Procedure for Rewinding Induction Motors

DATA

• Manufacturer: Motores Elétricos Brasil S.A.;
• No.: B418563;
• S.M. TYPE: 112;
• Voltage: 220/380;
• HP: 0.5 HP ≈ 0.37285 kW;
• R.P.M.: 2850 – 50 Hz (2.6 / 1.5 A);
• R.P.M.: 3450 – 60 Hz (2.6 / 1.5 A);
• Phase: Three-phase;
• Connection: Δ / Y;
• Number of Slots: 24;
• Rotor Diameter: 8.85 cm;
• Slot:
  • Approximate usable area: 1 cm²;
  • Approximate usable height: 1.2 cm;
  • Approximate average usable width: 0.82 cm;
  • Slot length: 5.8 cm;

CALCULATION METHODOLOGY

The clearest approach is to separate the process into two stages:

• Define the electrical geometry of the winding (which coil goes in which slot and with what direction). For this purpose, a hierarchy of priorities can be adopted in the rewinding process:
  • Reproduce the original design, if possible.
  • Minimize construction risk.
  • Only then optimize electromagnetic performance.
• Dimension the copper (number of turns and wire gauge).

ELECTRICAL GEOMETRY OF THE WINDING

Spatial distribution of the coils

Choice: Distributed winding.

Considering that the motor under study is a three-phase induction motor for industrial application, with a squirrel-cage rotor, and that this is a rewinding of a machine originally designed for continuous operation, the use of a distributed winding is chosen. This type of winding allows the coils of each phase to be distributed along the entire pole pitch, resulting in a rotating magnetic field with a waveform close to sinusoidal, lower harmonic content, reduced torque ripple, and lower levels of vibration and noise. Concentrated windings, although simpler, are typically used in permanent magnet motors or special applications, and are not the usual solution for industrial three-phase induction motors, especially in machines of classical or older design.

Integral or fractional winding

Result: Integral winding (q = 4)

Initially, the number of poles of the motor is calculated based on the synchronous speed equation of a three-phase induction motor: $$ n_{s} = \frac{120\cdot f}{P} = \frac{120\cdot 60}{2} = 3600\ rpm $$

It is concluded that the motor has 2 poles.

Now, the value of q (number of slots per pole per phase) is determined. This value defines the nature of the winding. $$ q = \frac{Q}{P\cdot m} = \frac{24}{2\cdot 3} = 4 $$

Thus, the winding is classified as an integral winding, in which each phase occupies an integer number of slots per pole. This characteristic enables a symmetrical distribution of the coils along the stator, favoring the formation of a more uniform rotating magnetic field, with lower harmonic distortion and better electromagnetic performance. Fractional windings, on the other hand, present greater distribution complexity and are more common in special machines or permanent magnet motors, and are not required in this case.

Full-pitch or short-pitch winding

Choice: Short-pitch (1 slot)

From this, $ \tau_s $ (pole pitch) is determined: $$ \tau_{s} = \frac{Q}{P} = \frac{24}{2} = 12 $$

The pole pitch of the motor is given by 12 slots. Thus, a full-pitch winding would correspond to a coil pitch of $ y = 12 $. However, in industrial three-phase induction motors, especially in classical designs and rewinding applications, it is common to adopt a short-pitch winding, typically reduced by one slot relative to the pole pitch (for example, $ y = 11 $).

Shortening the coil pitch allows the reduction of spatial harmonics of the magnetic field, particularly lower-order harmonics (such as the 5th and 7th), in addition to reducing the coil overhang length, resulting in lower copper consumption and reduced losses. The reduction in the induced voltage of the fundamental component is small and does not compromise motor performance, being largely compensated by the improvement in magnetic field quality and mechanical operation of the machine.

Single-layer or double-layer

Choice: Double-layer

Regarding the number of winding layers, the double-layer configuration is adopted, in which each stator slot accommodates two coil sides belonging to different coils, one in each layer (upper and lower). This configuration is the most common in industrial three-phase induction motors, as it offers greater design and assembly flexibility, allowing the adoption of short-pitch windings and better spatial distribution of the MMF along the stator. In addition, the double-layer arrangement tends to produce a rotating magnetic field with lower harmonic content, improved waveform, and reduced torque ripple when compared to single-layer windings.

Phase sequence

Choice: A+ | B- | C+ | A- | B+ | C-

The phase sequence in the stator must be defined in such a way as to ensure the formation of a continuous three-phase rotating magnetic field, respecting the correspondence between the temporal phase displacement of the three-phase currents and the spatial distribution of the winding.

The $ \alpha_{e} $ (electrical angle per slot): $$ \alpha_{e} = \frac{180{^\circ} \cdot P}{Q} = \frac{180{^\circ} \cdot 2}{24} = 15{^\circ} $$

Each phase occupies 4 slots per pole, forming an angular span of: $$ \Delta\theta_{e} = q\cdot \alpha_{e} = 4\cdot 15 = 60{^\circ} $$

Thus, the stator can be divided into six consecutive blocks of 60 electrical degrees, according to the following scheme:

Totaling the 360 electrical degrees corresponding to one complete rotation of the magnetic field for a 2-pole motor.

Balanced three-phase currents can be described by: $$ i_{A}(t) = I\cdot cos(\omega t) $$ $$ i_{B}(t) = I\cdot cos(\omega t – 120) $$ $$ i_{C}(t) = I\cdot cos(\omega t – 240) $$

These equations indicate that phases A, B, and C are displaced by 120 electrical degrees in time. For this temporal phase displacement to be reflected in space, the stator phases must follow the sequence A → B → C, with a spatial displacement of 120°, which corresponds to two consecutive 60° zones.

In a three-phase induction motor, the magnetic poles of the stator result from the vector sum of the magnetomotive forces (MMFs) generated by the distributed coils of the three phases, and can be described by the expression: $$ F\left( \theta_{m},t \right) = F_{\max}cos(p\theta_{m} – \omega t) $$

This represents a sinusoidal MMF wave that travels through space. In this equation, the term $ p\theta_{m} $ indicates that the number of pole pairs defines the spatial periodicity of the magnetic field, while the term $ \omega t $ represents the temporal variation imposed by the electrical supply frequency. The sign inversion of the same phase occurs every 180 electrical degrees, corresponding to the transition between North and South poles, a condition required for the closure of the magnetic flux lines.

Since the three-phase currents are displaced by 120 electrical degrees in time, this displacement must be reflected in space through the succession of phase zones in the stator. In this case, where each zone occupies 60 electrical degrees, the spatial separation between phases is obtained with a displacement of two consecutive zones (120°), while the alternation of polarity ensures the correct formation of the poles. Thus, the spatial sequence A⁺ | B⁻ | C⁺ | A⁻ | B⁺ | C⁻ simultaneously ensures the three-phase displacement, the alternation of magnetic poles, and the generation of a continuous rotating magnetic field.

Assembly and interconnection of stator coils

The formation of the stator coils was carried out with the aid of the SWAT-EM software, which allows configuring the geometric and electrical parameters of the motor and automatically provides the winding layout, as well as the slot pairs and the connection direction of each coil. Since a double-layer winding was adopted, each stator slot contains two distinct coil sides, one in each layer (Layer 1 and Layer 2).

The figures and tables presented below illustrate the stator winding layout and the formation of the coils, considering a distributed, integral, double-layer winding with short pitch.

Layout and Distribution of Windings in the Slots:

Legend of elements

• Phases (1, 2 and 3): The numbers 1, 2, and 3 indicate the electrical phase and correspond, respectively, to phases R, S, and T;
  • Layers (Layer 1 and Layer 2): Each slot has two layers. Each layer accommodates a distinct coil side. Layer 2 is the upper layer, while Layer 1 is the lower layer of the slot.
• Direction – Sign (+ / −): Geometric direction of the coil, where +1 indicates the clockwise direction and −1 indicates the counter-clockwise direction.

Next, a table is presented with the step-by-step electrical connection, to facilitate the interconnection of the windings in the stator.

The correct connection of the coils of each phase is achieved by always connecting the electrical end of one coil to the electrical start of the next coil of the same phase, forming a continuous series circuit. This type of connection ensures that the MMFs produced by all coils of the phase add up properly, resulting in the correct formation of the rotating magnetic field.

According to the connection table generated by the “SWAT-EM” software, the electrical start of a coil is defined by the fields (from_slot, from_layer), while the electrical end of the coil is defined by the fields (to_slot, to_layer). Thus, the practical rule for series connection consists of connecting the point (to_slot, to_layer) of one coil to the point (from_slot, from_layer) of the next coil belonging to the same phase, until all coils of that phase are interconnected.

Physical orientation of the coils in the stator (critical point): For this connection rule to work, all coils must be installed in the stator with the same physical orientation, that is, without inversions or 180° rotations between them. The physical inversion of a coil results in the inversion of the direction of its MMF, causing it to subtract from the magnetic field of the others instead of adding to it, compromising motor operation (torque reduction, increased current, and overheating).

Electrical reference and current direction:

• The “from_slot” column defines one side of the coil as the positive reference for electromagnetic modeling purposes. In practice, this field identifies which side of the coil should be considered as the current entry point when establishing a reference direction. During rewinding, this information is used only to identify the electrical start of each coil, requiring no additional action beyond respecting the indicated connections.

• The Direction column indicates the geometric direction of the current path around the stator, considering the spatial position of the coil, that is, clockwise (1) and counter-clockwise (-1). This information serves a project validation purpose and does not need to be interpreted or manually applied during the physical connection of the coils.

External electrical connections of the stator (star and delta)

After the series connection of the coils of each phase, the stator presents six external electrical terminals, corresponding to the starts and ends of each phase:

• U1 and U2 – phase 1
• V1 and V2 – phase 2
• W1 and W2 – phase 3

These terminals allow the motor to be configured in different connection schemes, according to the supply voltage and the intended application.

• Star connection (Y) – 380 V:
  • Interconnect: U2 – V2 – W2
  • Supply applied at: U1, V1, and W1
• Delta connection (Δ) – 220 V:
  • Connect: U1 with W2, V1 with U2, and W1 with V2.
  • Supply applied at the three connection points

COPPER SIZING

Number of Turns Design

At this stage, the flux per pole $ \phi $ is estimated from the average air-gap flux density $ B_{av} $ and the air-gap area per pole $ A_{p} $. Then, the classical $ FEM_{RMS} $ equation is used to obtain the total number of turns per phase $ N_{\text{phase}} $. Finally, the reduction due to distribution and short pitch is incorporated via $ k_{w} $.

Magnetic area per pole ($ A_{p} $)

Represents the approximate air-gap area corresponding to one pole. This area is estimated by imagining the area of a cylindrical surface. $$ A_{p} = \frac{2\pi\cdot \left( \frac{D}{2} \right)\cdot L}{P} = \frac{\pi\cdot D\cdot L}{P} = \frac{\pi\cdot 0.0885\cdot 0.058}{2} = 0.008063\ m^{2} $$

Flux per pole $ \phi $

An average air-gap flux density of $ B_{av} = 0.65\ T $ was adopted. This represents a good compromise between magnetic material utilization and saturation of the laminated silicon steel of the stator and rotor, whose saturation levels typically occur above 1.5–1.6 T. With this choice, the local flux densities in the teeth and yoke remain within safe ranges, while avoiding an excessive number of turns and high copper losses. $$ \phi = B_{av}\cdot A_{p} = 0.65\cdot 0.0080629 = 0.00524\ Wb $$

Factors $ k_{d} $, $ k_{p} $, $ k_{w} $

The winding factor $ k_{w} $ must be obtained, which measures the geometric efficiency of the winding. For this purpose, the distribution factor ($ k_{d} $) and the pitch factor ($ k_{p} $) are initially estimated:

$$ k_{d} = \frac{\sin\left( q\cdot \frac{\alpha_{e}}{2} \right)}{q\cdot \sin\left( \frac{\alpha_{e}}{2} \right)} = \frac{\sin\left( 4\cdot \frac{15{^\circ}}{2} \right)}{4\cdot \sin\left( \frac{15{^\circ}}{2} \right)} = 0.958 $$

$$ k_{p} = \sin\left( y\cdot \frac{\alpha_{e}}{2} \right) = \sin\left( 11\cdot \frac{15{^\circ}}{2} \right) = \sin\left( 11\cdot \frac{15{^\circ}}{2} \right) = 0.991 $$

$$ k_{w} = k_{d}\cdot k_{p} = 0.958\cdot 0.991 = 0.949 $$

Number of turns per phase ($ N_{phase} $)

Obtained from the $ FEM_{RMS} $ equation for a winding with sinusoidal excitation:

$$ E_{\mathrm{RMS}}=\frac{2\pi f N \phi_{\max} k_w}{\sqrt{2}} ;\rightarrow; N=\frac{\sqrt{2},E_{\mathrm{RMS}}}{2\pi f \phi_{\max} k_w} $$

$$ N = \frac{\sqrt{2}\cdot 220}{2\pi\cdot 60\cdot 0.00524\cdot 0.949} \approx 166 $$

We round to a convenient number to facilitate coil division: $$ N_{phase}= 168 $$

Number of turns per coil ($ N_{coil} $)

From the coil distribution layout, 8 coils per phase were selected: $$ N_{coil}=\frac{N_{phase}}{8}= 21\ turns\ per\ coil $$

Copper wire sizing

Here, the conductor cross-sectional area is sized based on the current density $ J $, an $ AWG $ gauge is selected, the slot fill is checked via the fill factor $ FF $, and resistance and $ I^{2}R $ losses are estimated. Finally, a thermal estimate is performed.

Minimum conductor area

In three-phase motors, the wire is sized based on the phase current considering the worst case: $$ I_{\phi}(\Delta) = \frac{I_{L}(\Delta)}{\sqrt{}3} = \frac{2.6}{\sqrt{3}} = 1.5\ A $$ The minimum conductor area is defined based on an admissible current density limit, which classically lies within: $$ 3 \leq J \leq 6\ A/mm^{2} $$ Thus, a value is chosen to balance copper losses, heating, and fill factor: $$ J = 5\ A/mm^{2} $$ The minimum copper area is given by: $$ A_{Cu,min} = \frac{I_{\phi}(\Delta)}{J} = \frac{1.5}{5} = 0.30\ mm^{2} $$

Selection of copper wire gauge

To ensure that the copper wire gauge is larger than $ 0.30\ mm^{2} $, reduce losses, and still maintain a good fill factor, AWG 19 wire is selected:

• AWG number: 19
• Diameter (mm): 0.9116
• Cross-sectional area (mm²): 0.65
• Resistance (Ohms/km): 26.15
• Current capacity (A): 2.0

Slot fill factor calculation (FF)

The fill factor FF checks whether the copper physically fits into the slot: $$ FF = \frac{A_{Cu,slot}}{A_{slot}} = \frac{2\cdot N_{coil}\cdot A_{wire}}{A_{slot}} = \frac{2\cdot 21\cdot 0.65}{100} = \frac{27.3}{100} = 0.273 $$

This is an excellent FF value, which leaves room to use smaller AWG numbers if it physically facilitates rewinding. Typical FF values lie within $ 0.25 \leq FF \leq 0.40 $.

Mean length of turn (MLT) calculation

Here it is defined that the turn consists of 4 sides: two sides have length $ C $ (stator stack length), and the other 2 sides have the equivalent length $ L_{overhang} $ (coil overhang arc). To estimate the MLT with margin, a factor $ k $ is applied, typically between 1.2 and 1.4, representing a design safety margin. $$ MLT = k\cdot (2\cdot C + 2\cdot L_{overhang}) $$

It is known that the mechanical slot angle is 15°, and since a short-pitch winding is used, the overhang arc spans 11·15 = 165°. The arc length formula is given below: $$ L_{overhang} = \frac{\theta\pi r}{180{^\circ}} = \frac{165\cdot \pi\cdot (8.85/2)}{180{^\circ}} = 12.74\ cm \approx 13.0\ cm $$

Assuming a factor $ k = 1.3 $: $$ MLT = 1.3\cdot (2\cdot 5.8 + 2\cdot 13.0) = 48.88\ cm\ \approx 50\ cm $$

Total wire length

The required wire length is estimated by multiplying the MLT by the number of turns.

• Length per coil: $$ l_{coil} = N_{coil}\cdot MLT = 21\cdot 0.50 = 10.5\ m $$
• Length per phase: $$ l_{\phi} = N_{phase}\cdot MLT = 168\cdot 0.50 = 84\ m $$
• Total length: $$ l_{tot} = 3\cdot l_{\phi} = 3\cdot 84 = 252\ m $$

Resistance, losses, and temperature rise of the winding per phase

We begin by calculating the resistance of each phase winding: $$ R_{\phi,AWG19} = \frac{26.15}{1000}\cdot 84 \approx 2.20\ \Omega $$ Copper losses: $$ P_{Cu} = 3\cdot I_{\phi}^{2}\cdot R_{\phi,AWG19} = 3\cdot {1.5}^{2}\cdot 2.20 \approx 14.8\ W $$

For a preliminary estimate, detailed construction data (ventilation, housing, insulation class, etc.) are not available, so a thermal resistance range is assumed. $$ R_{\theta} = 2\ to\ 4\ {^\circ}C/W $$ Thus: $$ \Delta T = R_{\theta}\cdot P_{Cu} = (2\ to\ 4)\cdot 14.8 = 30\ to\ 60\ {^\circ}C $$

WINDING TOPOLOGY

The analysis of the winding topology of a three-phase electric motor is a fundamental step for the calculation and execution of a correct rewinding process, as it defines the electromagnetic and construction characteristics of the machine.

Spatial Distribution of the Coils

Integral Winding and Fractional Winding

Full-Pitch and Short-Pitch

Single Layer vs Double Layer

FORM

Definition of the main variables

• $ Q $ – Total number of stator slots.
• $ P $ – Total number of magnetic poles of the motor.
• $ p $ – Number of pole pairs of the motor, given by $ p = P/2 $.
• $ m $ – Number of motor phases (for three-phase motors, $ m = 3 $).
• $ f $ – Supply electrical grid frequency (Hz).
• $ n_{s} $ – Synchronous speed of the rotating magnetic field of the stator (rpm).
• $ n $ – Mechanical speed of the rotor (rpm).
• $ s $ – Motor slip, a dimensionless quantity that expresses the relative difference between $ n_{s} $ and $ n $.
• $ q $ – Number of slots per pole per phase. Classifies the winding as integral or fractional.
• $ \tau_{s} $ – Pole pitch, expressed as the number of slots per magnetic pole.
• $ y $ – Coil pitch, defined as the distance (in slots) between the two active sides of a coil.
• $ \alpha_{e} $ – Electrical angle per slot (electrical degrees), representing the electrical advance when moving from one slot to the next.
• $ \theta_{m} $ – Mechanical angle (degrees or radians), associated with the physical position in the stator or rotor.
• $ \theta_{e} $ – Electrical angle (degrees or radians), associated with the position of the magnetic field.
• $ k_{d} $ – Winding distribution factor, which quantifies the spatial distribution of the coils.
• $ k_{p} $ – Pitch factor, which quantifies the effect of coil pitch shortening.
• $ k_{w} $ – Winding factor, given by the product.
• $ N $ – Total number of turns per phase of the winding.
• $ \varphi $ – Magnetic flux per pole (Weber).
• $ B $ – Magnetic flux density in the air gap (Tesla).
• $ A $ – Effective magnetic area per pole.
• $ D $ – Inner diameter of the stator (air-gap diameter).
• $ L $ – Axial length of the stator.
• $ V $ – Rated line or phase voltage, depending on the connection type (Δ or Y).
• $ I $ – Rated phase current of the motor.
• $ P_{out} $ – Useful mechanical output power of the motor.
• $ \eta $ – Motor efficiency.
• $ cos(\varphi) $ – Motor power factor.
• $ J $ – Current density in the conductor (A/mm²).
• $ A_{Cu} $ – Cross-sectional area of the copper conductor.
• $ A_{Cu_{tot}} $ – Total copper area occupying the slot (sum of all conductors).
• $ A_{slot} $ – Useful area of the stator slot.
• $ FF $ – Slot fill factor, defined as the ratio between the total copper area and the useful slot area.