Power Electronics
Winter 2012
Lecture 11
Dr. Walid Atef Omran
Today’s Lecture
Pulse Magnitude Modulation Technique
Pulse Width Modulation Technique
Single PWM
Multiple PWM
Sinusoidal PWM
Modified Sinusoidal PWM
Three Phase Inverters
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Methods of Control of Output Voltage
The magnitude of the output voltage of the inverter can be controlled
by two main techniques:
Pulse Amplitude Modulation technique(PAM), where the amplitude
of the input DC voltage is varied.
Pulse Width Modulation (PWM), where the width of the signal
pulse of the electronic switch is varied.
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Pulse Amplitude Modulation Technique
In this technique the input DC voltage is varies by one of the following
methods:
1) Controlled Rectifier: The input voltage is AC which is rectified by a
controlled rectifier to control the average output voltage. This average
voltage is then fed to the inverter.
2) DC Chopper: The input voltage is DC which is controlled by the use
of a DC chopper to generate a variable DC voltage. This variable DC
voltage is then fed to the inverter.
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Pulse Width Modulation Technique
Pulse width modulation (PWM) is the most commonly used technique
for controlling the output voltage of the inverter. It can be also used to
reduce the harmonics in the output voltage.
In this technique, the output voltage is a pulse width modulated wave
and the voltage is varied by varying the pulse width.
This technique can be applied with half bridge and full bridge inverters.
The commonly used methods are:
Single pulse width modulation
Multiple pulse width modulation
Sinusoidal pulse width modulation
Modified sinusoidal pulse width modulation 5
Single Pulse Width Modulation
In this method of control, the output voltage waveform has a single
pulse in each half period and the width of this pulse is varied to control
the magnitude of the voltage.
The gating pulses of the electronic switches are generated by
comparing a rectangular signal called the “reference signal” with a
triangular signal called the “carrier signal”.
The reference signal has an amplitude Ar and frequency fr, while the
carrier signal has an amplitude Ac and frequency fc.
The frequency of the reference signal fr determines the frequency of the
frequency of the output AC voltage fo and the frequency of its
fundamental component f1.
fr = fo = f1
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Single Pulse Width Modulation
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Single Pulse Width Modulation
The ratio of Ar to Ac is controlled to control the width of the pulse, and
hence, control the magnitude of the output voltage. This ratio is known
as the Modulation Index, M.
𝐴𝑟
𝑀=
𝐴𝑐
The rms output voltage is:
1
𝜋+𝛿 2
2
2 𝛿
𝑉𝑜 𝑟𝑚𝑠 = 𝑉𝑠2 𝑑𝜔𝑡 = 𝑉𝑠
2𝜋 𝜋
𝜋−𝛿
2
By varying the modulation index (M) from 0 → 1, the pulse width of
the voltage (δ) changes from 0 → π (i.e., 0 →T/2). Thus, the rms value
of the output voltage varies from 0 → Vs. 8
Single Pulse Width Modulation
To find the relation between the pulse width, δ, and the modulation
index, M, the following figure can be used:
From the similarity of triangles:
𝐴𝑐 𝜋/2
=
𝐴𝑟 𝛿/2
𝐴𝑐
𝛿=𝜋
𝐴𝑟
∴ 𝛿 = 𝜋𝑀 9
Single Pulse Width Modulation
The Fourier series of the output voltage is:
4𝑉𝑠 𝑛𝛿
𝑣𝑜 𝑡 = sin sin 𝑛𝜔𝑡
𝑛𝜋 2
𝑛=1,3,5,…
4𝑉𝑠 𝛿 4𝑉𝑠 3𝛿 4𝑉𝑠 5𝛿
𝑉1 𝑟𝑚𝑠 = sin , 𝑉3 𝑟𝑚𝑠 = sin , 𝑉5 𝑟𝑚𝑠 = sin
𝜋 2 2 3𝜋 2 2 5𝜋 2 2
It is possible to remove certain harmonic orders from the output voltage
2𝜋
by adjusting δ. For the third harmonic will be removed if 𝛿 = .
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The Fourier series of the output current is:
4𝑉𝑠 𝑛𝛿
𝑖𝑜 𝑡 = sin sin 𝑛𝜔𝑡 − 𝜃𝑛
𝑛𝜋 𝑍𝑛 2
𝑛=1,3,5,…
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Multiple Pulse Width Modulation
The harmonic content in the output voltage can be reduced by
generating a voltage that has multiple equal pulses each half cycle.
This is called multiple pulse width modulation and is also known as
uniform pulse width modulation.
The number of pulses per in each half cycle of the output voltage is
specified by the ration between the frequency of the carrier signal, fc
and the frequency of the reference signal, fr.
The frequency of the output voltage, fo, is specified by the frequency of
the reference signal, fr (fo = fr).
The rms value of the output voltage is controlled by the modulation
index, M.
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Multiple Pulse Width Modulation
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Multiple Pulse Width Modulation
To find the number pulses in each half cycle:
The number of pulses in each half cycle, p, can be found by the aid of
the previous waveforms of the reference and carrier signals as follows:
𝑇𝑟
= 𝑝𝑇𝑐
2
where Tr is the period of the reference signal (Tr=1/fr) and Tc is the
period of the carrier signal (Tc = 1/fc).
1 𝑝
=
2𝑓𝑟 𝑓𝑐
𝑓𝑐 𝑚𝑓
𝑝= =
2𝑓𝑟 2
𝑓𝑐
where 𝑚𝑓 = is the frequency modulation ratio.
𝑓𝑟
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Multiple Pulse Width Modulation
To find the relation between the pulse width, δ, and the modulation
index, M, the following figure can be used:
From the similarity of triangles:
𝐴𝑐 𝜋/2𝑝
=
𝐴𝑟 𝛿/2
𝜋 𝐴𝑐
𝛿=
𝑝 𝐴𝑟
𝜋
∴𝛿= 𝑀
𝑝 14
Multiple Pulse Width Modulation
The rms output voltage is:
1
𝛼𝑚 +𝛿 2
2𝑝 𝑝𝛿
𝑉𝑜 𝑟𝑚𝑠 = 𝑉𝑠2 𝑑𝜔𝑡 = 𝑉𝑠 = 𝑉𝑠 𝑀
2𝜋 𝜋
𝛼𝑚
By varying the modulation index (M) from 0 → 1, the pulse width of
the voltage (δ) changes from 0 → π/p (i.e., 0 →T/2p). Thus, the rms
value of the output voltage varies from 0 → Vs.
The main advantage of this method of modulation is the lower
harmonic content in the output voltage as compared to the single pulse
modulation method. However, the disadvantage of this method is the
increased switching losses due to the multiple switching of the
electronic switches. 15
Sinusoidal Pulse Width Modulation
The harmonic content in the output voltage can be further reduced by
using a sinusoidal reference signal rather than a rectangular signal.
The width of the pulses of the output voltage will not be uniform as in
the case of the multiple pulse width modulation method.
This is one of the most used methods in industrial applications.
The number of pulses per in each half cycle of the output voltage is
specified by the ration between the frequency of the carrier signal, fc
and the frequency of the reference signal, fr.
The frequency of the output voltage, fo, is specified by the frequency of
the reference signal, fr (fo = fr).
The rms value of the output voltage is controlled by the modulation
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index, M.
Sinusoidal Pulse Width Modulation
𝑝
𝛿𝑚
𝑉𝑜 𝑟𝑚𝑠 = 𝑉𝑠
𝜋
𝑚=1
where δm is the width of the mth pulse of the output voltage
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p is the number of pulses in half cycle
Modified Sinusoidal Pulse Width
Modulation
In this method, the carrier signal exists only during the first and last 60o
intervals per half cycle (i.e., 0 → 60o and 120o → 180o).
The advantages of this method is:
The fundamental component is increased.
The harmonic components are reduced.
The switching losses as reduced as compared to multiple and sinusoidal
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pulse width modulation methods.
Three phase Inverters
The three phase inverter consists of 3 half bridge inverter units, where
each unit produces voltage that is phase shifted by 120o from the other
voltages.
The electronic switches are switched periodically in a specific sequence
to obtain the required phase sequence of the output voltage (positive
sequence or negative sequence).
The magnitude of the output voltage can be controlled by using the
sinusoidal pulse width modulation method, where three sinusoidal
reference signals that are phase shifted by 120o are compared with a
carrier signal.
Each reference signal is responsible for operating the switches of one
unit of the three half bridge inverters.
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Three phase Inverters
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Three phase Inverters
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