**3.1. AC drive with 3-phase asynchronous motor**

The AC drive consists of an AC machine supplied by a converter. The variables of AC machine (an asynchronous motor in our case) like electrical quantities (supply voltages and currents), magnetic variables (magnetic fluxes), and mechanical variables (motor torque and rotor angular speed) are usually to be investigated in:


## **3.2. Asynchronous motor model**

For dynamic properties investigation of asynchronous motor (influence of non-harmonic supply to properties of the AC drive, etc.) a dynamical model of AC machine is used. The AC machine is described by set differential equations. For their derivation some generally accepted simplifications are used (not listed here) concerning physical properties, construction of the machine, electromagnetic circuit, and supply source.

In order to simplify mathematical model of the squirrel cage motor, the multiphase rotor is replaced by an equivalent three-phase one and its parameters are re-calculated to the stator. Equations describing behavior of the machine are transformed from three- to two-phase system what yields to decreased number of differential equations. The quantities in equations are transformed into reference systems.

To derive dynamic model of asynchronous motor, the three-phase system is to be transformed into the two-phase one. In the fact, this transformation presents a replacement of the three-phase motor by equivalent two-phase one. The stator current space vector having real and imaginary components is defined by the equation:

$$\overline{\mathbf{i}} = \frac{2}{3} (\dot{i}\_a + \mathbf{a}\dot{i}\_b + \mathbf{a}^2 \dot{i}\_c)$$

where

$$\mathbf{a} = \mathbf{e}^{\dagger 20^{\circ}} = -\frac{1}{2} + \mathbf{j}\frac{\sqrt{3}}{2}, \ \mathbf{a}^2 = \mathbf{e}^{\dagger 240^{\circ}} = -\frac{1}{2} - \mathbf{j}\frac{\sqrt{3}}{2}.$$

Basic equations of the AC machine with complex variables (denoted by a line over the symbol of the variable) in the reference frame rotating by general angular speed k are:

$$
\overline{\mu}\_1 = R\_1 \overline{\dot{i}}\_1 + \frac{d\Psi\_1}{dt} + j\,\phi\_k \,\overline{\Psi}\_1 \tag{1}
$$

$$
\overline{\mu}\_2 = R\_2 \overline{\dot{i}}\_2 + \frac{d\Psi\_2}{dt} + j(o\_k - o)\overline{\Psi}\_2 \tag{2}
$$

$$\frac{1}{p}\frac{d\alpha}{dt} = \frac{3p}{2}\operatorname{Im}(\overline{\Psi}\_{1k}^c \overline{i}\_{1k}) - m\_z \tag{3}$$

where the nomenclature is as follows:


For manipulation between various reference frames in the motor model the transformation formulas are used as listed in Tab. 1. All rotor parameters and variables are re-calculated to the stator side.

After inserting real and imaginary components into the complex of variables (e.g. for stator voltage 11 1 *x y u u ju* in synchronously rotating reference frame {*x, y*}), we get the AC motor


mathematical model whose equations are listed in Tab. 2 and a block diagram shown in Fig. 1 where 1 1 *K L* 1/( ) , 2 2 *K L* 1/( ) , 1 2 / ( ) *K L LL <sup>h</sup>* .

having real and imaginary components is defined by the equation:

<sup>o</sup> j120 1 3 ae j 2 2

equations are transformed into reference systems.

where




1 *,*

the stator side.


system what yields to decreased number of differential equations. The quantities in

To derive dynamic model of asynchronous motor, the three-phase system is to be transformed into the two-phase one. In the fact, this transformation presents a replacement of the three-phase motor by equivalent two-phase one. The stator current space vector

> <sup>2</sup> <sup>2</sup> ( ) <sup>3</sup> *ab c* **i aa** *ii i*

, <sup>o</sup> <sup>2</sup> j240 1 3 ae j 2 2

Basic equations of the AC machine with complex variables (denoted by a line over the symbol of the variable) in the reference frame rotating by general angular speed k are:

> 1 1 11 *k* 1 *<sup>d</sup> u Ri j dt*

2 2 22 <sup>2</sup> ( ) *<sup>k</sup>*

*J d <sup>p</sup> i m*


*<sup>d</sup> u Ri j dt*



– magnetic field angular speed, rotor angular speed, where 1

For manipulation between various reference frames in the motor model the transformation formulas are used as listed in Tab. 1. All rotor parameters and variables are re-calculated to

After inserting real and imaginary components into the complex of variables (e.g. for stator voltage 11 1 *x y u u ju* in synchronously rotating reference frame {*x, y*}), we get the AC motor

*<sup>k</sup>* – angular speed of a general rotating reference frame *<sup>k</sup>* <sup>1</sup>

12 1 ( )/ *<sup>h</sup> LL L L*

*p dt*

where the nomenclature is as follows:

calculated to stator quantities)


– (rotor) mechanical angular speed

 

1 1 <sup>3</sup> Im( ) <sup>2</sup> *c*

*kk z*

(3)

 or *0*

> 

(1)

(2)

**Table 1.** Transformation relations between three-phase system and two-phase reference frame and between {*x, y*} and {*α, β*} reference frames

 

cos sin sin cos

<sup>1</sup> *t*

*i i i i*

*x y*

{*α, β*} {*x, y*} from stator reference frame {

}into the synchronously rotating frame {*x, y*} (inverse Park transform)

*, β*}


**Table 2.** Equations of windings of asynchronous motor model in {x,y} reference frame

The corresponding Simulink model is drawn in Fig. 1. The model of squirrel cage motor (rotor voltages = 0) contains 4 inputs and 10 outputs (Tab. 3).

**Figure 1.** Simulink model of 3 phase squirrel cage asynchronous motor (the variables are denoted in the magnetic field reference frame {*x, y*})


**Table 3.** Notation of inputs and outputs of the asynchronous motor model

### *3.2.1. Modeling of supply source*

The asynchronous motor can be set into motion by various supply modes and control platforms:


Restrict our considerations to supply from indirect converter with the *Voltage-Source Inverter*  (VSI). Based on the inverter control mode the output voltage can be:


Developing inverter simulation schemes we have in mind two facts:

 the *constant stator flux* (i.e. fulfilling condition of constant ratio: *U1*/*f1* = const.) should be preserved at all modes of motor control

322 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

**Figure 1.** Simulink model of 3 phase squirrel cage asynchronous motor (the variables are denoted in the

**AM model inputs AM model outputs** 

The asynchronous motor can be set into motion by various supply modes and control

by *frequency starting* (with continuously increasing frequency of the supply voltage from

Restrict our considerations to supply from indirect converter with the *Voltage-Source Inverter* 

current i (4 components)

 motor torque *M* rotor angular speed

magnetic fluxes (4 components)

magnetic field reference frame {*x, y*})

 *U1* input voltage (axis x or ) *U2* input voltage (axis y or )

*3.2.1. Modeling of supply source* 

the frequency converter)

modulated by PWM

k reference frame angular speed

**Table 3.** Notation of inputs and outputs of the asynchronous motor model

by *direct connection* to the supply network or to the frequency converter

unmodulated (with 120 deg. switching in the power semiconductor devices)

(VSI). Based on the inverter control mode the output voltage can be:

Developing inverter simulation schemes we have in mind two facts:

*Mz* load torque

platforms:

 in range of very low frequency there should be kept an increased stator voltage (due to the voltage drop across the stator resistor) – so called V-curves (presenting a dependency of the supply voltage from the frequency). The V-curve can be modeled simply by a linear piecewise line.

The model of the motor supply source taking into consideration all described features is shown in Fig. 2 (signals denoted as SL and op are control signals from the GUI buttons). It has 4 inputs: supply frequency and voltage magnitude, ramp frequency and voltage (to simulate frequency starting). The switches "step/ramp" are controlled by pushbuttons from the GUI control panel.

**Figure 2.** Simulink model of various modes of supply source (DC, harmonic, frequency converter and PWM)

#### **Model of VSI converter (with constant output frequency)**

We start to model the inverter output voltage based on a similarity of output converter voltage with the perpendicular harmonic voltages (Fig. 3a). The VSI voltage vector changes its position 6 times per period, after every 60° (Fig. 3b).

Proper switching instants are realized by comparators and switches (Fig. 4). Harmonic oscillator creates a core of the inverter model. Generation of six switching states during period of the output voltage is adjusted by comparing values of the sin/cos signals with preset values of *sin* 60 = 3 /2 for the voltage 1 *u* and value of *cos* 60 = 1/2 for the voltage 1 *u* . The amplitudes of output voltage are adjusted by constants with values 1; 0,5 for 1 *u* and 0,866 = 3 /2 for 1 *u* .

**Figure 3.** Simulink model of inverter

**Figure 4.** Simulation scheme realizing rectangle voltages 1 u , <sup>1</sup> u of the inverter

#### **Model of PWM source**

The simplest way to generate a PWM signal uses the intersective method. The three-phase PWM voltage is generated directly in two axes {, } as shown in Fig. 5. The courses of the inverter PWM voltages 1 *u* and 1 *u* are shown in Fig. 6. In frequency starting mode of the asynchronous motor, the frequency of supply voltage increases from zero to required final value. To get the stator flux constant, the voltage across the motor has to increase linearly with frequency (*U/f* = const.), except of very low frequency range (due to voltage drop across the stator resistor). For this purpose, the connection must be completed by a compensating circuit which increases the value of supply voltage keeping the ratio *U/f* = const (Fig. 7). Up to the frequency of approx. 5 Hz the input voltage is kept constant on 10 % of its nominal value.

**Figure 5.** Model of voltages 1 u and 1 u from the inverter with PWM

**Figure 4.** Simulation scheme realizing rectangle voltages 1 u , <sup>1</sup> u of the inverter

The simplest way to generate a PWM signal uses the intersective method. The three-phase

(a) (b)

inverter PWM voltages 1 *u* and 1 *u* are shown in Fig. 6. In frequency starting mode of the asynchronous motor, the frequency of supply voltage increases from zero to required final value. To get the stator flux constant, the voltage across the motor has to increase linearly with frequency (*U/f* = const.), except of very low frequency range (due to voltage drop across the stator resistor). For this purpose, the connection must be completed by a compensating circuit which increases the value of supply voltage keeping the ratio *U/f* = const (Fig. 7). Up to the frequency of approx. 5 Hz the input voltage is kept constant on 10 % of its nominal value.

, 

} as shown in Fig. 5. The courses of the

**Figure 3.** Simulink model of inverter

**Model of PWM source** 

PWM voltage is generated directly in two axes {

**Figure 6.** Output voltages and 1 *u* and 1 *u* from the frequency converter: a) without and b) with PWM

**Figure 7.** The model of converter realizing the frequency starting under consideration of the law of constant stator flux (*U/f* = const.)

The model supposes that amplitude of the DC link voltage is changed in the frequency converter. This solution is suitable for drives with low requirements to motor dynamics. The DC link contains a large capacitor what causes the DC link voltage cannot be changed stepby-step. The output inverter voltage can change faster if the PWM control is used. Output voltages of the inverter model with linear increasing frequency and voltage are shown in Fig. 8 (observe a non-zero amplitude of the voltage that at the starting what is consequence of described V-curve block).

**Figure 8.** VSI output voltages 1 *u* and 1 *u* at increased frequency (the frequency time course is on the top figure)

#### *3.2.2. Model verification*

The AC induction motor model was simulated using following motor parameters: *R1=*1,8 Ω; *R2*=1,85 Ω; *p=*2; *J=*0,05 kgm2*, K1=*59,35; *K2=*59,35; *K=*56,93.

Time courses of mechanical variables are shown in Fig. 10 (they are the same regardless the used reference frame). Motor dynamical characteristics ��(�) at various modes of supply are compared in Fig. 11.

**Figure 9.** Time responses of asynchronous motor speed and torque at harmonic voltage supply at starting and loading the motor

**Figure 10.** Dynamic characteristic of the asynchronous motor *ω* = *f*(*M*) supplied: a) by harmonic voltage, b) from frequency converter, c) from frequency converter with PWM

## *3.2.3. GUI design and realisation*

326 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

of described V-curve block).

top figure)

u1a [V]

*3.2.2. Model verification* 

supply are compared in Fig. 11.

starting and loading the motor

*R2*=1,85 Ω; *p=*2; *J=*0,05 kgm2*, K1=*59,35; *K2=*59,35; *K=*56,93.

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -300

t [s]

u [V]

used reference frame). Motor dynamical characteristics

The model supposes that amplitude of the DC link voltage is changed in the frequency converter. This solution is suitable for drives with low requirements to motor dynamics. The DC link contains a large capacitor what causes the DC link voltage cannot be changed stepby-step. The output inverter voltage can change faster if the PWM control is used. Output voltages of the inverter model with linear increasing frequency and voltage are shown in Fig. 8 (observe a non-zero amplitude of the voltage that at the starting what is consequence

**Figure 8.** VSI output voltages 1 *u* and 1 *u* at increased frequency (the frequency time course is on the


<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

t [s]

u1b [V]

The AC induction motor model was simulated using following motor parameters: *R1=*1,8 Ω;

Time courses of mechanical variables are shown in Fig. 10 (they are the same regardless the

**Figure 9.** Time responses of asynchronous motor speed and torque at harmonic voltage supply at

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -250

t [s]

 ��(�) at various modes of

After debugging the motor model (Fig. 11), development of GUI continues with careful design of the program flowchart and design of GUI screen.

**Figure 11.** Arrangement of asynchronous motor subsystems in the Simulink GUI model

Description of the GUI functionality

The GUI screen (Fig. 13) consists of several panels. Their functionality is as follows:

	- in the synchronously rotating reference frame
	- in the reference frame associated with the stator

	- supply voltage time courses and in two coordinates
	- mechanical variables motor torque and speed
	- stator currents or magnetic fluxes
	- rotor currents or magnetic fluxes
	- Direct connection to the supply the button *Step*. The voltage *U*1 (effective rms value) and frequency *f*1 can be pre-set in the editing boxes.
	- Frequency starting the button *Linear* enables to pre-set the frequency time rise starting from zero.

**Figure 12.** GUI screen of the AC drive with induction machine

#### Screen outputs

Samples of the screens displaying variables in the stator reference frame {,} are shown in Fig. 13:

a. time courses at supplying motor by frequency converter – button *Time* )

b. chracteristics ( ) *M f* , 1 1 *i fi*( ) , 1 1 *f*( ) - button Rectangular )

328 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

 The graph to be displayed can be chosen by pushing radio button in the menu *Graphs*. Time courses are chosen by the button *Time*; dependency of one variable on other is

Direct connection to the supply – the button *Step*. The voltage *U*1 (effective rms

Frequency starting – the button *Linear* enables to pre-set the frequency time rise

 Using the buttons in the panel *Mode* we start *Simulation*, at pressing *Default* (original) parameters are set, and the Simulink scheme is shown by pushing the button *Model* .

Samples of the screens displaying variables in the stator reference frame {,} are shown in

a. time courses at supplying motor by frequency converter – button *Time* )

 at nonharmonics supply from the VSI with PWM *Output graphs*. Output variables are displayed in four graphs: supply voltage time courses and in two coordinates mechanical variables - motor torque and speed

*Mode of starting* the motor can be selected in the panel *Motor supply*:

value) and frequency *f*1 can be pre-set in the editing boxes.

at harmonic supply

 stator currents or magnetic fluxes rotor currents or magnetic fluxes

chosen by the button *Rectangular*.

starting from zero.

**Figure 12.** GUI screen of the AC drive with induction machine

Screen outputs

Fig. 13:

at nonharmonics supply from the VSI

c. time courses 1 *i ft*( ) , 2 *i ft*( ) at supplying from the PWM frequency converter

**Figure 13.** Examples of diplaying various graphs in the GUI for asynchronous motor
