EE361-Electromechanical Energy Conversion

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# EE-361
# PRACTICAL TRANSFORMERS

## Ozan Keysan

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

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

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# Review: Losses in Magnetic Circuits

## Hysteresis Loss: Proportional to Frequency

<img src="http://www.coolmagnetman.com/images/hysteresis.png" alt="Drawing" style="width: 600px;"/>
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# Losses in Magnetic Circuits

## Eddy Current Loss: Proportional to Frequency\$^2\$

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# Eddy Current Losses

### Sometimes Useful (e.g. Eddy current brakes)
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<img src="https://i5.walmartimages.com/asr/6e0716e0-aa91-4b6b-b730-8e0d49acab67_1.cc98a42515430713a00bfd344d22b7e3.jpeg" alt="Drawing" style="width: 300px;"/>

### How does [elliptical machine work](https://youtu.be/ekZkN_jQkZk?t=3m21s)?

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# Eddy Current Losses

### Sometimes NOT Useful (e.g. reduces efficiency in motors)
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## How can you minimize it? ([Hint](https://www.youtube.com/watch?v=Ttl7_q--CYI))

<img src="./images/ee361/eddy_lamination.png" alt="Drawing" style="width: 400px;"/>
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# Eddy Current Losses

## Use laminations instead of a solid core 
### Thin laminations increases the resistance of eddy current path
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### [Transformer Lamination Stamping](https://www.youtube.com/watch?v=e0MbosqTnk8)
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# Practical Transformers

- ## Losses
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 (Copper and Core)
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- ## Leakage flux
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- ## Magnetizing current
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- ## Hysteresis

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# Realistic Equivalent Circuit of a Transformer

## = Ideal Transformer + External Impedances
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![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Transformer_equivalent_circuit-2.svg/786px-Transformer_equivalent_circuit-2.svg.png)

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# Winding Resistances: (\$R_1, R_2\$) or (\$R_p, R_s\$)

- ## Resistance of the copper
   [Winding machine](http://www.youtube.com/watch?v=I_kJCLWwI2Y), [Winding for large transformers](http://youtu.be/yYFm3vxOVgw?t=24s), [Toroidal winding](https://www.youtube.com/shorts/4ibYv2cnPy4)
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- ### Question: In a step-down transformer which one is larger?
###  \$R_1\$ or \$R_2\$?

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- ## Copper Losses: \$I^2 R \$

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[Overloaded Transformers](https://youtu.be/D8EQPx-ptKk?t=41)

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# Leakage Flux
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## Leakage Reactance: \$ X_{leakage} \$
![](http://www.electrical4u.com/images/transformer-core-winding.gif)

- ## Leakage through Air (Linear B-H, no hysteresis)

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# How to minimize leakage flux?
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- ### Windings are wound on top of each other

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# Magnetizing Flux
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## A term represents the necessary flux in the core.

##Magnetizing Reactance: \$ X_{m} \$

![](http://www.electrical4u.com/images/transformer-core-winding.gif)

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# Core Losses

- ## Hysteresis Loss (\$\propto\$ Frequency)

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# Core Losses

- ## Eddy Current Loss (\$\propto\$ Frequency\$^2\$)

![](https://cdn2.magcraft.com/images/content/eddy-current-braking.png)

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## How to minimize eddy current loss?

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### Use Laminated Core
<img src="http://www.electrical-forensics.com/Transformers/L05-04/TransIronCore-LG.jpg" alt="Drawing" style="width: 700px;"/>
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## How to minimize eddy current loss?
## Use Laminated Core

<img src="./images/ee361/transformer_core.webp" alt="Drawing" style="width: 900px;"/>
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#Excitation Current

## $$I_e(t) = I_c(t) + I_m(t)$$
### Excitation Current = Core Loss Component + Magnetizing Component
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# Realistic Equivalent Circuit of a Transformer

![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Transformer_equivalent_circuit-2.svg/750px-Transformer_equivalent_circuit-2.svg.png)

### \$V_p , V_s, I_p, I_s \$: Primary, secondary voltages and currents
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### \$R_p , R_s\$: Primary, secondary coil resistances
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### \$X_p , X_s\$: Leakage Reactances
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# Realistic Equivalent Circuit of a Transformer

![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Transformer_equivalent_circuit-2.svg/750px-Transformer_equivalent_circuit-2.svg.png)

### \$R_c \$: Core Loss Resistance (to model hystresis/eddy current losses)

### \$X_m\$: Magnetizing Reactance (due to non-zero \$\int H dl\$ in the core)
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### \$E_p , E_s\$: Induced voltages in the ideal transformer

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## There are a few ways to make analyze easier:

![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Transformer_equivalent_circuit-2.svg/750px-Transformer_equivalent_circuit-2.svg.png)
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# Transfer Secondary Parameters to the Primary Side

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## Transfer Secondary Parameters to the Primary Side

## Multiply with \$(N_p/N_s)^2\$

<img src="./images/Transformer_equivalent_circuit_referred.svg" alt="Drawing" style="width: 800px;"/>
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## Transfer Secondary Parameters to the Primary Side

## Multiply with \$(N_p/N_s)^2\$

### Optional: Transfer voltage and current to get rid of the ideal transformer

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# Simplification#2:
## Move parallel branch to the source side
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# Simplification#2:
## Move parallel branch to the source side

### Assumption voltage drop in \$X_p\$ and \$R_p\$ is negligible

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# Simplification:
## Combine Primary and Secondary Series Impedances
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## Combine Primary and Secondary Series Impedances

### \$R_{eq}=R_1+R'_2 = R_1 + (\frac{N_p}{N_s})^2 R_2\$
### \$X_{eq}=X_1+X'_2 = X_1 + (\frac{N_p}{N_s})^2 X_2\$

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# Even More Simplification:
## Neglect Core Loss and Magnetizing Branch
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## Assumption: \$I_c\$ + \$I_m\$ small enough compared to \$I_p\$

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# For Large Transformers (>100 kVA)
## Efficiency > 99% (see [Siemens brochure](https://assets.new.siemens.com/siemens/assets/api/uuid:6eb3018983d08ec8ff18f87c5f5f14b70d82669f/version:1506684852/power-transformers-from-30-to-over-1300mva-en.pdf))
## Copper resistances (\$R_{eq}\$) can be neglected
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# For Large Transformers (>100 kVA)
## Efficiency > 99% (see [Siemens brochure](http://www.energy.siemens.com/hq/pool/hq/power-transmission/Transformers/Power%20Transformers/power-transformers-from-30-to-over-1300MVA_EN.pdf))
## Copper resistances (\$R_{eq}\$) can be neglected

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#Examples:

# Problem 1
### At no-load a transformer has a loss of 50 W and draws a current of 2 A, when the applied voltage is 230 Vrms.

### Assuming the parallel branch is moved to the primary side, find:

### a) No-load power factor,

### b) Core-loss current (\$I_c\$),

### c) Magnetizing current (\$I_m\$),

### d) Xm, Rc

# Phasors for Transformers

# Resistive Load

# Inductive Load

#Problem 2

### A 2400-V:460-V transformer has a series leakage reactance of 37.2 Ω as referred to the high-voltage side. Asssume the parallel branch is neglected.

### A load connected to the low-voltage side is absorbing 25 kW at unity power factor, and the voltage at the secondary-side is 450 V.

### Calculate the corresponding voltage and power factor as measured at the high-voltage terminals.

# Problem 3

### The resistances and leakage reactances of a 60-kVA, 60-Hz, 2400-V:240-V distribution transformer are

### R1 = 0.68 Ω  R2 = 0.0068 Ω
### Xl1 = 7.8 Ω  Xl2 = 0.0780 Ω

### a. Draw the equivalent circuit referred to (i) the high- and (ii) the low-voltage sides. Neglect the parallel branch.
# Problem 3

### The resistances and leakage reactances of a 60-kVA, 60-Hz, 2400-V:240-V distribution transformer are

### R1 = 0.68 Ω  R2 = 0.0068 Ω
### X1 = 7.8 Ω  X2 = 0.0780 Ω

### b. The transformer delivers rated kVA to a load on the low-voltage side with a lagging power factor of 0.85. If the voltage on the load is  230V.

### Find the terminal voltage of the high-voltage side.

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## You can download this presentation from: [keysan.me/ee361](http://keysan.me/ee361)