# Four-Quadrant Single-Phase Rectifier DC Drive

Implement single-phase dual-converter DC drive with circulating current

## Library

Electric Drives/DC drives

## Description

The high-level schematic shown below is built from six main blocks. The DC motor, the two single-phase full converters, and the bridge firing unit are provided with the SimPowerSystems™ library. More details on these blocks are available in the reference sections of these blocks. The two other blocks are specific to the Electric Drives library. These blocks are the speed controller and the current controller. They allow speed or torque regulation. A "regulation switch" block allows you to toggle from one type of regulation to the other. During torque regulation the speed controller is disabled. It is possible to use a simplified version of the drive, which contains an average-value model of the single-phase dual-converter and allows faster simulation.

 Note   In SimPowerSystems software, the Four-Quadrant Single-Phase Rectifier DC Drive block is commonly called the `DC2` motor drive.

## Speed Controller

The speed regulator shown below uses a PI controller. The controller outputs the armature current reference (in pu) used by the current controller in order to obtain the electromagnetic torque needed to reach the desired speed. During torque regulation, the speed controller is disabled.

The controller takes the speed reference (in rpm) and the rotor speed of the DC machine as inputs. The speed reference change rate will follow user-defined acceleration and deceleration ramps in order to avoid sudden reference changes that could cause armature over-current and destabilize the system. The speed measurement is filtered by a first-order low-pass filter.

The current reference output is limited between symmetrical lower and upper limits defined by the user.

## Current Controller

The armature current regulator shown below is based on a second PI controller. The regulator controls the armature current by computing the appropriate thyristor firing angles of the two full converters. This generates the converter output voltages needed to obtain the desired armature current and thus the desired electromagnetic torque.

The controller takes the current reference (in pu) and the armature current flowing through the motor as inputs. The current reference is either provided by the speed controller during speed regulation or computed from the torque reference provided by the user during torque regulation. This is managed by the "regulation switch" block.

The armature current input is filtered by a first-order low-pass filter. An arccosine function is used to linearize the control system. The firing angle can vary between 0 and 180 degrees. You can limit the lower and upper limits to intermediate values.

Both converters operate simultaneously, and the two firing angles are controlled so that the sum of their values stays equal to 180 degrees. This produces opposite average voltages at the converter DC output terminals and thus identical average voltages at the DC motor armature, the converters being connected in antiparallel. One converter is working in rectifier mode while the other is in inverter mode.

## Average-Value Single-Phase Rectifier

The average-value single-phase rectifier with circulating current is shown in the following figure.

It is composed of one controlled current source on the AC side and one controlled voltage and current source on the DC side. The AC current source allows the representation of the fundamental single-phase current behavior according to the following equation

${I}_{a}=\sqrt{2}{I}_{d}\mathrm{sin}\left(2\pi ft+\alpha +{\alpha }_{0}\right),$

with α being the firing angle value, α0 the phase angle of the AC side, f the AC frequency, and Id the rectified output current value. The DC voltage source represents the average voltage value of the rectified voltage waveform according to the next equation

${V}_{d}=\frac{2\sqrt{2}}{\pi }{V}_{\text{rms}}\mathrm{cos}\alpha -4fL{I}_{d},$

with Vrms being the input RMS voltage value and L being the source inductance value. The controlled current source represents the circulation current due to the instantaneous voltage difference between both converter outputs of the dual-converter. This current is equal to

${I}_{\text{circ}}=\frac{\sqrt{2}}{{\pi }^{2}f{L}_{\text{circ,tot}}}{V}_{\text{rms}}\left(\mathrm{sin}\alpha -\alpha \mathrm{cos}\alpha \right),$

with Lcirc,tot being the total circulation inductance of the circulation current loop.

## Bridge Firing Unit

The bridge firing unit converts the firing angles, provided by the current controller, to two series of four pulses applied respectively to the thyristor gates of each converter. The bridge firing unit block contains a band-pass filter on voltage measurement to remove voltage harmonics. Two discrete synchronized pulse generator blocks generate the pulses of each converter. Their architecture is based on the Discrete Synchronized 6-Pulse Generator block. Refer to the Synchronized 6-Pulse Generator for more information on this block. When using the average-value converter, the bridge firing unit simply outputs the firing angle value needed by the converter.

## Remarks

The machine is separately excited with a constant DC field voltage source. There is thus no field voltage control. By default, the field current is set to its steady-state value when a simulation is started.

The armature voltage is provided by two single-phase antiparallel-connected converters controlled by two PI regulators. The circulating current produced by the instantaneous voltage difference at the terminal of both converters is limited by inductors connected between these terminals. Armature current oscillations are reduced by a smoothing inductance connected in series with the armature circuit.

The average-value converter represents the average behavior of a single-phase rectifier for continuous armature current in a dual-converter topology. This model is thus not suitable for simulating DC drives under discontinuous armature current conditions. The converter outputs a continuous voltage value equal to the average-value of the real-life rectified voltage. The armature voltage, armature current and electromagnetic torque ripples are thus not represented. The input currents have the frequency and amplitude of the fundamental current component of the real-life input currents.

The model is discrete. Good simulation results have been obtained with a 4 µs time step. The control system (speed and current controllers) samples data following a user-defined sample time in order to simulate a digital controller device. Keep in mind that this sampling time has to be a multiple of the simulation time step.

The average-value converter allows the use of bigger simulation time steps since it does not generate small time constants (due to the RC snubbers) inherent to the detailed converter. For a controller sampling time of 100 µs good simulation results have been obtained for a simulation time step of 100 µs. This time step cannot be higher than the controller time step.

## Dialog Box and Parameters

### DC Machine Tab

The DC Machine tab displays the parameters of the DC Machine block of the Fundamental Blocks (powerlib) library.

Output bus mode

Select how the output variables are organized. If you select Multiple output buses, the block has three separate output buses for motor, converter, and controller variables. If you select Single output bus, all variables output on a single bus.

Model detail level

Select between the detailed and the average-value inverter.

Mechanical input

Select between the load torque, the motor speed and the mechanical rotational port as mechanical input. If you select and apply a load torque, the output is the motor speed according to the following differential equation that describes the mechanical system dynamics:

${T}_{e}=J\frac{d}{dt}{\omega }_{r}+F{\omega }_{r}+{T}_{m}$

This mechanical system is included in the motor model.

If you select the motor speed as mechanical input, then you get the electromagnetic torque as output, allowing you to represent externally the mechanical system dynamics. The internal mechanical system is not used with this mechanical input selection and the inertia and viscous friction parameters are not displayed.

For the mechanical rotational port, the connection port S counts for the mechanical input and output. It allows a direct connection to the Simscape™ environment. The mechanical system of the motor is also included in the drive and is based on the same differential equation.

### Converter Tab

Smoothing Inductance

The smoothing inductance value (H).

Field DC Source

The DC motor field voltage value (V).

Circulating Current Inductors

The four circulating current inductors inductance value (H).

#### Converter sections

The Converter 1 and Converter 2 sections of the Converter tab display the parameters of the Universal Bridge block of the Fundamental Blocks (powerlib) library. Refer to the Universal Bridge for more information on the Universal Bridge block parameters.

RMS Voltage

RMS voltage of the single-phase voltage source connected to the A+,A− terminals of the drive (V). This parameter is not used when using the detailed rectifier.

Frequency

Frequency of the single-phase voltage source connected to the A+,A− terminals of the drive (Hz). This parameter is not used when using the detailed rectifier.

Source Inductance

Source inductance of the single-phase voltage source connected to the A+,A− terminals of the drive (H). This parameter is not used when using the detailed rectifier.

Phase Angle

Phase angle of the single-phase voltage source connected to the A+,A− terminals of the drive (deg.). This parameter is not used when using the detailed rectifier.

### Controller Tab

Schematic Button

When you press this button, a diagram illustrating the speed and current controllers schematics appears.

Regulation Type

This pop-up menu allows you to choose between speed and torque regulation.

Sampling Time

The controller (speed and current) sampling time (s). The sampling time has to be a multiple of the simulation time step.

#### Controller — Speed Controller Subtab

Nominal Speed

The nominal speed value of the DC motor (rpm). This value is used to convert motor speed from rpm to pu (per unit).

Initial Speed Reference

The initial speed reference value (rpm). This value allows the user to start a simulation with a speed reference other than 0 rpm.

Low-Pass Filter Cutoff Frequency

Cutoff frequency of the low-pass filter used to filter the motor speed measurement (Hz).

Proportional Gain

The proportional gain of the PI speed controller.

Integral Gain

The integral gain of the PI speed controller.

Acceleration

The maximum change of speed allowed during motor acceleration (rpm/s). Too great a value can cause armature over-current.

Deceleration

The maximum change of speed allowed during motor deceleration (rpm/s). Too great a value can cause armature over-current.

#### Controller — Current Controller Subtab

Power and Voltage nominal values

The DC motor nominal power (W) and voltage (V) values.

The nominal power and voltage values are used to convert armature current from amperes to pu (per unit).

Proportional Gain

The proportional gain of the PI current controller.

Integral Gain

The integral gain of the PI current controller.

Low-Pass Filter Cutoff Frequency

Cutoff frequency of the low-pass filter used to filter the armature current measurement (Hz).

Symmetrical Reference Limit

Symmetrical current reference (pu) limit around 0 pu. 1.5 pu is a common value.

#### Controller — Bridge Firing Unit Subtab

Alpha Min

Minimum firing angle value (deg.). 20 degrees is a common value.

Alpha Max

Maximum firing angle value (deg.). 160 degrees is a common value.

Frequency of Synchronization Voltages

Frequency of the synchronization voltages used by the discrete synchronized pulse generator block (Hz). This frequency is equal to the line frequency of the single-phase power line. This parameter is not used when using the average-value converter.

Pulse Width

The width of the pulses applied to the thyristor gates (deg.). This parameter is not used when using the average-value converter.

## Block Inputs and Outputs

`SP`

The speed or torque set point. The speed set point can be a step function, but the speed change rate will follow the acceleration / deceleration ramps. If the load torque and the speed have opposite signs, the accelerating torque will be the sum of the electromagnetic and load torques.

`Tm` or `Wm`

The mechanical input: load torque (Tm) or motor speed (Wm). For the mechanical rotational port (S), this input is deleted.

`A+, A- `

The single-phase electric connections. The applied voltage must be adequate for the motor size.

`Wm`, `Te` or `S`

The mechanical output: motor speed (Wm), electromagnetic torque (Te) or mechanical rotational port (S).

When the Output bus mode parameter is set to Multiple output buses, the block has the following three output buses:

`Motor`

The motor measurement vector. It is composed of two elements:

• The armature voltage

• The DC motor measurement vector (containing the speed, armature current, field current, and electromagnetic torque values). Note that the speed signal is converted from rad/s to rpm before output.

`Conv`

The single-phase converter measurement vector. This vector includes

• The output voltage of converter 1

• The output voltage of converter 2

• The output current of converter 1

• The output current of converter 2

Note that all current and voltage values of the detailed bridges can be visualized with the Multimeter block.

`Ctrl`

The controller measurement vector. This vector contains:

• The armature current reference

• The firing angle computed by the current controller

• The speed or torque error (difference between the speed reference ramp and actual speed or between the torque reference and actual torque)

• The speed reference ramp or torque reference

When the Output bus mode parameter is set to Single output bus, the block groups the Motor, Conv, and Ctrl outputs into a single bus output.

## Model Specifications

The library contains a 5 hp drive parameter set. The specifications of the 5 hp drive are shown in the following table.

5 HP Drive Specifications

Drive Input Voltage

Amplitude

320 V

Frequency

60 Hz

Motor Nominal Values

Power

5 hp

Speed

1750 rpm

Voltage

240 V

## Example

The `dc2_example` example illustrates the single-phase dual-converter drive used with the 5 hp drive parameter set during speed regulation. The converters are fed by a 230 V AC 60 Hz voltage source followed by a linear transformer to boost the voltage to a sufficient value.

During this simulation, the motor is coupled to a linear load, which means that the mechanical torque produced by the load is proportional to the speed.

The speed reference is set at 1200 rpm at t = 0 s. Observe that the firing angles are symmetrical around 90 degrees and that the converter average output DC voltages are of opposite signs. The armature current is supplied by converter 1, and the total current in this converter is the sum of load current and circulating current. Converter 2 simply carries the circulating current.

Observe that the motor speed follows the reference ramp accurately (+250 rpm/s) and reaches steady state after 5.5 s. The armature current follows the current reference very well and stabilizes around 12 A.

At t = 6 s, speed reference drops to −600 rpm. The current reference decreases to reduce the electromagnetic torque, and the load torque causes the motor to decelerate. Around t = 8.5 s, the armature current becomes negative and the electromagnetic torque reverses in order to brake the motor down to 0 rpm, the load torque being insufficient to decelerate the motor. The armature current is now provided by converter 2, converter 1 only handling the circulating current. At t = 10.8 s, the motor reaches 0 rpm and the load torque becomes negative. The electromagnetic torque now produces an accelerating torque to allow the motor to follow the negative speed ramp (−250 rpm/s). The armature current is now provided by converter 2, converter 1 only handling the circulating current.

At t = 14 s, speed stabilizes at −600 rpm.

DC2 Example Waveforms (Blue/Green : Detailed Converter, Red : Average- Value Converter)

## References

[1] Sen, P.C., Thyristor DC Drives, J.Wiley and Sons, 1981.