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# EE-568 Selected Topics in Electrical Machines


## Main Flux Paths

## Ozan Keysan

[keysan.me](http://keysan.me)

Office: C-113 <span class="meta">&#8226;</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)

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