Converters#

Inverter#

The figure below shows a three-phase two-level inverter and its equivalent model, where ideal switches are assumed. In the equivalent model, each changeover switch is connected to either negative or positive potential of the DC bus. The switchings are assumed to be infinitely fast. The inverter model is provided in the class gritulator.model.Inverter.

Three-phase two-level inverter

Three-phase two-level inverter: (left) main circuit; (right) equivalent model. The DC-bus voltage and currents are \(u_\mathrm{dc}\) and \(i_\mathrm{dc}\), respectively.#

Carrier Comparison#

The figure below shows an inverter equipped with a generic three-phase load. In pulse-width modulation (PWM), carrier comparison is commonly used to generate instantaneous switching state signals \(q_\mathrm{a}\), \(q_\mathrm{b}\), and \(q_\mathrm{c}\) from duty ratios \(d_\mathrm{a}\), \(d_\mathrm{b}\), and \(d_\mathrm{c}\). The duty ratios are continuous signals in the range [0, 1} while the switching states are either 0 or 1.

Inverter and carrier comparison

Instantaneous switching states are defined by the carrier comparison. In this example, the switching states are \(q_\mathrm{a}=1\), \(q_\mathrm{b}=0\), and \(q_\mathrm{c}=0\).#

The figure below shows the principle of carrier comparison. The logic shown in the figure is implemented in the class gritulator.model.CarrierComparison, where the switching instants are explicitly computed in the begininning of each sampling period (instead of searching for zero crossings), allowing faster simulations.

Carrier comparison

Carrier comparison. The duty ratios are \(d_\mathrm{a}\), \(d_\mathrm{b}\), and \(d_\mathrm{c}\) are constant over the sampling period \(T_\mathrm{s}\) (or, optionally, over the the switching period \(T_\mathrm{sw}=2T_\mathrm{s}\)). The carrier signal is the same for all three phases and varies between 0 and 1.#

The zero-sequence voltage does not affect the phase currents if the neutral wire is not connected. Therefore, the reference potential of the phase voltages can be freely chosen when computing the space vector of the converter output voltage. The converter voltage vector in stationary coordinates is

(1)#\[\begin{split}\boldsymbol{u} &= \frac{2}{3}\left(u_\mathrm{an} + u_\mathrm{bn}\mathrm{e}^{\mathrm{j}2\pi/3} + u_\mathrm{cn}\mathrm{e}^{\mathrm{j} 4\pi/3}\right) \\ &= \frac{2}{3}\left(u_\mathrm{aN} + u_\mathrm{bN}\mathrm{e}^{\mathrm{j} 2\pi/3} + u_\mathrm{cN}\mathrm{e}^{\mathrm{j} 4\pi/3}\right) \\ &= \underbrace{\frac{2}{3}\left(q_\mathrm{a} + q_\mathrm{b}\mathrm{e}^{\mathrm{j} 2\pi/3} + q_\mathrm{c}\mathrm{e}^{\mathrm{j} 4\pi/3}\right)}_{\boldsymbol{q}}u_\mathrm{dc}\end{split}\]

where \(\boldsymbol{q}\) is the switching state space vector.

Note

The carrier comparison is compatible with all standard pulse-width modulation (PWM) methods, such as space-vector PWM (see gritulator.control.PWM) and discontinuous PWM methods [1], [2].

The sampling period \(T_\mathrm{s}\) is returned by the control method, and it does not need to be constant.

If the zero sequence is of interest, it could be easily added to the inverter model.

Since the converter models are invariably in stationary coordinates, the space vectors are not marked with the superscript s here to simplify the notation.

Switching-Cycle Averaging#

If the switching ripple is not of interest in simulations, the carrier comparison can be replaced with zero-order hold (ZOH) of the duty ratios. In this case, the output voltage vector is

(2)#\[\boldsymbol{u} = \underbrace{\frac{2}{3}\left(d_\mathrm{a} + d_\mathrm{b}\mathrm{e}^{\mathrm{j} 2\pi/3} + d_\mathrm{c}\mathrm{e}^{\mathrm{j} 4\pi/3}\right)}_{\boldsymbol{d}}u_\mathrm{dc}\]

where \(\boldsymbol{d}\) is the duty ratio space vector. This ZOH is the default option in most of Examples.

References