**Meshgin-Kelk Homayoun Ph.D.**

*Tafresh University*

The magnetic equivalent circuit (MEC) method introduces another approach for modeling electric machines. In fact, the approach can be considered as a reduced order finite element (FE) method. By taking into account approximately accurate machine geometry, stator and rotor slots effects, skewing, type of winding connections, stator and rotor leakages, and linear or nonlinear magnetic characteristics of machine coress it is a more accurate method with respect to the winding function approach (WFA). Therefore it can be helpful for design engineers and also it may be applied to find and to study more reliable algorithms for fault-detection strategies. Neglecting core property MEC is very similar to WFA.

Although WFA is based on calculating machine inductances, the magnetic equivalent circuit method can be used in two ways, indirect and one direct one. In the indirect way where linear magnetic core is considered, it may be applied to calculate machine inductances as the first step in analyzing the machine performances. Since magnetic properties of core parts can be incorporated in calculating machine inductances, it can provide a more accurate way to calculate these inductances. On the other hand, it may be applied directly without calculating machine inductances to analyze machine behavior in most conditions. MEC usually provides a deep understanding about effects of the machine geometry and design data on its parameters and performance.

Most conventional machines, such as induction and synchronous machines, are divided into three main parts: the stator, the rotor, and the air-gap. The stator consists of a stator core and stator windings, and the rotor consists of a rotor core and rotor windings. Stator or rotor cores are divided into yoke and teeth. Stator windings are located in the stator slots and rotor windings are located in the rotor slots. Figure 4.1 shows a portion of the main parts of a typical electric machine. Several magnetic flux lines are also shown. The flux lines that link both the stator and the rotor windings are the useful flux linkages, and the flux lines that link only the stator or the rotor windings are called leakage fluxes. In Figure 4.1, three lines of fluxes are shown.

**FIGURE 4.1**

A portion of the main parts of a typical electric machine.

An electric machine is assumed to be a quasi-stationary device; that is, any change of current that builds the flux is followed by an immediate change of flux. In other words, the time needed for an electromagnetic wave to pass through the machine is negligible compared to the period of the wave. Such a space may be partitioned into flux tubes. The flux tubes are the basis of the magnetic equivalent circuit method. A flux tube is a geometrical space in which all lines of flux are perpendicular to their bases and no lines of flux cut their sides. Lines of equal magnetic scalar potential, *u*, are perpendicular to lines of flux,φ. Therefore the bases of a flux tube are equipotential planes. Magnetic scalar potential, *u*, is a very useful quantity in magnetic equivalent circuit theory. But it has no physical meaning like the electrical scalar potential, *v*.

Lines of constant scalar magnetic potential lie perpendicular to the *H* vector. Scalar magnetic potential [6,7] is defined by

$$\overrightarrow{H}=-\nabla u\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.1\right)$$

The electrical scalar potential, *v*, in an electrostatic field is defined by

$$\overrightarrow{E}=-\nabla v\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.2\right)$$

A similarity exists between the definitions of these potentials. If the reluctance of a flux tube with magnetic scalar potentials *u*_{1} and *u*_{2} at their bases is *R*, and the flux through it is φ, then

$$F={u}_{2}-{u}_{1}=\varphi .R\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.3\right)$$

which states that *u*_{2} − *u*_{1} is the magnetomotive force (MMF) drop on the reluctance *R*.

To construct the magnetic equivalent circuit of an electric machine, all yoke parts, teeth, slots, air-gap tubes between the stator teeth and rotor teeth, and windings are modeled. If all teeth and yokes reluctances and all slot leakages are neglected and only air-gap permeances are considered, then the magnetic equivalent circuit approach is the same as the classic WFA. In this case, there exist MMF drops across the air-gap permeances due to currents flowing in the windings.

Currents flowing in the windings are the MMF sources in the magnetic equivalent circuit of an electric machine. These MMF sources are placed in the teeth. A reluctance/permeance and an unknown flux are assigned to each yoke part or each tooth. Slots are modeled by their leakage permeances. In the air-gap, a flux tube is defined when any of the stator teeth come face to face with any rotor teeth. A permeance is assigned to each air-gap flux tube. Geometries of these flux tubes vary due to the rotation of rotor teeth with respect to stator teeth and also depend on the air-gap length between any two facing stator and rotor teeth. Any air-gap asymmetry will have an effect on the height of these flux tubes.

To find the magnetic equivalent circuit of an electric machine we need first to compute the permeance between any two pairs of face-to-face stator and rotor teeth. Figure 4.2 shows three positions of *j*th rotor tooth with respect to the fixed *i*th stator tooth. The permeance between these two teeth is called *G*_{ij}. At positions *a* and *c, G*_{ij} is zero because there is no flux tube (no crossing flux) between these two teeth. If the widths of these teeth are equal, the *G*_{ij} reaches its maximum only at position *b*.

**FIGURE 4.2**

Three positions of jth rotor tooth with respect to the fixed ith stator tooth.

**FIGURE 4.3**

A flux tube with rectangular bases between jth rotor tooth and ith stator tooth for nonskewed stator and rotor slots.

For un-skewed stator and rotor slots, the bases of air-gap flux tubes are rectangles. In this case, the flux tubes have a general shape as shown in Figure 4.3 and the air-gap permeance *G*_{ij} can be calculated by

$${G}_{ij}={\text{\mu}}_{0}\frac{l.w\left(\text{\theta}\right)}{g}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.4\right)$$

where *l* is the length, *w*(θ) is the width, and *g* is the height of the flux tube. *w*(θ) varies with the angular position of the stator tooth *i* and rotor tooth *j*. The airgap length between the stator tooth *i* and the rotor tooth *j, g*, is constant for a symmetric air-gap. For a nonsymmetric air-gap, *g* is a function of θ.

If the widths of these teeth are not equal, the maximum of *G*_{ij} occurs during an interval. Figure 4.4 shows the air-gap permeance for two conditions. It is necessary to note that due to the fringing flux, the real shape of *G*_{ij} is smoother than what are shown in Figure 4.4.

If either the stator slots or the rotor slots, or both, are skewed, then the bases of the air-gap flux tubes will not be rectangles any more. Depending on the width of the stator and the rotor teeth and the amount of skewing and the angular position of rotor teeth to stator teeth, the bases of flux tubes have different shapes. Figure 4.5. shows the different shapes of bases for different positions of a typical skewed rotor tooth with respect to a typical rectangular stator tooth. Shaded areas are the bases of flux tube as the rotor tooth moves.

**FIGURE 4.4**

Air-gap permeance for two conditions: (a) equal width of stator and rotor teeth, (b) different width of stator and rotor teeth.

Skewing is a method to get better performance of an electric machine. By skewing either the stator or the rotor, the shape of air-gap permeance function and its space derivative are smoother. The derivative of air-gap permeance *G*_{ij} when multiplied by the square of the MMF drop over the element *G*_{ij} gives the value of electromagnetic force between the *i*th stator and the *j*th rotor teeth. Figure 4.6 shows the air-gap permeance and its space derivative between the *i*th stator and the *j*th rotor teeth for unskewed and skewed conditions. It is seen that by skewing, the magnitude of the air-gap permeance decreases. However, its space derivative is smoother [1].

**FIGURE 4.5**

Different shapes of bases for positions of a skewed rotor tooth with respect to a stator tooth. Shaded area is the basis of flux tube as the rotor tooth moves.

**FIGURE 4.6**

Air-gap permeance for (a) unskewed rotor and (b) its derivative; (c) skewed rotor and (d) its derivative.

The well-known equations of synchronous or induction machines that relates the flux linkages, stator and rotor currents, and self and mutual inductances are

$$\begin{array}{c}{L}_{ss}{i}_{s}+{L}_{sr}{i}_{r}={\text{\lambda}}_{s}\\ {L}_{rs}{i}_{s}+{L}_{rr}{i}_{r}={\text{\lambda}}_{r}\end{array}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.5\right)$$

Using the magnetic equivalent circuit of synchronous and induction machines [1], the machines inductances *L*_{ss}, *L*_{sr}*, L*_{rs}, and *L*_{rr} will be determined. Neglecting iron saturation and using the magnetic equivalent circuit of synchronous machines, the inductance coefficients of the machines are obtained. These inductances can be used for analysis and study of the machine behavior under healthy and faulty conditions.

A part of magnetic equivalent circuit of a typical salient-pole synchronous machine [2] is shown in Figure 4.7.

**FIGURE 4.7**

A part of the magnetic equivalent circuit of a salient pole synchronous machine, including all permeances and reluctances.

For the analysis the following assumptions are made:

Flux orientation from stator to rotor crosses the air-gap in radial direction.

Linear magnetic characteristic is considered.

For simplification no damper winding effect is considered.

Slot effects are considered.

Eddy current is neglected.

Each rotor pole core is divided into two parts. Damper windings are located in the outer part and field windings are wound around the inner part of each rotor pole. Here, damper windings are not considered. Therefore, only outer part of each rotor pole is considered having two segments represented by their reluctances. As shown *R*_{rsj} and *R*_{rsj+1} are the reluctances of these sections. *R*_{rt} is the rotor shank reluctance and *R*_{st} is the stator tooth reluctance. *R*_{sl} and *R*_{rl} are tooth leakage reluctances of stator and rotor, respectively. *G*_{sσ} is the stator openings leakage permeance and *G*_{sy} is the stator yoke permeance. *G*_{rσ} is the rotor openings leakage permeance and *G*_{ry} is the rotor yoke permeance. *G*_{sh} indicates the permeance of the rotor shaft, which is in parallel with the rotor yoke permeances.

There are five levels of MMF nodes defined as *u*_{1}, *u*_{2}, *u*_{3}, *u*_{4}, and *u*_{5} vectors, which are used in writing the nodal equations of the machine. The number of rotor poles is *p*, number of sections on each rotor pole is *m*, and number of stator slots is *n*.

According to the magnetic equivalent circuit of Figure 4.7, the nodal equations are written as

$${A}_{11}{u}_{1}+{A}_{sl}{u}_{2}=-{\Phi}_{st}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.6\right)$$

$${A}_{sl}{u}_{1}+{A}_{22}{u}_{2}+{A}_{23}{u}_{3}={\Phi}_{st}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.7\right)$$

$${A}_{32}{u}_{2}+{A}_{33}{u}_{3}+{A}_{34}{u}_{4}=0\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.8\right)$$

$${A}_{43}{u}_{3}+{A}_{44}{u}_{4}+{A}_{rl}{u}_{5}={\Phi}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.9\right)$$

$${A}_{rl}{u}_{4}+{A}_{55}{u}_{5}=-{\Phi}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.10\right)$$

The vectors Φ_{st} and Φ_{rt} contain the stator and rotor teeth fluxes. *A*_{11}, *A*_{22}*, A*_{33}, *A*_{44}, *A*_{55}, *A*_{23}, *A*_{32}, *A*_{34} and *A*_{43}, are the node permeance matrices, which are constructed according to the magnetic equivalent circuit theory concepts. The first two matrices *A*_{11} and *A*_{22} are *n* × *n*, *A*_{44} and *A*_{55} are *p* × *p* matrices, and *A*_{33} is an *m* × *m* matrix. *A*_{sl} and *A*_{rl} are *n* × *n* and *p* × *p* matrices, respectively, and contain the leakage permeances of stator teeth and rotor shank. *A*_{23} is an *n* × *p* matrix containing the air-gap permeances, *G*_{ij}. *A*_{32} is the transpose of *A*_{23}. *A*_{34} is an *m* × *n* matrix containing the rotor-sections permeances, and its transpose is *A*_{43}.

The two following equations relate the magnetic potentials of yoke parts and teeth in the stator and the rotor sides, respectively,

$${u}_{2}={u}_{1}-{R}_{st}{\Phi}_{st}+{F}_{st}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.11\right)$$

$${u}_{4}={u}_{5}-{R}_{rt}{\Phi}_{rt}+{F}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.12\right)$$

where *F*_{st} and *F*_{rt} are the MMF vectors of the stator and the rotor. *F*_{st} is related to stator current vector by

$${F}_{st}={W}_{s}{i}_{s}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.13\right)$$

and *F*_{rt} is related to the rotor current vector by

$${F}_{rt}={W}_{r}{i}_{r}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.14\right)$$

*W*_{s} is *n* × 3 and *W*_{r} is *p* × *p* matrices. *W*_{s} describes the stator winding configuration of stator windings of the machine. *W*_{s} is called the MMF transform matrix. It has an important role in the calculation of machine inductance coefficients. Each column of *W*_{s} corresponds to the winding function (winding distribution in slots) of one of the stator phases. *W*_{r} is a diagonal matrix describing the winding turns per rotor poles.

Using Equation (4.6) *u*_{2} is calculated as the following:

$${u}_{2}=-{A}_{sl}^{-1}{A}_{11}{u}_{1}-{A}_{sl}^{-1}{\Phi}_{st}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.15\right)$$

Combining Equations (4.11), (4.13), and (4.15) leads to

$$\begin{array}{l}-{A}_{sl}^{-1}{A}_{11}{u}_{1}-{A}_{sl}^{-1}{\Phi}_{st}-{u}_{1}+{R}_{st}{\Phi}_{st}={W}_{s}{i}_{s}\hfill \\ \left(-{A}_{sl}^{-1}{A}_{11}-{I}_{n\times n}\right){u}_{1}=-\left({A}_{sl}^{-1}+{R}_{st}\right){\Phi}_{st}+{W}_{s}{i}_{s}\hfill \end{array}$$

By defining *B*_{n×n} and *N*_{n×n} as

$$\begin{array}{l}{B}_{n\times n}=\left(-{A}_{sl}^{-1}{A}_{11}-{I}_{n\times n}\right)\hfill \\ {N}_{n\times n}=-{B}^{-1}\left(-{A}_{sl}^{-1}+{R}_{st}\right)\hfill \end{array}$$

*u*_{1} and *u*_{2} are related to the stator currents and stator teeth fluxes as

$$\begin{array}{c}{u}_{1}=-{B}^{-1}\left(-{A}_{\text{s}l}^{-1}+{R}_{st}\right){\Phi}_{st}+{B}^{-1}{W}_{s}{i}_{s}\\ \Rightarrow {u}_{1}=N{\Phi}_{st}+{B}^{-1}{W}_{s}{i}_{s}\end{array}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.16\right)$$

$${u}_{2}=\left[-{B}^{-1}\left(-{A}_{sl}^{-1}+{R}_{st}\right)-{R}_{st}\right]{\Phi}_{st}+\left({B}^{-1}{W}_{s}+{W}_{s}\right){i}_{s}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.17\right)$$

By defining *C* and *F* by

$$\begin{array}{l}{C}_{n\times n}=-{B}^{-1}\left(-{A}_{sl}^{-1}+{R}_{st}\right)-{R}_{st}\hfill \\ {F}_{n\times 3}=\left({B}^{-1}{W}_{s}+{W}_{s}\right)\hfill \end{array}$$

Equation (4.17) simplifies to

$${u}_{2}=C{\Phi}_{st}+F{i}_{s}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.18\right)$$

By the same procedure for rotor Equations (4.9) and (4.10), starting from Equation (4.10), and by defining the following matrices:

$$\begin{array}{l}{D}_{m\times m}=\left(-{A}_{rl}^{-1}{A}_{55}-{I}_{m\times m}\right)\hfill \\ {M}_{m\times m}=-{D}^{-1}\left(-{A}_{rl}^{-1}+{R}_{rt}\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{E}_{m\times m}=-{D}^{-1}\left(-{A}_{rl}^{-1}+{R}_{rt}\right)-{R}_{rt}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{G}_{m\times m}=\left({D}^{-1}{W}_{r}+{W}_{r}\right)\hfill \end{array}$$

*u*_{4} and *u*_{5} are related to the rotor currents and rotor teeth fluxes as the following:

$${u}_{4}=E{\Phi}_{rt}+G{i}_{r}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.19\right)$$

$${u}_{5}=M{\Phi}_{rt}+{D}^{-1}{W}_{r}{i}_{r}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.20\right)$$

By inserting *u*_{2} from Equation (4.18) and *u*_{4} from Equation (4.19) into Equation (4.8), *u*_{3} is obtained as the following:

$$\begin{array}{l}{u}_{3}=-{A}_{33}^{-1}({A}_{32}C{\Phi}_{st}+{A}_{32}F{i}_{s}+{A}_{34}E{\Phi}_{rt}+{A}_{34}G{i}_{r}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Rightarrow {u}_{3}=\left(-{A}_{33}^{-1}{A}_{32}C\right){\Phi}_{st}+\left(-{A}_{33}^{-1}{A}_{32}F\right){i}_{s}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(-{A}_{33}^{-1}{A}_{34}E\right){\Phi}_{rt}+\left(-{A}_{33}^{-1}{A}_{34}G\right){i}_{r}\hfill \end{array}$$

where by defining the following matrices

$$\begin{array}{l}{H}_{m\times m}=\left(-{A}_{33}^{-1}{A}_{32}C\right)\hfill \\ {J}_{m\times 3}=\left(-{A}_{33}^{-1}{A}_{32}F\right)\hfill \\ {K}_{m\times m}=\left(-{A}_{33}^{-1}{A}_{34}E\right)\hfill \\ {L}_{m\times m}=\left(-{A}_{33}^{-1}{A}_{34}G\right)\hfill \end{array}$$

*u*_{3} is related to the stator and rotor currents and stator teeth fluxes as the following:

$${u}_{3}=H{\Phi}_{st}+j{i}_{s}+K{\Phi}_{rt}+L{i}_{r}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.21\right)$$

Replacing Equations (4.16), (4.18), and (4.21) into Equation (4.7) gives

$$\begin{array}{l}{A}_{sl}N{\Phi}_{st}+{A}_{sl}{B}^{-1}{W}_{s}{i}_{s}+{A}_{22}C{\Phi}_{st}+{A}_{22}F{i}_{s}\hfill \\ +{A}_{23}H{\Phi}_{st}+{A}_{23}J{i}_{s}+{A}_{23}K{\Phi}_{rt}+{A}_{23}L{i}_{r}={\Phi}_{st}\hfill \end{array}$$

Rearranging the terms in the preceding relation leads to

$$\begin{array}{l}\left[{I}_{n\times n}-{A}_{sl}N-{A}_{22}C-{A}_{23}C-{A}_{23}H\right]{\Phi}_{st}-{A}_{23}K{\Phi}_{rt}\\ =\left[{A}_{sl}{B}^{-1}{W}_{s}+{A}_{22}F+{A}_{23}J\right]{i}_{s}+{A}_{23}L{i}_{r}\end{array}$$

where by defining the following matrices

$$\begin{array}{l}{P}_{n\times n}={I}_{n\times n}-{A}_{sl}N-{A}_{22}C-{A}_{23}H\\ {Q}_{n\times 3}={A}_{sl}{B}^{-1}{W}_{s}+{A}_{22}F+{A}_{23}J\end{array}$$

leads to

$$\begin{array}{l}P{\Phi}_{st}-{A}_{23}K{\Phi}_{rt}=Q{i}_{s}+{A}_{23}L{i}_{r}\\ \Rightarrow {\Phi}_{st}-{P}^{-1}{A}_{23}K{\Phi}_{rt}={P}^{-1}Q{i}_{s}+{P}^{-1}{A}_{23}L{i}_{r}\end{array}$$

Now by multiplying both sides of the preceding relation by ${W}_{S}^{T}$

$${W}_{s}^{T}{\Phi}_{st}-{W}_{s}^{T}{P}^{-1}{A}_{23}K{\Phi}_{rt}={W}_{s}^{T}{P}^{-1}Q{i}_{s}+{W}_{s}^{T}{P}^{-1}{A}_{23}L{i}_{r}$$

and defining the left-hand side of the result as stator flux linkage λ_{s} results in

$${\text{\lambda}}_{\text{s}}={W}_{s}^{T}{P}^{-1}Q{i}_{s}+{W}_{s}^{T}{P}^{-1}{A}_{23}L{i}_{r}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.22\right)$$

Now using the same procedure and replacing Equations (4.19), (4.20), and (4.21) into Equation (4.9) gives

$$\begin{array}{l}{A}_{43}H{\Phi}_{st}+{A}_{43}J{i}_{s}+{A}_{43}K{\Phi}_{rt}+{A}_{43}L{i}_{r}+{A}_{44}E{\Phi}_{rt}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{A}_{44}G{i}_{r}+{A}_{ri}M{\Phi}_{rt}+{A}_{rl}{D}^{-1}{W}_{r}{i}_{r}={\Phi}_{rt}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({A}_{43}H\right){\Phi}_{st}+\left[{A}_{43}K+{A}_{44}E+{A}_{rl}M-{I}_{m\times m}\right]{\Phi}_{rt}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(-{A}_{43}J\right){i}_{s}+\left[-{A}_{43}L-{A}_{44}G-{A}_{rl}G-{A}_{rl}{D}^{-1}{W}_{r}\right]{i}_{r}\hfill \end{array}$$

where by defining the following matrices

$$\begin{array}{l}{R}_{m\times m}={A}_{43}K+{A}_{44}E+{A}_{rl}M-{I}_{m\times m}\hfill \\ {S}_{m\times 3}=-{A}_{43}L-{A}_{44}G-{A}_{rl}{D}^{-1}{W}_{r}\hfill \end{array}$$

it simplifies to

$$\begin{array}{c}\left({A}_{43}H\right){\text{\Phi}}_{st}+R{\text{\Phi}}_{rt}=\left(-{A}_{43}J\right){i}_{s}+S{i}_{r}\\ \Rightarrow {R}^{-1}\left({A}_{43}H\right){\text{\Phi}}_{st}+{\text{\Phi}}_{rt}={R}^{-1}\left(-{A}_{43}J\right){i}_{s}+{R}^{-1}S{i}_{r}\end{array}$$

Now by multiplying both sides of the previous relation by ${W}_{r}^{T}$

$$\begin{array}{c}{W}_{r}^{T}{R}^{-1}\left({A}_{43}H\right){\text{\Phi}}_{st}+{W}_{r}^{T}{\text{\Phi}}_{rt}\\ ={W}_{r}^{T}{R}^{-1}\left(-{A}_{43}J\right){i}_{s}+{W}_{r}^{T}{R}^{-1}S{i}_{r}\end{array}$$

and defining the left-hand side of the result as rotor flux linkage λ_{r} results in

$${\text{\lambda}}_{\text{s}}={W}_{r}^{T}{R}^{-1}\left(-{A}_{43}J\right){i}_{s}+{W}_{s}^{T}{R}^{-1}S{i}_{r}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.23\right)$$

Now by comparing Equation (4.22) and Equation (4.23) with Equation (4.5) the following equations are obtained for the calculation of inductance coefficients of a salient pole synchronous machine

$${L}_{ss}={W}_{s}^{T}{P}^{-1}Q\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.24\right)$$

$${L}_{sr}={W}_{s}^{T}{P}^{-1}{A}_{23}L\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.25\right)$$

$${L}_{sr}={W}_{r}^{T}{R}^{-1}\left(-{A}_{43}J\right)\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.26\right)$$

$${L}_{rr}={W}_{r}^{T}{R}^{-1}S\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.27\right)$$

It should be noted that due to the inclusion of all stator and rotor reluctances it is possible to study the effect of the magnetic properties on all inductances by this model. Here only linear magnetic curve is studied.

Using Equations (4.24) to (4.27) all inductance coefficients are calculated for a 9 kVA, three-phase, four-pole salient pole synchronous machine as an example. Effects of conventional air-gap asymmetries such as static, dynamic, and mixed eccentricities on machine inductances are studied. These inductances are known as static eccentricity (SE), dynamic eccentricity (DE), and mixed eccentricity (ME), respectively. It is assumed that the reference points for both SE and DE are the same. This reference point is at zero angle. The middle point of the first stator tooth and first rotor pole are set at zero angles.

**FIGURE 4.8**

Air-gap curves for (a) symmetric, (b) 30% static eccentric, and (c) 30% dynamic eccentric machine from the viewpoint of a fixed point on the rotor (left) and a fixed point on the stator (right).

The left panel of Figure 4.8 shows air-gap variations seen from the reference point on the rotor side for eccentric and non-eccentric conditions. The right panel of Figure 4.8 shows these variations seen from the reference point on the stator side.

Figure 4.9 depicts the effective air-gap width for the simulated machine. Stator slot openings, rotor saliency effects, and the shape of rotor poles are considered. All these effects are included in the simulation. Maximum width corresponds with the rotor openings. Exact modeling of the air-gap is the most important task for calculating machine inductances. Figure 4.10 shows the air-gap variation for mixed air-gap eccentricity. There are 30% static and 30% dynamic eccentricities. The effect of the rotor pole shape can be seen in this figure [2].

**FIGURE 4.9**

Effective air-gap width for the simulated machine for healthy condition.

**FIGURE 4.10**

Zoomed effective air-gap width for the simulated machine with mixed air-gap eccentricity.

In the following we calculate machine inductances for symmetric and non-symmetric air-gap conditions. When there is any type of individual static or dynamic eccentricities, the amount of each eccentricity is considered to be 30%. Therefore, in mixed air-gap eccentricity there exists 30% static and 30% dynamic eccentricities.

Figure 4.11 shows the stator self-inductances for a symmetric air-gap and for a mixed air-gap eccentricity. Dashed curve is related to healthy machine and solid curve is related to faulty machine. Whereas the maximum magnitude of stator self-inductance in this figure mostly depends on the air-gap width under the rotor pole arc, the minimum magnitude depends on the rotor depth between the rotor poles. It can be seen that under mixed air-gap eccentricity, maximum points have more variation with respect to the minimum points. The reason is that the percentage variation of air-gap width under rotor pole arc is much more than the percentage variation of rotor depth between rotor poles. In the case of individual static or dynamic eccentricities no significant variation is observed.

**FIGURE 4.11**

Stator self-inductances for symmetric and ME (SE: 30% and DE: 30%) conditions.

**FIGURE 4.12**

Self-inductance of one of the rotor windings for healthy (SE: 30%, DE: 30%) and ME (SE: 30% and DE: 30%) conditions.

Figure 4.12 shows the rotor self-inductances for a symmetric air-gap, for an individual static or dynamic air-gap eccentricity, and for a mixed air-gap eccentricity. The effects on individual static eccentricity and dynamic eccentricity are the same. Both just increase the rotor self-inductance. However, in mixed air-gap eccentricity the rotor self-inductance varies with the rotor position. It is concluded that this may be a suitable measure for the existence of mixed air-gap eccentricity in synchronous machine. The effect of the stator slot openings is apparent in the rotor self-inductance.

Figure 4.13 shows the stator mutual inductances for a symmetric air-gap and for a mixed air-gap eccentricity. A dashed curve is related to a healthy machine and a solid curve is related to a faulty machine. As expected, mutual inductances have negative values. In this figure, the most negative points of the stator mutual inductance are mostly dependent on the air-gap width under rotor pole arc and the least negative points are dependent on the rotor depth between the rotor poles.

**FIGURE 4.13**

Stator mutual inductances compared for healthy and ME (SE: 30% and DE: 30%) conditions.

**FIGURE 4.14**

Mutual inductances between one of the stator phases and rotor winding for the healthy machine and faulty machine with mixed air-gap eccentricity (SE: 30% and DE: 30%).

It can be seen that under mixed air-gap eccentricity the most negative points have more variation with respect to the least negative points. The reason is the same as for the stator self-inductance. Again the curves of stator mutual inductances for static and dynamic eccentricities are approximately the same and are very close to the healthy case.

Figure 4.14 represents the mutual inductance curves of one of the stator phases and the rotor winding for a symmetric air-gap and for a mixed airgap eccentricity. Dashed curve is related to healthy machine and solid curve is related to faulty machine. As expected, the mutual inductance has both positive and negative values. To show the effects of air-gap fault on this inductance a zoomed part of top plot is also shown. It is seen that the effect of the same value of mixed eccentricity on stator to rotor mutual inductance is less than the effect on previous inductances.

Skewing of stator or rotor slots is a technical manufacturing method for better machine performance. With skewed stator or rotor slots, machine inductances vary more smoothly. In other words, slot opening degrades the mutual inductances. Skew can recover it again [4]. In the derived equation for calculating machine inductances, skew effect can be included.

In Figure 4.15, the plots of stator to rotor mutual inductances for a skewed and for an unskewed stator slots are shown. Comparison of the plots in Figure 4.15 shows how a skewed slot affects the shape of these mutual inductances. Magnitude of mutual inductances in the skewed case is a little less with respect to unskewed inductances. However, the ripples due to slot openings are suppressed in this plot. To further study the effect of skew, the derivative of mutual inductance is calculated and plotted in Figure 4.16. Effect of stator slot openings in an unskewed machine is clear. For a skewed slot the derivative of the mutual inductance variation is smooth.

**FIGURE 4.15**

Comparison of mutual inductances between one of the stator phases and the rotor winding for skewed and unskewed slots.

The experimental investigation was carried out to verify the theoretical and simulation findings. The same synchronous machine that is used for simulation is used in experiments. Design data for this machine are given in Appendix A. This machine is coupled to a three-phase induction motor.

**FIGURE 4.16**

Derivative of mutual inductances for skewed and unskewed conditions.

To measure the mutual inductance between windings *p* and *q, L*_{pq}, a direct current voltage is applied to the second winding. The synchronous machine is run by an induction motor. The current in the second winding is measured and saved by a digital oscilloscope. The induced voltage in the first winding is also measured and saved. Faraday's law implies that

$$e=\frac{d\text{\lambda}}{dt}=\frac{d\left({L}_{pq}i\right)}{dt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.28\right)$$

Integrating both sides of the preceding equation leads to

$$\text{\lambda =}{\displaystyle \int e.dt={\displaystyle \int d\left({L}_{pq}i\right)={L}_{pq}i}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.29\right)$$

Then

$${L}_{pq}=\frac{{\displaystyle \int e.dt}}{i}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.30\right)$$

To measure the self-inductance of windings *k, L*_{kk}, a direct voltage is applied to this winding. The synchronous machine is run by an induction motor. The current in the winding and the applied voltage are measured and saved.

The terminal voltage equation for the supplied winding is given by

$$v={R}_{k}i+e={R}_{k}i+\frac{d\left({L}_{kk}i\right)}{dt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.31\right)$$

Manipulating the preceding equation and integrating both sides leads to

$$\int \left(v-{R}_{ki}i\right).dt={\displaystyle \int e.dt=}{\displaystyle \int d\left({L}_{kk}i\right)={L}_{kk}i}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.32\right)$$

Then

$${L}_{kk}=\frac{{\displaystyle \int \left(v-{R}_{k}i\right)dt}}{i}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.33\right)$$

It should be noted that in the calculation of self-inductance it is necessary to measure winding resistance.

According to Equations (4.28) to (4.33), four inductances of synchronous machine are measured and plotted in Figures 4.17 to 4.20. It is necessary to note that the given experimental plots have been filtered. To see the effect of filtering, the unfiltered plot of stator self-inductance is shown in the top of Figure 4.17.

In the first view, comparison of inductance plots extracted from the model and the derived equations with inductance plots obtained from the experiment show very good agreements. More investigation into experimental plots shows that in the stator self-inductance, the magnitude of consecutive peaks is different. This may be due to some kind of rotor misalignment or nonlinear behavior of the machine core. In the two-dimensional model used in this textbook these conditions are not included. However, the method can be developed for the analysis of three-dimensional conditions.

**FIGURE 4.17**

Stator self inductances resulted from the experiment (a) before filtering and(b) after filtering.

It is seen that there are some ripples in the rotor self inductance in Figure 4.18. It may be due to rotor misalignment or nonlinear behavior of the machine core.

**FIGURE 4.18**

Rotor self-inductance curve.

**FIGURE 4.19**

Mutual inductance between two stator phases.

By using a simplified magnetic equivalent circuit of an induction machine we obtain the inductance coefficients of the machines [3,4]. These inductances can be used for analysis and study of the machine behavior in healthy and under faulty conditions.

In Figure 4.21, a part of a magnetic equivalent circuit of an induction machine is shown [3,4]. *u*_{1}, *u*_{2}, *u*_{3}, and *u*_{4} are the vectors of magnetic node potential in stator back iron, stator teeth, rotor teeth, and rotor back iron, respectively.

**FIGURE 4.20**

Mutual inductance between phase a of stator and rotor winding.

The node potential equations for the network of Figure (4.21) are given as

$${A}_{11}{u}_{1}=-{\Phi}_{st}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.34\right)$$

$${A}_{22}{u}_{2}+{A}_{23}{u}_{3}={\Phi}_{st}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.35\right)$$

$${A}_{32}{u}_{2}+{A}_{33}{u}_{3}={\Phi}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.36\right)$$

$${A}_{44}{u}_{4}=-{\Phi}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.37\right)$$

$${u}_{2}={u}_{1}-{R}_{st}{\Phi}_{st}+{F}_{st}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.38\right)$$

$${u}_{3}={u}_{4}-{R}_{rt}{\Phi}_{rt}+{F}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.39\right)$$

where Φ_{st} and Φ_{rt} are the vectors of stator and rotor teeth fluxes. *R*_{st} and *R*_{rt} are stator and rotor teeth reluctance matrices. *F*_{st} and *F*_{rt} are the vectors of MMF sources in the stator side and rotor side, respectively. *A*_{11}, *A*_{22}, *A*_{23}, *A*_{32}, *A*_{33}, and *A*_{44} are the node permeance matrices and are given in Appendix B. The element of *A*_{11} and *A*_{44} matrices depends only on stator and rotor back iron segment permeances, respectively. In Figure 4.21, these permeances are shown by *G*_{sy,i} and *G*_{ry,j}. The elements of *A*_{22}, *A*_{23}, *A*_{32}, and *A*_{33} depend on stator slot, rotor slot, and air-gap permeances. *G*_{sσ} is the stator slot openings leakage permeance and is constant. *G*_{rσ,j} is the rotor slot leakage permeance of slot *j*, and for closed slot, due to saturation effect, it is a nonlinear permeance. *G*_{ij} is the air-gap permeance between stator tooth *i* and rotor tooth *j*. *G*_{ij} is the most important parameter in the magnetic equivalent circuit modeling. Derivative of the air-gap permeance *G*_{ij} with respect to the rotor angle when multiplied by the square of the MMF drop over the same permeance gives the value of electromagnetic force between them.

**FIGURE 4.21**

A part of magnetic equivalent circuit of an induction machine.

*F*_{st} and *F*_{rt} vectors are related to the stator phase currents and rotor mesh currents through the following equations

$${F}_{st}={W}_{s}{i}_{s}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.40\right)$$

$${F}_{rt}={W}_{r}{i}_{r}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.41\right)$$

where *W*_{s} is generated using the same approaches as for the synchronous machine. *W*_{r} is an identity matrix and its size depends on the number of independent rotor mesh currents *i*_{r}.

Equation (4.34) and Equation (4.37) are written for the back iron portions of the stator and rotor, respectively. Figure 4.22 shows a portion of a typical rotor lamination of an induction machine. The geometric shape of back iron parts in stator and rotor are such that they have a large cross-sectional area and nearly short length with respect to the teeth segments of stator and rotor. As a result, the MMF drops in these back iron parts are generally several times smaller than MMF drops on teeth segments. It should be noted that in the MEC model, it is assumed that the directions of fluxes in a tooth and in a back iron segment are perpendicular to each other. Although back iron segments reluctances have certainly some effects on the values of machine inductances, simulation results show actually that neglecting the MMF drops in back iron segments has a very small effect on machine inductance coefficients. On the other hand, it is possible to change the values of teeth reluctances by some percentage to compensate for the removal of back iron reluctances.

**FIGURE 4.22**

A portion of a typical rotor lamination of an induction machine.

Neglecting the back iron reluctances in stator leads to equality of *u*_{1} elements and neglecting the back iron reluctances in rotor leads to equality of *u*_{4} elements. On the other hand, due to the fact that

$$\sum _{i=1}^{{n}_{s}}{\Phi}_{s{t}_{i}}=0}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.42\right)$$

Therefore, *u*_{1} = 0 and

$${u}_{2}=-{R}_{st}{\Phi}_{st}+{W}_{s}{i}_{s}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.43\right)$$

and

$$\sum _{j=1}^{{n}_{r}}{\Phi}_{r{t}_{j}}=0}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.44\right)$$

Hence, *u*_{4} = 0 and

$${u}_{s}=-{R}_{rt}{\Phi}_{rt}+{i}_{r}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.45\right)$$

The results of such assumptions lead to removal of Equations (4.34) and (4.37) from the system of algebraic equations of MEC model of induction machine shown by Equations (4.34) to (4.41).

By substituting Equations (4.38), (4.39), (4.43), and (4.45) in Equations (4.35) and (4.36), and then rearranging parameters, we have

$${A}_{22}{W}_{s}{i}_{s}+{A}_{23}{i}_{r}=\left({I}_{ns\times ns}+{A}_{22}{R}_{st}\right){\text{\Phi}}_{st}+{A}_{23}{R}_{rt}{\text{\Phi}}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.46\right)$$

$${A}_{32}{W}_{s}{i}_{s}+{A}_{33}{i}_{r}=\left({I}_{nr\times nr}+{A}_{33}{R}_{rt}\right){\text{\Phi}}_{rt}+{A}_{32}{R}_{st}{\text{\Phi}}_{st}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.47\right)$$

By introducing matrices *C* and *D* as follows:

$$C={\left({I}_{ns\times s}+{A}_{22}{R}_{st}\right)}^{-1}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.48\right)$$

$$D={\left({I}_{ns\times s}+{A}_{33}{R}_{rt}\right)}^{-1}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.49\right)$$

and by further simplification the following equations will be obtained

$$C\cdot {A}_{22}{W}_{s}{i}_{s}+C\cdot {A}_{23}{i}_{r}={\text{\Phi}}_{st}+C\cdot {A}_{23}{R}_{rt}{\text{\Phi}}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.50\right)$$

$$D\cdot {A}_{32}{W}_{s}{i}_{s}+D\cdot {A}_{33}{i}_{r}={\text{\Phi}}_{rt}+D\cdot {A}_{32}{R}_{st}{\text{\Phi}}_{st}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.51\right)$$

Multiplying both sides of Equation (4.50) by ${W}_{S}^{T}$ and defining the right-hand side of the result as λ_{s} and also defining the right-hand side of Equation (4.51) as λ_{r}, yield

$${W}_{S}^{T}C{A}_{22}{W}_{s}{i}_{s}+{W}_{S}^{T}C{A}_{23}{i}_{r}={\text{\lambda}}_{s}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.52\right)$$

$$D\cdot {A}_{32}{W}_{s}{i}_{s}+D\cdot {A}_{33}{i}_{r}={\text{\lambda}}_{r}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.53\right)$$

Comparing these equations with Equation (4.5) results in

$${L}_{ss}={W}_{s}^{T}C{A}_{22}{W}_{s}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.54\right)$$

$${L}_{sr}={W}_{s}^{T}C{A}_{23}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.55\right)$$

$${L}_{rs}=D{A}_{32}{W}_{s}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.56\right)$$

$${L}_{rr}=D{A}_{33}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.57\right)$$

In the preceding equations, the effects of all space harmonics, rotor skew, leakage path reluctances, and slot openings are taken into account for the calculation of inductance coefficients. Since there is no restriction concerning symmetry of stator windings, rotor bars, and air-gap length, this calculation may be applied in the study of asymmetrical effects and fault conditions on machine inductances. These fault conditions are short turns in stator windings and air-gap asymmetry such as the static and dynamic eccentricities. While short turns are reflected in the calculation of *W*_{s}, the air-gap asymmetries change the air-gap permeances and are included in matrices *A*_{22}, *A*_{23}, *A*_{32}, and *A*_{33}. It should be noted that due to the inclusion of stator and rotor teeth reluctances, it is possible to study the effect of the magnetic property of different cores on machine inductances by this model. Therefore, this model may be applied in the design of induction machines more efficiently.

Based on the equations derived for an induction machine, the inductance coefficients of a 3 hp, three-phase induction machine with the parameter given in Appendix A are calculated under different conditions. Table 4.1 and Figure 4.23 show some of the results of these calculations. The plots of mutual inductance between phase *a* of stator and one of the rotor loops, *L*_{ar}, for a healthy machine and for a machine with mixed air-gap eccentricity are shown in Figure 4.23. There is 5% dynamic eccentricity and 20% static eccentricity in faulty condition. Stator and rotor slot openings are considered, and the rotor is skewed by one rotor pitch. It is seen that mixed air-gap eccentricity causes an asymmetrical mutual inductance between stator phase and rotor loop. The effects of slot openings are clear in both plots.

**TABLE 4.1**

Calculation of Inductance Coefficient

**FIGURE 4.23**

Plots of mutual inductance between phase a of stator and one of the rotor loops, L_{ar}, for a healthy machine (top) and for a machine with 5% dynamic eccentricity and 20% static eccentricity (bottom).

**FIGURE 4.24**

The plots of L_{ar} from FE calculation (top) and from the magnetic equivalent circuit (bottom) method.

The plots of *L*_{ar} from FE calculation and from the magnetic equivalent circuit method are shown in Figure 4.24. Comparison of these plots shows that calculation of inductance coefficients by the MEC is in agreement with the FE method.

Comparison of rows 1 to 3 with rows 6 to 8 in Table 4.1 shows that as the stator slot opening increases, the magnitudes of mutual inductances between stator phases and rotor loops with skew and without skew effect have more differences. So for larger slot openings, increasing the skew causes more reduction in the magnitude of these mutual inductances.

Equations (4.54) and (4.57) show that teeth reluctances affect mutual inductances between stator phases and rotor loop. Also notice that *L*_{ar} is not the same as *L*_{ra}. However, comparing rows 2 and 3 with rows 9 and 10 in Table 4.1 shows that this effect is small.

Table 4.1 depicts the average of *L*_{aa} and *L*_{ab}, including slot width, rotor skew, and teeth reluctances effects for both healthy and eccentric rotors under different conditions. Again, it is seen that for both types of eccentricities there are considerable changes in stator inductances with respect to healthy condition, and this knowledge may be applied for the detection of these faults. According to the inductance values shown in Table 4.1 and the plots of Figure 4.23, the following general results can be concluded:

By increasing the values of core reluctances, the values of inductance coefficients decrease. So, the effect of magnetic property of different cores can be studied in the design of induction motors.

Rotor skew affects mutual inductances between stator phases and rotor loops considerably.

By increasing the slot opening, the entire inductance coefficient values decrease.

Static eccentricity affects all inductance values.

Dynamic eccentricity affects all inductance values considerably.

Reluctances of stator and rotor tooth depend on the flux through them. Therefore, when nonlinear magnetic characteristics have to be considered in the analysis, it is not possible to calculate the inductance coefficients before the calculation of machine variables. In this case, the system of algebraic equations of the machine is nonlinear. Therefore an iterative procedure has to be employed to solve these equations [3,5]. In order to apply the solution procedure to the algebraic machine equation, some algebraic manipulations are required. These equations are repeated for convenience:

$${A}_{22}{u}_{2}+{A}_{23}{u}_{3}={\text{\Phi}}_{st}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.58\right)$$

$${A}_{32}{u}_{2}+{A}_{33}{u}_{3}={\text{\Phi}}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.59\right)$$

$${u}_{2}=-{R}_{st}{\text{\Phi}}_{st}+{W}_{s}{i}_{s}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.60\right)$$

$${u}_{3}=-{R}_{rt}{\text{\Phi}}_{rt}+{i}_{r}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.61\right)$$

As the first step, Equations (4.60) and (4.61) are substituted in Equations (4.58) and (4.59). The results are arranged as

$${A}_{22}{W}_{s}{i}_{s}+{A}_{23}{i}_{r}-\left({I}_{{n}_{s}}+{A}_{22}{R}_{st}\right){\text{\Phi}}_{st}={R}_{rt}{\text{\Phi}}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.62\right)$$

$${A}_{32}{W}_{s}{i}_{s}+{A}_{33}{i}_{r}-{A}_{32}{R}_{st}{\text{\Phi}}_{st}=\left({I}_{{n}_{r}}+{A}_{33}{R}_{rt}\right){\text{\Phi}}_{rt}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\left(4.63\right)$$

Both sides of Equation (4.62) are multiplied by ${W}_{S}^{T}$ and by defining ${W}_{S}^{T}.{\Phi}_{st}$ as the stator flux linkage λ_{s} the following equation will be obtained: