EE464-Static Power Conversion-II

class: center, middle

# EE-464 STATIC POWER CONVERSION-II

# Three Phase Inverters

## Ozan Keysan

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

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

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# Three Phase Inverters

### Different Sized Variable Frequency Drives (VFD)

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# Three Phase Inverters

<img src="http://in.element14.com/productimages/large/en_GB/42270890.jpg" alt="Drawing" style="width:800px;">
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# Three Phase Voltage-Source Inverters
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### Three inverter legs are connected in parallel

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# Three Phase Voltage-Source Inverters
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- ## Do not close top and bottom switches at the same time
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- ## Point (o) is not needed put shown for simplicity in calculations
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- ## Current can flow through the switch or anti-parallel diodes.
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# PWM Techniques
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### There are many different PWM techniques that will be covered:

- ## Square-wave (Six-step) PWM

- ## Sinusoidal PWM (SPWM)

- ## Hysteresis (Bang-Bang) Control

- ## Space-Vector PWM (SVPWM)

- ## Third harmonic injection
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# Six-Step Inverter

### Commonly used in BLDC motor Drives

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# Six-Step Inverter

### Commonly used in BLDC motor Drives

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# Six-Step Inverter

- ### Each switch has 50% duty ratio.

- ### Each leg has a phase difference of 120 degrees

- ### One switching action takes place at every 60 degrees

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# Six-Step Inverter

<img src="./images/ee464/six_step.png" alt="Drawing" style="width: 800px;">
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# Six-Step Inverter

<img src="./images/ee464/six_step_switches.png" alt="Drawing" style="width: 750px;">
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# Six-Step Inverter

## Line-to-line voltage: \\(V\_{AB}=V\_{A0}-V\_{B0}\\)

<img src="./images/ee464/six_step_vab.png" alt="Drawing" style="width: 750px;">
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## Line-to-line voltages:

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# Square Wave Operation

### [BLDC Drive with square wave](https://www.youtube.com/watch?v=IiY01xIKg28)
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## Switching Sequence

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## Line-to-Line Voltages

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## Equivalent Phase Voltages

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## Line-to-Line Harmonics

## Fourier Coefficients

### \\(\hat{V}\_{n,l-l} = \dfrac{1}{n}\dfrac{4}{\pi}V\_{dc}cos(n \dfrac{\pi}{6})\\)

### For: \\(n = 6k \pm 1 = 1,5,7,11,13...\\)

- ### No even harmonics

- ### No third order harmonics
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## Line-to-Line Harmonics

## RMS of the fundamental component?

### \\({V}\_{1,l-l,rms} = \dfrac{1}{\sqrt{2}}\dfrac{4}{\pi}V\_{dc} \dfrac{\sqrt{3}}{2}=0.78 V\_{dc}\\)

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### Harmonics RMS:

### \\({V}\_{n,l-l,rms} = \dfrac{1}{n} 0.78 V\_{dc}\\)

### For: \\(n = 6k \pm 1 = 1,5,7,11,13...\\)

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## Line-to-Line Harmonics

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## Line-to-Neutral voltages:

### Neutral point is floating

### Voltage level changes every 60 degrees (that's why it's a six-step inverter!)

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## Line-to-Neutral Harmonics

## Fourier Coefficients

### \\(\hat{V}\_{n,l-N} = \dfrac{1}{n}\dfrac{2}{3\pi}V\_{dc}(2 +cos(\dfrac{\pi n}{3}) - cos(\dfrac{2\pi n}{3}))\\)

### For: \\(n = 6k \pm 1 = 1,5,7,11,13...\\)

### Simpler Form

### \\(\hat{V}\_{n,l-N} = \dfrac{1}{n}\dfrac{2}{\pi}V\_{dc}\\)
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## Line-to-Neutral Harmonics

### \\(\hat{V}\_{n,l-N} = \dfrac{1}{n}\dfrac{2}{\pi}V\_{dc}\\)

### For: \\(n = 6k \pm 1 = 1,5,7,11,13...\\)

- ### No even harmonics

- ### No third order harmonics

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##Example: (D. Hart . 8-12)
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### For the six-step three phase inverter shown below, Vin=100V, \\(f\_{out}=60Hz\\). The load is Y-connected to load with a phase load of \\(R=10 \Omega\\), \\(L=20 mH\\).

### Calculate the total harmonic distortion (THD) of the load current and voltage.

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##Example: (D. Hart . 8-12)

### Amplitude for load current at each frequency:

### \\(I\_n=\dfrac{V\_{n,L-N}}{Z\_n}\\)
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\\(=\dfrac{V\_{n,L-N}}{\sqrt{R²+(n \omega L)²}}\\)

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##Example: (D. Hart . 8-12)

### Amplitude for load current at each frequency:

### \\(I\_n=\dfrac{V\_{n,L-N}}{Z\_n} = \dfrac{V\_{n,L-N}}{\sqrt{10²+(n 2 \pi 60 \, 0.02)²}}\\)
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### Line-to-Neutral Harmonics

### \\(\hat{V}\_{n,l-N} = \dfrac{1}{n}\dfrac{2}{\pi}V\_{dc}\\)

### For: \\(n = 6k \pm 1 = 1,5,7,11,13...\\)

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### Voltage THD=
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#### \\(\dfrac{\sqrt{ \sum\_{n=2}^{\infty} V\_n²}}{V\_{1,rms}} \approx \dfrac{\sqrt{12.73²+9.09²+5.79²+4.90²}}{63.6}=0.31=31\%\\)

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### Current THD=
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#### \\(\dfrac{\sqrt{ \sum\_{n=2}^{\infty} I\_n²}}{I\_{1,rms}} \approx \dfrac{\sqrt{0.23²+0.12²+0.05²+0.04²}}{3.59}=0.07=7\%\\)

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### [Voltage Plot](https://www.google.com.tr/search?ei=1E_DWtwJhZLoBITenLgK&q=sin%28x%29%2Bsin%285x%29%2F5%2Bsin%287x%29%2F7%2Bsin%2811x%29%2F11%2Bsin%2813x%29%2F13&oq=sin%28x%29%2Bsin%285x%29%2F5%2Bsin%287x%29%2F7%2Bsin%2811x%29%2F11%2Bsin%2813x%29%2F13&gs_l=psy-ab.3...3238.6702.0.9540.3.3.0.0.0.0.154.392.0j3.3.0....0...1c.1.64.psy-ab..0.0.0....0.ut757zYR-Lg), [Current Plot](https://www.google.com.tr/search?ei=_tjEWrz2McHv6QSByb3IBQ&q=plot+5.08*sin%28x%29%2B0.33*sin%285x%29%2B0.17*sin%287x%29%2B0.07*sin%2811x%29%2B0.05*sin%2813x%29&oq=plot+5.08*sin%28x%29%2B0.33*sin%285x%29%2B0.17*sin%287x%29%2B0.07*sin%2811x%29%2B0.05*sin%2813x%29&gs_l=psy-ab.3...6798.8060.0.8402.5.5.0.0.0.0.298.740.0j1j2.3.0....0...1c.1.64.psy-ab..2.0.0....0.ijL6lnnGqOQ)

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# Three Phase Voltage-Source Inverter
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# Sinusoidal PWM (SPWM)
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### A triangular carrier wave is generated and compared with each phase.
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<img src="./images/ee464/3ph_carrier.png" alt="Drawing" style="width: 750px;">
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# Sinusoidal PWM (SPWM)
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### Vd or 0 voltage is generated at \\(V\_{AN}\\) depending on the comparison.
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# Sinusoidal PWM (SPWM)
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### Line to line voltage (\\(V\_{AB}=V\_{AN}-V\_{BN}\\))
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# Sinusoidal PWM (SPWM)

## Harmonics in the line voltage
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### Harmonics at the side bands,
### Like the unipolar but starts at mf.
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# Sinusoidal PWM (SPWM)

## Harmonics in the line voltage
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### If mf is small, it is better to use synchronized PWM, and mf should be an odd interger, preferably multiple of 3 to reduce harmonics.

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# Sinusoidal PWM (SPWM)

## Harmonics in the line voltage
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# Voltage Levels?
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## Linear Region (\\(m_a \lt 1\\))
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### \\(\hat{V}\_{AN1}=m\_a\dfrac{V\_d}{2}\\)
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### \\(V\_{l-l, rms}= \dfrac{\sqrt{3}}{\sqrt{2}}m\_a\dfrac{V\_d}{2}\\)
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### \\(V\_{l-l, rms}= 0.612 V\_d\\) (max in linear region)

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# Voltage Levels?
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## Overmodulation (\\(m_a \gt 1\\))
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## Square-Wave Operation?
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### \\(V\_{l-l, rms}= \dfrac{\sqrt{3}}{\sqrt{2}}\dfrac{4}{\pi}m\_a\dfrac{V\_d}{2}\\)
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### \\(V\_{l-l, rms}= 0.78 V\_d\\)
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### \\(V\_{l-l, rms,h}= \dfrac{0.78}{h} V\_d\\)  for  \\(h=6n \pm 1\\)

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# Voltage Levels?
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## You can download this presentation from: [keysan.me/ee464](http://keysan.me/ee464)

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## Harici Slaytlar

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# Push-Pull Inverter
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## Similar to Push-Pull Converter

<img src="./images/ee464/push_pull.png" alt="Drawing" style="width: 600px;">
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## But without the rectifying diodes

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# Push-Pull Inverter
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<img src="./images/ee464/push_pull_inverter.png" alt="Drawing" style="width: 600px;">
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## T1, T2 operates in sequence
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# Push-Pull Inverter
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## Voltage output can be adjusted by the turns-ratio
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## \\(\hat{V}\_{o1} = m\_a \dfrac{V\_d}{n}\\)
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# Push-Pull Inverter

## Advantages?
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- ### Only single transistor conduct at a time,
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small voltage drop

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- ### Especially important for low voltage applications (e.g. fed from a battery)
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- ### There are a few PV applications as well
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- ### Transistors have common ground (no isolation required for gate drives)

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# Push-Pull Inverter

## Disadvantages?
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- ### What is the required voltage rating of the transistors?
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### \\(V_T = 2 V_d\\)

### Therefore may not be practical for higher input voltages
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- ### A good transformer, with high coupling is required (to reduce energy in the leakage inductance)

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# Switch Utilization in Single Phase Inverters
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## Ratio of the output power to max. power capacity of the switches
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## Assume highly inductive load, no current harmonics (just the fundamental)
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### \\(= \dfrac{V\_{o1}I\_{o,max}}{q V\_T I\_T}\\)

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## Maximum utilization occurs at square wave

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# Switch Utilization in Single Phase Inverters
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## Half Bridge Inverter
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### Voltage rating?
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### \\(V\_T = V\_{d,max}\\)
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### Current rating?
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### \\(I\_T = \sqrt{2} I\_{o,max}\\)
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# Switch Utilization in Single Phase Inverters
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## Half Bridge Inverter
### Output Voltage: \\(  V\_{o1,max} = \dfrac{4}{\pi \sqrt{2}} \dfrac{V\_{d,max}}{2}\\)
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### Max. Switch Utilization
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### q=2

### \\(= \dfrac{1}{2\pi} \approx 0.16\\) 
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# Switch Utilization in Full Bridge Inverter
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## Voltage, Current Ratings?
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: Same with Half Bridge
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## q=
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4
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## Voltage output =
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twice of the half bridge
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## Switch utilization=?
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### \\(= \dfrac{1}{2\pi} \approx 0.16\\)

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# Switch Utilization in Push-Pull Inverter
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## Voltage, Current Ratings?
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### \\(V\_T = 2 V\_{d,max}\\)
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### \\(I\_T = \sqrt{2} \dfrac{I\_{o,max}}{n}\\)
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### Output Voltage: \\(  V\_{o1,max} = \dfrac{4}{\pi \sqrt{2}} \dfrac{V\_{d,max}}{n}\\)

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# Switch Utilization in Push-Pull Inverter
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## q =2
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## Switch Utilization
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### \\(= \dfrac{1}{2\pi} \approx 0.16\\)

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# Switch Utilization in Linear Region
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### \\(= \dfrac{1}{2\pi}\dfrac{\pi}{4} m\_a = \dfrac{1}{8} m\_a\\)
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## Linear Region

### \\(= 0.125 \\) when  \\(m\_a=1\\)