C.  Pulse Width Modulated Inverter Model.

There are several circuit topologies and control methods used to convert a dc input into a 3-phase ac output.  A common circuit-topology is a voltage-source inverter which is shown in Fig. PWM-1. Fig. PWM-1:  Inverter circuit diagram

The inverter is fed by a dc voltage and has three phase-legs each consisting of two transistors and two diodes (labeled with subscripts a, b,c).  A common inverter control method covered in both beginning undergraduate and graduate power electronics or machines courses is sine-triangle pulse width modulation (STPWM) control. With  STPWM control, the switches of the inverter are controlled based on a comparion of  a sinusoidal control signal and a triangular switching signal. The sinusoidal control waveform establishes the desired fundamental frequency of the inverter output, while the triangular waveform establishes the switching frequency of the inverter.  The ratio between the frequencies of the triangle wave and the sinusoid is referred to as the modulation frequency ratio.  The switches of the phase legs are controlled based on the following comparison: A graphical representation of the switch control is shown in Fig. PWM-2. In theory, the switches in each leg are never both on or off simultaneously; therefore, the voltages Vag, Vbg,  and Vcg fluctuate between the input voltage (Vdc) and zero. By controlling the switches in this manner, the line-line inverter output voltages are ac, with a fundamental frequency corresponding to the frequency of the sinusoidal control voltage.  In most instances the magnitude of the triangle wave is held fixed.  The amplitude of the inverter output voltages is therefore controlled by adjusting the amplitude of the sinusoidal control voltages.  The ratio of the amplitude of the sinusoidal waveforms relative to the amplitude of the triangle wave is the amplitude modulation ratio.  In systems in which the inverter sources inductive loads, the inverter must source power in all four quadrants.  The diodes provide paths for current when a transistor is gated on but cannot conduct the polarity of the load current. For example, if the load current is negative at the instant the  the upper transistor is gated on, the diode in parallel with the upper transistor will conduct until the load current becomes positive at which time the upper transistor will begin to conduct. Fig. PWM-2:  Sine-triangle, pulse-width-modulated control waveforms (upper left),
phase voltages Vag and Vbg, and line voltage Vab

An ACSL/GM model of a sine-triangle PWM controlled, voltage-source inverter has been created to facilitate undergraduate and graduate education of both basic and advanced concepts.  Fig. PWM-3 shows the Graphic Modeller simulation of the PWM connected to an inductive load.  It is assumed that all switches are ideal, and thus have zero voltage across them when on and zero current through them when off. Fig. PWM-3:  ACSL/GM model of STPWM

To model a system in ACSL/GM, the dynamic equations of the system must be established.  In the case of a PWM inverter sourcing an inductive load, the differential equations describing the system can be expressed as From Fig. PWM-1, using Kirchoff’s voltage laws , the phase voltages can be expressed as: Since the load is wye-connected, the sum of the phase currents is zero.  Therefore, from equations PWM-1 to PWM-3, the sum of the phase voltages is also zero.  Summing equations PWM-4 to PWM-6, the voltage from the neutral to the negative rail of the inverter, Vng,  can be expressed as: The phase voltages can therefore be expressed in terms of the state of the bottom phase leg switches.  In particular, when the bottom switches are on, regardless of the direction of load current, the voltage across the switch is zero (ideal switch); when the switch is off, the voltage across it is the dc input voltage, Vdc.  In the ACSL/GM model the voltages Vag, Vbg, and Vcg are determined from the PWM control logic  These voltages are then used to determine the phase voltages.  The phase voltages provide inputs to integrators that calculate the phase currents, which are then used to determine whether the diodes or transistors are conducting.

The ACSL/GM model is useful for teaching both basic and advanced STPWM theory.   For basic concepts such as the control method or the conduction of the switches, the model can be used to generate waveforms which highlight specific points of interest.  For example, plots of the phase-a  current, the switching signal of the upper transistor, along with the upper transistor and diode currents are shown in Fig. PWM-4.  Therein, it can be seen that if the transistor is gated on while the load current is negative, the upper diode will conduct current (lower right) until either the lower transistor is gated on, or the load current becomes positive, at which time the upper transistor will conduct (upper right).  Through multiple studies, the length of time the diode or transistor conducts (conduction angle) can be observed as a function of the power factor of the load.  As the power factor increases (more resistive), the diode-conduction angle  will decrease while the transistor conduction angle will increase. Fig. PWM-4:  Phase-a inverter variables (top left-upper transistor gating signal, bottom left-phase a current, top right-transistor current, bottom-right-diode current)

Another use of the model is to determine the amplitude of the fundamental frequency of the line-line voltage as a function of the amplitude modulation ratio.  The determination of such a function requires a means of  transforming the simulated time-domain line-line voltage to the frequency domain.  Built-in algorithms for performing such transformations, such as an FFT, are not available within the Graphic Modeler run-time environment; however, ACSL provides a direct interface to Matlab or ACSL-Math from the run-time command line.  ACSL-Math is a mathematical software package provided with the full version of ACSL/GM.  Thus variables are easily loaded into either mathematical package for post processing of simulation results. To illustrate this utility, the STPWM inverter (with Vdc = 350 volts, Rs = 0.5 ohm, and Ls = 70 mH ) was simulated repeatedly with increasing values of the amplitude modulation ratio.  The frequency modulation ratio was held constant at mf = 15.  The resulting line-line voltages were input to Matlab where an FFT was performed to determine the amplitude of the fundamental frequency (60 Hz) component.  A plot of the amplitude of the 60 Hz component versus the amplitude modulation ratio is shown in Fig. PWM-5.  Therein it is seen that the line-line voltage increases linearly for ma less than or equal to one.  For ma > 1.0 (overmodulation), the line-line voltage continues to increase nonlinearly until  eventually any increase in amplitude modulation ratio has no effect on the line-line voltage.  Therein  STPWM degenerates into square-wave inverter operation. Fig. PWM-5:  Line-Line Voltage vs. Amplitude Modulation Ratio

An important aspect of STPWM design is the minimization of the system harmonics.   The harmonic performance of the system is readily obtained from the inverter model using the same simulation postprocessing described above.  An example frequency response of the output line-line voltage of an inverter was calculated using an FFT and is shown in Fig. PWM-6.  By performing repeated studies, comparisons of harmonic performance can be made with regard to the modulation frequency (synchronous or asynchronous) or the operating region of the inverter.  In addition to providing design information, such exercises are an excellent means of encouraging students to use tools  such as the FFT in their engineering analysis. Fig. PWM-6 Frequency Spectrum of Inverter Line-Line Voltage