class: center, middle # EE-362 ELECTROMECHANICAL ENERGY CONVERSION-II ## Ozan Keysan [ozan.keysan.me](http://ozan.keysan.me) Office: C-113
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Tel: 210 7586 --- # Review: -- ## Distributed Winding
--- # Review: ## Three Phase Winding -- # A -- , -C -- , B -- , -A -- , C -- , -B [Rotating MMF animation with discrete windings](http://people.ucalgary.ca/~aknigh/electrical_machines/fundamentals/f_ac_rotation.html) --- # Review: Number of Poles
--- #Review ###Electrical Angle is not equal to Mechanical Angle! ## \\(\theta\_{elec} = (\dfrac{p}{2}) \theta\_{mech} \\) ### \\(p\\) : Number of poles (always even number) ### \\(\dfrac{p}{2}\\) : Number of pole pairs ### \\(\omega\_{elec} = (\dfrac{p}{2}) \omega\_{mech} \\) (radians/second) --- # Winding Factors ### Two constants that help to estimate induced voltage characteristics in the machine windings analytically. -- - ## Distribution Factor (\\(k_d\\)) -- - ## Pitch Factor (\\(k_p\\)) --- # Distribution Factor
--- # Distribution Factor ## Vector sum of voltages in a distributed coil is less than the algebraic sum of the voltage in each coil. -- ## \\(k_d = \dfrac{\mathrm{Vector\,Sum\,of\,Voltages}}{\mathrm{Algebraic\,Sum\,of\,Voltages}}\\) [Animation](http://cusp.umn.edu/electric_drives/animations/electric_drives_animation_1_distributed_winding.php) --- # Distribution Factor ### Q1: What is the distribution factor of a concentrated coil? -- ### Q2: What is the distribution factor of two coils seperated by \\(\pi/3\\)? -- ### Q3: What is the distribution factor of three coils seperated by \\(\pi/6\\)? -- ### Generalized equation --- # Distribution Factor ## \\(k_d = \dfrac{sin(q \dfrac{\alpha}{2})}{q sin(\dfrac{\alpha}{2})}\\) ## \\(q\\): Number of coils ## \\(\alpha\\): Angle between each coil (electrical angle!) --- # Pitch-Factor -- ## Review of Definitions: - # Full-pitched (= \\(\pi\\) electrical) -- - # Under-pitched (< \\(\pi\\) electrical) -- - # Over-pitched (> \\(\pi\\) electrical) --- ## A 2-pole machine with sinusoidal \\(B_{gap}\\)
--- # Can you plot: -- - ## Airgap flux density? -- - ## Flux Linkage in the coil? -- - ## Induced voltage? --- ## A 2-pole machine with sinusoidal \\(B_{gap}\\)
--- # Full-pitched vs Fractional-pitched Coil - ## Full-pitch = 180 degrees (\\(\pi\\)) -- - ## Under pitch < 180 degrees (\\(\pi\\)) -- - ## \\(V\_{full-pitch} > V\_{under-pitch}\\) -- - # If so, what is the point? --- # A big Paranthesis ## Fourier Series -- > ## All waveforms, no matter what you scribble or observe in the universe, are actually just the sum of simple sinusoids of different frequencies. --- # Fourier Series  [Fourier Series using Circles](https://www.youtube.com/watch?v=LznjC4Lo7lE), [Complex Orbits](https://www.youtube.com/watch?v=QVuU2YCwHjw), [Useful applets](http://www.falstad.com/fourier), [Fourier examples](http://ptolemy.eecs.berkeley.edu/eecs20/week8/examples.html) [More Useful Links on Fourier Series](http://keysan.me/explained/) --- # Fourier Series [Google Plot](https://www.google.com/search?q=square+wave+fourier+transform&ie=utf-8&oe=utf-8#q=plot+%28sin%28x%29%2B0.25*sin%283*x%29%2B0.1*sin%285*x%29%29) -- - ## Let's see what happens with a 2/3 under pitched coil? -- - ## Under-Pitched coils are very useful for the elimination of harmonics --- #Pitch Factor ## Voltage in a short-pitched coil is less than a full-pitched coil ## \\(k_p = \dfrac{\mathrm{Short-Pitched\;Coil\;Flux}}{\mathrm{Full-Pitched\;Coil\;Flux}}\\) --- #Pitch Factor # \\(k_p = sin(\dfrac{\lambda}{2})\\) # \\(\lambda\\): Coil-pitch in electrical degrees --- # Pitch and Distribution Factor for Harmonics ## For harmonic number n: ## \\(k_p(n) = sin(\dfrac{n\lambda}{2})\\) -- ## \\(k_d(n) = \dfrac{sin(q n \dfrac{\alpha}{2})}{q sin(\dfrac{n\alpha}{2})}\\) --- # Winding Factor: ## is the combination of distribution and pitch factor -- #\\(k_w = k_d \times k_p\\) -- ## gives an idea about the magnitude of the coil voltage (or MMF) compared to concentrated and full-pitched winding. --- ## You can download this presentation from: [keysan.me/ee362](http://keysan.me/ee362)