This document summarizes inverters and their operation. It begins with an introduction that defines inverters as devices that convert DC to AC power by switching the DC input voltage in a predetermined sequence. It then discusses the basic principles of inverters including single-phase half-bridge and full-bridge inverter circuits. Fourier series analysis is introduced as a tool to analyze the output waveforms of inverters in terms of harmonic components. The document concludes with a discussion of total harmonic distortion as a measure of output waveform quality.
Pulse width modulation (PWM) is a method of changing the duration of a pulse with respect to the analog input. The duty cycle of a square wave is modulated to encode a specific analog signal level. This pulse width modulation tutorial gives you the basic principle of generation of a PWM signal. The PWM signal is digital because at any given instant of time, the full DC supply is either ON or OFF completely. PWM method is commonly used for speed controlling of fans, motors, lights in varying intensities, pulse width modulation controller etc. These signals may also be used for approximate time-varying of analogue signals. Below you can see the pulse width modulation generator circuit diagram (pulse width modulator) using op amp. PWM is employed in a wide variety of applications, ranging from measurement and communications to power control and conversion. Pulse width modulation dc motor control is one of the popular circuits in Robotics.
Part of Lecture series on EE321N, Power Electronics-I delivered by me during Fifth Semester of B.Tech. Electrical Engg., 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Please comment and feel free to ask anything related. Thanks!
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively
Introduction
Band Pass Amplifiers
Series & Parallel Resonant Circuits & their Bandwidth
Analysis of Single Tuned Amplifiers
Analysis of Double Tuned Amplifiers
Primary & Secondary Tuned Amplifiers with BJT & FET
Merits and de-merits of Tuned Amplifiers
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
Pulse width modulation (PWM) is a method of changing the duration of a pulse with respect to the analog input. The duty cycle of a square wave is modulated to encode a specific analog signal level. This pulse width modulation tutorial gives you the basic principle of generation of a PWM signal. The PWM signal is digital because at any given instant of time, the full DC supply is either ON or OFF completely. PWM method is commonly used for speed controlling of fans, motors, lights in varying intensities, pulse width modulation controller etc. These signals may also be used for approximate time-varying of analogue signals. Below you can see the pulse width modulation generator circuit diagram (pulse width modulator) using op amp. PWM is employed in a wide variety of applications, ranging from measurement and communications to power control and conversion. Pulse width modulation dc motor control is one of the popular circuits in Robotics.
Part of Lecture series on EE321N, Power Electronics-I delivered by me during Fifth Semester of B.Tech. Electrical Engg., 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Please comment and feel free to ask anything related. Thanks!
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively
Introduction
Band Pass Amplifiers
Series & Parallel Resonant Circuits & their Bandwidth
Analysis of Single Tuned Amplifiers
Analysis of Double Tuned Amplifiers
Primary & Secondary Tuned Amplifiers with BJT & FET
Merits and de-merits of Tuned Amplifiers
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
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students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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2. CONTENTS
۩ Introduction
۩ Basic Principles of Inverter
۩ Single-phase Half-Bridge Square-Wave Inverter
۩ Single-phase Full-Bridge Square-Wave Inverter
۩ Quasi Inverter
۩ Three-phase inverter
۩ Fourier Series and Harmonics Analysis
۩ Pulse-Width Modulation (PWM)
3. Definition:
Converts DC to AC power by switching the DC input
voltage in a pre-determined sequence so as to
generate AC voltage.
Applications:
Induction motor drives, traction, standby power
supplies, and uninterruptible ac power supplies (UPS).
INTRODUCTION
5. Three types of inverter:
INTRODUCTION
Vdc Vac
+
-
+
-
AC
Load
Iac
Inverter
switching
control
“DC LINK”
Vdc
+
-
AC
Load
ILOAD
Inverter
switching
control
L
IDC
(a) Voltage source inverter (VSI) (b) Current source inverter (CSI)
6. Three types of inverter: (cont.)
INTRODUCTION
Vdc
+
-
AC
Load
Iac
Inverter
switching
control
Comparison
circuit
Reference
Waveform
Output current
sensing circuit
(c) Current regulated inverter
7. BASIC PRINCIPLES
The schematic of single-phase full-bridge square-wave
inverter circuit
D1
D2
D3
D4
T1 T3
T4 T2
+ V0 -
I0
VDC
10. The total RMS value of the load output voltage,
The instantaneous output voltage is: (refer to slide 30/pp:88)
The fundamental rms output voltage (n=1)is
Single-phase Half-bridge
Square-wave Inverter
2
2
2 2
/
0
2
DC
T
DC
O
V
dt
V
T
V
2,4,....
n
for
0
sin
2
,...
5
,
3
,
1
n
DC
O t
n
n
V
v
DC
DC
O V
V
V 45
.
0
2
1
2
1
11. In the case of RL load, the instantaneous load current io ,
where
The fundamental output power is
Single-phase Half-bridge
Square-wave Inverter
,..
5
,
3
,
1
2
2
)
sin(
)
(
2
n
n
DC
o t
n
L
n
R
n
V
i
)
/
(
tan 1
R
L
n
n
2
2
2
1
1
1
1
1
)
(
2
2
cos
L
R
V
R
I
I
V
P
DC
o
o
o
o
12. The total harmonic distortion (THD),
Single-phase Half-bridge
Square-wave Inverter
,....
7
,
5
,
3
2
1
1
n
n
O
V
V
THD
2
1
2
1
1
o
o
O
V
V
V
THD
13. Example 3.1
The single-phase half-bridge inverter has a resistive load of R =
2.4Ω and the DC input voltage is 48V. Determine:
(a) the rms output voltage at the fundamental frequency
(b) the output power
(c) the average and peak current of each transistor.
(d) the THD
Single-phase Half-bridge
Square-wave Inverter
14. Solution
VDC = 48V and R = 2.4Ω
(a) The fundamental rms output voltage,
Vo1 = 0.45VDC = 0.45x48 = 21.6V
(b) For single-phase half-bridge inverter, the output voltage
Vo = VDC/2
Thus, the output power,
Single-phase Half-bridge
Square-wave Inverter
W
R
V
P o
o
240
4
.
2
)
2
/
48
(
/
2
2
15. Solution
(c) The transistor current Ip = 24/2.4 = 10 A
Because each of the transistor conducts for a 50% duty cycle,
the average current of each transistor is IQ = 10/2 = 5 A.
(d)
Single-phase Half-bridge
Square-wave Inverter
2
1
2
1
1
o
o
O
V
V
V
THD
%
34
.
48
45
.
0
2
45
.
0
1 2
2
DC
DC
DC
V
V
xV
16. The switching in the second leg is delayed by 180
degrees from the first leg.
The maximum output voltage of this inverter is twice
that of half-bridge inverter.
Single-phase Full-bridge
Square-wave Inverter
17. The output RMS voltage
And the instantaneous output voltage in a Fourier series is
The fundamental RMS output voltage
In the case of RL load, the instantaneous load current
Single-phase Full-bridge
Square-wave Inverter
DC
DC
O V
dt
V
T
V
2
2
,...
5
,
3
,
1
sin
4
n
DC
O t
n
n
V
v
DC
DC
V
V
V 9
.
0
2
4
1
n
n
DC
o t
n
L
n
R
n
V
i
sin
4
,...
5
,
3
,
1
2
2
18. Example 3.2
A single-phase full-bridge inverter with VDC = 230 and
consist of RLC in series. If R = 1.2Ω, ωL = 8 Ω and
1/ωC = 7 Ω, find:
(a) The amplitude of fundamental rms output current,
io1
(b) The fundamental component of output current in
function of time.
(c) The power delivered to the load due to the
fundamental component.
Single-phase Full-bridge
Square-wave Inverter
19. Example 3.3
A single-phase full-bridge inverter has an RLC load with R
= 10Ω, L = 31.5mH and C = 112μF. The inverter frequency
is 60Hz and the DC input voltage is 220V. Determine:
(a) Express the instantaneous load current in Fourier series.
(b) Calculate the rms load current at the fundamental
frequency.
(c) the THD of load current
(d) Power absorbed by the load and fundamental power.
(e) The average DC supply current and
(f) the rms and peak supply current of each transistor
Single-phase Full-bridge
Square-wave Inverter
21. Viewed as extensions of the single-phase bridge circuit.
The switching signals for each switches of an inverter leg are
displaced or delayed by 120o.
With 120o conduction, the switching pattern is T6T1 – T1T2 –
T2T3 – T3T4 – T4T5 – T5T6 – T6T1 for the positive A-B-C
sequence.
When an upper switch in an inverter leg connected with the
positive DC rail is turned ON, the output terminal of the leg
(phase voltage) goes to potential +VDC/2.
When a lower switch in an inverter leg connected with the
negative DC rail is turned ON, the output terminal of that leg
(phase voltage) goes to potential -VDC/2.
Three-Phase Inverter
22.
23. The line-to-neutral voltage can be expressed in Fourier series
The line voltage is vab = √3van with phase advance of 30o
Three-Phase Inverter
,..
5
,
3
,
1
,..
5
,
3
,
1
,..
5
,
3
,
1
6
7
sin
3
sin
2
2
sin
3
sin
2
6
sin
3
sin
2
n
dc
cn
n
dc
bn
n
dc
an
t
n
n
n
V
v
t
n
n
n
V
v
t
n
n
n
V
v
,..
5
,
3
,
1
,..
5
,
3
,
1
,..
5
,
3
,
1
sin
3
sin
3
2
3
sin
3
sin
3
2
3
sin
3
sin
3
2
n
dc
ca
n
dc
bc
n
dc
ab
t
n
n
n
V
v
t
n
n
n
V
v
t
n
n
n
V
v
24. Fourier series is a tool to analyze the wave shapes of the
output voltage and current in terms of Fourier series.
Fourier Series and
Harmonics Analysis
2
0
1
d
v
f
ao
2
0
cos
1
d
n
v
f
an
2
0
sin
1
d
n
v
f
bn
Inverse Fourier
1
0 sin
cos
2
1
n
n n
bn
n
a
a
v
f
Where
t
25. If no DC component in the output, the output voltage and
current are
The rms current of the load can be determined by
Fourier Series and
Harmonics Analysis
1
sin
)
(
n
n
n
o t
n
V
t
v
1
sin
)
(
n
n
n
o t
n
I
t
i
2
1
1
2
,
2
n
n
n
rms
n
rms
I
I
I
Where
n
n
n
Z
V
I
26. The total power absorbed in the load resistor can be
determined by
Fourier Series and
Harmonics Analysis
1
2
,
1 n
rms
n
n
n R
I
P
P
27. Since the objective of the inverter is to use a
DC voltage source to supply a load that
requiring AC voltage, hence the quality of
the non-sinusoidal AC output voltage or
current can be expressed in terms of THD.
The harmonics is considered to ensure that
the quality of the waveform must match to
the utility supply which means of power
quality issues.
This is due to the harmonics may cause
degradation of the equipments and needs to
be de-rated.
Total Harmonics
Distortion
29. The THD of the load voltage is expressed as,
The current THD can be obtained by replacing the
harmonic voltage with harmonic current,
Total Harmonics
Distortion
rms
rms
rms
rms
n rms
n
v
V
V
V
V
V
THD
,
1
2
,
1
2
,
1
2
2
, )
(
rms
n rms
n
i
I
I
THD
,
1
2
2
, )
(
30. Harmonics of Square-
wave Waveform
2
t
DC
V
DC
V
0
1
0
2
0
DC
DC V
d
V
a
0
)
cos(
)
cos(
0
2
d
n
d
n
V
a DC
n
n
n
n
n
n
n
V
n
n
n
V
d
n
d
n
V
b
DC
DC
DC
n
cos
1
2
)
cos
2
(cos
)
cos
0
(cos
)
cos(
)
cos(
)
sin(
)
sin(
2
0
0
2
31. When the harmonics number, n of a waveform is
even number, the resultant of
Therefore,
When n is odd number,
Hence,
Harmonics of Square-wave
Waveform
1
cos
n
0
n
b
1
cos
n
n
V
b DC
n
4
35. Example 3.4
The full-bridge inverter with DC input voltage of 100V,
load resistor and inductor of 10Ω and 25mH
respectively and operated at 60 Hz frequency.
Determine:
(a) The amplitude of the Fourier series terms for the
square-wave load voltage.
(b) The amplitude of the Fourier series terms for load
current.
(c) Power absorbed by the load.
(d) The THD of the load voltage and load current for
square-wave inverter.
36. Amplitude and
Harmonics Control
2
t
VDC
-VDC
S2
S4
S1
S2
S1
S3
S3
S4
S2
S4
0 VDC 0 0
-VDC
S1 Closed Opened
S2
S3
S4
Vo
The output voltage of the full-
bridge inverter can be controlled
by adjusting the interval of on
each side of the pulse as zero .
37. The rms value of the voltage waveform is
The Fourier series of the waveform is expressed as
The amplitude of half-wave symmetry is
Amplitude and
Harmonics Control
2
1
)
(
1 2
DC
DC
rms V
t
d
V
V
odd
n
n
O t
n
V
t
v
,
)
sin(
)
(
)
cos(
4
)
(
)
sin(
2
n
n
V
t
d
t
n
V
V DC
DC
n
38. The amplitude of the fundamental frequency is controllable by
adjusting the angle of α.
The nth harmonic can be eliminated by proper choice of
displacement angle α if
Amplitude and
Harmonics Control
cos
4
1
DC
V
V
0
cos
n
OR
n
90
39. Pulse-width modulation provides a way to decrease
the total harmonics distortion (THD).
Types of PWM scheme
Natural or sinusoidal sampling
Regular sampling
Optimize PWM
Harmonic elimination/minimization PWM
SVM
Pulse-Width Modulation
(PWM)
40. Several definition in PWM
(i) Amplitude Modulation, Ma
If Ma ≤ 1, the amplitude of the fundamental
frequency of the output voltage, V1 is linearly
proportional to Ma.
(ii) Frequency Modulation, Mf
Pulse-Width Modulation
(PWM)
tri
,
sin
,
carrier
,
,
m
e
m
m
reference
m
a
V
V
V
V
M
e
tri
carrier
f
f
f
f
f
M
sin
reference
41. Bipolar Switching of PWM
Pulse-Width Modulation
(PWM)
vtri (carrier)
vsine (reference)
VDC
-VDC
(a)
(b)
S1 and S2 ON when Vsine > Vtri
S3 and S4 ON when Vsine < Vtri
43. Unipolar Switching
of PWM
Pulse-Width Modulation
(PWM)
vtri (carrier)
vsine (reference)
(a)
(b)
(c)
S4
S2
Vo
Vdc
0
Vdc
0
S1 is ON when Vsine > Vtri
S2 is ON when –Vsine < Vtri
S3 is ON when –Vsine > Vtri
S4 is ON when Vsine < Vtri
45. ■ Advantages of PWM switching
- provides a way to decrease the THD of load current.
- the amplitude of the o/p voltage can be controlled with the
modulating waveform.
- reduced filter requirements to decrease harmonics.
■ Disadvantages of PWM switching
- complex control circuit for the switches
- increase losses due to more frequent switching.
Pulse-Width Modulation
(PWM)
46. Harmonics of Bipolar PWM
Assuming the PWM output is symmetry, the
harmonics of each kth PWM pulse can be
expressed
Finally, the resultant of the integration is
PWM Harmonics
k
k
k k
k
t
d
t
n
-V
t
d
t
n
V
t
d
t
n
t
v
V
DC
DC
T
nk
1
k
)
(
)
sin(
)
(
)
(
)
sin(
2
)
(
)
sin(
)
(
2
0
)
(
cos
2
cos
cos
2
1 k
k
k
k
DC
nk n
n
n
n
V
V
48. Harmonics of Bipolar PWM
The Fourier coefficient for the PWM waveform is the
sum of Vnk for the p pulses over one period.
The normalized frequency spectrum for bipolar
switching for ma = 1 is shown below
PWM Harmonics
p
k
nk
n V
V
1
ma = 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
n=1 1.00 0.90 0.80 0.70 0.60 0.50 o.40 0.30 0.20 0.10
n = mf 0.60 0.71 0.82 0.92 1.01 1.08 1.15 1.20 1.24 1.27
n=mf +2 0.32 0.27 0.22 0.17 0.13 0.09 0.06 0.03 0.02 0.00
Normalized Fourier Coefficients Vn/Vdc for Bipolar PWM
49. Example 3.5
The inverter has a resistive load of 10Ω and
inductive load of 25mH connected in series with
the fundamental frequency current amplitude of
9.27A. The THD of the inverter is not more than
10%. If at the beginning of designing the inverter,
the THD of the current is 16.7% which is does
not meet the specification, find the voltage
amplitude at the fundamental frequency, the
required DC input supply and the new THD of the
current.
50. Example 3.6
The single-phase full-bridge inverter is used to produce
a 60Hz voltage across a series R-L load using bipolar
PWM. The DC input to the bridge is 100V, the
amplitude modulation ratio is 0.8, and the frequency
modulation ratio is 21. The load has resistance of R =
10Ω and inductance L = 20mH. Determine:
(a) The amplitude of the 60Hz component of the output
voltage and load current.
(b) The power absorbed by the load resistor
(c) The THD of the load current