Theoretical and computational design of AC induction motor

This project deals with magnetic design for a new electric machine in the wastewater aeration equipment market. The required conditions to meet are

• rated speed of 10,000 RPM

• rated power of 50 kW

• electric frequency under 600 Hz

• three phases

• use surface mounted permanent magnets on the rotor

• use forced air cooling.

To design the machine, we need to determine the number of slots, poles, the winding layout, and various dimensions of the machine. The main objectives are to minimize the design cost, losses, and torque ripple. The product will go off the shelf to the customers only if it has an efficiency over 90% and a torque ripple less than 50%.

To do this, we will use various FEMM models and several functions to link them to Matlab’s multiobjective optimization algorithm, and finally running the optimization.

Design specification

• Rated power: 50 kw

• Rated speed: 10,000 RPM

• Frequency: < 600 Hz

Constants and Materials

• Conductor current density: 5 Arms/mm^2

• Magnets: NdFeB (N40 grade)

• Steel: M19, 29 Gauge

• Wire: 18 AWG wire

• Slot fill factor: k = to be determined based on the type of winding you select

Free variables: (to be determined)

• 𝑑m

• 𝛿

• 𝛼st

• 𝑑sy

• 𝑑st

• 𝑤st

Dependent and fixed variables

• 𝑑mp = 𝑑m

• 𝑑so = 2 mm

• 𝑑sp = 4 mm

• 𝛼m = 180∘/p, where p is the number of poles or the rotor

• Stator radius 𝑟s = 180 mm

• 𝑑ri = 𝑟si − 𝛿 − 𝑑m (only iron resides inside the magnets), where 𝑟si= 𝑟s − 𝑑st − 𝑑sp

• Number of turns in a coil Zq: determined from available slot area and copper fill factor

• Active length of motor 𝑙: to be determined through FEA analysis.

Allowable free variable range

• 0.25 mm < 𝑑m < 5mm

• 1mm < 𝛿 < 5mm

• 5mm < 𝑑sy < 50mm

• 5mm < 𝑑st < 100mm

• 5mm < 𝑤st < 100mm

• Determine appropriate range for: 𝛼st

Constraints

• Surface speed of the rotor 𝑣tip < 175 [m/s]

• 𝑤st < 𝛼st * 𝑟si, where 𝑟si = 𝑟s − 𝑑st − 𝑑sp

• 𝐴rms < 80 kArms/m (constrain electric loading to a range that allows for air cooling)

• Valid tooth tip geometry

Objectives

• Maximize efficiency by minimizing model losses (magnet, iron, and winding loss)

  • Hysteresis loss coefficient, Ch = 0.0186 [W/kgT^2Hz]

  • Eddy current loss coefficient, Ce = 6.887 × 10^(-5) [W/kgT^2Hz^2]

  • Conductivity of copper = 5.77 × 10^7 [S/m]

  • Conductivity of selected magnets = 5.55 × 10^5 [S/m]

• Minimize torque ripple

• Minimize the following material costs (rotor steel, rotor magnets, stator copper)

  • M19 steel: 14.03 × 10^3 [$/m3]

  • Copper: 0.06 [$/m] of AWG 18 magnet wire

  • Magnets: 708.5 × 10^3 [ $/m3]

To perform a winding layout, we decided to make a double layer winding. Below we provide a winding schematic for the design that shows the connections of phase u.

Tasks performed

An appropriate winding fill factor for the winding is decided

𝛼o (mechanical rotor angle) is determined to create the maximum torque per ampere of current. The currents are given as

𝐼u = 𝐼 cos 𝛼w

𝐼v = 𝐼 cos(𝛼w − 2pi/3)

Iw = I cos(𝛼w − 4pi/3)

Assuming that slot 1 is aligned with 𝛼 = 0.

WORK FLOW

• Dependent, fixed variables and allowable free variable are calculated along with number of turns in a coil Zq and an appropriate range for 𝛼st.

• An expression for active material cost (rotor steel, rotor magnets, stator copper) in terms of the free and dependent variables is developed that will be used to evaluate the objectives.

• For the copper, we determine the length of magnet wire based on the same expression for length that we used for loss calculations

• Initial analytic designs are performed for the motor to determine a set of values for the free variables that satisfy the allowable variable range and the constraint. Further the following information is also indicated

  • Selected magnetic loading, electric loading, and rotor volume

  • Number of turns in a coil

  • The maximum field in the stator teeth

  • The maximum field in the stator yoke

  • All dimensions of the machine cross-section (free variables and the dependent variables)

• MATLAB function 'evaluateDesign' is built to determine the required axial length to achieve a specified rated torque. This is done by performing the complete FEMM analysis of our machine with an axial length of 1mm, extract the field and torque values at each rotor location and then the length required to obtain the rated torque is calculated. This newly calculated length is then used for the loss calculations

• Next, we create a Matlab function 'evaluateConstraints' that will be called by Matlab’s 'gamultiobj'. This function will be responsible for evaluating the design constraints. In addition we have a Matlab function which will indicate whether the free variables will yield valid tooth-tip geometry. The function returns 0 if the geometry is valid and 1 if the geometry is invalid. This function is used inside 'evaluateConstraints' to constrain the optimization to valid.

• Another Matlab function called 'evaluateObjectives' is created that will be called by Matlab’s 'gamultiobj'. This function will receive the optimization algorithm’s free variables as its arguments, formulate the arguments for 'evaluateDesign' and then convert the returned values of 'designEval' into the objectives that this function will return. To determine the appropriate settings.steps value, we use our analytic design from previous steps and evaluate the iron and magnet losses when the number of steps is swept over a range of even integers from 10 to 30. A plot is created where the y axis is the total iron and magnet losses and the x axis is the value of steps used.

• Analysis of initial design

  • Matlab functions are used to evaluate the objectives of this initial design.

  • FEMM is used to audit the assumptions for maximum field in the stator teeth, stator yoke, and magnetic loading.

• Matlab function called 'optimizeMotor.m' is developed that uses 'gamultiobj' to optimize our motor design. A population size of at least 20 is used and appropriate values for settings.lowestHarmonic and settings.steps are determined.

• Optimization is ran to find a set of designs that meet customer’s requirements. The optimization is ran till a set of suitable designs meet the customer’s requirements.

As seen from the above value, the losses remain almost saturated with increase in value of steps. So it is wise to take a small step of 10. Since p* is 1 so, the harmonics are integer. Therefore the required harmonic is 2.

Therefore, settings.steps = 10; and n = 2

The value of the objectives of initial design

After FFT

Genetic algorithm is performed for optimisation using MATLAB

The design recommended satisfies all the customer’s requirement . The efficiency is nice and well above 90% and ripple is less than 50 %. The reason it has been chosen is because it offers high efficiency and low ripple with reasonable cost .

There are many advantages of using Optimization methods. One of them is it gives a varied number of options available for a given set of constraints. With some fixed parameters, optimization can help in reaching the best of the objectives. As seen here, the initial design had an efficiency of just 77.17% but as we have performed the optimization, there are possibilities to get better designs and higher efficiency based on the given constraints. Hence, it is advantageous to use optimization techniques.