# Model-oriented design. Building an active rectifier (based on a mathematical model)

Continuation of a series of articles on model-oriented design. In the previous series:

- Model-oriented design - how not to repeat Chernobyl
- Model-oriented design. Electric drive with brushless DC motor
- Model-oriented design. Creating a reliable model, using the example of an aviation heat exchanger
- Shine and poverty of model-oriented design according to aviation standards DO-331

In this series, authors Yu. N. Kalachev and A.G. Alexandrov, represent a mathematical model of an active rectifier in a structural modeling environment.

Active rectifiers are widely used in converting technology to ensure the active nature of the exchange of energy with the network. They are a bi-directional **AC-DC **converter with a unit power factor and low nonharmonic distortion. The basis of the device is a three-phase bridge inverter connected to the network through a three-phase reactor (see Fig. 1).

*Fig. 1 - Schematic diagram of an active rectifier*

### 1. How does it work?

The inverter in this device works as a step-up converter that supports a given voltage in the DC link ( *U _{dc }*) by controlling the amplitude and phase of the inductor current.

In this case, the charged capacity *C _{dc & lt;/sub }> the inverter can be considered as a voltage source, from which the inverter with the help of PWM is able to generate a three-phase voltage of various amplitudes and phases (naturally, due to the pulse control we are talking about the average voltage). This voltage, together with the mains voltage, forms a voltage that determines the phase and amplitude of its current on a three-phase inductor. *

**At idle **(if there is no current consumption from the network), the inverter generates a voltage that matches the voltage of the input network in amplitude and phase. Naturally, the current in the inductor does not flow.

In **active consumption mode **, a voltage is generated on the inductor, whose phase is ahead of the mains phase by π/2. In this case, the current in the inductance lagging behind the voltage by π/2 will coincide in phase with the mains voltage. Its amplitude, necessary to maintain a given voltage in the DC link, is determined by the amplitude of the voltage across the inductor.

In the **recovery mode **(when energy is supplied to the network), a voltage is generated on the inductor that is lagging behind the mains by π/2, which leads to the flow of the inductor current in antiphase with the mains voltage. The specified voltage in the DC link is thus supported by the amplitude of the voltage across the inductor.

Fig. 2 shows vector diagrams explaining the above.

*Fig.2 - vector diagrams in various modes*

In the charts:

- voltage vector input network

- voltage vector formed by inverter

- throttle voltage vector

- network current vector

Coordinate System **ABC **- Fixed, Three Phase

The coordinate system **XY **is a rotating coordinate system whose X axis coincides with the rotating network voltage vector.

* NOTE *Looking at the diagrams in Fig. 2, you can notice an interesting detail: the slightest difference between the phase of the inverter voltage vector and the phase of the mains voltage vector leads to a phase jump in voltage across the inductor by ± 90º, and accordingly, to a change in the idling mode to recovery or active consumption.

So, as already mentioned, a step-up converter is built on a three-phase inductor and inverter, which ensures the maintenance of a given voltage of the DC link (U _{dc }). This maintenance is done by controlling the input current vector. Due to PWM control and increasing the switching frequency of the IGBT keys of the inverter, it is possible to reduce the inductance of the input inductor to reasonable values when obtaining a sinusoidal shape of the input current and ensuring its active character.

### 2. Mathematical description of the operation of an active rectifier

For the circuit in Fig. 1, you can write the following expression:

R - active resistance of the inductor;

L is the inductance of the inductor.

For the XY rotating coordinate system associated with the input voltage vector, you can write:

where:

for 50 Hz

- active component of the input current ( coincides with the phase of the network);

- reactive component of the input current ( lags or is 90º ahead of the network phase).

In order for the corrector's consumption pattern to be active, it is necessary to maintain .

In addition, the corrector must provide the functions of a rectifier, that is, maintain the specified value , regardless of the load current.

### 3. Active rectifier control system structure

Consider the structure of the system based on its model in SimInTech (Fig. 3).

*Fig. 3 - block diagram of the model*

The control system is built in a coordinate system **XY **that rotates synchronously with the input vector voltage vector by a two-loop structure.The external voltage circuit, using the voltage regulator, generates a task for the active component of the inductor current ( ) needed to maintain the specified .

The internal current loop provides testing the task of the active component of the current ( ) with a zero reactive component ( ).

The system blocks are listed and briefly described below: **Uset **- voltage regulator Udc,. **VF **- a network phase calculator that determines the angle of rotation of the voltage vector of the input network in a fixed coordinate system and generates signals necessary for coordinate transformations. **ABC=> XY **- coordinate converter, makes the transition from a fixed three-phase coordinate system to a rotating rectangular coordinate system XY associated with the input voltage vector. **XY=> ABC **- the coordinate transformer makes the transition from the rotating coordinate system XY to the stationary three-phase. **PU **- voltage regulator (PI), converts the error signal into the signal for setting the active component of the input current . **РIx **- current regulator, converts the active current error signal into a signal for setting voltage along the X axis of the rotating coordinate system. **РIy **- current regulator, converts a reactive current error signal into a signal for setting voltage along the Y axis of a rotating coordinate system. **Ogr.U **- the voltage limiter limits the voltage vector modulo to the maximum possible value with priority Y - component. **KPS **- the compensation block is designed to compensate for cross-connections between coordinate currents (see the last terms in equations 1.1). Compensation is not performed along the X coordinate, since the current is assumed to equal zero on the Y axis. **ФЗМ **- Shaper of the Modulation Law - a block of PWM algorithm with full use of DC link voltage.

*ATTENTION :*- Having eyes - let him see a somewhat unusual construction of subtracting blocks at the input of current regulators. In them, the current reference signal is subtracted from the feedback signal, and not vice versa, as is usually the case. This is due to the fact that during coordinate transformations, the phase current flowing from the voltage source is considered positive. For the ABC=> XY conversion, the source is the network, and for the inverse, XY=> ABC, the source is an inverter. Since the phase currents are opposite from the point of view of the network and the inverter, it is necessary to invert the reference and feedback in the current circuit, which is realized in the structure of the subtracting blocks at the input of its regulators.
- I don’t dwell on the description of block mathematics in detail, since it is present in the internal structures of blocks available to the SimInTech user (see internal structure, read help for elements).

### 4. The structure of the power section of the rectifier

The structure of the power part of the rectifier is considered on the basis of its mathematical model in presented in Fig. 4.

*Fig. 4 - block diagram of the model of the power part of the rectifier*

The power circuit parameters are as follows:

L=0.0015 Hn

C=10,000 uF

PWM Frequency - 8.33 kHz

### 5. Model work

The model package consists of two projects, the schemes of which are given above.The integration time of the project of the control system is equal to the PWM cycle - 160 µs. The integration time of the project of the power unit is 1 µs. Synchronization of projects with the frequency of calculation of the project of the control system models the temporal discreteness of the real (digital) control system.

Below is a description of the algorithm of the model package and the graphics explaining its operation (Fig. 5, 6 and 7).

**In section 1 **- the inverter transistors are turned off, the DC link capacitance is charged through the inverter diodes with current limiting on the charging resistor. **In section 2 **- the charging resistor is shunted by the relay and at idle the voltage rises to the set voltage (700V). **In section 3 **- the load current (50A) is consumed from the network. **On site 4 **- the load current (-50A) is recovered to the network.

*Fig. 5*

*Fig. 6*

*Fig. 7*

Network currents are active and sinusoidal in nature. The content of higher harmonics in the current according to the data of the spectrum analyzer of the graph in SimInTech is 4.4%.

The phase A current and voltage waveform and the mode measurement screen taken during operation of a real active rectifier as a DC link of the frequency converter are shown in Fig. 8 below.

*Fig. 8*

It can be stated that the current and voltage coincide in phase and Cosφ=1

To contact the author Yuri Nikolaevich Kalachev (Kalachev_i@mail.ru)

More information on the Electric Drive toolbox at:

http://3v-services.com/books/978-5-97060-766-4/? Yclid=3971894245794548684

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