**4. Determining the equivalent global change parameters for the power rotor fed by the static frequency converter**

52 Induction Motors – Modelling and Control

the following expression can be written:

One can notice the following:

k

and including the fundamental:

where:

Making the relations (31) and (32) equal, it results:

As in the previous case, the stator winding reactance corresponding to the fundamental, X1(1) (determined at the fundamental's frequency f1(1)) and the stator winding reactances, corresponding to all higher time harmonics X1() (determined at frequencies f1()=f1 where Jmf±k) are replaced by an equivalent reactance, X1(CSF), determined at fundamental's frequency. This equivalent reactance, traversed by the current I1(CSF), conveys the same reactive power, QCu1(CSF) as in the case of considering "" reactances X1(), (each of them determined at f1() frequency and traversed by the current I1()). Following the equalization,

 22 2 2 2 2 1 CSF 1 1 1 11 11 1 1 1 1 1 1

1 CSF

X

X

the factor that highlights the changes that the reactants of the stator phase value suffer in the case of a machine supplied through a power frequency converter, compared to sinusoidal supply, both calculated at the fundamental's frequency. From relations (25) and (33) it follows:

\*

*X*

*sc*

1

 

1 1 U U 1 1 1 1

 

\* 2 \*

\* 2 2 \* 1 1 1r sc 1 1 1r sc 1 1

f x U U f x

 

 

> (1) *sc*

*X*

*Z*


<sup>X</sup> 1 1 U U 1 1 1 1

X f x U U f x

1 CSF 1 1 1r sc 1 1 1r sc 1 1 X1 2 2 2 2 <sup>1</sup> 1 1

X1

k

Cu1 CSF 1 CSF 1 CSF 1 CSF 1 1 1

11 1 X I I X I XI X I I (33)

2 2 <sup>2</sup>

Q 3X I 3X I I (32)

1

2 2 2 2 1 1

1 CSF Z R jX R jk X 1 CSF 1 CSF 1 CSF X1 1 (35)

 

  (34)

Further, it is considered a winding with multiple cages whose bars (in number of "c") are placed in the same notch of any form, electrically separated from each other (see Fig. 3). These bars are connected at the front by short-circuiting rings (one ring may correspond to several bars notch). This "generalized" approach, pure theoretically in fact, has the advantage that by its applying the relations of the two equivalent factors kr(CSF) and kx(CSF), valid for any notch type and multiple cages, are obtained. The rotor notch shown in Fig. 3 is the height hc and it is divided into "n" layers (strips), each strip having a height hs = hc/n. The number of layers "n" is chosen so that the current density of each band should be considered constant throughout the height hs (and therefore not manifesting the skin effect in the strip). The notch bars are numbered from 1 to c, from the bottom of the notch. The lower layer of each bar is identified by the index "i" and the top layer by the index "s". Thus, for a bar with index characterized by a specific resistance and an absolute magnetic permeability, the lower layer is noted with Ni and the extremely high layer with Ns. The current that flows through the bar is noted with ic (Ic - rated value). The length of the bar, over which the skin effect occurs, is L. For the beginning, let us consider only the presence of the fundamental in the power supply, which corresponds to the supply pulsation, ω1(1)=ω1=2πf1. In this case:

$$\mathbf{k}\_{\text{v5(1)}} = \frac{\mathbf{R}\_{\text{s}(1)}}{\mathbf{R}\_{\text{s}}} = \frac{1}{\mathbf{I}\_{\text{c5(1)}}^2} \cdot \sum\_{\text{s}=\text{N}\_{\text{il}}}^{\text{N}\_{\text{ls}}} \frac{\mathbf{I}\_{\text{c}(1)}^2}{\mathbf{b}\_{\text{c}}} \cdot \sum\_{\text{s}=\text{N}\_{\text{il}}}^{\text{N}\_{\text{ls}}} \mathbf{b}\_{\text{c}} \,\prime \tag{36}$$

$$\mathbf{k}\_{\text{sol}(1)} = \frac{\mathbf{L}\_{\text{face}(1)} \mathbf{\bar{\phantom{x}}}}{\mathbf{L}\_{\text{face}\text{ }\text{---}}} = \frac{\left| \text{Re} \left[ \underbrace{\underline{\underline{\text{\scriptsize{\tiny}}}} \underline{\text{\phantom{\scriptsize{\text{\scriptsize{\tiny}}}}} \mathbf{\bar{\phantom{\scriptscriptstyle{\scriptsize{\text{\scriptsize{\tiny}}}}}}}} \right] \right| \left( \sum\_{\iota=\text{N}\_{\text{a}}}^{\text{N}\_{\text{la}}} \mathbf{b}\_{\iota} \right)^{2}}\_{\mathbf{\overline{\phantom{\scriptsize{\text{\scriptsize{\tiny}}}}}\text{cs}} = \frac{\left| \text{Re} \left[ \underbrace{\mathbf{L}\_{\text{sa}} \right] \right| \mathbf{\bar{\phantom{\scriptsize{\tiny}}}}^{2} \mathbf{b}\_{\iota}}{\mathbf{b}\_{\iota} \left| \left( \sum\_{\iota=\text{N}\_{\text{a}}}^{\text{N}\_{\text{a}}} \mathbf{b}\_{\iota} \right) \left( \sum\_{\iota=\text{N}\_{\text{a}}}^{\text{N}\_{\text{a}}} \mathbf{b}\_{\iota} \right) + \frac{\mathbf{b}\_{\iota}^{2}}{\mathbf{3}} \right|} \tag{37}$$

where b and bε are the width of and ε order strips and Ψδnσ(1) is the bar flux corresponding to the fundamental of the own magnetic field, assuming that for the order strip, the magnetic linkage corresponds to a constant repartition of the fundamental current density on the strip.

**Figure 3.** Notch generalized for multiple cages.

If in the motor power supply one considers only the order harmonic which corresponds to the supply pulsation 1()=1, the relations (36) and (37) remain valid with the following considerations: index "1" is replaced by index "" and the rotor phenomena are with the pulsation 2() given by the relation:

$$\mathbf{co}\_{2(\mathbf{v})} = \mathbf{s}\_{(\mathbf{v})} \cdot \mathbf{oo}\_{1(\mathbf{v})} = \left(\mathbf{1} \mp \frac{\mathbf{1}}{\mathbf{v}} \pm \frac{\mathbf{s}}{\mathbf{v}}\right) \cdot \mathbf{v} \cdot \mathbf{oo}\_1 \tag{38}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 55

I II (45)

p RI I (46)

1

qq q , (49)

1~ x 1 1 n c 1 q kL I (50)

~ 1 n ~c 1 n <sup>x</sup> <sup>c</sup> q L I kL I (51)

<sup>q</sup> , (48)

1 c 1

I

  2

(47)

I

 

 2 2 c CSF c 1 c

1

 2 2 CSF c1 c

1

is the rated value of the current which runs through the bar, in the case of a motor

By replacing the relations (43) and (46) in (39) one obtains the expression for the global equivalent factor of the a.c. increasing resistance in the bar , kr (CSF), in case of the presence

<sup>c</sup> 2 2 r1 r c1 r1 c r <sup>1</sup> CSF ~ c 1 <sup>1</sup>

 

 CSF ~

 CSF ~ 1 ~ <sup>~</sup>

1

CSF

where q(CSF)~ is the a.c. total reactive power, in the bar, and q(CSF)- is the total reactive power for a uniform current distribution in the bar. Applying the superposition in the case

A.c. reactive power corresponding to the fundamental is calculated using the following

<sup>2</sup>

In the same way, the expression of the a.c. reactive power in the bar corresponding to the

2 2

By replacing the relations (50) and (51) in the relation (28), the expression for calculating the

<sup>p</sup> RI I <sup>I</sup>

r CSF 2 CSF 2 2 <sup>c</sup> c1 c 1

The global equivalent change of a.c. bar inductance modification has the expression:

<sup>q</sup> <sup>k</sup>

x CSF

of a.c. total reactive power, the following relationship is obtained:

 

k k RI k I k <sup>p</sup> <sup>I</sup>

supplied by a frequency converter. By replacing the relation (45) in relation (44):

of all harmonics in the motor power:

 

k

relation:

order harmonic is obtained:

total a.c. reactive power in the bar is obtained:

Subsequently we shall consider the real case, where in the bar both the fundamental and order time harmonics are present. For this, the equivalent d.c. global factor of the bar resistance modification is calculated with the relation:

$$\mathbf{k}\_{\text{v\\$\text{(CSF)}}} = \frac{\mathbf{P}\_{\text{\\$\text{(CSF)}}} - \mathbf{R}\_{\text{\\$\text{(CSF)}}}}{\mathbf{P}\_{\text{\\$\text{(CSF)}}} - \mathbf{R}\_{\text{\\$\text{(CSF)}}}} = \frac{\mathbf{R}\_{\text{\\$\text{(CSF)}}}}{\mathbf{R}\_{\text{\\$\text{(CSF)}}}} \; \; \; \tag{39}$$

where p(CSF)~ represents the total a.c. losses in bar (considering the appropriate skin effect for all harmonics) and p(CSF)- represents the bar total losses, without considering the repression phenomenon. The a.c. total losses in the bar are obtained by applying the effects superposition principle by adding all the bar a.c. losses caused by each order time, including the fundamental. Therefore one can obtain:

$$\mathbf{p}\_{\mathbb{A}(\text{CSF})-} = \mathbf{p}\_{\mathbb{A}(1)} + \sum\_{\mathbb{v} \neq 1} \mathbf{p}\_{\mathbb{A}(\text{v})- \text{ } \text{'} }\tag{40}$$

The a.c. loss in bar, corresponding to the fundamental, p(1)~, is calculated with the following relation:

$$\mathbf{P}\_{\delta(1)-} = \mathbf{I}\_{c\delta(1)}^2 \cdot \mathbf{k}\_{c\delta(1)} \cdot \mathbf{R}\_{\delta-} \tag{41}$$

In the same way, the expression of the bar a.c. losses produced by some order time harmonic is obtained:

$$\mathbf{p}\_{\delta(\mathbf{v}) - \mathbf{ }} = \mathbf{I}\_{\mathrm{c\beta(v)}}^2 \cdot \mathbf{R}\_{\delta(\mathbf{v}) - \mathbf{ }} = \mathbf{I}\_{\mathrm{c\beta(v)}}^2 \cdot \mathbf{k}\_{\mathrm{r\beta(v)}} \cdot \mathbf{R}\_{\delta - \mathbf{ }} \tag{42}$$

By replacing the relations (41) and (42) in relation (40), it results:

$$\mathbf{P}\_{\mathsf{sl}(\text{CSF})} = \mathbf{I}\_{\mathsf{c}\mathbf{i}(1)}^{2} \cdot \mathbf{k}\_{\mathsf{z}\mathbf{i}(1)} \cdot \mathbf{R}\_{\mathfrak{d}-} + \sum\_{\mathsf{v}\star 1} \mathbf{I}\_{\mathsf{c}\mathbf{i}(\mathbf{v})}^{2} \cdot \mathbf{k}\_{\mathsf{z}\mathbf{i}(\mathbf{v})} \cdot \mathbf{R}\_{\mathfrak{d}-} = \mathbf{R}\_{\mathfrak{d}-} \left( \mathbf{I}\_{\mathsf{c}\mathbf{i}(1)}^{2} \cdot \mathbf{k}\_{\mathsf{z}\mathbf{i}(1)} + \sum\_{\mathsf{v}\star 1} \mathbf{I}\_{\mathsf{c}\mathbf{i}(\mathbf{v})}^{2} \cdot \mathbf{k}\_{\mathsf{z}\mathbf{i}(\mathbf{v})} \right). \tag{43}$$

The bar losses without considering the repression phenomenon in the bar are calculated using the following relationship:

$$\mathbf{P}\_{\text{\ $\text{(CSF)} - \text{ }}} = \mathbf{I}\_{\text{ct}\text{(CSF)}}^2 \cdot \mathbf{R}\_{\text{\$  -- \text{ }} \text{ }} \tag{44}$$

where:

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 55

$$\mathbf{I}\_{\text{c\\$\{CSF\}}} = \sqrt{\mathbf{I}\_{\text{c\\$\{1\}}}^2 + \sum\_{\mathbf{v} \neq \mathbf{1}} \mathbf{I}\_{\text{c\\$\{v\}}}^2} \tag{45}$$

is the rated value of the current which runs through the bar, in the case of a motor supplied by a frequency converter. By replacing the relation (45) in relation (44):

54 Induction Motors – Modelling and Control

pulsation 2() given by the relation:

If in the motor power supply one considers only the order harmonic which corresponds to the supply pulsation 1()=1, the relations (36) and (37) remain valid with the following considerations: index "1" is replaced by index "" and the rotor phenomena are with the

2 1 <sup>1</sup>

Subsequently we shall consider the real case, where in the bar both the fundamental and order time harmonics are present. For this, the equivalent d.c. global factor of the bar

> 

where p(CSF)~ represents the total a.c. losses in bar (considering the appropriate skin effect for all harmonics) and p(CSF)- represents the bar total losses, without considering the repression phenomenon. The a.c. total losses in the bar are obtained by applying the effects superposition principle by adding all the bar a.c. losses caused by each order time,

> CSF ~ 1 <sup>~</sup>

The a.c. loss in bar, corresponding to the fundamental, p(1)~, is calculated with the

<sup>2</sup>

In the same way, the expression of the bar a.c. losses produced by some order time

2 2

The bar losses without considering the repression phenomenon in the bar are calculated

<sup>2</sup>

CSF ~ c 1 r 1 c r c1 r1 c r

 2 2 2 2

1 1 p I k R I k R R I k I k . (43)

1

p R <sup>k</sup>

 CSF ~ CSF ~

CSF CSF

1 s s 1 , (38)

p R , (39)

p p p , (40)

1~ c 1 r 1 p I k R (41)

CSF c CSF P I R , (44)

~c ~c r p I R I k R (42)

 

r CSF

resistance modification is calculated with the relation:

including the fundamental. Therefore one can obtain:

By replacing the relations (41) and (42) in relation (40), it results:

following relation:

harmonic is obtained:

using the following relationship:

where:

$$\mathbf{P}\_{\mathbb{A}(\text{CSF})-} = \mathbf{R}\_{\mathbb{A}-} \left( \mathbf{I}\_{\text{c\ $}(1)}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{I}\_{\text{c\$ }(\mathbf{v})}^2 \right) \tag{46}$$

By replacing the relations (43) and (46) in (39) one obtains the expression for the global equivalent factor of the a.c. increasing resistance in the bar , kr (CSF), in case of the presence of all harmonics in the motor power:

$$\mathbf{k}\_{\text{ci(CSF)}} = \frac{\mathbf{P}\_{\text{i(CSF)}} - \mathbf{P}\_{\text{o(CSF)}}}{\mathbf{P}\_{\text{i(CSF)}} - \mathbf{P}\_{\text{i(CSF)}}} = \frac{\mathbf{R}\_{\text{o}} \left( \mathbf{I}\_{\text{c\ $}\{\text{t\$ }\}}^2 \cdot \mathbf{k}\_{\text{ri\ $}\{\text{t\$ }\}} + \sum\_{\text{v}\in\text{1}} \mathbf{I}\_{\text{c\ $}\{\text{v\$ }\}}^2 \cdot \mathbf{k}\_{\text{vi\ $}\{\text{v\$ }\}} \right)}{\mathbf{R}\_{-\text{i}} \left( \mathbf{I}\_{\text{c\ $}\{\text{t\$ }\}}^2 + \sum\_{\text{v}\in\text{1}} \mathbf{I}\_{\text{c\ $}\{\text{v\$ }\}}^2 \right)} = \frac{\mathbf{k}\_{\text{i\ $}\{\text{t\$ }\}} + \sum\_{\text{v}\in\text{1}} \mathbf{k}\_{\text{vi\ $}\{\text{v\$ }\}} \left( \frac{\mathbf{I}\_{\text{c\ $}\{\text{v\$ }\}}}{\mathbf{I}\_{\text{c\ $}\{\text{t\$ }\}}} \right)^2}{\mathbf{1} + \sum\_{\text{v}\in\text{1}} \left( \frac{\mathbf{I}\_{\text{c\ $}\{\text{v\$ }\}}}{\mathbf{I}\_{\text{c\ $}\{\text{t\$ }\}}} \right)^2} \tag{47}$$

The global equivalent change of a.c. bar inductance modification has the expression:

$$\mathbf{k}\_{\text{\ $}k\text{\$ (CSF)}} = \frac{\mathbf{q}\_{k\text{\ $(CSF)}}}{\mathbf{q}\_{k\text{\$ (CSF)}}} \, \text{},\tag{48}$$

where q(CSF)~ is the a.c. total reactive power, in the bar, and q(CSF)- is the total reactive power for a uniform current distribution in the bar. Applying the superposition in the case of a.c. total reactive power, the following relationship is obtained:

$$\mathbf{q}\_{\mathbb{A}(\text{CSF}) - \ } = \mathbf{q}\_{\mathbb{A}(1) - \ } + \sum\_{\mathbf{v} \neq 1} \mathbf{q}\_{\mathbb{A}(\text{v}) - \ } \tag{49}$$

A.c. reactive power corresponding to the fundamental is calculated using the following relation:

$$\mathbf{q}\_{\text{i}\delta(1)-} = \boldsymbol{\alpha}\_{1} \cdot \mathbf{k}\_{\text{v}\delta(1)} \cdot \mathbf{L}\_{\text{foc}\text{e}\_{-}-} \cdot \mathbf{l}\_{\text{c\delta(1)}}^{2} \tag{50}$$

In the same way, the expression of the a.c. reactive power in the bar corresponding to the order harmonic is obtained:

$$\mathbf{q}\_{\text{l}\text{\textquotedblleft(v)\textquotedblright }-} = \boldsymbol{\alpha}\_{\text{l}\text{\textquotedblleft(v)\textquotedblright}} \mathbf{L}\_{\text{\textquotedblleft(v)\textquotedblright}} \cdot \mathbf{I}\_{\text{c\textquotedblleft(v)}}^2 = \mathbf{v} \cdot \boldsymbol{\alpha}\_1 \cdot \mathbf{k}\_{\text{x\textquotedblleft(v)}} \cdot \mathbf{L}\_{\text{\textquotedblleft(v)}} \cdot \mathbf{I}\_{\text{c\textquotedblleft(v)}}^2 \tag{51}$$

By replacing the relations (50) and (51) in the relation (28), the expression for calculating the total a.c. reactive power in the bar is obtained:

$$\begin{split} \mathbf{q}\_{\text{sl}(\text{CSF})-} &= \mathbf{0}\_{1} \cdot \mathbf{k}\_{\text{sd}(1)} \cdot \mathbf{L}\_{\text{ltor}-} \cdot \mathbf{I}\_{\text{c\%}(1)}^{2} + \left( \mathbf{k}\_{\text{sd}(1)} \cdot \mathbf{I}\_{\text{c\%}(1)}^{2} + \sum\_{\mathbf{v}\sim 1} \mathbf{v} \cdot \mathbf{k}\_{\text{s\%}(\mathbf{v})} \cdot \mathbf{I}\_{\text{c\%}(\mathbf{v})}^{2} \right) = \\ &= \mathbf{0}\_{1} \cdot \mathbf{L}\_{\text{ltor}-} \left( \mathbf{k}\_{\text{s\%}(1)} \cdot \mathbf{I}\_{\text{c\%}(1)}^{2} + \sum\_{\mathbf{v}\sim 1} \mathbf{v} \cdot \mathbf{k}\_{\text{s\%}(\mathbf{v})} \cdot \mathbf{I}\_{\text{c\%}(\mathbf{v})}^{2} \right) \end{split} \tag{52}$$

The total reactive power for an uniform current repartition in the bar, in the case of a motor supplied through a frequency converter, is calculated by the relation:

$$\mathbf{q}\_{\mathbb{A}(\text{CSF})-} = \mathbf{q}\_{\mathbb{A}(1)-} + \sum\_{\mathbf{v} \neq 1} \mathbf{q}\_{\mathbb{A}(\text{v})-} \quad \text{ } \tag{53}$$

The Behavior in Stationary Regime of an Induction Motor Powered by Static Frequency Converters 57

s (58)

' ' U IZ e 1 21 21 , (59)

I I (60)

U RI , =1, 2, …, c , (61)

(62)

(63)

reduced to the stator is used. For this, the rotor with multiple bars is replaced by a rotor with a single bar on the pole pitch. Initially only the fundamental present in the power supply

> '

Knowing that the induced EMF by the fundamental component of the main magnetic field

In the relation (60), the number of the cages and respectively the rotor bars/ pole pitch is equal to "c". In the case of motors with the power up to 45 [kW], c=1 (simple cage or high bars) or c=2 (double cage). Δ(1) is the determinant corresponding to the equation system:

 c c ' ' e 1 21 c 1 1 1 1 1 U

11 1 1n 1

R ... R . . . . R ... R

R ... R 1 R ... R . . . . R ... R 1 R ...R

111 1, 1 1 1, 11 1n 1

n1 1 n, 1 1 n, 1 1 nn 1

Because in the first phase the steady-state regime is under focus, the phenomenon in the rotor corresponding to the fundamental has the pulsation ω2(1)=sω1, where s is the motor slip for the sinusoidal power supply in the steady-state regime. If the relation (63) is introduced

n1 1 nn 1

Δδ(1) is the determinant corresponding to the fundamental obtained from Δ(1), where column

2 1 ' ' 2 1 2 1 1

R Z jX

of the motor is considered. The rotor impedance reduced to the stator has the equation:

 c e 1 1 c1 1

1

where, for the general case of multiple cages is valid the relation:

from the machine in the pole pitch bars is:

having the expression:

δ is replaced by a column of 1:

1

where q(1)- is the reactive power corresponding to the fundamental, in case of an uniform current distribution Ic(1) in the bar, while q()- is the reactive power corresponding to the harmonic in case of a uniform current distribution Ic() in the bar:

$$\mathbf{q}\_{\text{i}\delta(1)} = \alpha \mathbf{o}\_{\text{i}(1)} \mathbf{L}\_{\text{\ $one\$ }-} \cdot \mathbf{I}\_{\text{c\ $}(1)}^2 = \alpha \mathbf{o}\_1 \cdot \mathbf{L}\_{\text{\$ one\ $}-} \cdot \mathbf{I}\_{\text{c\$ }(1)}^2 \tag{54}$$

Similarly, for the reactive power corresponding to the harmonic, in the case of an uniform current Ic() repartition in the bar, the following relation is obtained:

$$\mathbf{q}\_{\text{i}\text{s}\text{(v)}\text{ --}} = \boldsymbol{\alpha}\_{\text{1}\text{(v)}} \cdot \mathbf{L}\_{\text{\textdegree aco}\text{--}} \cdot \mathbf{I}\_{\text{c\textdegree}\text{(v)}}^{2} = \mathbf{v} \cdot \boldsymbol{\alpha}\_{\text{1}} \cdot \mathbf{L}\_{\text{\textdegree aco}\text{--}} \cdot \mathbf{I}\_{\text{c\textdegree}\text{(v)}}^{2} \tag{55}$$

By replacing the relations (54) and (55) in relation (53), the expression for the total reactive power for a uniform current distribution in the bar becomes:

$$\mathbf{q}\_{\text{sl}(\text{CSF})-} = \boldsymbol{\alpha}\_{1} \cdot \mathbf{L}\_{\text{tore}-} \cdot \mathbf{I}\_{\text{c\%}(1)}^{2} + \sum\_{\text{v}\neq 1} \mathbf{v} \cdot \boldsymbol{\alpha}\_{1} \cdot \mathbf{L}\_{\text{tore}-} \cdot \mathbf{I}\_{\text{c\%}(\text{v})}^{2} = \boldsymbol{\alpha}\_{1} \cdot \mathbf{L}\_{\text{tore}-} \left(\mathbf{I}\_{\text{c\%}(1)}^{2} + \sum\_{\text{v}\neq 1} \mathbf{v} \cdot \mathbf{I}\_{\text{c\%}(\text{v})}^{2}\right) \tag{56}$$

By replacing the relations (52) and (56) in relation (48), the expression for the global equivalent factor of the a.c. modifying inductance is obtained:

$$\mathbf{k}\_{\text{sol}(\text{CSF})} = \frac{\mathbf{q}\_{\text{sl}(\text{CSF})} - \mathbf{c}}{\mathbf{q}\_{\text{sl}(\text{CSF})} - \mathbf{c}} = \frac{\alpha\_1 \mathbf{L}\_{\text{lon}} - \left(\mathbf{k}\_{\text{sol}(1)} \cdot \mathbf{I}\_{\text{sol}(1)}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{v} \cdot \mathbf{k}\_{\text{sol}(\mathbf{v})} \cdot \mathbf{I}\_{\text{sol}(\mathbf{v})}^2\right)}{\alpha\_1 \mathbf{L}\_{\text{lon}} - \left(\mathbf{I}\_{\text{sol}(1)}^2 + \sum\_{\mathbf{v} \neq 1} \mathbf{v} \cdot \mathbf{I}\_{\text{sol}(\mathbf{v})}^2\right)} = \frac{\mathbf{k}\_{\text{sol}(1)} + \sum\_{\mathbf{v} \neq 1} \left[\mathbf{v} \cdot \left(\frac{\mathbf{I}\_{\text{sol}(\mathbf{v})}}{\mathbf{I}\_{\text{sol}(\mathbf{v})}}\right)^2 \cdot \mathbf{k}\_{\text{sol}(\mathbf{v})}\right]}{\mathbf{1} + \sum\_{\mathbf{v} \neq 1} \left[\mathbf{v} \cdot \left(\frac{\mathbf{I}\_{\text{sol}(\mathbf{v})}}{\mathbf{I}\_{\text{sol}(\mathbf{v})}}\right)^2\right]} \tag{57}$$
