**2. Single-phase induction motor model**

The commercial SPIM commonly used in low power applications is usually a two-phase induction machine with asymmetrical windings, whose equivalent circuit without the permanent split-capacitor can be represented as in Fig. 1.

**Figure 1.** Equivalent circuit of SPIM.

As in [14], in this chapter the squirrel cage SPIM mathematical model is described in a stationary reference-frame by the following equations

2 Will-be-set-by-IN-TECH

parameter estimation, for adaptive systems and self-tuning controllers due to the fact that the parameters of induction machine change with temperature, saturation, and frequency ([22]). Differently from the three-phase induction motors, the SPIM is an asymmetrical and coupled machine; these features make the electrical parameter estimation by classical methods difficult, and these characteristics complicate the use of high performance techniques, such as vector and sensorless control. Thus, the use of Recursive Least Square (RLS) algorithm can be a solution for the parameter estimation or self-tuning and adaptive controllers, such as presented in [28] and [30]. Other studies have also been reported in literature describing the

The aim of this chapter is to provide a methodology to identify a set of parameters for an equivalent SPIM model, and to obtain an improved SPIM representation, as consequence it is possible to design a high performance sensorless SPIM controllers. Here, from the machine model, a classical RLS algorithm is applied at *q* and *d* axes based on the current measurements and information of fed voltages with a standstill rotor. The automatized test with standstill rotor can be a good alternative in some applications, such as hermetic compressor systems, where the estimation by conventional methods is a hard task due the fact of the machine is

An equivalent SPIM behavior representation is obtained with this methodology in comparison with the SPIM model obtained by classical tests. However, some types of SPIM drives, for instance a hermetic system, it is impossible to carried out classical tests. In addition, the

This chapter is organized as follows: Section 2 presents the SPIM model, Section 3 gives the RLS parameter algorithm, Section 4 presents and discusses the experimental results obtained with the proposed methodology, and Section 5 gives the main conclusions of this study.

The commercial SPIM commonly used in low power applications is usually a two-phase induction machine with asymmetrical windings, whose equivalent circuit without the

*Lmq*

*Lmd*

*q* ' *d N <sup>N</sup> r rd Llrq*

<sup>+</sup> \_

<sup>+</sup> \_

' *d q N <sup>N</sup> r rq Llrd* *Rrq*

*Rrd*

*Llsq*

*Llsd*

parameter estimation of SPIM ([2, 11, 17, 25]).

proposed methodology has a simple implementation.

**2. Single-phase induction motor model**

*Vsq sq <sup>i</sup>* <sup>+</sup>

*Vsd sd i* <sup>+</sup>

\_

**Figure 1.** Equivalent circuit of SPIM.

\_

permanent split-capacitor can be represented as in Fig. 1.

*Rsq*

*Rsd*

sealed.

$$
\begin{bmatrix} V\_{sq} \\ V\_{sd} \end{bmatrix} = \begin{bmatrix} R\_{sq} & 0 \\ 0 & R\_{sd} \end{bmatrix} \begin{bmatrix} i\_{sq} \\ i\_{sd} \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \phi\_{sq} \\ \phi\_{sd} \end{bmatrix} \tag{1}
$$

$$
\begin{aligned}
\begin{bmatrix} V\_{rq} \\ V\_{rd} \end{bmatrix} &= \begin{bmatrix} R\_{rq} & 0 \\ 0 & R\_{rd} \end{bmatrix} \begin{bmatrix} i\_{rq} \\ i\_{rd} \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \phi\_{rq} \\ \phi\_{rd} \end{bmatrix} \\ &+ \omega\_r \begin{bmatrix} 0 & -1/\_{\mathcal{H}} \\ n & 0 \end{bmatrix} \begin{bmatrix} \phi\_{rq} \\ \phi\_{rd} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{aligned} \tag{2}
$$

$$
\begin{bmatrix} \phi\_{sq} \\ \phi\_{sd} \end{bmatrix} = \begin{bmatrix} L\_{sq} & 0 \\ 0 & L\_{sd} \end{bmatrix} \begin{bmatrix} i\_{sq} \\ i\_{sd} \end{bmatrix} + \begin{bmatrix} L\_{mq} & 0 \\ 0 & L\_{md} \end{bmatrix} \begin{bmatrix} i\_{rq} \\ i\_{rd} \end{bmatrix} \tag{3}
$$

$$
\begin{bmatrix}
\phi\_{rq} \\
\phi\_{rd}
\end{bmatrix} = \begin{bmatrix}
L\_{mq} & 0 \\
0 & L\_{md}
\end{bmatrix} \begin{bmatrix}
i\_{sq} \\
i\_{sd}
\end{bmatrix} + \begin{bmatrix}
L\_{rq} & 0 \\
0 & L\_{rd}
\end{bmatrix} \begin{bmatrix}
i\_{rq} \\
i\_{rd}
\end{bmatrix} \tag{4}
$$

$$T\_{\varepsilon} = p(L\_{mq}i\_{sq}i\_{rd} - L\_{md}i\_{sd}i\_{rq}) \tag{5}$$

$$p(T\_\ell - T\_L) = f \frac{d\omega\_r}{dt} + B\_{ll}\omega\_r \tag{6}$$

where, the indexes *q* and *d* represent the main winding and auxiliary winding, respectively, the indexes *sq* and *sd* represent the stator variables, and the indexes *rq* and *rd* are used for the rotor variables. *Vsq*, *Vsd*, *Vrq*, *Vrd*, *isq*, *isd*, *irq*, *ird*, *φsq*, *φsd*, *φrq*, and *φrd* are the stator and rotor voltages, currents, and flux; *Rsq*, *Rsd*, *Rrq*, and *Rrd* are the stator and rotor resistances; *Llsq*, *Llsd*, *Llrq*, and *Llrd* are the leakage inductances; *Lmq* and *Lmd* are the mutual inductances; *Lsq*, *Lsd*, *Lrq*, and *Lrd* are the stator and rotor inductances, and are given by: *Lsq* = *Llsq* + *Lmq*, *Lsd* = *Llsd* + *Lmd*, *Lrq* = *Llrq* + *Lmq*, and *Lrd* = *Llrd* + *Lmd*; *Nq* and *Nd* represent the number of turns for the main and auxiliary windings, respectively; *p* is the pole pair number and *ωr* is the rotor speed, and *n* is the relationship between the number of turns for auxiliary and for main winding *Nd*/*Nq*. *Te* is the electromagnetic torque, *TL* is the load torque, *Bn* is the viscous friction coefficient, and *J* is the inertia coefficient.

From (1) - (4) it is possible to obtain the differential equations that express the dynamical behavior of the SPIM, as follows,

$$\frac{d}{dt}\dot{i}\_{sq} = -\frac{R\_{sq}L\_{rq}}{\overline{\sigma}\_{q}}\dot{i}\_{sq} - \omega\_{r}\frac{1}{n}\frac{L\_{mq}L\_{md}}{\overline{\sigma}\_{q}}\dot{i}\_{sd} + \frac{R\_{rq}L\_{mq}}{\overline{\sigma}\_{q}}\dot{i}\_{rq} - \omega\_{r}\frac{1}{n}\frac{L\_{rd}L\_{mq}}{\overline{\sigma}\_{q}}\dot{i}\_{rd} + \frac{L\_{rq}}{\overline{\sigma}\_{q}}V\_{sq} \tag{7}$$

$$\frac{d}{dt}\mathbf{i}\_{\rm sd} = \omega\_{\rm r}\mathbf{n}\frac{\mathbf{L}\_{\rm md}\mathbf{L}\_{\rm mq}}{\overline{\sigma}\_{\rm d}}\mathbf{i}\_{\rm sq} - \frac{\mathbf{L}\_{\rm rd}\mathbf{R}\_{\rm sd}}{\overline{\sigma}\_{\rm d}}\mathbf{i}\_{\rm sd} + \omega\_{\rm r}\mathbf{n}\frac{\mathbf{L}\_{\rm rq}\mathbf{L}\_{\rm md}}{\overline{\sigma}\_{\rm d}}\mathbf{i}\_{\rm rq} + \frac{\mathbf{R}\_{\rm rd}\mathbf{L}\_{\rm md}}{\overline{\sigma}\_{\rm d}}\mathbf{i}\_{\rm rd} + \frac{\mathbf{L}\_{\rm rd}}{\overline{\sigma}\_{\rm d}}\mathbf{V}\_{\rm sd} \tag{8}$$

$$\frac{d}{dt}\dot{\mathbf{i}}\_{rq} = \frac{\mathbf{L}\_{mq}\mathbf{R}\_{sq}}{\overline{\sigma}\_{q}}\dot{\mathbf{i}}\_{sq} + \omega\_{r}\frac{\mathbf{1}}{n}\frac{\mathbf{L}\_{sq}\mathbf{L}\_{md}}{\overline{\sigma}\_{q}}\dot{\mathbf{i}}\_{sd} - \frac{\mathbf{L}\_{sq}\mathbf{R}\_{rq}}{\overline{\sigma}\_{q}}\dot{\mathbf{i}}\_{rq} + \omega\_{r}\frac{\mathbf{1}}{n}\frac{\mathbf{L}\_{sq}\mathbf{L}\_{rd}}{\overline{\sigma}\_{q}}\dot{\mathbf{i}}\_{rd} - \frac{\mathbf{L}\_{mq}}{\overline{\sigma}\_{q}}\mathbf{V}\_{sq} \tag{9}$$

$$\frac{d}{dt}\dot{\mathbf{i}}\_{rd} = -\omega\_r n \frac{\mathbf{L}\_{\rm sd} \mathbf{L}\_{m\eta}}{\overline{\sigma}\_d} \mathbf{i}\_{sq} + \frac{\mathbf{L}\_{\rm md} \mathbf{R}\_{\rm sd}}{\overline{\sigma}\_d} \mathbf{i}\_{sd} - \omega\_r n \frac{\mathbf{L}\_{\rm sd} \mathbf{L}\_{r\eta}}{\overline{\sigma}\_d} \mathbf{i}\_{rq} - \frac{\mathbf{L}\_{\rm sd} \mathbf{R}\_{\rm rd}}{\overline{\sigma}\_d} \mathbf{i}\_{rd} - \frac{\mathbf{L}\_{\rm md}}{\overline{\sigma}\_d} V\_{\rm sd} \tag{10}$$

where *<sup>σ</sup><sup>q</sup>* <sup>=</sup> *LsqLrq* <sup>−</sup> *<sup>L</sup>*<sup>2</sup> *mq*, *<sup>σ</sup><sup>d</sup>* <sup>=</sup> *LsdLrd* <sup>−</sup> *<sup>L</sup>*<sup>2</sup> *md*.

The transfer functions in the axes *q* and *d* at a standstill rotor (*ω<sup>r</sup>* = 0) are obtained from (7)-(10), where these functions are decoupled and presented in (11) and (12).

$$H\_q(\mathbf{s}) = \frac{i\_{sq}(\mathbf{s})}{V\_{sq}(\mathbf{s})} = \frac{s\overline{\sigma}\_q^{-1}L\_{rq} + \overline{\sigma}\_q^{-1}\tau\_{rq}^{-1}L\_{rq}}{s^2 + sp\_q + \overline{\sigma\_q}^{-1}R\_{rq}R\_{sq}}\tag{11}$$

by RLS Algorithm 5

*sd*(*s*) <sup>=</sup> *sb*1*<sup>d</sup>* <sup>+</sup> *<sup>b</sup>*0*<sup>d</sup>*

, *<sup>a</sup>*1*<sup>q</sup>* <sup>=</sup> *RsqLrq* <sup>+</sup> *RrqLsq*

, *<sup>a</sup>*1*<sup>d</sup>* <sup>=</sup> *RsdLrd* <sup>+</sup> *RrdLsd*

In order to obtain the regression linear model the transfer functions of (18) and (19) can be

*dt* <sup>+</sup> *<sup>a</sup>*0*qisq* <sup>=</sup> *<sup>b</sup>*1*<sup>q</sup>*

*dt* <sup>+</sup> *<sup>a</sup>*0*disd* <sup>=</sup> *<sup>b</sup>*1*<sup>d</sup>*

*dVsq dt Vsq* �

*dVsd dt Vsd* �

> *dVsq dt Vsq* �

*dVsd dt Vsd* �

*a*1*<sup>q</sup> a*0*<sup>q</sup> b*1*<sup>q</sup> b*0*<sup>q</sup>*

*a*1*<sup>d</sup> a*0*<sup>d</sup> b*1*<sup>d</sup> b*0*<sup>d</sup>*

The estimation of coefficients *b*1*q*, *b*0*q*, *a*1*q*, *a*0*q*, *b*1*d*, *b*0*d*, *a*1*<sup>d</sup>* and *a*0*<sup>d</sup>* is done by using RLS estimation algorithm described in the equations (13)-(17). The linear regression model form

**<sup>Y</sup>***q*(*k*) = *<sup>d</sup>*2*isq*

**<sup>Y</sup>***d*(*k*) = *<sup>d</sup>*<sup>2</sup>*isd*

<sup>−</sup> *disq dt* <sup>−</sup>*isq*

<sup>−</sup> *disd dt* <sup>−</sup>*isd*

*s*<sup>2</sup> + *sa*1*<sup>d</sup>* + *a*0*<sup>d</sup>*

Electrical Parameter Identi cation of Single-Phase Induction Motor by RLS Algorithm 279

, *<sup>a</sup>*0*<sup>q</sup>* <sup>=</sup> *RsqRrq σq*

, *<sup>a</sup>*0*<sup>d</sup>* <sup>=</sup> *RsdRrd σd*

*dt* <sup>+</sup> *<sup>b</sup>*0*qVsq* (22)

*dt* <sup>+</sup> *<sup>b</sup>*0*dVsd* (23)

*σq*

*σd*

*dVsq*

*dVsd*

⎡ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎣

*a*1*<sup>d</sup> a*0*<sup>d</sup> b*1*<sup>d</sup> b*0*<sup>d</sup>* ⎤ ⎥ ⎥ ⎦

*dt*<sup>2</sup> (26)

*dt*<sup>2</sup> (29)

� (28)

� (31)

*a*1*<sup>q</sup> a*0*<sup>q</sup> b*1*<sup>q</sup> b*0*<sup>q</sup>* ⎤ ⎥ ⎥ ⎦ (19)

(20)

(21)

(24)

(25)

(27)

(30)

*Hd*(*s*) = *<sup>i</sup>*

*σqτrq*

*σdτrd*

*disq*

*disd*

� <sup>−</sup> *disq dt* <sup>−</sup>*isq*

� <sup>−</sup> *disd dt* <sup>−</sup>*isd*

, *<sup>b</sup>*0*<sup>q</sup>* <sup>=</sup> *Lrq*

, *<sup>b</sup>*0*<sup>d</sup>* <sup>=</sup> *Lrd*

*d*2*isq dt*<sup>2</sup> <sup>+</sup> *<sup>a</sup>*1*<sup>q</sup>*

*d*<sup>2</sup>*isd dt*<sup>2</sup> <sup>+</sup> *<sup>a</sup>*1*<sup>d</sup>*

Solving for the second derivative of the stator current,

*d*2*isq dt*<sup>2</sup> <sup>=</sup>

*d*<sup>2</sup>*isd dt*<sup>2</sup> <sup>=</sup>

(13) is given for the *q* axis by the following equations,

*φT <sup>q</sup>* (*k*) = �

*φT <sup>d</sup>* (*k*) = �

*θT <sup>q</sup>* (*k*) = �

*θT <sup>d</sup>* (*k*) = �

*<sup>b</sup>*1*<sup>q</sup>* <sup>=</sup> *Lrq σq*

*<sup>b</sup>*1*<sup>d</sup>* <sup>=</sup> *Lrd σd*

generalized and rewritten as,

*s sd*(*s*) *Vs*

and

where

and

and

and

And, for the *d* axis,

$$H\_d(\mathbf{s}) = \frac{i\_{\rm sd}(\mathbf{s})}{V\_{\rm sd}(\mathbf{s})} = \frac{s\overline{\sigma}\_d^{-1}L\_{rd} + \overline{\sigma}\_d^{-1}\tau\_{rd}^{-1}L\_{rd}}{s^2 + sp\_d + \overline{\sigma}\_d^{-1}R\_{rd}R\_{sd}}\tag{12}$$

where *pq* = *RsqLrq* + *RrqLsq<sup>σ</sup><sup>q</sup>* and *pd* = (*RsdLrd* + *RrdLsd*) *σd*.

### **3. Parameter identification of single-phase induction machine**

In section 2, the decoupled transfer functions of the SPIM were obtained assuming a standstill rotor (*ω<sup>r</sup>* = 0). Thus, in this section the parameter identification is achieved with SPIM at a standstill rotor by a RLS algorithm. The identification with a standstill rotor is appropriated in some cases such as hermetic refrigeration compressors ([28]). The RLS identification algorithm requires the plant model in a discrete time linear regression form. Assuming the actual sampling index *k*, the regression model is given by

$$
\hat{\mathbf{Y}}(k) = \boldsymbol{\phi}^T(k)\boldsymbol{\theta}(k)\tag{13}
$$

The recursive algorithm is achieved with the equations (14)-(17).

$$
\varepsilon(k) = \mathbf{Y}(k) - \mathbf{\hat{Y}}(k) \tag{14}
$$

$$\mathbf{K}(k) = \frac{\mathbf{P}(k-1)\phi(k)}{1 + \phi^T(k)\mathbf{P}(k-1)\phi(k)}\tag{15}$$

$$\theta(k) = \theta(k-1) + \mathbf{K}(k)e(k) \tag{16}$$

$$\mathbf{P}(k) = \left(I - \mathbf{K}(k)\boldsymbol{\phi}^T(k)\right)\mathbf{P}(k-1) \tag{17}$$

where *dim* **<sup>Y</sup>** <sup>=</sup> *<sup>M</sup>*¯ <sup>×</sup> *<sup>N</sup>*¯ , *dim <sup>φ</sup>T*(*k*) = *<sup>M</sup>*¯ <sup>×</sup> *<sup>r</sup>*¯

*dim <sup>θ</sup>*(*k*) = *<sup>r</sup>*¯ <sup>×</sup> *<sup>N</sup>*¯ , *dim e*(*k*) = *<sup>M</sup>*¯ <sup>×</sup> *<sup>N</sup>*¯

$$\dim \mathbf{K}(k) = \begin{array}{c} \vec{r} \times \vec{M} \end{array}, \dim I = \dim \mathbf{P}(k) = \vec{r} \times \vec{r}$$

From the equations (11) and (12) it is possible to reformulate the estimation parameter problem based on a linear regression model. Here, the parameter estimation method is divided into two steps:

### *First step: estimation of (18) and (19) vide equations (20) and (21) :*

This step consists into obtaining a linear-time-invariant model of the SPIM. The identification of *b*1, *b*0, *a*<sup>1</sup> and *a*<sup>0</sup> is done by performing a standstill test. The coefficients presented in (11) and (12) are functions of the machine parameters. For simplicity, the transfer functions given in (11) and (12) are rewritten in two transfer functions given by (18) and (19).

$$H\_q(\mathbf{s}) = \frac{i\_{sq}^s(\mathbf{s})}{V\_{sq}^s(\mathbf{s})} = \frac{sb\_{1q} + b\_{0q}}{s^2 + sa\_{1q} + a\_{0q}} \tag{18}$$

278 Induction Motors – Modelling and Control Electrical Parameter Identification of Single-Phase Induction Motor by RLS Algorithm 5 Electrical Parameter Identi cation of Single-Phase Induction Motor by RLS Algorithm 279

and

4 Will-be-set-by-IN-TECH

The transfer functions in the axes *q* and *d* at a standstill rotor (*ω<sup>r</sup>* = 0) are obtained from

*<sup>q</sup> Lrq* + *<sup>σ</sup>*−<sup>1</sup>

*<sup>d</sup> Lrd* <sup>+</sup> *<sup>σ</sup>*−<sup>1</sup>

*s*<sup>2</sup> + *spd* + *σ*−<sup>1</sup>

*s*<sup>2</sup> + *spq* + *σ*−<sup>1</sup>

*<sup>q</sup> τ*−<sup>1</sup> *rq Lrq*

*<sup>d</sup> <sup>τ</sup>*−<sup>1</sup> *rd Lrd*

*<sup>q</sup> RrqRsq*

*<sup>d</sup> RrdRsd*

 *σd*.

**Yˆ**(*k*) = *φT*(*k*)*θ*(*k*) (13)

*<sup>e</sup>*(*k*) = **<sup>Y</sup>**(*k*) <sup>−</sup> **Yˆ**(*k*) (14)

<sup>1</sup> <sup>+</sup> *<sup>φ</sup>T*(*k*)**P**(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>)*φ*(*k*) (15)

**P**(*k* − 1) (17)

*θ*(*k*) = *θ*(*k* − 1) + **K**(*k*)*e*(*k*) (16)

(11)

(12)

(18)

*md*.

*Vsq*(*s*) <sup>=</sup> *<sup>s</sup>σ*−<sup>1</sup>

*Vsd*(*s*) <sup>=</sup> *<sup>s</sup>σ*−<sup>1</sup>

*<sup>σ</sup><sup>q</sup>* and *pd* = (*RsdLrd* + *RrdLsd*)

In section 2, the decoupled transfer functions of the SPIM were obtained assuming a standstill rotor (*ω<sup>r</sup>* = 0). Thus, in this section the parameter identification is achieved with SPIM at a standstill rotor by a RLS algorithm. The identification with a standstill rotor is appropriated in some cases such as hermetic refrigeration compressors ([28]). The RLS identification algorithm requires the plant model in a discrete time linear regression form. Assuming the actual

**<sup>K</sup>**(*k*) = **<sup>P</sup>**(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>)*φ*(*k*)

*<sup>I</sup>* <sup>−</sup> **<sup>K</sup>**(*k*)*φT*(*k*)

From the equations (11) and (12) it is possible to reformulate the estimation parameter problem based on a linear regression model. Here, the parameter estimation method is divided into

This step consists into obtaining a linear-time-invariant model of the SPIM. The identification of *b*1, *b*0, *a*<sup>1</sup> and *a*<sup>0</sup> is done by performing a standstill test. The coefficients presented in (11) and (12) are functions of the machine parameters. For simplicity, the transfer functions given

*sq*(*s*) <sup>=</sup> *sb*1*<sup>q</sup>* <sup>+</sup> *<sup>b</sup>*0*<sup>q</sup>*

*s*<sup>2</sup> + *sa*1*<sup>q</sup>* + *a*0*<sup>q</sup>*

*mq*, *<sup>σ</sup><sup>d</sup>* <sup>=</sup> *LsdLrd* <sup>−</sup> *<sup>L</sup>*<sup>2</sup>

*Hq*(*s*) = *isq*(*s*)

*Hd*(*s*) = *isd*(*s*)

*RsqLrq* + *RrqLsq*

sampling index *k*, the regression model is given by

where *dim* **<sup>Y</sup>** <sup>=</sup> *<sup>M</sup>*¯ <sup>×</sup> *<sup>N</sup>*¯ , *dim <sup>φ</sup>T*(*k*) = *<sup>M</sup>*¯ <sup>×</sup> *<sup>r</sup>*¯ *dim <sup>θ</sup>*(*k*) = *<sup>r</sup>*¯ <sup>×</sup> *<sup>N</sup>*¯ , *dim e*(*k*) = *<sup>M</sup>*¯ <sup>×</sup> *<sup>N</sup>*¯

*dim* **<sup>K</sup>**(*k*) = *<sup>r</sup>*¯ <sup>×</sup> *<sup>M</sup>*¯ , *dim I* <sup>=</sup> *dim* **<sup>P</sup>**(*k*) = *<sup>r</sup>*¯ <sup>×</sup> *<sup>r</sup>*¯

*First step: estimation of (18) and (19) vide equations (20) and (21) :*

The recursive algorithm is achieved with the equations (14)-(17).

**P**(*k*) =

in (11) and (12) are rewritten in two transfer functions given by (18) and (19).

*i s sq*(*s*) *Vs*

*Hq*(*s*) =

(7)-(10), where these functions are decoupled and presented in (11) and (12).

**3. Parameter identification of single-phase induction machine**

where *<sup>σ</sup><sup>q</sup>* <sup>=</sup> *LsqLrq* <sup>−</sup> *<sup>L</sup>*<sup>2</sup>

where *pq* =

two steps:

$$H\_d(\mathbf{s}) = \frac{i\_{sd}^s(\mathbf{s})}{V\_{sd}^s(\mathbf{s})} = \frac{sb\_{1d} + b\_{0d}}{s^2 + sa\_{1d} + a\_{0d}} \tag{19}$$

where

$$b\_{1q} = \frac{L\_{rq}}{\overline{\sigma\_q}}, b\_{0q} = \frac{L\_{rq}}{\overline{\sigma\_q}\tau\_{rq}}, \quad a\_{1q} = \frac{R\_{sq}L\_{rq} + R\_{rq}L\_{sq}}{\overline{\sigma\_q}}, a\_{0q} = \frac{R\_{sq}R\_{rq}}{\overline{\sigma\_q}}\tag{20}$$

and

$$b\_{1d} = \frac{L\_{rd}}{\overline{\sigma}\_d}, b\_{0d} = \frac{L\_{rd}}{\overline{\sigma}\_d \tau\_{rd}}, \quad a\_{1d} = \frac{R\_{sd}L\_{rd} + R\_{rd}L\_{sd}}{\overline{\sigma}\_d}, \\ a\_{0d} = \frac{R\_{sd}R\_{rd}}{\overline{\sigma}\_d} \tag{21}$$

In order to obtain the regression linear model the transfer functions of (18) and (19) can be generalized and rewritten as,

$$\frac{d^2 \mathbf{i}\_{sq}}{dt^2} + a\_{1q}\frac{d\mathbf{i}\_{sq}}{dt} + a\_{0q}\mathbf{i}\_{sq} = b\_{1q}\frac{dV\_{sq}}{dt} + b\_{0q}V\_{sq} \tag{22}$$

and

$$\frac{d^2\mathbf{i}\_{sd}}{dt^2} + a\_{1d}\frac{d\mathbf{i}\_{sd}}{dt} + a\_{0d}\mathbf{i}\_{sd} = b\_{1d}\frac{dV\_{sd}}{dt} + b\_{0d}V\_{sd} \tag{23}$$

Solving for the second derivative of the stator current,

$$\frac{d^2 i\_{sq}}{dt^2} = \left[ -\frac{di\_{sq}}{dt} - i\_{sq}\frac{dV\_{sq}}{dt} \; V\_{sq} \right] \begin{bmatrix} a\_{1q} \\ a\_{0q} \\ b\_{1q} \\ b\_{0q} \end{bmatrix} \tag{24}$$

and

$$\frac{d^2 i\_{sd}}{dt^2} = \left[ -\frac{di\_{sd}}{dt} \begin{array}{c} -i\_{sd} \ \frac{dV\_{sd}}{dt} \ V\_{sd} \end{array} \right] \begin{bmatrix} a\_{1d} \\ a\_{0d} \\ b\_{1d} \\ b\_{0d} \end{bmatrix} \tag{25}$$

The estimation of coefficients *b*1*q*, *b*0*q*, *a*1*q*, *a*0*q*, *b*1*d*, *b*0*d*, *a*1*<sup>d</sup>* and *a*0*<sup>d</sup>* is done by using RLS estimation algorithm described in the equations (13)-(17). The linear regression model form (13) is given for the *q* axis by the following equations,

$$\mathbf{Y}\_q(k) = \frac{d^2 \mathbf{i}\_{sq}}{dt^2} \tag{26}$$

$$\boldsymbol{\phi}\_q^T(\boldsymbol{k}) = \left[ -\frac{d\boldsymbol{i}\_{sq}}{dt} \ -\boldsymbol{i}\_{sq}\ \frac{d\boldsymbol{V}\_{sq}}{dt}\ \boldsymbol{V}\_{sq} \right] \tag{27}$$

$$\boldsymbol{\theta}\_q^T(k) = \begin{bmatrix} \boldsymbol{a}\_{1q} \ \boldsymbol{a}\_{0q} \ \boldsymbol{b}\_{1q} \ \boldsymbol{b}\_{0q} \end{bmatrix} \tag{28}$$

And, for the *d* axis,

$$\mathbf{Y}\_d(k) = \frac{d^2 \mathbf{i}\_{sd}}{dt^2} \tag{29}$$

$$\boldsymbol{\phi}\_d^T(\boldsymbol{k}) = \left[ -\frac{d\boldsymbol{i}\_{sd}}{dt} \ -\boldsymbol{i}\_{sd} \ \frac{d\boldsymbol{V}\_{sd}}{dt} \ \boldsymbol{V}\_{sd} \right] \tag{30}$$

$$\theta\_d^T(k) = \begin{bmatrix} a\_{1d} \ a\_{0d} \ b\_{1d} \ b\_{0d} \end{bmatrix} \tag{31}$$

where, let us assume that the derivatives presented in (26)-(27) and (29)-(30) are measurable quantities. In the implementation, these quantities are obtained by State Variable Filters (SVF) such as in [6]. Four SVF filters were developed by discretization of the continuous-time transfer function given by,

$$\frac{V\_{sqf}}{V\_{sq}} = \frac{V\_{sdf}}{V\_{sd}} = \frac{\mathbf{i}\_{sqf}}{\mathbf{i}\_{sq}} = \frac{\mathbf{i}\_{sdf}}{\mathbf{i}\_{sd}} = \mathbf{G}\_{svf}(\mathbf{s}) = \frac{\omega\_{svf}^3}{(\mathbf{s} + \omega\_{svf})^3} \tag{32}$$

by RLS Algorithm 7

The electrical parameters of the SPIM are obtained combining the identified coefficients of *θq*(*k*) and *θd*(*k*) in (28) and (31) with the coefficients of the transfer functions (20) and (21), respectively, after the convergence of the RLS algorithm by the equations (34) and (35). In the numerical solution, the stator and rotor inductances are considered to have the same values

<sup>−</sup> *<sup>R</sup>*ˆ*sq*

<sup>−</sup> *<sup>R</sup>*ˆ*sd*

The RLS parameter identification algorithm presented in this chapter is implemented in a DSP based platform using TMS320F2812 DSP and a three-leg voltage source inverter. Figure 3 shows the diagram of the system to obtain the experimental results. The machine used for the validation of this methodology is a SPIM of a commercial hermetic refrigeration compressor of an air conditioning. The SPIM was removed from the hermetic compressor to achieve classical tests. The machine is two-pole, 220 V type. In the implementation of RLS identification parameter algorithm, the DC bus was limited in 177V/5A, and the sampling time of 400*μs*

As described in Sections 2 and 3, the experiment for the parameter identification is achieved with a standstill rotor. In the first step, a square wave with variable frequency and reduced voltage supplies the main winding, while the auxiliary winding is opened. The stator current in the main winding is measured using hall effect sensor. The voltage used in this algorithm is estimated by the product of modulation and DC bus indexes. After the identification for the main winding, the same procedure is repeated for the auxiliary winding. The square wave is

The frequency of the supply voltage is 5 Hz in the estimation of resistances to minimize the

<sup>1</sup>*qR*ˆ*rq* <sup>−</sup> *<sup>b</sup>*0*<sup>q</sup>*

<sup>1</sup>*dR*ˆ*rd* <sup>−</sup> *<sup>b</sup>*0*<sup>d</sup>*

*b*0*<sup>d</sup>*

*b*1*<sup>d</sup> b*0*<sup>d</sup>* *b*0*<sup>q</sup>*

*b*1*<sup>q</sup> b*0*<sup>q</sup>* Electrical Parameter Identi cation of Single-Phase Induction Motor by RLS Algorithm 281

(34)

(35)

*<sup>R</sup>*ˆ*sq* <sup>=</sup> *<sup>a</sup>*0*<sup>q</sup> b*0*<sup>q</sup>*

*<sup>R</sup>*ˆ*rq* <sup>=</sup> *<sup>a</sup>*1*<sup>q</sup> b*1*<sup>q</sup>*

> *<sup>R</sup>*ˆ*rq b*2

*L*ˆ *sq* = *L*ˆ *rq* = *R*ˆ*rq*

*<sup>R</sup>*ˆ*sd* <sup>=</sup> *<sup>a</sup>*0*<sup>d</sup> b*0*<sup>d</sup>*

*<sup>R</sup>*ˆ*rd* <sup>=</sup> *<sup>a</sup>*1*<sup>d</sup> b*1*<sup>d</sup>*

> *<sup>R</sup>*ˆ*rd b*2

*L*ˆ *sd* = *L*ˆ *rd* = *R*ˆ*rd*

*L*ˆ *md* =

*L*ˆ *mq* =

in each winding.

**4. Results and discussions**

used for better excitation of the plant.

skin effect, and it is 30 Hz for the estimation of inductances.

was used.

where, *ωsvf* is the filter bandwidth defined at around 5 to 10 times the input frequency signal. Here, *ωsvf* = *ω*, and *ω* is signal frequency, and the signals *Vsq*, *Vsd*, *isq*, and *isd* are used to obtain the filtered signals *Vsqf* , *Vsdf* , *isqf* and *isdf* .

The discretized transfer function, using the Euler method and sampling time of *Ts*, can be performed in state-space as

$$\mathbf{x}\_{\text{svf}(k+1)} = \left(1 + \mathbf{A}\_{\text{svf}} T\_{\text{s}}\right) \mathbf{x}\_{\text{svf}(k)} + T\_{\text{s}} \mathbf{B}\_{\text{svf}} \boldsymbol{\mu}\_{\text{svf}(k)} \tag{33}$$

where, **A***svf* = ⎡ ⎣ 0 10 0 01 <sup>−</sup>*ω*<sup>3</sup> *svf* <sup>−</sup>3*ω*<sup>2</sup> *svf* 3*ωsvf* ⎤ ⎦, **B***svf* = ⎡ ⎣ 0 0 *ω*3 *svf* ⎤ ⎦, *xsvf* = ⎡ ⎣ *x*1 *x*2 *x*3 ⎤ ⎦ The variable *usvf*

represents the input signal, while the state variables *x*1, *x*<sup>2</sup> and *x*<sup>3</sup> represent the input filtered signal, first derivative signal and second derivative signal, respectively.

With the SVF it is possible to avoid the use o low-pass filters in signals of the currents and the voltages, due the fact of the SVF has behavior of a low-pass filter. In addition, due this characteristic the SVF atenues the impact of pulsed and noise signals on the measurements of stator currents. For instance, to demonstrate the response of SVF Figure 2 presents the Module and the Phase Bode diagram with bandwidth of *ωsvf* = 5 ∗ 2*π* ∗ 30*Hz* [*rad*/*s*]. It is possible to observer that the high frequency signals are attenuated.

**Figure 2.** Bode diagram of the State Variable Filter.

*Second step: identification of machine parameters, Rsq, Rsd, Rrq, Rrd, Lmq, Lmd, Lsq, Lsd, Lrq and, Lrd*:

The electrical parameters of the SPIM are obtained combining the identified coefficients of *θq*(*k*) and *θd*(*k*) in (28) and (31) with the coefficients of the transfer functions (20) and (21), respectively, after the convergence of the RLS algorithm by the equations (34) and (35). In the numerical solution, the stator and rotor inductances are considered to have the same values in each winding.

$$\begin{aligned} \dot{R}\_{sq} &= \frac{a\_{0q}}{b\_{0q}} \\ \dot{R}\_{rq} &= \frac{a\_{1q}}{b\_{1q}} - \dot{R}\_{sq} \\ \dot{L}\_{mq} &= \frac{\sqrt{\dot{R}\_{rq} \left(b\_{1q}^2 \dot{R}\_{rq} - b\_{0q}\right)}}{b\_{0q}} \\ \dot{L}\_{sq} &= \dot{L}\_{rq} = \dot{R}\_{rq} \frac{b\_{1q}}{b\_{0q}} \end{aligned} \tag{34}$$
 
$$\begin{aligned} \dot{R}\_{sd} &= \frac{a\_{0d}}{b\_{0d}} \\ \dot{R}\_{rd} &= \frac{a\_{1d}}{b\_{1d}} - \dot{R}\_{sd} \\ \dot{L}\_{rd} &= \frac{\sqrt{\dot{R}\_{rd} \left(b\_{1d}^2 \dot{R}\_{rd} - b\_{0d}\right)}}{b\_{0d}} \\ \dot{L}\_{sd} &= \dot{L}\_{rd} = \dot{R}\_{rd} \frac{b\_{1d}}{b\_{0d}} \end{aligned} \tag{35}$$

### **4. Results and discussions**

6 Will-be-set-by-IN-TECH

where, let us assume that the derivatives presented in (26)-(27) and (29)-(30) are measurable quantities. In the implementation, these quantities are obtained by State Variable Filters (SVF) such as in [6]. Four SVF filters were developed by discretization of the continuous-time

where, *ωsvf* is the filter bandwidth defined at around 5 to 10 times the input frequency signal. Here, *ωsvf* = *ω*, and *ω* is signal frequency, and the signals *Vsq*, *Vsd*, *isq*, and *isd* are used to

The discretized transfer function, using the Euler method and sampling time of *Ts*, can be

�

⎡ ⎣

0 0 *ω*3 *svf* ⎤

⎦, *xsvf* =

⎦, **B***svf* =

represents the input signal, while the state variables *x*1, *x*<sup>2</sup> and *x*<sup>3</sup> represent the input filtered

With the SVF it is possible to avoid the use o low-pass filters in signals of the currents and the voltages, due the fact of the SVF has behavior of a low-pass filter. In addition, due this characteristic the SVF atenues the impact of pulsed and noise signals on the measurements of stator currents. For instance, to demonstrate the response of SVF Figure 2 presents the Module and the Phase Bode diagram with bandwidth of *ωsvf* = 5 ∗ 2*π* ∗ 30*Hz* [*rad*/*s*]. It is possible

101 102 103 104 105

*Second step: identification of machine parameters, Rsq, Rsd, Rrq, Rrd, Lmq, Lmd, Lsq, Lsd, Lrq and,*

=*Gsvf*(*s*) =

*ω*3 *svf*

(*<sup>s</sup>* <sup>+</sup> *<sup>ω</sup>svf*)<sup>3</sup> (32)

**x***svf*(*k*) + *Ts***B***svf usvf*(*k*) (33)

⎤

⎦ The variable *usvf*

⎡ ⎣ *x*1 *x*2 *x*3

transfer function given by,

performed in state-space as

⎡ ⎣

<sup>−</sup>*ω*<sup>3</sup>

−80 −60 −40 −20 0

−180 −135 −90 −45 0

**Figure 2.** Bode diagram of the State Variable Filter.

Magnitude

Phase (deg)

*Lrd*:

 (dB)

where, **A***svf* =

*Vsqf Vsq*

obtain the filtered signals *Vsqf* , *Vsdf* , *isqf* and *isdf* .

**x***svf*(*k*+1) =

0 10 0 01

to observer that the high frequency signals are attenuated.

*svf* <sup>−</sup>3*ω*<sup>2</sup>

<sup>=</sup> *Vsdf Vsd*

<sup>=</sup> *isqf isq*

�

*svf* 3*ωsvf*

signal, first derivative signal and second derivative signal, respectively.

1 + **A***svf Ts*

⎤

<sup>=</sup> *isdf isd*

> The RLS parameter identification algorithm presented in this chapter is implemented in a DSP based platform using TMS320F2812 DSP and a three-leg voltage source inverter. Figure 3 shows the diagram of the system to obtain the experimental results. The machine used for the validation of this methodology is a SPIM of a commercial hermetic refrigeration compressor of an air conditioning. The SPIM was removed from the hermetic compressor to achieve classical tests. The machine is two-pole, 220 V type. In the implementation of RLS identification parameter algorithm, the DC bus was limited in 177V/5A, and the sampling time of 400*μs* was used.

> As described in Sections 2 and 3, the experiment for the parameter identification is achieved with a standstill rotor. In the first step, a square wave with variable frequency and reduced voltage supplies the main winding, while the auxiliary winding is opened. The stator current in the main winding is measured using hall effect sensor. The voltage used in this algorithm is estimated by the product of modulation and DC bus indexes. After the identification for the main winding, the same procedure is repeated for the auxiliary winding. The square wave is used for better excitation of the plant.

> The frequency of the supply voltage is 5 Hz in the estimation of resistances to minimize the skin effect, and it is 30 Hz for the estimation of inductances.

by RLS Algorithm 9

Electrical Parameter Identi cation of Single-Phase Induction Motor by RLS Algorithm 283

0 2 4 6 8 10

(a)

0 2 4 6 8 10

time (s)

5 Hz 30 Hz

(b)

*a*1*<sup>d</sup>* 246.15 779.65 *a*0*<sup>d</sup>* 559.4 3304.5 *b*1*<sup>d</sup>* 9.83 17.62 *b*0*<sup>d</sup>* 43.67 2448.17

The electrical parameters of the induction machine are obtained combining the final value of coefficients in Table 1 and Table 2 with the equations (34) and (35), respectively. Table 3 presents the identified electrical parameters of SPIM. In this study we also make a comparison with the results obtained by classical methods, thus, the SPIM identified by RLS algorithm is

time (s)

a1<sup>q</sup> a0<sup>q</sup>

b1<sup>q</sup>

b0<sup>q</sup>

−100

−50

**Figure 4.** Coefficients convergence.

**Table 2.** Final convergence for coefficients of the auxiliary winding.

0

50

100

150

200

0

**Figure 3.** System diagram for the experimental results.

The first test for the identification of stator and rotor resistances is achieved in the main winding. The convergence of coefficients *θq* for this winding is presented in Fig. 4. Fig. 4 (a) presents the coefficients *a*1*<sup>q</sup>* and *a*0*q*, while Fig. 4 (b) gives the coefficients *b*1*<sup>q</sup>* and *b*0*q*. In this experiment the frequency of the supply voltage is selected on 5 Hz. As presented in figures, the coefficient convergence is fast and it is excited by the reset of the covariance matrix (**P**). Here, the reset of the covariance matrix (**P**) is used to avoid that this matrix reach zero and consequently loses the ability of to update the parameter matrix (*θq*(*k*)). Some oscillations are introduced every time that the covariance matrix (**P**) is reset, but the convergence of coefficients *θq*(*k*) is stable around a region.

The aim of the algorithm (13)-(17) is to identify a set of parameters that produce a good dynamic response such as real dynamic response of the induction machine. If it is necessary to ensure the parameter convergence to the true values the Lemma 12.5.2 (Persistent Excitation) of [16] must be satisfied. However, in practical implementations the designer normally unknown the exactly parameter values due the assumptions and approximations made to develop the mathematical model of the plant, and due the presence of unmodulated dynamics in the system, such as the measurement and drive systems.

The coefficients used in the electrical parameter calculation are the final values of Fig 4. The coefficient convergence for *d* axis has behavior similar to coefficient convergence for *q* axis in Fig. 4.

Table 1 shows the final obtained coefficients *a*1*q*, *a*0*q*, *b*1*q*, and *b*0*<sup>q</sup>* in a *q* axis when the voltage frequency is 5 Hz and 30 Hz. Table 2 shows the final obtained coefficients *a*1*d*, *a*0*d*, *b*1*d*, and *b*0*<sup>d</sup>* in a *d* axis when the voltage frequency is 5 Hz and 30 Hz. From Fig. 4 it can be observed that the coefficient convergence is fast.


**Table 1.** Final convergence for coefficients of the main winding.

**Figure 4.** Coefficients convergence.

8 Will-be-set-by-IN-TECH

The first test for the identification of stator and rotor resistances is achieved in the main winding. The convergence of coefficients *θq* for this winding is presented in Fig. 4. Fig. 4 (a) presents the coefficients *a*1*<sup>q</sup>* and *a*0*q*, while Fig. 4 (b) gives the coefficients *b*1*<sup>q</sup>* and *b*0*q*. In this experiment the frequency of the supply voltage is selected on 5 Hz. As presented in figures, the coefficient convergence is fast and it is excited by the reset of the covariance matrix (**P**). Here, the reset of the covariance matrix (**P**) is used to avoid that this matrix reach zero and consequently loses the ability of to update the parameter matrix (*θq*(*k*)). Some oscillations are introduced every time that the covariance matrix (**P**) is reset, but the convergence of

The aim of the algorithm (13)-(17) is to identify a set of parameters that produce a good dynamic response such as real dynamic response of the induction machine. If it is necessary to ensure the parameter convergence to the true values the Lemma 12.5.2 (Persistent Excitation) of [16] must be satisfied. However, in practical implementations the designer normally unknown the exactly parameter values due the assumptions and approximations made to develop the mathematical model of the plant, and due the presence of unmodulated dynamics

The coefficients used in the electrical parameter calculation are the final values of Fig 4. The coefficient convergence for *d* axis has behavior similar to coefficient convergence for *q* axis in

Table 1 shows the final obtained coefficients *a*1*q*, *a*0*q*, *b*1*q*, and *b*0*<sup>q</sup>* in a *q* axis when the voltage frequency is 5 Hz and 30 Hz. Table 2 shows the final obtained coefficients *a*1*d*, *a*0*d*, *b*1*d*, and *b*0*<sup>d</sup>* in a *d* axis when the voltage frequency is 5 Hz and 30 Hz. From Fig. 4 it can be observed that

> 5 Hz 30 Hz *a*1*<sup>q</sup>* 244.27 707.64 *a*0*<sup>q</sup>* 447.96 8084.14 *b*1*<sup>q</sup>* 24.82 27.73 *b*0*<sup>q</sup>* 128.58 2913.7

**Figure 3.** System diagram for the experimental results.

coefficients *θq*(*k*) is stable around a region.

the coefficient convergence is fast.

Fig. 4.

in the system, such as the measurement and drive systems.

**Table 1.** Final convergence for coefficients of the main winding.


**Table 2.** Final convergence for coefficients of the auxiliary winding.

The electrical parameters of the induction machine are obtained combining the final value of coefficients in Table 1 and Table 2 with the equations (34) and (35), respectively. Table 3 presents the identified electrical parameters of SPIM. In this study we also make a comparison with the results obtained by classical methods, thus, the SPIM identified by RLS algorithm is also tested using no-load and standstill classical methods. The electrical parameters estimated when the SPIM is tested by classical methods are shown in Table 4.

by RLS Algorithm 11

0 0.2 0.4 0.6 0.8 1 1.2

experimental

experimental

classical

RLS

Electrical Parameter Identi cation of Single-Phase Induction Motor by RLS Algorithm 285

time (s)

0 0.2 0.4 0.6 0.8 1 1.2

time (s)

(b)

**Figure 5.** Comparison among measured and simulated currents for the first test. (a) Simulated *isq* using parameters estimated with RLS algorithm and measured *isq* current. (b) Simulated *isq* current using

measured *isd* current. Fig. 7 (b) gives the simulated current using parameters estimated by

Fig. 8 shows the comparison between *isd* currents in steady-state condition, Fig. 8 (a) gives the comparison between the measured *isd* current and the simulated *isd* current using SPIM parameters estimated by the RLS algorithm, whereas, Fig. 8 (b) presents the comparison

(a)

−6

−6

parameters estimated by classical tests and measured *isq* current.

classical tests and the measured current.

−4

−2

isq (A)

0

2

4

6

−4

−2

isq (A)

0

2

4

6


**Table 3.** Experimental identified electrical parameters by RLS algorithm.


**Table 4.** Electrical parameters estimated by classical tests.

Aiming the model validation, two experiments are carried out. In the first experiment, the SPIM is driven by a *v/f* strategy at no-load operation, thus, the main and the auxiliary windings are supplied by controlled voltages varying the frequency from zero until a steady-state condition. The rotor speed and the stator currents are also measured. These measurements are recorded for posterior comparison with simulated values.

The model of single-phase induction machine presented in equations (7)-(10) is simulated using the estimated parameters of the machine by RLS algorithm given in Table 3. The model is also simulated using the parameters estimated by the classical methods presented in Table 4. The simulated stator currents are compared with the measured stator currents. The induction motor model equations (7)-(10) are discretized using the Euler Method in the same frequency of the experimental commutation for this test at 5 kHz. The recorded rotor speed is used in the simulated model to make it independent of the mechanical parameters.

In the first experimental result, the frequency of stator voltages varies from zero until 25 Hz and it is fixed in 25 Hz, by the *v/f* method. The recorded voltage values are used to supply the SPIM model in the simulation. Thus, from the same input voltages, Fig. 5 shows currents simulated and measured for the main winding in the first experiment. Fig. 5 (a) presents the comparison between the simulation of *isq* current with parameters estimated by RLS algorithm, and the experimental measurement of *isq* current, and Fig 5 (b) presents the comparison of the *isq* current when the simulations is carried out with parameters estimated by classical tests.

Fig. 6 presents a detail of the comparison between the measured and the simulated currents in steady-state condition for the main winding. In Fig. 6 (a) the comparison between experimental current and simulated current is presented when the SPIM parameters are estimated by RLS algorithm, while Fig. 6 (b) presents the comparison between experimental current and simulated current when the SPIM parameters are estimated by classical tests.

Fig. 7 presents a comparison among *isd* currents on the first experiment. Fig. 7 (a) shows the simulated current using parameters estimated by RLS algorithm in the SPIM model and the

10 Will-be-set-by-IN-TECH

also tested using no-load and standstill classical methods. The electrical parameters estimated

Identified 3.62Ω 6.27Ω 0.2186H 0.2376H

Identified 12.79Ω 12.23Ω 0.2849H 0.3142H

Estimated 3.95Ω 5.1506Ω 0.2149H 0.2292H

Estimated 11.95Ω 8.6463Ω 0.382H 0.401H

Aiming the model validation, two experiments are carried out. In the first experiment, the SPIM is driven by a *v/f* strategy at no-load operation, thus, the main and the auxiliary windings are supplied by controlled voltages varying the frequency from zero until a steady-state condition. The rotor speed and the stator currents are also measured. These

The model of single-phase induction machine presented in equations (7)-(10) is simulated using the estimated parameters of the machine by RLS algorithm given in Table 3. The model is also simulated using the parameters estimated by the classical methods presented in Table 4. The simulated stator currents are compared with the measured stator currents. The induction motor model equations (7)-(10) are discretized using the Euler Method in the same frequency of the experimental commutation for this test at 5 kHz. The recorded rotor speed is used in

In the first experimental result, the frequency of stator voltages varies from zero until 25 Hz and it is fixed in 25 Hz, by the *v/f* method. The recorded voltage values are used to supply the SPIM model in the simulation. Thus, from the same input voltages, Fig. 5 shows currents simulated and measured for the main winding in the first experiment. Fig. 5 (a) presents the comparison between the simulation of *isq* current with parameters estimated by RLS algorithm, and the experimental measurement of *isq* current, and Fig 5 (b) presents the comparison of the *isq* current when the simulations is carried out with parameters estimated

Fig. 6 presents a detail of the comparison between the measured and the simulated currents in steady-state condition for the main winding. In Fig. 6 (a) the comparison between experimental current and simulated current is presented when the SPIM parameters are estimated by RLS algorithm, while Fig. 6 (b) presents the comparison between experimental current and simulated current when the SPIM parameters are estimated by classical tests.

Fig. 7 presents a comparison among *isd* currents on the first experiment. Fig. 7 (a) shows the simulated current using parameters estimated by RLS algorithm in the SPIM model and the

measurements are recorded for posterior comparison with simulated values.

the simulated model to make it independent of the mechanical parameters.

*Rsq Rrq Lmq Lsq* = *Lrq*

*Rsd Rrd Lmd Lsd* = *Lrd*

*Rsq Rrq Lmq Lsq* = *Lrq*

*Rsd Rrd Lmd Lsd* = *Lrd*

when the SPIM is tested by classical methods are shown in Table 4.

**Table 3.** Experimental identified electrical parameters by RLS algorithm.

**Table 4.** Electrical parameters estimated by classical tests.

by classical tests.

**Figure 5.** Comparison among measured and simulated currents for the first test. (a) Simulated *isq* using parameters estimated with RLS algorithm and measured *isq* current. (b) Simulated *isq* current using parameters estimated by classical tests and measured *isq* current.

measured *isd* current. Fig. 7 (b) gives the simulated current using parameters estimated by classical tests and the measured current.

Fig. 8 shows the comparison between *isd* currents in steady-state condition, Fig. 8 (a) gives the comparison between the measured *isd* current and the simulated *isd* current using SPIM parameters estimated by the RLS algorithm, whereas, Fig. 8 (b) presents the comparison

by RLS Algorithm 13

0 0.2 0.4 0.6 0.8 1 1.2

experimental

experimental

classical

RLS

Electrical Parameter Identi cation of Single-Phase Induction Motor by RLS Algorithm 287

time (s)

0 0.2 0.4 0.6 0.8 1 1.2

time (s)

(b)

Fig. 9 presents a comparison between measured and simulated *isq* currents in steady-stead condition. In Fig. 9 (a) shows the measured *isq* current and the simulated *isq* current using parameters estimated by RLS algorithm on the SPIM model, while Fig. 9 (b) gives the measured *isq* and the simulated *isq* current using parameters estimated by classical tests.

Fig. 10 presents a comparison between *isd* currents in steady-state condition for the second

**Figure 7.** Comparison among *isd* currents in first test. (a) Simulated *isd* current by RLS parameter estimation and measured *isd* current. (b) Simulated *isd* current by classical tests and measured *isd* current.

(a)

−3

−3

test.

−2

−1

isd (A)

0

1

2

3

−2

−1

isd (A)

0

1

2

3

**Figure 6.** Comparison between measured and simulated currents at 25 Hz. (a) Simulated (RLS - Parameters) and measured *isq* currents. (b) Simulated (Classical - Parameters) and measured *isq* currents.

between the measured *isd* current and the simulated *isd* current using classical method for parameter estimation.

From Figures 5-8 it is possible to observer the good matching between the simulated and experimental currents for the *q* and *d* axes. In addition, some small discrepancies are found in these figures due the parameter inaccuracies and unmodulated effects (for instance the measurement and drive systems).

The advantage of the methodology presented in this chapter employing the RLS algorithm is that some types of applications, such as hermetic compressor, it is impossible carried out classical tests for estimation of the electrical parameters of the SPIM. In addition, the methodology has simplicity in implementation.

In the second experiment, the frequency of the supplied voltages of SPIM varies from zero until 30 Hz, and it is fixed at 30 Hz. As in the previous test, this is a no-load test and the SPIM is driven by a *v/f* strategy. The stator currents and rotor speed are measured and recorded.

12 Will-be-set-by-IN-TECH

1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12

(a)

1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12

(b)

between the measured *isd* current and the simulated *isd* current using classical method for

From Figures 5-8 it is possible to observer the good matching between the simulated and experimental currents for the *q* and *d* axes. In addition, some small discrepancies are found in these figures due the parameter inaccuracies and unmodulated effects (for instance the

The advantage of the methodology presented in this chapter employing the RLS algorithm is that some types of applications, such as hermetic compressor, it is impossible carried out classical tests for estimation of the electrical parameters of the SPIM. In addition, the

In the second experiment, the frequency of the supplied voltages of SPIM varies from zero until 30 Hz, and it is fixed at 30 Hz. As in the previous test, this is a no-load test and the SPIM is driven by a *v/f* strategy. The stator currents and rotor speed are measured and recorded.

**Figure 6.** Comparison between measured and simulated currents at 25 Hz. (a) Simulated (RLS - Parameters) and measured *isq* currents. (b) Simulated (Classical - Parameters) and measured *isq* currents.

time (s)

time (s)

experimental

experimental

simulated − classical

simulated − RLS

−3 −2 −1 0 1 2 3

−3 −2 −1 0 1 2 3

isq (A)

parameter estimation.

measurement and drive systems).

methodology has simplicity in implementation.

isq (A)

**Figure 7.** Comparison among *isd* currents in first test. (a) Simulated *isd* current by RLS parameter estimation and measured *isd* current. (b) Simulated *isd* current by classical tests and measured *isd* current.

Fig. 9 presents a comparison between measured and simulated *isq* currents in steady-stead condition. In Fig. 9 (a) shows the measured *isq* current and the simulated *isq* current using parameters estimated by RLS algorithm on the SPIM model, while Fig. 9 (b) gives the measured *isq* and the simulated *isq* current using parameters estimated by classical tests.

Fig. 10 presents a comparison between *isd* currents in steady-state condition for the second test.

by RLS Algorithm 15

1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12

(a)

1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12

(b)

**Figure 9.** Comparison between *isq* currents at 30 Hz in steady-state condition. (a) Simulated (RLS parameter) and measured *isq* currents. (b) Simulated (classical tests) and measured *isq* currents.

time (s)

time (s)

experimental

Electrical Parameter Identi cation of Single-Phase Induction Motor by RLS Algorithm 289

experimental

simulated − classical

simulated − RLS

−3 −2 −1 0 1 2 3

−3 −2 −1 0 1 2 3

isq (A)

isq (A)

**Figure 8.** Comparison between *isd* currents at 25 Hz. (a) Simulated (RLS - parameter) and measured *isd* currents. (b) Simulated (classical tests) and measured *isd* currents.

14 Will-be-set-by-IN-TECH

1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12

(a)

1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12

(b)

**Figure 8.** Comparison between *isd* currents at 25 Hz. (a) Simulated (RLS - parameter) and measured *isd*

time (s)

experimental

simulated − classical

time (s)

experimental

simulated − RLS

−1.5 −1 −0.5 0 0.5 1 1.5

−1.5 −1 −0.5 0 0.5 1 1.5

currents. (b) Simulated (classical tests) and measured *isd* currents.

isd (A)

isd (A)

**Figure 9.** Comparison between *isq* currents at 30 Hz in steady-state condition. (a) Simulated (RLS parameter) and measured *isq* currents. (b) Simulated (classical tests) and measured *isq* currents.

by RLS Algorithm 17

Electrical Parameter Identi cation of Single-Phase Induction Motor by RLS Algorithm 291

*Federal University of Pampa - UNIPAMPA, Federal University of Santa Maria - UFSM, Power*

*Federal University of Rio Grande - FURG, Federal University of Santa Maria - UFSM, Power*

*Federal University of Pampa - UNIPAMPA, Federal University of Santa Maria - UFSM, Power*

*Federal University of Santa Maria - UFSM, Power Electronics and Control Research Group - GEPOC,*

[1] Amin, A., El Korfally, M., Sayed, A. & Hegazy, O. [2009]. Efficiency optimization of two-asymmetrical-winding induction motor based on swarm intelligence, *IEEE*

[2] Azzolin, R. Z., Gastaldini, C. C., Vieira, R. P. & Gründling, H. A. [2011]. *A RMRAC Parameter Identification Algorithm Applied to Induction Machines, Electric Machines and*

[3] Azzolin, R. Z. & Gründling, H. A. [2009]. A MRAC parameter identification algorithm for three-phase induction motors, *IEEE International Electric Machines and Drives Conference,*

[4] Blaabjerg, F., Lungeanu, F., Skaug, K. & Tonnes, M. [2004]. Two-phase induction motor

[5] Bose, B. [2009]. Power electronics and motor drives recent progress and perspective,

[6] Câmara, H. T., Cardoso, R. C., Azzolin, R. Z., Pinheiro, H. & Gründling, H. A. [2006]. Low-cost sensorless induction motor speed control, *IEEE 32nd Annual Conference on*

[7] de Rossiter Corrêa, M., Jacobina, C., Lima, A. & da Silva, E. [2000]. Rotor-flux-oriented control of a single-phase induction motor drive, *IEEE Transactions on Industrial Electronics*

[8] Donlon, J., Achhammer, J., Iwamoto, H. & Iwasaki, M. [2002]. Power modules for

[9] Finch, J. & Giaouris, D. [2008]. Controlled AC electrical drives, *IEEE Transactions on*

[10] Holtz, J. [2005]. Sensorless control of induction machines: With or without signal

[11] Hrabovcova, V., Kalamen, L., Sekerak, P. & Rafajdus, P. [2010]. Determination of single phase induction motor parameters, *International Symposium on Power Electronics Electrical*

[12] Jemli, M., Ben Azza, H., Boussak, M. & Gossa, M. [2009]. Sensorless indirect stator field orientation speed control for single-phase induction motor drive, *IEEE Transactions on*

appliance motor control, *IEEE Industry Applications Magazine* 8(4): 26–34.

injection?, *IEEE Transactions on Industrial Electronics* 53(1): 7 – 30.

*Drives Automation and Motion (SPEEDAM)*, pp. 287 –292.

**Author details**

Rodrigo Padilha Vieira

Rodrigo Zelir Azzolin

Cristiane Cauduro Gastaldini

Hilton Abílio Gründling

**6. References**

*Brazil*

*Electronics and Control Research Group - GEPOC, Brazil*

*Electronics and Control Research Group - GEPOC, Brazil*

*Electronics and Control Research Group - GEPOC, Brazil*

*Transactions on Energy Conversion* 24(1): 12 –20.

drives, *IEEE Industry Applications Magazine* 10(4): 24–32.

*IEEE Transactions on Industrial Electronics* 56(2): 581 –588.

*Industrial Electronics, IECON 2006*, pp. 1200 –1205.

*Drives, Miroslav Chomat (Ed.)*, InTech.

*Industrial Electronics* 55(2): 481 –491.

*Power Electronics* 24(6): 1618–1627.

*IEMDC '09.*, pp. 273 –278.

47(4): 832–841.

**Figure 10.** Comparison between *isd* currents at 30 Hz. (a) Simulated (RLS - parameter) and measured *isd* currents. (b) Simulated (classical tests) and measured *isd* currents.

### **5. Conclusion**

A methodology for single-phase induction machine parameter identification was presented and discussed in this chapter. The machine tested was a SPIM used in a hermetic compressor of air conditioning. Using the proposed methodology it is possible to obtain all electrical parameters of SPIM for simulation and design of high performance vector control and sensorless SPIM drives. The main contribution of this study is the development of an automatized procedure for the identification of all electrical parameters of SPIM, such as the SPIM used in hermetic conditioning compressor. Experimental results demonstrate the effectiveness of the method. Some experimental comparisons among measurements and simulations using parameters estimated by classical tests and simulations using parameters obtained by RLS algorithm are presented. From Table 3 and Table 4 it is possible to observe that the parameters obtained with RLS algorithm converge to different values compared to classical tests. However, the results in Fig. 5 - 10 show that the parameters estimated with RLS algorithm present equivalent dynamical behavior compared with parameters estimated by classical methods. The methodology proposed in this chapter can be extended to be applied in other SPIM drives and three-phase induction motor drives.
