EE564-Design of Electrical Machines

class: center, middle

# EE-564 Design of Electrical Machines

## Ozan Keysan

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

Office: C-113 <span class="meta">•</span> Tel: 210 7586

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# Estimation of Machine Parameters
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## Some definitions:
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- ## Effective Core length

- ## Carter's Coefficient

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# Equivalent Core Length

### Section 3.2 of the textbook
<img src="./images/L_eq.png" alt="Drawing" style="width: 600px;"/>

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# Carter's Coefficient
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## Way to estimate the flux density by converting slotted rotor/stator to a perfect cylinder.

### Ref: Section 3.1.1 of the textbook

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# Carter's Coefficient

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# Carter's Coefficient
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- ## First assume the rotor is smooth to find \$k\_{cs}\$:

## \$ \delta\_e = k\_{cs} \delta \$

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- ## 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} \$:

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# Carter's Coefficient

## \$k\_c = \dfrac{\tau\_u}{\tau\_u - K b\_1}\$

## where:

## \$K = \dfrac{b\_1 / \delta }{5 + b\_1 / \delta }\$

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# Example 3-1: 
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### Calculate the Carter's coefficient for an air gap (\$\delta = 0.8 \$) mm. The slot opening \$b_1\$ is 3 mm. Rotor slots are closed and the stator slot pitch is 10 mm.

### How much current is required to magnetize the airgap to 0.9 T (fundamental), if Ns=100, pole number=4, q=3, full-pitched winding.

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## Back to Equivalent Core Length
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## Equivalent Core Length with Cooling

### Larger machines requires ducts for cooling
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## Need to calculate the effective length of the opening
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## Use the same technique
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## Example:
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### In a machine, stator core is 990 mm. There are 25 stacks of 30 mm long, with 10 mm cooling ducts (24 ducts). Airgap is 3 mm.

### Calculate the effective core length if the rotor surface is smooth.
 
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# Flux Density in a Slot
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## \$\tau_u = \dfrac{\pi D}{Q_s}\$

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# Flux Density in a Slot
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# Example

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# Back Core Flux
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# D-Q Axis

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# D-Q Axis

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# D-Q Axis

<img src="./images/ee564/dq_axis.png" alt="Drawing" style="width: 550px;"/>
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# Back Core Flux

### For magnetic voltage calculations refer to pg. 178

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# Example

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## Magnetizing Current

### Can be linearized by Carter's coefficient (use \$\delta_e\$)  (see section 3.5 for details)
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# Magnetizing Inductance

## What is inductance?
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# Magnetizing Inductance

## \$\Phi\_{pole} = \int B dS\$
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\$=\dfrac{2}{\pi}\hat{B}\tau\_{pole}l'\$

### if B is sinusoidal

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# Magnetizing Inductance (Per-phase)
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### \$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\$
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### Writing pole pitch in terms of diameter
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### \$L\_{m (ph)} = \dfrac{2 \mu\_0 D}{\pi p^2 \delta\_{ef}} l' (k\_{ws}N\_s)^2  \$

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# Magnetizing Inductance (Total)
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### For 3-phase machines

### \$L\_{m} = \dfrac{3}{2} L\_{m (ph)} \$

### \$L\_{m} = \dfrac{3 \mu\_0 D}{\pi p^2 \delta\_{ef}} l' (k\_{ws}N\_s)^2  \$

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# Magnetizing Inductance
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- ## Increases with number of turns
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- ## Reduces with large airgap
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- ## Reduces with the number of poles (power factor is worse at machines with higher number of poles)
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# Magnetizing Inductance
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- ## Reduces with increasing voltage:
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Saturation

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- ## Reduces with torque: Why?

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# Flux Lines vs Torque

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# 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;"/>

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# Magnetizing Inductance

#### Magnetizing inductance variation of a 4-pole machine 
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# Leakage Flux (Ch4)
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## Flux that does not cross the airgap
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## Flux crosses the airgap but does not link the winding

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# Leakage Flux

## Flux that does not cross the airgap
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- ### Pole Leakage Flux
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- ### Slot Leakage Flux
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- ### Tooth Tip Leakage Flux
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- ### End Winding (Overhang) Leakage Flux

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# Pole Leakage Flux

### In salient pole machines (i.e. synchronous machine)

<img src="./images/pole_leakage.png" alt="Drawing" style="width: 700px;"/>
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# Slot-Tooth Tip Leakage Flux

<img src="./images/slot_leakage.png" alt="Drawing" style="width: 700px;"/>
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# End Winding Leakage Flux

<img src="./images/end_winding_leakage.png" alt="Drawing" style="width: 600px;"/>
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# End Winding Leakage Flux

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# Air-Gap Leakage Inductance (\$L\_{\delta}\$)

### Models the higher order harmonics in the induced voltage 
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### \$E\_v = \dfrac{\mu\_0}{\pi} \omega \dfrac{m}{\delta} D l' (\dfrac{N}{p})^2 I\_m (\dfrac{k\_{wv}}{v})^2\$

### \$E = \sum E_v \$
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### \$E = \omega I\_m (L\_m + L\_\delta)\$

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# Air-Gap Leakage Inductance (\$L\_{\delta}\$)

### Models the higher order harmonics in the induced voltage 
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### \$L\_\delta = \dfrac{\mu\_0}{\pi}\dfrac{m}{\delta} D l' (\dfrac{N}{p})^2 \sum\_{v \neq 1 }(\dfrac{k\_{wv}}{v})^2\$

### \$L\_\delta = \sigma\_\delta L\_m \$
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### \$\sigma\_\delta \$: leakage factorof the airap inductance

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# Air-Gap Leakage Inductance (\$L\_{\delta}\$)

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# Slot Leakage Inductance
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<img src="./images/ee564/slot_leakage.png" alt="Drawing" style="width: 700px;"/>
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# Slot Leakage Inductance
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### for the bottom part (\$L\_{u1}\$)

### \$B(h) = \mu\_0 H (h) = \mu\_0 \dfrac{z\_Q I \dfrac{h}{h\_4}}{b\_4} \$
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### \$L\_{u1} = \dfrac{l' b\_4}{\mu\_0 I^2} \int_0^{h\_4} B^2(h) dh\$

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# Slot Leakage Inductance
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### repeat for the upper part (\$L\_{u2}\$)

### \$B(h) = \mu\_0 \dfrac{z\_Q I}{b\_1} \$   (Constant B)
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# Slot Leakage Inductance
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### 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)\$

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# Total Slot Leakage Inductance
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# Total Slot Leakage Inductance
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### \$L\_u = \dfrac{Q}{am}\dfrac{1}{a} L\_{u1}\$
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### \$L\_u = \mu\_0 l' \dfrac{Q}{m} (\dfrac{z\_Q}{a})^2 \lambda\_{u}\$
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### \$N= \dfrac{Q}{2am} z\_Q\$
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### \$L\_u = \mu\_0 l' \dfrac{4m}{Q} N^2 \lambda\_{u}\$
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# Ex. 4.2:

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# Slot Shapes

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# Tooth Tip Leakage Flux
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# Leakage Inductance

#### Leakage inductance variation of a 4-pole machine

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# Reading Assignment: Chapter 4

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# Losses in Magnetic Circuits
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- ## Hysteresis Losses
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- ## Eddy Current Losses
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# Hysteresis Losses

## Steinmetz's [Equation](http://web.eecs.utk.edu/~dcostine/ECE482/Spring2015/materials/magnetics/CoreLossTechniques.pdf)
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## \$P\_{hy}= \eta V f B\_{max}^k\$

- ### \$f\$: Frequency
- ### \$V\$: Volume
- ### \$B\_{max}\$: Maximum flux density
- ### \$\eta\$,\$k\$: Constants depending on the material properties. k is typically 1.6 (or between 1.5 and 2.5)
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# Eddy Current Losses
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### [Catalogues](http://www.sura.se/Sura/hp_products.nsf/vOpendocument/03A8B2433FAE16C4C1256AA8002280E6/$FILE/NO-08.pdf) usually give combined eddy and hysteresis losses
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# Magnetic Losses
## Beware of harmonics (especially with an inverter)

### With Sinusoidal Supply
<img src="https://raw.githubusercontent.com/ozank/ozank.github.io/master/presentations/images/loss_sinusoidal.png" alt="Drawing" style="width: 400px;"/>
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# Magnetic Losses
## Beware of harmonics (especially with an inverter)

### With Inverter
<img src="https://raw.githubusercontent.com/ozank/ozank.github.io/master/presentations/images/loss_inverter.png" alt="Drawing" style="width: 400px;"/>

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## You can download this presentation from: [keysan.me/ee564](http://keysan.me/ee564)