to and bisecting the resultant vector v⃗ r , effectively dividing the original triangle v⃗a v⃗b v⃗ r into two equal, small right angled triangles. The magnitude of the resultant vector, ∣v⃗ r∣ may now be evaluated from Pythagoras' theorem by summation of the sides of both triangle as

∣v⃗ r∣= √∣v⃗a∣−∣v⃗a∣cos(60o)+√∣v⃗b∣−∣v⃗b∣cos(60o) 38.1

∣v⃗ r∣= 2 √0.75 = √4×0.75 = √3 38.2

This result can obviously also be obtained as

∣v⃗ r∣= (∣v⃗a∣+∣v⃗b∣)sin (60o)

= 2∣v⃗a∣sin (60o)

38.3

PWM waveform fundamental (first harmonic) amplitude

The Fourier expansion of an x-axis symmetrical square wave v sq (ωt ) of peak magnitude vdc is

v sq(ω t) =4V sq ( pk)

π ∑n=1,3,5,...

∞1n

sin (nω t ) 38.4

Thus the bn coefficients of the Fourier series approximation to the square wave function are

bn =4

n π{ 0 n even

1 n odd 38.5

Therefore the peak amplitude of the fundamental sinusoidal component of a square wave voltage stimulus is

b1 =4π

38.6

The peak amplitude of the fundamental sinusoidal voltage as a function of an x-axis symmetrical square wave input of peak voltage V sq( pk) can be written as

V pF=

4V sq ( pk)π

38.7

State Space-Vector Modulation Fundamentals 1 Brendan Graham, μSys Integral, 3/2015

Half-bridge fundamental (first harmonic) amplitude

The peak value of the first harmonic (the fundamental) sinusoidal component V pF of a 50% duty cycle half H-bridge

totem-pole output as a function of the DC supply voltage V dc is

V P (HB)

F=

4π

V dc

2=

2π V dc 38.8

For a PWM modulated, three phase, 6-switch converter composed of three similar half-bridge totem pole output stages switching between a DC link bus of voltage V dc , the peak per phase fundamental harmonic output voltage obtainable is 63.7% of V dc and identical to the above.

Six-step modulation characteristic voltages

Within a PWM modulator where a reference sine waveform is compared to a periodic triangular carrier, the magnitude of the fundamental component of the PWM output waveform is proportional to the pulse width duty cycle, which may not exceed unity and often refered to as the modulation index, such that

m pwm =mref

mcarrier

:m pwm<1 38.9

Inspection of any of the state vector traingles in Figure 1 below shows that the maximum inter-state, output phase voltage obtainable by a three-phase 6-switch SVM converter due to an applied voltage V Z (load ) across a phase impedance is

V svm(max) = sin(2 π

6 )∣V Z (max)∣

=√32

∣V Z (max)∣38.9.1

From Figure 2, it is clear that the maximum possible voltage applied across either of the three load impedance is

V Z (max) =±23

V dc 38.9.2

Combining (38.9.1) and (38.9.2), the maximum SVM output phase voltage obtainable with a DC bus input V dc is the radius of the largest inscribed circle within the state vector hexagon and the optimum state space vector trajectory

V svm(max ) =√32

23

V dc =1

√3V dc 38.9.3

The maximum SVM line-to-line (inter-phase) voltage for 3-ϕ, 2π

3 operation and using (38.2), is

V svm(LL) = √3V svm( max) = V dc 38.9.4

This result shows that SVM can attain 3-ϕ line-to-line output voltages equal to the DC supply bus, thus providing 100% DC bus voltage utilization while minimizing switching currents.

The ratio of the maximum SVM output phase voltage V svm(max ) to the maximum first harmonic fundamental of a 50% duty cycle PWM waveform, is known as the fundamental modulation index which is

msvm( max )=

1

√3V dc

2π V dc

= π2 √3

= 0.907 38.9.5


State Space-Vector Modulation

Consider a DC to sinusoidal 3-ϕ AC grid tied power converter for which the model reference waveform is the output from a PLL locked to the frequency and phase of the grid voltage. A state machine may be configured to output three temporally spaced sinusoidal voltage command vectors. The state machine command vector outputs control three half-bridge power switches that can switch either the input DC bus voltage V dc or the ground potential on either of three outputs phases. Such a state machine can thus be considered analogous to the stator of a physical generator with the rotor being spun at the reference frequency. It has the ability to synthesize an arbitrary magnitude 3-ϕ sine wave output synchronized to the command reference input.

Figure 1: 3-ϕ switch output driven impedances for state vectors 100 and 011

For a three phase power converter comprising three half-bridge switches where the upper and lower switches are switched1800 out of phase with one another, six unique active state vectors may be applied to the switch inputs plus two null

vectors that support no voltage across either of the three phase neutral point coupled output load impedance. The six active state vector operation of this configuration is referred to as six-step. It can easily be verified from consideration of Figure 1, that the maximum voltage applied to any output phase load impedance is and occurs across the specific phase indicated by the minority one or zero of any non null Gray coded active state vector.

With this configuration, the aforementioned state machine comprises six discrete states each of which is responsible for operation over width sectors and computing the PWM duty cycle of each of the three half-bridge switches, on a per cycle basis at the PWM switching frequency, from the advancing phase angle of the reference input. As the phase angle of advances to the border of the current sector, the state machine transitions to the following state where the operation is repeated. Each of the six states, by virtue of the three-bit Gray coded state variable (vector), has the ability to produce a voltage on each of the three output phases, within each of the six sectors of a full cycle at. As the six states are synchronizedto and sequenced through completely during one full cycle at the radial frequency of the reference vector, three phase voltage outputs are synthesized.

Figure 2: SVM voltage hexagon

Space Vector Modulation relies on the principle that any vector inside the dashed hexagon can be expressed as the time or pulse duration weighted average combination of any two adjacent active space vectors and the null-state vectors 0 (000) and7 (111). In order to obtain optimum harmonic performance and the minimum switching frequency for each of the power switching devices, the state sequence is arranged such that the transition from one state to the next is performed by


switching only one half-bridge inverter leg at a time. This effectively means that the sequential state vector space is Gray coded.

As the weight of adjacent active state vectors always differs by one, then either of the null vectors (000) or (111) can be judiciously inserted within active state vector time-weighted switching sequences, whilst still preserving Gray coding and thus minimum switching, and enabling either inter sector time averaged synthesis of a desired voltage or equivalently, optimum vector angular positioning. For instance, within sector III, the synthesis of an optimum intermediate voltage may be achieved by switching the symmetrical state vector sequence 7340437 while within sector IV the state sequence is 7540457. In this example both the two null state vectors 7 (111) and 0 (000) were judiciously inserted at both ends and middle respectively of the switching sequence comprising state vectors 3,4 and 4,5 whilst preserving Gray coding. Note thatsector to sector state vector transition sequences always begin and end with a null state vector. Also, in order to preserve Gray coding, active state vector sequences are either forward or reversed depending on whether the sector is even or odd and the placement of the null state vectors 0 and 7. As a result, the state vector sequences for sectors III and IV of the previous example could alternatively have been 0437340 and 0457540 respectively.

Six-step hexagonal state space vector switching operation alone will not produce sufficiently low THD as this output waveform does not sufficiently approximate a sine. Indeed the sharp vertices of the state space hexagon define an imperfect approximation to a smooth circle. In order to remove the sharp hexagonal vertices, the optimum state space vector trajectoryis that which encompass the largest inscribed circle of radius within the bounds of the state space hexagon, where the absolute magnitude of any active state vector is 2

3 V dc . It is for this reason that the active state vectors are not switched during symmetric center aligned conventional SVM modulation at the hexagon vertices. Inter-sector pulse duration or PWMduty-cycle weighted vector averaging is thus used to more accurately approximate a perfect circular state space trajectory resulting in a more perfect sine output waveform with sufficiently low THD.

a b c Ua Ub Uc UAB UBC UCA Vector

0 0 0 0 0 0 0 0 0 O000

1 0 0 23 V dc − 1

3 V dc − 13 V dc

V dc 0 −V dc U0

1 1 0 13 V dc

13 V dc − 2

3 V dc0 V dc −V dc U60

0 1 0 − 13 V dc

23 V dc − 1

3 V dc−V dc V dc 0 U120

0 1 1 − 23 V dc

13 V dc

13 V dc

−V dc 0 V dc U240

0 0 1 − 13 V dc − 1

3 V dc23 V dc

0 −V dc V dc U300

1 0 1 13 V dc − 2

3 V dc13 V dc

V dc −V dc 0 U360

1 1 1 0 0 0 0 0 0 O111

Table 1: 3-ϕ Gray coded state vectors

Center aligned conventional SVM modultaion is defined by symmetrical inter-sector PWM of adjacent active state vectors and an appropriate null vector whilst maintaining a constant average volt-second product. Clearly, as the weight of adjacent active state vectors differs by one, the three permissible inter-sector voltage values applied that can be sequentially to any phase impedance during the inter-sector interval T S are v t1=

13 V dc , v t2=

23 V dc and the null vector voltage

v t0 /7=0 .

If the total inter-sector interval T S is devoted to voltage averaging by switching each of these three state vectors, then

T S=t 1+t 2+t0 /7 38.9.6

From (38.9.2) the maximum value of each of the six active state vectors defining the vertices of the SVM hexagon is23 V dc thus the active state vectors can be written as


V⃗ k =23

V dc ej( k−1) π

3 : k ∈{1,2, ... ,6}

=23

V dc(cos(k−1) π3

+ j sin (k−1) π3 )

38.9.7

As the volt-second product of all phase impedances are equal and constant then, an avarage or mean inter-state vectorV⃗ ref can also be defined such that

V⃗ ref T S = v⃗ k t k+v⃗k +1 t k+1+v⃗0 /7t 0 /7

= v⃗ k t k+v⃗ k+1 t k+1

38.9.8

Figure 3: Average vector V⃗ ref within active state sector 1

The maximum inter-sector angular range of motion of the vector V⃗ ref is π3 . A sectorial bounding triangle as in Figure

1, with the vector V⃗ ref T S pointing outwards from the origin with π3 radians of freedom can be invisaged composed of

two active state vectors v⃗k t k and v⃗ k +1 t k+1 . If in state sector 1, vector v⃗k t k is placed at the origin entirely on the x-axis then vector v⃗k +1 t k+1 lies at the end of v⃗k t k at an angle π

3 to the x-axis.

Defining the complex sinusoidal vector V⃗ ref =∣V⃗ ref∣ejω t =∣V⃗ ref∣(cosΨ +sinΨ ) where for simplicity Ψ =ω t and

writing the vectors V⃗ ref T S , v⃗k t k and v⃗k +1 t k+1 in terms of their orthogonal components we have

V⃗ ref T S =∣V⃗ ref∣T S (cosΨ + jsinΨ ) 38.9.9

v⃗ k t k = ∣⃗vk∣t k (cos(k −1)θ + j sin (k−1)θ ) 38.9.10

v⃗ k +1 t k+1 =∣⃗vk+1∣t k+1 (cos kθ + j sin kθ ) 38.9.11

Separating the x-axis and y-axis components we have

∣V⃗ ref∣T S cosΨ = ∣⃗v k∣t k cos(k −1)θ + ∣⃗vk +1∣t k+1cos kθ 38.9.12

∣V⃗ ref∣T S sinΨ = ∣⃗v k∣t k sin (k −1)θ + ∣⃗vk +1∣t k+1 sin kθ 38.9.13

As an example for sector 1, with k = 1 , ∣⃗v k∣= ∣⃗v k +1∣=23 V dc , θ= π

3 , equation (38.9.13) can thus be written as


∣V⃗ ref∣T S sinΨ =23

V dc t k+1√32

38.9.14

t k +1

T S

= √3∣V⃗ ref∣V dc

sinΨ 38.9.15

Equation (38.9.12) can then be written as

∣V⃗ ref∣T S cosΨ =23

V dc(t k+12

t k+1) 38.9.16

32

∣V⃗ ref∣V dc

T S cosΨ = t k+12

t k+1 38.9.17

Using the sine of sum of angles identity we can further write

t k

T S

=32∣V⃗ ref∣V dc

cosΨ −12

t k +1

T S

= √3∣V⃗ ref∣V dc

(√32

cosΨ −12

sinΨ )= √3

∣V⃗ ref∣V dc

(sin π3

cosΨ −cos π3

sinΨ )= √3

∣V⃗ ref∣V dc

sin(π3−Ψ )

38.9.18

More generally we require a canonical solution. Representing the system of equations in (38.9.12) and (38.9.13) in matrix form, we have

∣V⃗ ref∣T S [cosΨsinΨ ]=

23 V dc(t k[cos(k −1)θ

sin (k −1)θ ]+ t k +1[cos kθsin kθ ]) 38.9.19

Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix.

∣V⃗ ref∣T S [cosΨsinΨ ]=

23 V dc[cos(k−1)θ cos kθ

sin (k−1)θ sin kθ ][ t k

t k+1] 38.9.20

32

∣V⃗ ref∣T S

V dc[cosΨ

sinΨ ]=[cos(k−1)θ cos k θsin (k−1)θ sin k θ ][ t k

t k+1] 38.9.21

A relationship for the vector [ t k

t k+1] is required so multiplication of both sides of equation (39.9.21) by the inverse matrix

[cos(k −1)θ cosk θsin (k −1)θ sin k θ ]

−1

is required. Using the inversion formula [a bc d ]=

1ad −bc [ d −b

−c a ] , we have

[cos(k−1)θ cos kθsin(k−1)θ sin kθ ]

−1

=1

sin kθ cos(k−1)θ − coskθ sin(k−1)θ [sin kθ sin(k−1)θcos kθ cos(k−1)θ ] 38.9.22

[cos(k−1)θ cos kθsin(k−1)θ sin kθ ]

−1

=1

sinθ [ sin kθ −cos kθ−sin(k−1)θ cos(k−1)θ ]=

2

√3 [ sin kθ −cos kθ−sin(k−1)θ cos(k−1)θ ] 38.9.23


Replacing Ψ =ω t and θ= π3 , equation (39.9.21) can thus be written in the computationally compact form

[ t k

t k+1]=

√3∣V⃗ ref∣T S

V dc [sin k π

3−cosk π

3−sin (k−1) π

3cos(k−1) π

3 ][cos ωtsin ωt ] 38.9.24


state space-vector modulation fundamentals

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