Converters#

Voltage-Source Converter#

The figure below shows a three-phase two-level voltage-source converter, where \(u_\mathrm{dc}\) is the DC-bus voltage, \(i_\mathrm{dc}\) is the external DC current, \(i'_\mathrm{dc}\) is the converter DC current and \(C_\mathrm{dc}\) is the DC-bus capacitance. This converter can operate both as an inverter and a rectifier, depending on the direction of the power flow.

Three-phase two-level voltage-source converter

Three-phase two-level voltage-source converter.#

The figure below then shows the equivalent model for the three-phase voltage-source converter, where ideal switches are assumed. In the equivalent model, each changeover switch is connected to either negative or positive potential of the DC bus and the switching phenomena are assumed to be infinitely fast. This converter model is provided in the class motulator.drive.model.VoltageSourceConverter.

Equivalent model of three-phase voltage-source converter

Equivalent model of a three-phase voltage-source converter.#

By default, the DC-bus voltage is stiff, i.e. the capacitor is replaced by an ideal voltage source. Alternatively, the capacitive dynamics of the DC-bus can be simulated. The model is implemented as

(1)#\[C_{dc}\frac{\mathrm{d}u_\mathrm{dc}}{\mathrm{d} t} = i_\mathrm{dc} - i'_\mathrm{dc}\]

where the converter DC current is calculated from the converter phase currents and switching states as

(2)#\[i'_\mathrm{dc} = q_\mathrm{a} i_\mathrm{a} + q_\mathrm{b} i_\mathrm{b} + q_\mathrm{c} i_\mathrm{c}\]

Six-Pulse Diode Bridge#

The figure below shows a six-pulse diode bridge rectifier, where the inductor \(L_\mathrm{dc}\) is placed in the DC link. For simplicity, a three-phase supply voltage is assumed to be stiff. The voltage-source converter described above is extended with this diode bridge model in the class motulator.drive.model.FrequencyConverter.

Six-pulse diode bridge rectifier an three-phase supply voltage

Six-pulse diode bridge rectifier.#

The model is implemented as

(3)#\[L_{\mathrm{dc}} \frac{\mathrm{d}i_{L}}{\mathrm{d}t} = u_{\mathrm{di}} - u_\mathrm{dc}\]

where \(i_\mathrm{L}\) is the DC-bus current, \(u_\mathrm{di}\) is the voltage over the diode bridge, \(u_\mathrm{dc}\) is the DC-bus voltage, and \(L_{\mathrm{dc}}\) is the DC link inductance.

Examples using the six-pulse diode bridge can be found in 2.2-kW induction motor, diode bridge and 2.2-kW PMSM, diode bridge

Carrier Comparison#

The figure below shows a converter 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.

Voltage-source converter 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 motulator.common.model.CarrierComparison, where the switching instants are explicitly computed in the beginning 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 of the load 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

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

where \(\boldsymbol{q}_\mathrm{c}^\mathrm{s}\) 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 motulator.common.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 converter model.

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

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

where \(\boldsymbol{d}_\mathrm{c}^\mathrm{s}\) is the duty ratio space vector. This ZOH is the default option in most of Drives and Grid Converters.

References