3-Phase Index-Based Modulation

Purpose

Generate the modulation index for a three-phase reference voltage

Library

Control / Modulators

Description

This block generates the modulation index for a three-phase reference voltage defined by the input \(V^*\) with the chosen modulation strategy. This block may be used in series with a PWM generation block for the control of three-phase inverters without neutral point connection.

A universal representation of the three-phase modulation index output is as follows (phase \(i = \{a, b, c\}\)):

\[m_i(t) = m_i^*(t) + m_0(t)\]

where \(m_0(t)\) represents the zero-sequence harmonic injected into all three phases, dependent on the selected modulation strategy. \(m_i^*(t)\) represents three-phase fundamental sinusoidal modulation indices calculated using the block inputs, i.e. \(V^*\) divided by \(V_{dc}/2\). It can be written as follows:

\[\begin{split}m^*_a(t) &= M \sin(\omega t) \\ m^*_b(t) &= M \sin(\omega t - 2\pi/3) \\ m^*_c(t) &= M \sin(\omega t + 2\pi/3)\end{split}\]

\(M\) stands for the modulation depth of the three-phase converter, where \(M = 1\) defines the limit of the linear modulation range by the Sinusoidal PWM modulation strategy (i.e. the formed voltage vector modulus equals to \(V_{dc}/2\)).

The modulation strategies demonstrated below all showcase the same \(M = 1\) for a three-phase reference voltage at 50 Hz fundamental frequency. The modulation index pattern of only phase A is depicted in each case, i.e. \(m_a(t)\).

Sinusoidal PWM

Sinusoidal modulation without adding a zero sequence:

\[m_0(t) = 0\]

Fig. 163 Sinusoidal PWM

Space Vector PWM (Symmetrical)

Symmetrical SVPWM pattern where the injected zero sequence is:

\[m_0(t) = \frac{1}{2}(1 - m^*_{\max}(t)) + \frac{1}{2}(-1 - m^*_{\min}(t))\]

Note that \(m^*_{\max}(t)\) and \(m^*_{\min}(t)\) represent the instantaneous maximum and minimum of the three-phase \(m_i^*(t)\) signals respectively.

Fig. 164 Space Vector PWM (Symmetrical)

Space Vector PWM (DPWM1)

For the Discontinuous PWM pattern 1, the one with the largest magnitude between \(m^*_{\max}(t)\) and \(m^*_{\min}(t)\) determines the injected zero sequence. \(m_0(t)\) can be expressed by:

\[\begin{split}m_0(t) = \begin{cases} 1 - m^*_{\max}(t), & \text{if } |m^*_{\max}(t)| \geq |m^*_{\min}(t)| \\ -1 - m^*_{\min}(t), & \text{if } |m^*_{\max}(t)| < |m^*_{\min}(t)| \end{cases}\end{split}\]

It represents a 60-degree max/min modulation index clamping interval, centered at each half fundamental period.

Fig. 165 Space Vector PWM (DPWM1)

Space Vector PWM (DPWM0)

For the Discontinuous PWM pattern 0, the zero-sequence signal is generated based on the one in DPWM1. The only difference is that the three-phase \(m_i^*(t)\) signals are phase-shifted by 30-degree leading.

Fig. 166 Space Vector PWM (DPWM0)

Space Vector PWM (DPWM2)

For the Discontinuous PWM pattern 2, the zero-sequence signal is generated based on the one in DPWM1. The only difference is that the three-phase \(m_i^*(t)\) signals are phase-shifted by 30-degree lagging.

Fig. 167 Space Vector PWM (DPWM2)

Space Vector PWM (DPWM3)

For the Discontinuous PWM pattern 3, the one with smallest magnitude between \(m^*_{\max}(t)\) and \(m^*_{\min}(t)\) determines the injected zero sequence. It represents a 30-degree max/min modulation index clamping interval, centered at every quarter of the fundamental period.

Fig. 168 Space Vector PWM (DPWM3)

Space Vector PWM (DPWMMIN)

For the Discontinuous PWM MIN pattern, \(m^*_{\min}(t)\) determines the injected zero sequence. It clamps the modulation index to -1 for a 120-degree interval, centered inside each negative half fundamental period.

Fig. 169 Space Vector PWM (DPWMMIN)

Space Vector PWM (DPWMMAX)

For the Discontinuous PWM MAX pattern, \(m^*_{\max}(t)\) determines the injected zero sequence. It clamps the modulation index to 1 for a 120-degree interval, centered inside each positive half fundamental period.

Fig. 170 Space Vector PWM (DPWMMAX)

Parameters

Modulation strategy

The modulation strategy can be set to Sinusoidal PWM, Space Vector PWM (Symmetrical), Space Vector PWM (DPWM0), Space Vector PWM (DPWM1), Space Vector PWM (DPWM2), Space Vector PWM (DPWM3), Space Vector PWM (DPWMMIN), Space Vector PWM (DPWMMAX) using the combo box.

Inputs and Outputs

DC voltage

The input signal \(V_{\rm dc}\) is the voltage measured on the dc side of the inverter, specified as a scalar.

Reference voltage

This input, labeled \(V_{\rm abc}^{*}\), is a three-dimensional vector signal comprising the three-phase voltage reference \([V_{\rm a}^{*}, V_{\rm b}^{*}, V_{\rm c}^{*}]\).

Modulation index output

The output labeled $m$ is formed from three-phase modulation indices \([m_{\rm a}, m_{\rm b}, m_{\rm c}]\) of the selected modulation strategy, which is also a three-dimensional vector signal. For the linear modulation range of each modulation strategy it varies within \([-1,1]\). A PWM generator block can be connected in series with this block to generate control signals for the three-phase inverter legs A, B, and C.

Probe Signals

3-phase modulation index

A vector signal consisting of the three-phase modulation indices, \([m_{\rm a}, m_{\rm b}, m_{\rm c}]\) for the selected modulation strategy.

References

• K. Zhou and D. Wang, “Relationship between space-vector modulation and three-phase carrier-based PWM: a comprehensive analysis three-phase inverters,” in IEEE Transactions on Industrial Electronics, vol. 49, no. 1, pp. 186-196, Feb. 2002.

• A. M. Hava, R. J. Kerkman and T. A. Lipo, “Simple analytical and graphical methods for carrier-based PWM-VSI drives,” in IEEE Transactions on Power Electronics, vol. 14, no. 1, pp. 49-61, Jan. 1999.