**3. Virtual model development of a single-phase induction machine**

Single-phase induction motors are widely used in applications, and they excel in its simplicity, undemandingness, and reliability. They are also affordably priced. Usually, they are of smaller powers, approximate in the range of tens of watts to kilowatts units, and typically they are used in household appliances (because usually the single-phase supply system is available in the home distribution of electrical energy).

#### **3.1. Principle of operation**

The mechanical construction of the single-phase induction motor does not differ much from the three-phase one. Different is the way of arrangement of the stator winding and supply: one winding in the stator of the single-phase induction motor cannot create rotating magnetic field necessary for running the machine, but it produces only a vibrant field. The rotor will not start without any auxiliary winding. This is why the motor has two stator windings – the main and the auxiliary ones. The motor power is transmitted through the main winding, which is stored in two-thirds of the stator slots and is supplied directly from the single-phase network. The auxiliary winding is rated for lower current than the main winding. In interactivity having the main winding, its aim is to develop rotating magnetic field with the form approaching as the most to the circular shape.

The single-phase induction motors are distinguished by the structure of the rotor motor with wound rotor and squirrel cage rotor. The cage motor is used only for small performances.

#### **3.2. Modes of starting**

To get the rotating magnetic field, the current in the auxiliary winding shall be displaced by 90° from the current in the main winding. This is achieved by connecting an inductor or a capacitor in series with the auxiliary winding and parallel connection of the circuit to the main winding connected to the supply. For this reason, a suitable reactance is connected into the auxiliary winding (Figure 1). Here we recognize the following:


**Figure 1.** Starting modes of one-phase induction machine.

The capacitor *C*r connected into the auxiliary phase is optimally calculated according to the following equation:

Support for Learning of Dynamic Performance of Electrical Rotating Machines by Virtual Models http://dx.doi.org/10.5772/60723 7

$$\text{C}\_{\text{r}} = 2200 \frac{P\_{\text{N}}}{\text{U}\_{\text{N}}^2} \left[ \mu \text{F; W, V} \right] \tag{1}$$

The one-phase induction motor is supplied by an alternating sinusoidal harmonic voltage that is generated by a simple harmonic oscillator in the model, having on its output the signal of harmonic voltage with the amplitude *U*1*<sup>n</sup>* and frequency *f* <sup>1</sup> corresponding to the supply net. To achieve the starting torque, which is equal to the nominal, two to three times larger capacitor *C*s is required, i.e.,

$$\mathbf{C}\_s = \begin{pmatrix} \mathbf{2} \ \mathbf{-3} \end{pmatrix} \mathbf{C}\_r \tag{2}$$

#### **3.3. The one-phase induction motor model**

powers, approximate in the range of tens of watts to kilowatts units, and typically they are used in household appliances (because usually the single-phase supply system is available in

The mechanical construction of the single-phase induction motor does not differ much from the three-phase one. Different is the way of arrangement of the stator winding and supply: one winding in the stator of the single-phase induction motor cannot create rotating magnetic field necessary for running the machine, but it produces only a vibrant field. The rotor will not start without any auxiliary winding. This is why the motor has two stator windings – the main and the auxiliary ones. The motor power is transmitted through the main winding, which is stored in two-thirds of the stator slots and is supplied directly from the single-phase network. The auxiliary winding is rated for lower current than the main winding. In interactivity having the main winding, its aim is to develop rotating magnetic field with the form approaching as the

The single-phase induction motors are distinguished by the structure of the rotor motor with wound rotor and squirrel cage rotor. The cage motor is used only for small performances.

To get the rotating magnetic field, the current in the auxiliary winding shall be displaced by 90° from the current in the main winding. This is achieved by connecting an inductor or a capacitor in series with the auxiliary winding and parallel connection of the circuit to the main winding connected to the supply. For this reason, a suitable reactance is connected into the

**a.** Starting with inductor (Figure 1a). This is seldom used (it results in a lower efficiency)

The capacitor *C*r connected into the auxiliary phase is optimally calculated according to the

**b.** Starting with capacitor (Figure 1b): connected full time or by a double capacitor.

auxiliary winding (Figure 1). Here we recognize the following:

**Figure 1.** Starting modes of one-phase induction machine.

following equation:

the home distribution of electrical energy).

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**3.1. Principle of operation**

most to the circular shape.

**3.2. Modes of starting**

The motor mathematical model consists of models of electrical and mechanical parts. The electrical part is represented by the scheme in Figure 2.

**Figure 2.** Equivalent circuits: (a) the main winding and (b) the auxiliary winding.

Based on the equivalent circuits and dynamic equation of the motor, the mathematical equations with their representations are shown in Table 1.


**Table 1.** Mathematical and simulation models of subsystems of the one-phase induction machine

Combining all schemes together, we get the block diagram of the motor (Figure 3), which presents a core of the virtual model enabling deeper understanding of the phenomena in the motor at various modes of operation and supply.

Support for Learning of Dynamic Performance of Electrical Rotating Machines by Virtual Models http://dx.doi.org/10.5772/60723 9

**Figure 3.** Simulink block diagram of one-phase induction motor.

**Equation**

**rotor agnetic flux**

**stator currents**

**rotor currents**

**motor torque**

**dynamic**

**equation**

**for Mathematical model (equations) Simulation model in Simulink**

A similar simulation scheme with corresponding parameters and variables

A similar simulation scheme with corresponding parameters and variables

A similar simulation scheme with corresponding parameters and variables

*ψr<sup>α</sup>* = *L <sup>m</sup>αi*

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*ψr<sup>β</sup>* = *L <sup>m</sup>βi*

*i <sup>s</sup><sup>α</sup>* <sup>=</sup> <sup>1</sup> *L <sup>s</sup><sup>α</sup>*

*i <sup>s</sup><sup>β</sup>* <sup>=</sup> <sup>1</sup> *L <sup>s</sup><sup>β</sup>*

*i <sup>s</sup><sup>α</sup>* <sup>=</sup> <sup>1</sup> *L <sup>s</sup><sup>α</sup>*

*i <sup>r</sup><sup>β</sup>* <sup>=</sup> <sup>1</sup> *L <sup>r</sup><sup>β</sup>*

*Mm* <sup>=</sup> *<sup>p</sup>*( *<sup>N</sup>*<sup>1</sup> *N*2 *ψrβi <sup>r</sup><sup>α</sup>* − *N*2 *N*1 *ψrαi rβ*)

> *dω dt* <sup>=</sup> <sup>1</sup>

motor at various modes of operation and supply.

*<sup>s</sup><sup>α</sup>* + *L <sup>r</sup>αi rα*

*<sup>s</sup><sup>β</sup>* + *L <sup>r</sup>βi rβ*

(*ψs<sup>α</sup>* − *L <sup>m</sup>αi*

(*ψs<sup>β</sup>* − *L <sup>m</sup>βi*

(*ψs<sup>α</sup>* − *L <sup>m</sup>αi*

(*ψr<sup>β</sup>* − *L <sup>m</sup>βi*

*<sup>J</sup>* <sup>⋅</sup> (*Mm* <sup>−</sup>*Mload* )

**Table 1.** Mathematical and simulation models of subsystems of the one-phase induction machine

Combining all schemes together, we get the block diagram of the motor (Figure 3), which presents a core of the virtual model enabling deeper understanding of the phenomena in the

*rα*)

*rβ*)

*rα*)

*sβ*)

In investigating various modes of starting and operation, it is suitable to complete the model by supplementary blocks, like harmonic oscillator generating sinus and cosinus voltages, by the switches switching the capacitors according to the chose mode of operation, and by block generating the load torque in the optional time instant of loading the machine. The final scheme, used for the virtual model, is shown in Figure 4.

**Figure 4.** The model of the motor with connected inputs and outputs enabling to simulate capacitor run and double capacitor starting the motor.

#### **3.4. Verification of the motor model with permanently connected capacitor**

The speed can be changed by the frequency, by changing the number of poles, and-in a small scale-by change of the voltage or value of the capacitor. The change of the direction of rotation is simply done by the pole change of the auxiliary winding.

Time courses of the motor basic variables, motor torque, speed, and currents in both windings, are shown in Figure 5.

Simulation parameters:

$$\text{If } \mathcal{U}\_1 = 230 \text{ V, } f\_1 = 50 \text{ Hz, } M\_{\text{load}} = 5 \text{ Nm}, \mathcal{C}\_r = 32 \text{ } \mu\text{F, } T\_{\text{sim}} = 0.8 \text{ s}, T\_{\text{load}} = 0.5 \text{ s}.$$

**Figure 5.** Time responses of the one-phase induction motor with the permanently connected capacitor in the stator ref‐ erence frame {*α,β*} at starting and loading the motor in the time 0.5 s: (a) supply voltages *u*sα, *u*sβ of the motor mode; (b) motor torque *M* ; (c) angular speed *ω*; (d) static characteristic of the motor *ω/M;* and *(*d) stator currents: torque pro‐ ducing component *i*sα and magnetic flow component *i*sβ.

#### **3.5. One-phase motor with a double capacitor**

**3.4. Verification of the motor model with permanently connected capacitor**

is simply done by the pole change of the auxiliary winding.

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are shown in Figure 5.

Simulation parameters:

The speed can be changed by the frequency, by changing the number of poles, and-in a small scale-by change of the voltage or value of the capacitor. The change of the direction of rotation

Time courses of the motor basic variables, motor torque, speed, and currents in both windings,

<sup>1</sup> <sup>1</sup> load sim load = 230 V, = 50 Hz, = 5 Nm, = 32 µF, = 0.8 s, = 0.5 s. *U f M C TT <sup>r</sup>*

**Figure 5.** Time responses of the one-phase induction motor with the permanently connected capacitor in the stator ref‐ erence frame {*α,β*} at starting and loading the motor in the time 0.5 s: (a) supply voltages *u*sα, *u*sβ of the motor mode; (b) motor torque *M* ; (c) angular speed *ω*; (d) static characteristic of the motor *ω/M;* and *(*d) stator currents: torque pro‐

ducing component *i*sα and magnetic flow component *i*sβ.

To improve the motor performance during start period, a higher capacity is required in the auxiliary phase circuit. This is done by a capacitor *C*<sup>s</sup> connected in parallel to the existing one (Figure 1b) up to the time instant the motor runs by speed about 70% *ω*N, which is followed by a centrifugal switch. After disconnecting, only the capacitor *C*<sup>r</sup> of a lower value remains connected permanently. The next figure (Figure 6) shows performance graphs of the motor with double capacitor.

Simulation parameters: *U*<sup>1</sup> = 230 V, *f*<sup>1</sup> = 50 Hz, *M*load = 5 Nm, *C*<sup>s</sup> = 53 µF, *C*<sup>r</sup> = 32 µF, *ω*<sup>n</sup> = 110 rad/ s, *T*sim = 0.8 s, and *T*load = 0.5 s.

**Figure 6.** Time responses of the one-phase induction motor with the double-starting capacitor in the stator reference frame {*α,β*} while starting and loading the motor at time 0.5 s: (a) supply voltages *u*sα, *u*sβ of the motor mode; (b) motor torque *M* ; (c) angular speed *ω;* (d) static characteristic of the motor *ω* = *f* (*M* ) ; (e, f) stator currents: torque produc‐ ing component *i <sup>s</sup>α* and magnetic flow component *i sβ*

#### **3.6. Comparison of the motor performance supplied by the voltages of different frequencies**

To get the best motor performance, the constant stator flux must be preserved at various supply frequencies. From this condition, it follows up that *U/f* = const. (Figure 7).

**Figure 7.** Time courses of the angular speed *ω* and motor torque at various frequencies of the supply voltage: (a) *U* = 230 V, *f* = 50 Hz, *C*s= 53 µF *C*r = 32 µF; (b) *U* = 115 V, *f* = 25 Hz, *C*sb = 28 µF, *C*r = 14 µF.

#### **3.7. GUI application for the one-phase induction motor**

After the verification of the mathematical model of the single-phase motor by simulation, we can design and develop the motor virtual model-graphical user interface in the MATLAB program. The application is designed so that the user is granted simple handling in adjusting motor parameters, and waveforms show typical values of the motor. Graphical user interface (GUI) allows you to visualize four different graphs: the input supply voltage, torque, angular velocity of the rotor, components of stator and rotor currents, and components of magnetic fluxes motor when starting the motor, during steady-state operation, and after loading it the chosen time. The ergonomics of handling and pedagogical respects should be strictly taken into consideration, as widely analyzed in the previous publications [1-3]. Moreover, these aspects of using the model should be also considered-they should be suitable.


#### **3.8. GUI screen description**

**3.6. Comparison of the motor performance supplied by the voltages of different frequencies**

To get the best motor performance, the constant stator flux must be preserved at various supply

**Figure 7.** Time courses of the angular speed *ω* and motor torque at various frequencies of the supply voltage: (a) *U* =

After the verification of the mathematical model of the single-phase motor by simulation, we can design and develop the motor virtual model-graphical user interface in the MATLAB program. The application is designed so that the user is granted simple handling in adjusting motor parameters, and waveforms show typical values of the motor. Graphical user interface (GUI) allows you to visualize four different graphs: the input supply voltage, torque, angular velocity of the rotor, components of stator and rotor currents, and components of magnetic fluxes motor when starting the motor, during steady-state operation, and after loading it the chosen time. The ergonomics of handling and pedagogical respects should be strictly taken into consideration, as widely analyzed in the previous publications [1-3]. Moreover, these

**1.** for explanation at lectures the phenomena and motor behavior during various operating

aspects of using the model should be also considered-they should be suitable.

**2.** for preparation of the students for the experimentation in the labs.

230 V, *f* = 50 Hz, *C*s= 53 µF *C*r = 32 µF; (b) *U* = 115 V, *f* = 25 Hz, *C*sb = 28 µF, *C*r = 14 µF.

**3.7. GUI application for the one-phase induction motor**

modes and

frequencies. From this condition, it follows up that *U/f* = const. (Figure 7).

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The developed GUI for the one-phase induction motor consists of two screens:

	- **a.** for explanation it displays equivalent diagram of the motor on the left side and its mathematical model on the right side. The basic differential equations describing the motor that are in boxes, the color of which corresponds to the color of the graphs
	- **b.** for inputting the motor basic parameters.

**Figure 8.** The screen for inputting parameters for the model of single-phase motor displaying motor equivalent dia‐ gram, equations of the model, and input motor parameters.

The panel *Motor Parameters* (Figure 10a) enable to input motor parameters and thus to verify and compare behavior of various motors. After pushing the return button, the main program starts model simulation with the actual parameters.

In the bottom-right corner, there are three buttons: the *model* button displays the motor model in the Simulink program, as shown in Figure 4 (and inside the block of the motor, the full scheme appears according to the Figure 3). The *default* button sets the preset values of the parameters, and the *return* button causes return to the second screen with the graphs, and immediately, the simulation starts with the set parameters.

**Figure 9.** The GUI main screen with the graphs and control modes of calculation and visualization.

**Figure 10.** (a) The panel for input and changes basic parameters of the motor. (b) The panel for calculation of value of the capacitor connected into the auxiliary phase.

#### **3.9. Description of the control panels**

The panel *Coordinate system* (Figure 11a) contains three switches to choose the supply mode: (1) supply by the harmonic voltage in the {*α, β*} coordinate system connected with the stator, (2) supply with the permanently connected capacitor, and (3) supply with the double capacitor.

The panel *Graphs*(Figure 11b) consists of two parts: the upper one contains four buttons serving to choose the graphs to display stator/rotor currents or stator/rotor magnetic fluxes. By the buttons in the lower part, the time graphs ( *i* = *f* <sup>1</sup> (*t*), *ψ*= *f* <sup>2</sup> (*t*)) or mutual dependence of the variables is chosen (*i <sup>α</sup>* = *g*1(*i <sup>β</sup>*), *ψα* = *g*1(*ψβ*)). The variables of the motor we are intend to display (stator or rotor) are selected by the above-mentioned buttons.

Support for Learning of Dynamic Performance of Electrical Rotating Machines by Virtual Models http://dx.doi.org/10.5772/60723 15

**Figure 11.** The GUI panels: (a) Coordinate system; (b) Graphs; (c) Load and Stop; (d) Model.

**Figure 9.** The GUI main screen with the graphs and control modes of calculation and visualization.

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the capacitor connected into the auxiliary phase.

**3.9. Description of the control panels**

variables is chosen (*i*

buttons in the lower part, the time graphs ( *i* = *f* <sup>1</sup>

(stator or rotor) are selected by the above-mentioned buttons.

*<sup>α</sup>* = *g*1(*i*

**Figure 10.** (a) The panel for input and changes basic parameters of the motor. (b) The panel for calculation of value of

The panel *Coordinate system* (Figure 11a) contains three switches to choose the supply mode: (1) supply by the harmonic voltage in the {*α, β*} coordinate system connected with the stator, (2) supply with the permanently connected capacitor, and (3) supply with the double capacitor. The panel *Graphs*(Figure 11b) consists of two parts: the upper one contains four buttons serving to choose the graphs to display stator/rotor currents or stator/rotor magnetic fluxes. By the

(*t*), *ψ*= *f* <sup>2</sup>

*<sup>β</sup>*), *ψα* = *g*1(*ψβ*)). The variables of the motor we are intend to display

(*t*)) or mutual dependence of the

The panel *Load and Stop* (Figure 11c) shows the edit boxes for setting the values of the capacitors: permanently connected *Cr*, starting capacitor *Cs*, load torque *Mload* , time of loading *Tload* , time of simulation *Tsim*, and supply voltage *U* and its frequency *f* .

The panel *Model* (Figure 11d) contains the possibility to finish simulation and exit from the program (*Close*), displaying motor model in Simulink (*Model*), and after pushing the button *Parameters*, we switch to the screen with parameters (Figures 8 and 10a).

The panel of tools (Table 2) makes the work with the GUI comfortable. The *Tools* panel consists of five icons. The *Context* menu enables to set up line widths for the simulation courses (Line Width), to make a copy of the screen (screenshot) by saving it (Save), to run help (Help), and to close the graphical interface (Close).


**Table 2.** Panel of tools and menu of the graphical user Interface
