# Two-Quadrant Single-Phase Rectifier DC Drive

Implement two-quadrant single-phase rectifier DC drive

## Library

Electric Drives/DC drives

## Description

The high-level schematic shown below is built from five main blocks. The DC motor, the single-phase full converter, and the bridge firing unit are provided with the Simscape™ Power Systems™ library. More details are available in the reference pages for 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 containing an average-value model of the single-phase converter and allowing faster simulation.

 Note   In Simscape Power Systems software, the Two-Quadrant Single-Phase Rectifier DC Drive block is commonly called the `DC1` motor drive.

## Speed Controller

The speed regulator in the following figure 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 0 pu and an upper limit defined by the user.

## Current Controller

The armature current regulator in the following figure is based on a second PI controller. The regulator controls the armature current by computing the appropriate thyristor firing angle. This generates the rectifier output voltage 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 during continuous conduction. To compensate nonlinearities appearing during discontinuous conduction, a feedforward term is added to the firing angle. This improves the system's response time. The firing angle can vary between 0 and 180 degrees. You can limit the lower and upper limits to intermediate values.

## Average-Value Single-Phase Rectifier

The average-value converter is shown in the following figure.

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

## Bridge Firing Unit

The bridge firing unit converts the firing angle, provided by the current controller, to four pulses applied to the thyristor gates. The bridge firing unit block contains a band-pass filter on voltage measurement to remove voltage harmonics. The discrete synchronized pulse generator block generates the pulses. Its 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 a single-phase rectifier controlled by two PI regulators. 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. 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 25 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 can of course not be higher than the controller time step.

## Dialog Box

### 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.

### Rectifier section

The Rectifier section of the Converter tab displays 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.

Smoothing Inductance

The smoothing inductance value (H).

Field DC Source

The DC motor field voltage value (V).

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. These values are used to convert armature current values 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).

Reference Limit

Maximum current reference value (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 four 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. This vector 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. It includes the converter output voltage. The output current is not included since it is equal to the DC motor armature current.

Note that all current and voltage values of the 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

50 Hz

Motor Nominal Values

Power

5 hp

Speed

1750 rpm

Voltage

240 V

## Example

The `dc1_example` example illustrates the single-phase rectifier drive used with the 5 hp drive parameter set during speed regulation.

The rectifier is fed by a 220 V AC 50 Hz voltage source followed by a linear transformer to boost the voltage to a sufficient value.

The speed reference is set at 1750 rpm at t = 0 s. Initial load torque is 15 N.m.

Observe that the motor speed follows the reference ramp accurately (+250 rpm/s) and reaches steady state around t = 8.5 s.

The armature current follows the current reference very well, and the firing angle stays below 90 degrees, the converter being in rectifier mode (first quadrant operating mode). The lower limit of the firing angle has been set to 20 degrees.

At t = 8.75 s, the load torque passes from 15 N.m to 20 N.m. The motor speed recovers fast and is back at 1750 rpm at t = 10 s. The current reference rises to about 17.5 A to generate a higher electromagnetic torque to maintain the needed speed. As observed before, the armature current follows its reference perfectly.

The following figure illustrates the results obtained respectively with the detailed and the average-value converters. Average voltage, current, torque and speed values are identical for both models. The average firing angle values however are slightly different. Notice that the higher frequency signal components are not represented with the average-value converter.

DC1 Example Waveforms (Blue: Detailed Converter, Red: Average-Value Converter)

## References

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

[2] Nondahl, Thomas A., Microprocessor Control of Motor Drives and Power Converters, tutorial course, IEEE Industry Application Society, October 1993, pp. 7.1-7.26.