< class: center, middle # EE-568 Selected Topics in Electrical Machines ## Main Flux Paths ## Ozan Keysan [keysan.me](http://keysan.me) Office: C-113 <span class="meta">•</span> Tel: 210 7586 --- # Main Flux Paths & Machine Parameters -- ## Some definitions: -- - ## Carter's Coefficient - ## Effective Core length --- # Carter's Coefficient -- ## Way to estimate the flux density by converting slotted rotor/stator to a perfect cylinder. ### Ref: Section 3.1.1 of the textbook (Pyrhonen) --- # Carter's Coefficient <img src="./images/flux_distribution.png" alt="Drawing" style="width: 600px;"/> --- # Carter's Coefficient -- - ### First assume the rotor is smooth to find \\(k\_{cs}\\): ### \\( \delta\_e = k\_{cs} \delta \\) -- - ### Then assume the stator is smooth to find \\(k\_{cr}\\): -- - ### Total Carter coefficient is the product of two ### \\( k\_{c} = k\_{cs} \times k\_{cr} \\): ### Effective airgap is slightly larger than the actual gap (\\(k_c \gtrapprox 1 \\) ) --- # Carter's Coefficient ## \\(k\_c = \dfrac{\tau\_u}{\tau\_u - K b\_1}\\) ## where: ## \\(K = \dfrac{b\_1 / \delta }{5 + b\_1 / \delta }\\) --- ## Example (3.1) <img src="./images/ee564/carter_example.png" alt="Drawing" style="width: 800px;"/> --- ### Variations in flux density creates harmonics (and losses) <img src="./images/ee564/flux_airgap.png" alt="Drawing" style="width: 800px;"/> --- ### These losses can be minimized by using magnetic wedge or special tooth shape <img src="./images/ee564/magnetic_wedge.png" alt="Drawing" style="width: 800px;"/> --- # Equivalent Core Length <img src="./images/L_eq.png" alt="Drawing" style="width: 450px;"/> ### \\(l' \approx l + 2 \delta \\) #### Section 3.2 of the textbook (Pyrhonen) --- ## Equivalent Core Length with Cooling -- ### Larger machines requires ducts for cooling -- <img src="./images/L_eq_cooling.png" alt="Drawing" style="width: 750px;"/> --- <img src="./images/L_eq_cooling.png" alt="Drawing" style="width: 750px;"/> ### Use the Carter's coefficient to calculate effective length ### \\(l' \approx l - n\_{v} b\_{ve} + 2 \delta \\) ### \\(b_{ve} \\) can be calculated Carter's coefficient again ### or just assume \\(b_{ve} \lessapprox b_v \\) --- ### Cooling ducts both at the stator and rotor <img src="./images/ee564/rotor_stator_cooling.png" alt="Drawing" style="width: 750px;"/> --- ### Example <img src="./images/ee564/ex_3_2.png" alt="Drawing" style="width: 750px;"/> --- # Back Core Flux --- # D-Q Axis <img src="./images/dq_axis.png" alt="Drawing" style="width: 550px;"/> --- # D-Q Axis <img src="https://www.pantechsolutions.net/media/wysiwyg/article-images/synchronous-machine-and-the-main-principle-of-the-vector-control.jpg" alt="Drawing" style="width: 800px;"/> --- # D-Q Axis <img src="./images/ee564/dq_axis.png" alt="Drawing" style="width: 550px;"/> --- # Back Core Flux <img src="./images/ee564/back_core_flux.png" alt="Drawing" style="width: 500px;"/> ### \\(\hat{B}\_{back-core} = \dfrac{\hat{\Phi}\_{pole}}{2 A\_{back-core}} = \dfrac{\hat{\Phi}\_{pole}}{2 k\_{stacking} l' h\_{ys}}\\) --- # Magnetizing Inductance -- ## What is inductance? -- ## Flux Linkage Per Current: (\\(L = \dfrac{\lambda}{I}\\)) --- # Magnetizing Inductance ### If B is sinusoidal ## \\(\hat{\Phi}\_{pole} = \int B dS\\) -- \\(=\dfrac{2}{\pi}\hat{B}\tau\_{pole}l'\\) -- ### Flux Linkage ### \\( \lambda = N \Phi = k\_{w1} N\_s \hat{\Phi}\_{pole} \\) -- \\(= k\_{w1} N\_s \dfrac{2}{\pi}\hat{B}\tau\_{pole}l' \\) --- # Magnetizing Inductance -- ### \\(\hat{H}\_m \delta\_e = MMF = \dfrac{\hat{B}}{\mu_0} \delta\_e \\) ### \\(\delta\_e\\): Effective air-gap -- ### \\(MMF = \dfrac{4}{\pi} \dfrac{k\_{w1} N\_s}{2p} \sqrt{2} I\_{s,rms}\\) -- ### \\( \hat{B} = \dfrac{4}{\pi} \dfrac{k\_{w1} N\_s}{2p} \sqrt{2} I\_{s,rms} \dfrac{\mu_0}{\delta\_e}\\) --- # Magnetizing Inductance ### \\(\lambda = \dfrac{2}{\pi} \dfrac{\mu\_0}{\delta\_e} \dfrac{4}{\pi} \dfrac{(k\_{w1} N\_s)²}{2p} \tau\_{p}l' \sqrt{2} I\_{s,rms} \\) -- ### \\(L = \dfrac{\lambda}{I}\\) -- ### \\(L\_{m (ph)} = \dfrac{2}{\pi} \mu\_0 \dfrac{1}{2p}\dfrac{4}{\pi} \dfrac{\tau\_p}{\delta\_{ef}} l' (k\_{ws}N\_s)^2\\) --- # Magnetizing Inductance (Per-phase) -- ### \\(L\_{m (ph)} = \dfrac{2}{\pi} \mu\_0 \dfrac{1}{2p}\dfrac{4}{\pi} \dfrac{\tau\_p}{\delta\_{ef}} l' (k\_{ws}N\_s)^2\\) ### Writing pole pitch in terms of diameter -- ### \\(L\_{m (ph)} = \dfrac{2 \mu\_0 D}{\pi p^2 \delta\_{ef}} l' (k\_{ws}N\_s)^2 \\) ### Derivation in Pyrhonen Section 3.9 --- # Total Magnetizing Inductance ### \\(L\_m = \dfrac{3}{2} L\_{m (ph)} \\) -- \\( = \dfrac{3}{2} \dfrac{2 \mu\_0 D}{\pi p^2 \delta\_{ef}} l' (k\_{ws}N\_s)^2 \\) --- # Magnetizing Inductance -- - ## Increases with number of turns -- - ## Reduces with large airgap -- - ## Reduces with the number of poles (power factor gets worse with higher number of poles) --- # Magnetizing Inductance -- - ## Reduces with increasing voltage: -- Saturation -- - ## Reduces with torque: Why? --- # Flux Lines vs Torque <img src="https://raw.githubusercontent.com/ozank/ozank.github.io/master/presentations/images/flux_vs_torque.png" alt="Drawing" style="width: 750px;"/> --- # Magnetizing Inductance vs Torque <img src="https://raw.githubusercontent.com/ozank/ozank.github.io/master/presentations/images/inductance_vs_torque.png" alt="Drawing" style="width: 750px;"/> --- # Leakage Flux (Ch4) -- ## Flux that does not cross the airgap -- ## Flux crosses the airgap but does not link the winding --- # Leakage Flux ## Flux that does not cross the airgap -- - ### Pole Leakage Flux -- - ### Slot Leakage Flux -- - ### Tooth Tip Leakage Flux -- - ### End Winding (Overhang) Leakage Flux --- # Pole Leakage Flux ### In salient pole machines (i.e. synchronous machine) <img src="./images/ee564/pole_leakage.png" alt="Drawing" style="width: 700px;"/> --- # Pole Leakage Flux ### In PM machines (between adjacent magnets) <img src="./images/ee564/pm_leakage_1.jpg" alt="Drawing" style="width: 700px;"/> --- # Pole Leakage Flux ### In PM machines (between magnet and rotor core) <img src="./images/ee564/pm_leakage_2.png" alt="Drawing" style="width: 600px;"/> --- # Slot Leakage Flux & Tooth Tip Leakage Flux <img src="./images/slot_leakage.png" alt="Drawing" style="width: 700px;"/> --- # Slot Leakage Inductance -- <img src="./images/ee564/slot_leakage_2.png" alt="Drawing" style="width: 500px;"/> ### H increases as you go higher in the slot, as there is more coils are enclosed in \\(\int H dl \\) --- # Slot Leakage Inductance -- <img src="./images/ee564/slot_leakage.png" alt="Drawing" style="width: 700px;"/> --- # Slot Leakage Inductance -- ### for the bottom part (\\(L\_{u1}\\)) ### \\(B(h) = \mu\_0 H (h) = \mu\_0 \dfrac{z\_Q I \dfrac{h}{h\_4}}{b\_4} \\) -- ### \\(L\_{u1} = \dfrac{l' b\_4}{\mu\_0 I^2} \int_0^{h\_4} B^2(h) dh\\) --- # Slot Leakage Inductance -- ### repeat for the upper part (\\(L\_{u2}\\)) ### \\(B(h) = \mu\_0 \dfrac{z\_Q I}{b\_1} \\) (Constant B) --- # Slot Leakage Inductance -- ### Inductance for one slot: ### \\(W\_u = 1/2 L\_{u1} I^2 = 1/2 \mu_0 l' z\_Q^2 I^2 (\lambda_1 + \lambda_4)\\) --- # Total Slot Leakage Inductance -- <img src="./images/ee564/total_slot_leakage.png" alt="Drawing" style="width: 750px;"/> --- # Total Slot Leakage Inductance -- ### \\(L\_u = \dfrac{Q}{am}\dfrac{1}{a} L\_{u1}\\) -- ### \\(L\_u = \mu\_0 l' \dfrac{Q}{m} (\dfrac{z\_Q}{a})^2 \lambda\_{u}\\) -- ### \\(N= \dfrac{Q}{2am} z\_Q\\) -- ### \\(L\_u = \mu\_0 l' \dfrac{4m}{Q} N^2 \lambda\_{u}\\) --- # End Winding Leakage Flux <img src="./images/end_winding_leakage.png" alt="Drawing" style="width: 600px;"/> --- # End Winding Leakage Flux <img src="./images/end_winding_leakage2.png" alt="Drawing" style="width: 600px;"/> --- # Slot Shapes <img src="./images/slot_leakage_shapes.png" alt="Drawing" style="width: 700px;"/> --- # Tooth Tip Leakage Flux -- <img src="./images/ee564/tooth_leakage.png" alt="Drawing" style="width: 650px;"/> --- ## You can download this presentation from: [keysan.me/ee564](http://keysan.me/ee564)