< class: center, middle # EE-568 Selected Topics in Electrical Machines ## Airgap & Mechanical Constraints ## Ozan Keysan [keysan.me](http://keysan.me) Office: C-113 <span class="meta">•</span> Tel: 210 7586 --- # Suitable Airgap -- ### There is not a definite answer -- ### \\(\delta = 0.2 + 0.01 P^{0.4} \\)mm when p=1 ### P: power -- ### \\(\delta = 0.18 + 0.006 P^{0.4} \\)mm when p > 1 -- ### Smallest airgap is 0.2 mm --- # Suitable Airgap ### For heavy duty motors the gap may be increased by 60 %. -- ### For converter driven motors airgap can be increased by 60 % to reduce rotor surface losses. -- ### For high speed machines increase airgap (eqn. 6.25 of the textbook) -- ### For very large diameter machines airgap is approximate to D/1000. --- ##Ex: ## What should be the suitable airgap if the motor in the previous example (30 kW) is a heavy-duty motor? -- ### \\( \delta= 1.6 \; 0.8 \; 0.006 (30k)^{0.4}=0.88 \approx 0.9mm \\) --- # Mechanical Constraints ## Tip Speed ### What is the rotational speed of a machine with 0.5m diameter rotor, to reach the tip speed reach to the speed of sound (1 Mach)? -- ### Max allowable tip speed: 75m/s for high-strength non-magnetic alloy sleeves, and 100 m/s for carbon-fiber sleeves Reading: Section 6.1 of textbook --- # Mechanical Loadability -- ### Rotor material should withstand centrifugal forces (especially at high speeds). -- ## Centrifugal Stress: ## \\(\sigma\_{mech} = C' \rho r\_r^2 \Omega^2 \\) ### \\(\Omega\\): Mechanical speed in rad/s ### \\(\rho\\): Density of the material --- ## Centrifugal Stress: ## \\(\sigma\_{mech} = C' \rho r\_r^2 \Omega^2 \\) -- #### \\(C'= 1\\) for a thin cylinder -- #### \\(C'= (3+v)/8\\) for a smooth homogenous cylinder -- #### \\(C'= (3+v)/4\\) for a cylinder with a small bore -- ### \\( v \\): Poisson's ratio -- ### [Poisson's ratio](https://www.youtube.com/watch?v=hBnzrBhnzVo), [deflection of a golf ball](https://www.youtube.com/watch?v=aMqM13EUSKw), [deflection of a face, 2:15](https://www.youtube.com/watch?v=On1CsbTwlDs) -- ### Poisson Ratios of metals: Aluminium=0.34, Steel=0.29, Copper=0.34 --- # Ex. 6.3: ### Calculate the maximum diameter for a smooth steel sylinder having a small bore. The speed is 15.000 rpm. Yield strength is 300 MPa. The density of the material is 7860 kg/m³. --- # Other Mechanical Constraints ### Bending Modes <img src="https://qph.fs.quoracdn.net/main-qimg-9c0fb36366b6b11967494f294b47c246" alt="Drawing" style="width:450px;"/> --- # Dynamics of Mechanical Systems: Resonance <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/Resonant_frequency_amplitude.svg/524px-Resonant_frequency_amplitude.svg.png" alt="Drawing" style="width:300px;"/> ### [Transfer function and mathematical modelling](https://www.slideshare.net/vishalgohel12195/transfer-function-and-mathematical-modeling) -- ### [Tacoma Bridge](https://www.youtube.com/watch?v=lXyG68_caV4) ### [Forced vibration-1](https://www.youtube.com/watch?v=OaXSmPgl1os), [Resonant Freq.](https://www.youtube.com/watch?v=LV_UuzEznHs) ### [Torsional Resonance](https://www.youtube.com/watch?v=JLY-yQOpL20) --- # Resonant Modes ### [Cantilever Vibration](https://www.youtube.com/watch?v=lKT3wBIUFhA) ### [Resonant Modes](https://www.youtube.com/watch?v=uWoiMMLIvco) ### [Modal Shapes](https://www.youtube.com/watch?v=kvG7OrjBirI) ### [Modal Shapes](https://www.youtube.com/watch?v=d3U_m-4XOtg) --- # Critical Speeds ### The rotational speed should be below the critical speed (preferably with a safety factor) -- ### Usually the limiting factor for very high-speed machines: -- <img src="./images/ee564/critical_speed.png" alt="Drawing" style="width: 800px;"/> --- # Ex 6.5 ### Calculate the max. length with a safety factor k=1.5 for a smooth solid rotor when the rotor diameter is 0.15 m and the rotor speed is 20.000 rpm. --- # Review: Aspect Ratio -- ## \\(\chi = \dfrac{L'}{D}\\) --- # Typical Aspect Ratios -- ## Asynchronous Machines: ## \\(\chi \approx \dfrac{\pi}{2p} \sqrt[3]{p}\\\) -- ## Synchronous Machines: ## \\(\chi \approx \dfrac{\pi}{4p} \sqrt{p}\\\) -- --- # Define \\(D_i\\) and \\(L\\) -- ## Usually \\(0.5 < D_i/L < 2.5\\) -- ## Small diameter for high-speed or servo-type motors, why? -- ### Small inertia, -- ### Low tip speed! -- ### Bending Modes --- --- # How to define Outer Diameter \\(D_o\\)? -- <img src="./images/varying_number_of_poles.png" alt="Drawing" style="width: 800px;"/> --- # How to define \\(D_o\\)? -- <style type="text/css"> .tg {border-collapse:collapse;border-spacing:0;border-color:#bbb;margin:0px auto;} .tg td{font-family:Arial, sans-serif;font-size:32px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#bbb;color:#594F4F;background-color:#E0FFEB;} .tg th{font-family:Arial, sans-serif;font-size:32px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:#bbb;color:#493F3F;background-color:#9DE0AD;} </style> <table class="tg"> <tr> <th class="tg-031e">N Poles</th> <th class="tg-031e">2</th> <th class="tg-031e">4</th> <th class="tg-031e">6</th> <th class="tg-031e">8</th> <th class="tg-031e">10</th> <th class="tg-szh5">12</th> </tr> <tr> <td class="tg-031e">Do/Di</td> <td class="tg-031e">2 <br></td> <td class="tg-031e">1.88</td> <td class="tg-031e">1.78</td> <td class="tg-031e">1.66</td> <td class="tg-031e">1.54</td> <td class="tg-031e">1.43</td> </tr> </table> Source: T.Miller - Electric Machine Design Course, Lecture-5, Slide4 --- ## How to choose height of the slots? -- ### Slot Ratio (d): Ratio of the inner stator slot diameter to outer stator slot diameter -- <img src="./images/ee564/slot_ratio.png" alt="Drawing" style="width: 800px;"/> #### Reading Assignment: [The Rediscovery of Synchronous Reluctance and Ferrite Permanent Magnet Motors](https://link.springer.com/book/10.1007%2F978-3-319-32202-5) --- ## How to choose height of the slots? ### For the same outer diameter: -- ### As the slot ratio increases (i.e. higher slots): -- - ### Electric loading increases (more copper can be fit) -- - ### Diameter for the rotor gets smaller (less surface area & less torque) --- ## How to choose height of the slots? ### For the same outer diameter: ### As the slot ratio decreases (i.e. shorter slots): -- - ### Electric loading decreases (less area for copper) -- - ### Diameter (&rotor volume) gets larger ## There should be an optimum point! --- ## How to choose height of the slots? ### Assume parallel (rectangular) slots: copper area is proportional to slot height: -- ### \\( I \propto (1-d) \\) -- ### Electrical Loading is current per circumference -- ### \\(K_s \propto (1-d)/d \\) -- ### Torque can be expressed as: ### \\( T \propto \sigma . Vol_R\\) -- \\(\propto [(1-d)/d].d² \propto (1-d).d \\) --- ## How to choose height of the slots? ### Assume parallel (rectangular) slots: copper area is proportional to slot height: ### Torque can be expressed as: ### \\( T \propto \sigma . Vol_R\\) -- \\(\propto [(1-d)/d].d² \propto (1-d).d \\) ## Optimum point=? -- ## d=0.5 --- ## How to choose height of the slots? ### But parallel teeth are more common: Slots gets wider with diameter ### \\( I \propto (1-d²) \\) -- ### Electrical Loading is current per circumference -- ### \\(K_s \propto (1-d²)/d \\) ### Torque can be expressed as: ### \\( T \propto \sigma . Vol_R \; \propto [(1-d²)/d].d² \propto (1-d²).d \\) ### Optimum point= \\(d= 1/\sqrt{3} = 0.58\\) --- ## How to choose height of the slots? <img src="./images/ee564/d_optimum.png" alt="Drawing" style="width: 600px;"/> --- # Stator Slot Types: -- <img src="./images/ee564/slot_types.png" alt="Drawing" style="width: 800px;"/> -- ### Most Common Types: - ### Open Slots: Constant width, easy repair and assembly -- - ### Semi-closed Slots: Difficult to assembly but better magnetic characteristics -- - ### Tapered Slots: Varying width (constant tooth width) --- # Stator Slot Types: ### There are several other options. Depending on operating conditions, manufacturing constraints etc. <img src="./images/ee564/slot_10.png" alt="Drawing" style="width: 240px;"/> <img src="./images/ee564/slot_14.png" alt="Drawing" style="width: 240px;"/> <img src="./images/ee564/slot_12.png" alt="Drawing" style="width: 240px;"/> --- # Stator Slot Types: ### There are several other options. Depending on operating conditions, manufacturing constraints etc. <img src="./images/ee564/slot_15.png" alt="Drawing" style="width: 240px;"/> <img src="./images/ee564/slot_17.png" alt="Drawing" style="width: 240px;"/> <img src="./images/ee564/slot_18.png" alt="Drawing" style="width: 240px;"/> --- # Stator Slot Types: ### There are several other options. Depending on operating conditions, manufacturing constraints etc. <img src="./images/ee564/slot_27.png" alt="Drawing" style="width: 240px;"/> <img src="./images/ee564/slot_29.png" alt="Drawing" style="width: 240px;"/> <img src="./images/ee564/slot_13.png" alt="Drawing" style="width: 200px;"/> --- # Production of Electric Machines ### [TES Generators and Motors](https://www.youtube.com/watch?v=5Mu42TzHy8M) ### [Induction Motors: Overhauling a Motor](https://www.youtube.com/watch?v=yPvYd03cKJU) ### [Rewinding a Large Motor](https://www.youtube.com/watch?v=_65mXQ-GNVM) ### [Automatic Coil Insertion](https://www.youtube.com/watch?v=Kih3hyl8CUg) ### [E-propulsion System](https://www.youtube.com/watch?v=d5cEIGDg2Co) ### [BMW i-8](https://www.youtube.com/watch?v=oESBbRu32-E) --- ## Case Study: Ferrite vs. NdFeB -- ### Base Design - ### Di=100mm - ### L =100mm - ### Slot Ratio=0.7 - ### Airgap = 1.5mm - ### Magnet (NdFeB)= 4mm, Brem=1.1T #### Reading Assignment: [The Rediscovery of Synchronous Reluctance and Ferrite Permanent Magnet Motors](https://link.springer.com/book/10.1007%2F978-3-319-32202-5) --- ## Case Study: Ferrite vs. NdFeB ### Base Design - ### Current Density = 6.7 A/mm2 - ### Electrical Loading = 30kA/m - ### Shear Stress = 18 kPa - ### Output Torque = 28 Nm - ### Magnet (NdFeB)= 4mm, Brem=1.1T #### Reading Assignment: [The Rediscovery of Synchronous Reluctance and Ferrite Permanent Magnet Motors](https://link.springer.com/book/10.1007%2F978-3-319-32202-5) --- ### Base Design with NdFeB <img src="./images/ee564/ndfeb_design.png" alt="Drawing" style="width: 400px;"/> --- ### Magnets replaced with Ferrite (Brem=0.4 T) <img src="./images/ee564/ferrite_1.png" alt="Drawing" style="width: 400px;"/> --- ### Double Ferrite Thickness (4mm -> 8mm) <img src="./images/ee564/ferrite_2.png" alt="Drawing" style="width: 400px;"/> --- ### Reduce teeth width until saturation <img src="./images/ee564/ferrite_3.png" alt="Drawing" style="width: 400px;"/> ### Electric loading is 157% of the base design --- ### Reduce back-core (yoke) until saturation ### Total volume reduces to 83%. <img src="./images/ee564/ferrite_4.png" alt="Drawing" style="width: 400px;"/> --- ### Comparison of Electrical and Magnetic Loadings <img src="./images/ee564/ndfeb_ferrite_compare.png" alt="Drawing" style="width: 600px;"/> --- # Selection of number of stator slots -- ## Advantages of Low number of slots: - ### Reduced manufacturing cost - ### Less space lost due to insulation and slot opening -- ## Disadvantages of low number of slots - ### Increased leakage inductance - ### Reduced breakdown torque - ### Larger MMF harmonics --- # Selection of number of stator slots ## Advantages of High number of slots: - ### Reduced tooth pulsation - ### Higher overload capacity - ### Better Cooling -- ## Disadvantages of high number of slots - ### Increased magnetizing current - ### Poor Cooling - ### Difficult manufacturing --- # Number of Slots vs Winding Factor <img src="./images/fundamental_winding_factors_for_different_slot_numbers.png" alt="Drawing" style="width: 850px;"/> --- # Further Reading # Selection of Phases - Poles ## T.Miller Electric Machine Design Course, Lecture 10-12 ### [Ref](https://www.youtube.com/watch?v=uoJfVMynV44&list=PLR3pRvvCj_Y_jcg_Ia6ARvzefov05c7vf&index=11) --- ## You can download this presentation from: [keysan.me/ee568](http://keysan.me/ee568)