Lectures 9-12, Page 1
Engineering IIA, 3B3 Switch-Mode Electronics
9 Basic dc to dc converter circuits
Converter circuits are widely used to convert one dc voltage to another. The use
of switching circuits allows high voltage ratios without the power losses
associated with linear regulators.
9.1 Energy storage in Inductors and capacitors
The energy stored in a capacitor is given by 1/2CV2 . The energy stored in an
inductor is given by 1/2LI2.
In lectures 2 and 3, capacitors and inductors are used as smoothing components.
They perform the smoothing by storing energy at some points of the cycle and
delivering it up at others. This occurs very explicitly in dc to dc converter
circuits.
Most dc - dc converter circuits use both inductors and capacitors.
9.2 The step-down (buck) converter
The dc output voltage is always less than the dc input voltage.
converter is known also as a chopper.
The
Lectures 9-12, Page 2
Engineering IIA, 3B3 Switch-Mode Electronics
9.2.1 Basic operation with continuous current
Waveforms
The MOSFET is switched by appropriate gate drive circuitry and is on for a
portion ρT of a cycle of duration T , and off for a period (1-ρ)T . Provided
the time constant L/RL is much greater than T the variation in the load current is
small and the current is continuous.
Then the mean output voltage Vo is simply ρVDC ,
and the mean load current, I L , is
The analysis of current ripple is very similar to that of Section 3.1.
When the MOSFET is on, the current grows from I1 to I2 and the inductor
charges. When the MOSFET is off, the inductor discharges and the load current
flows through the freewheel diode.
∆i =
Lectures 9-12, Page 3
Engineering IIA, 3B3 Switch-Mode Electronics
As L/RL >> T , the current ripple is small, so the load voltage, iRL , is essentially
constant at the mean voltage level, ρVDC . Hence,
The peak-to-peak load voltage ripple, ∆V, is the current ripple multiplied by the
load resistance.
Low ripple is achieved without an unduly large inductor by using a high
switching frequency, 1/T , often around 100 kHz. Connecting a capacitor
(shown dotted) across the load further reduces ripple. Using Fourier
analysis as in section 3.1.4, the effect of the capacitor can be calculated.
Note that using very large value inductors and capacitors will lead to a slow
response to changes in ρ.
9.2.2 Switching device stress in dc-dc converters
The inductor maintains the load current in the transistor during turn off while
the drain-source voltage reaches the supply voltage. Then the freewheel diode
takes over. When the transistor turns on, the collector voltage cannot fall until
the transistor has taken over all the load current.
The losses per cycle are given by
A high switching frequency causes higher losses, giving a trade off with the
inductor and capacitor size.
Lectures 9-12, Page 4
Engineering IIA, 3B3 Switch-Mode Electronics
9.2.3 Onset of discontinuous current
If the load varies and its current reduces, the inductor may discharge completely
during the off-period of the MOSFET. If there is no capacitor across the load,
the output current will be a series of pulses. If the capacitor is present CRL >>
T, the output voltage may still have low ripple but the average output voltage is
no longer equal to ρV DC.
To obtain the condition for the onset of discontinuities conduction, consider the
inductor current at the boundary of continuous conduction.
During the rise and fall in current the di/dt 's are linear as the driving voltage
across the inductor is constant provided that the capacitor maintains VO
essentially constant.
During current build up the voltage across the inductor is VDC - VO , and during
discharge it is - VO . (Neglect the diode voltage for this analysis).
Build up ( Transistor on)
Decay (Diode on)
From above the current ripple, ∆I =
The average inductor current at the boundary, IOB =
Lectures 9-12, Page 5
Engineering IIA, 3B3 Switch-Mode Electronics
The highest critical average current, IOBmax , occurs at duty cycle, ρ = 0.5.
The normalised regions of operation can now be plotted.
An alternative presentation of this analysis is used to see what duty cycle is
needed for various input voltages as the load current varies.
This is useful in power supply applications, where the dc input voltage is
smoothed rectified ac. The switching converter is under closed loop
feedback control and the controller changes ρ to keep VO constant:
Lectures 9-12, Page 6
Engineering IIA, 3B3 Switch-Mode Electronics
9.3 The step up (boost) converter
As the name suggests, the output voltage is always greater than the input
voltage.
9.3.1 Basic circuit
When the MOSFET is turned on, the current builds up in the inductor. The
MOSFET is turned off, the inductor discharges into the load capacitor through
the diode. The time constant RLC is long compared to the switchiong period T
and all components are ideal.
9.3.2 Analysis for continuous conduction mode
The MOSFET is on for a period ρT and off for (1-ρ)T .
Lectures 9-12, Page 7
Engineering IIA, 3B3 Switch-Mode Electronics
If the load voltage is constant at VO, the inductor current is given by the
following equations:
Build up:
During discharge, the inductor voltage is (VDC - VO) . Again in integral form
Equating
VO increases from a minimum of VDC as ρ is increased. However, the
switching losses in the transistor limit the attainable ratio of output to input
voltage (see the losses equation above).
As with the step-down converter, the inductor current may become
discontinuous. With operation at a fixed duty cycle and light load, the
capacitor may become charged to a very high voltage, causing failure of the
circuit. Generally, ρ must be varied by a control circuit to maintain a
particular output voltage. Since feeback control is used, these converters are
often designed to only operate in discontinuous mode.
Lectures 9-12, Page 8
Engineering IIA, 3B3 Switch-Mode Electronics
8.2.3 Discontinuous conduction
Note that the average inductor current is the same as the average supply current.
At the boundary condition, the power in is
The power out is
Hence
This is plotted in the form for a switched mode power supply (i.e. Vo is kept
constant by the controller).
Lectures 9-12, Page 9
Engineering IIA, 3B3 Switch-Mode Electronics
9.4 The step-down-step-up (buck-boost) converter
This circuit is a combined step-down/step-up converter. It may also be called a
flyback converter. When the MOSFET is on, the current through the inductor
grows. When the MOSFET is turned off, the inductor charges the load
capacitor negatively.
As it happens, the output voltage is given by the combination of the expressions
for the step-down and step-up converters (shown by equating the ∆I as usual).
The output voltage magnitude may be greater or smaller in than that of VDC.
The circuit is often used to obtain a negative supply rail. Again a controller
is essential.
9.5 General comments on dc-dc converters
The step-down converter provides an efficient and compact method of obtaining
a variable dc voltage up to a maximum almost equal to the input voltage. It is
the most important circuit amongst switching power supplies.
Operating at high frequencies allows the use of small energy storage
components, but care must be taken to limit radio frequency interference as
there are complex regulations on 'electromagnetic compatibility'.
Lectures 9-12, Page 10
Engineering IIA, 3B3 Switch-Mode Electronics
10 Bridge converters
10.1 The full bridge dc-dc converter
The full bridge, or H bridge, converter permits reversible power flow, unlike the
converters described in lecture 8. The full bridge converter has several areas of
applications, namely dc motor drives, dc to ac inversion with a medium
frequency sine wave output and high frequency dc to ac conversion as part of a
switch mode power supply.
The circuit supplying a simple L/R load
There are two ‘legs’ and each leg is made up of two switches and two diodes in
inverse parallel. Here n-channel MOSFETs are shown and the diodes are the
body diodes of the MOSFETs, which are good enough if the switching speed is
modest.
A number of ways of controlling the bridge exist. It is usual to have one of
the two transistors in each leg conducting at a given time, although in
practice a dead time when both are off is allowed to avoid shorting the dc
rail. The effect of too short a dead time is a shoot-through current, which
leads to high losses, if not destroying the devices. It is possible for both
transistors to be turned off in a leg as the freewheel diodes will pass any
inductive load current, but this mode of operation is unattractive as the
output voltage of the leg would depend on the direction of the load current,
causing difficulties in the control of the bridge.
Lectures 9-12, Page 11
Engineering IIA, 3B3 Switch-Mode Electronics
10.1.1 Overview of switching in the full bridge
For an inductive load (L/RL >> 1/f )
T1 , T2 on
T1 , T2 off
Turning T1 , T2 on again turns D3 and D4 off.
vA is VDC when T1 is on, and 0V when T4 is switched on, even if it is D4
passing the current.
Similarly for reverse load cuurent:
When T1 is on, D1 passes the load current and vA is VDC; when T4 is on vA is 0V,
so the values of vA, and vB, are same whatever the direction of I L.
Lectures 9-12, Page 12
Engineering IIA, 3B3 Switch-Mode Electronics
Providing one transistor in a given leg is always on, and the current is
continuous, the mean output voltage is given switching pattern of the upper
and lower switches. This is the most practical and common way of
controlling such as bridge and is identical to the way logic gates work.
The average value of vA ,
VA =
where ρA is the fraction of the switching period for which T1 is on and T4 is off.
Similarly for the other leg,
VB =
where ρB is the duty cycle of T3.
So the full bridge leg outputs can be considered separately, as step down
converters and in a bridge they allow for full four quadant operation:
There are two possible switching strategies:
1. Pulse width modulation with T1 , T2, and T3 , T4, switching as pairs
2. Pulse width modulation with T1 , T4, and T3 , T2, switching independently
Lectures 9-12, Page 13
Engineering IIA, 3B3 Switch-Mode Electronics
10.2 dc-dc converter pulse width modulation
10.2.1 Output voltage with bipolar switching
Using the full bridge as a dc-dc converter, the average load voltage, VO is
Assuming that the dead time is negligible, the two duty cycles are related:
ρB =(1−ρΑ)
So
10.2.2 Generation of ρA for bipolar switching
The control of the pulse width modulation often uses a triangular waveform of
amplitude V̂ t :
Lectures 9-12, Page 14
Engineering IIA, 3B3 Switch-Mode Electronics
The duty cycle ρA is set by the time for which the triangular waveform does not
exceed a control voltage vc. The on time, t is
where T is the switching period.
The time t = v c T , so the duty cycle ρA is
1 V̂ 4
t
Substituting this in the earlier result for VO gives
The result shows that the output voltage is a linear function of the control
voltage.
Clearly the load voltage jumps between +VDC and -VDC , hence the name
bipolar voltage switching. The output current can be positive or negative with
either polarity of output voltage, so the power flow is reversible. As in the
chopper, inductance in the load circuit is attractive to smooth the current.
Various control methods are possible. Pulse width or duty ratio modulation
is preferred generally as the harmonic content is fairly stable and easy to
filter. A pwm frequency of greater than 20kHz has the advantage of being
out of the audible range. Many microcontrollers contain a pwm generation
block.
Applications:
Lectures 9-12, Page 15
Engineering IIA, 3B3 Switch-Mode Electronics
10.2.3 Output voltage for unipolar switching
Using the full bridge as a dc-dc converter with unipolar switching exploits the
fact that VO is zero if T1 and T3 are both on, or if T2 and T4 are both on.
As the two legs are operating independently, a simple example is T1 held in the
on-state and T2 used to modulate the voltage across the load.
More commonly the two legs do similar things with a phase displacement:
For example, the pulse width modulation strategy may use a triangle wave with
two control voltages +vc and -vc. The switches in leg A are controlled by vc and
those in leg B are controlled by -vc.
T1 is on if
T3 is on if
As before, ρA =
In this case (from the figures), ρB =(1−ρΑ)
The average voltage is therefore the same as for bipolar switching.
Lectures 9-12, Page 16
Engineering IIA, 3B3 Switch-Mode Electronics
Waveforms (Unipolar switching):
Period I:
Period III:
T1 and T2 on
T1 and T2 on
Period II: T4 and T2 on
Period IV: T1 and T3 on
The switching frequency of each leg is the same and the same as for bipolar
switching as it is given by the frequency of the triangle waveform.
Here the benefit of the triangle carrier can be appreciated. A clear phase
displacement in the leg switching is seen. The sawtooth carrier common in
microciontrollers would not give the effective doubling of the switching
frequency seen here by the load. Here, the ripple at the load is much lower.
Consequently, the boundary between continuous and discontinuous
conduction will be at a lower current.
Lectures 9-12, Page 17
Engineering IIA, 3B3 Switch-Mode Electronics
10.3 Full bridge circuit as a dc to ac inverter
The full bridge inverter can be operated in a square wave mode.
Fourier analysis gives the output at f 1 , equal to 1/T, as
Data book:
With bipolar switching, the amplitude of the ac output can only be controlled by
varying the dc input voltage, which could, for example, be derived from a
step-down converter. The frequency can be varied by changing the frequency of
the controlling triangular wave. The output contains all odd harmonics.
With Unipoar switching, the ac output and its harmonic current can be varied by
keeping the duty ratio ρB the same as ρA but the phase between the leg
switching is shifted away from 180O (or 0O).
The subject of dc - ac inverter bridges with a variable output voltage and
frequency will be considered again in a later Lecture.
Lectures 9-12, Page 18
Engineering IIA, 3B3 Switch-Mode Electronics
10.4 The half bridge
One leg of the half bridge can be replaced for ac operation, by two capacitances.
The voltage across the load is half the value obtained from the full bridge. The
mid-point of the capacitors is at VDC/2.
10.5 Summary
The Bridge converter circuits offer a great amount of flexibility. Their use in
dc-dc conversion is widespread at powers above one or two hundred watts. They
are widely used in dc ‘servo’ motor drives. The use of full and half bridges to
generate ac is also widespread, particularly for ac motor drives, isolated switch
mode power supplies and RF heating.
Operation in the 'square-wave' mode is simple, but the harmonic content of
the output is high. For some applications at high frequencies, > 20 kHz,
such as switch mode power supplies and RF heating, the square waveform is
acceptable or the load can be tuned to minimise harmonic currents (see later
Lectures). For ac motor drives, operating at around 50 Hz, filtering
components are very bulky and expensive and harmonics cause severe
losses. In these systems, pulse width modulation (pwm) schemes with
sinusoidal references are adopted to produce an output which is closely
sinusoidal.
Lectures 9-12, Page 19
Engineering IIA, 3B3 Switch-Mode Electronics
11 DC-DC converters with isolation
The converter circuits introduced in lecture 9 do not provide isolation between
input and output. Frequently isolation is needed to achieve split supply rails, to
avoid unwanted current loops in power supply connections or for safety. In
switched mode power supplies, isolation is provided by a transformer operating
at a high switching frequency. High frequency operation permits the use of a
physically small transformers. The use of a transformer implies that are two
stages of switching devices.
11.1 Transformer operation
11.1.1 Introduction to transformer design*
In broad terms, power frequency transformers and inductors use cores made up
of laminations of silicon steel. However, at the frequencies encountered in
switching converters, > 25 kHz, ferrites are used. Their high resistivity and
granular nature reduces eddy current losses, but the practical working flux
density is limited to about 0.25T compared to the 1.7 T possible in silicon steel.
Ferrites are sintered mixtures of ore Fe203 and oxides such as Zn0 and Mn0.
Ferrites are offered in various grades, reflecting different compositions. Ferrites
are available in a range of shapes; E and I cores, pot cores, rods, tubes, toroids
etc.
The design of a transformer takes into account the basic VA rating, VI , the
working peak flux density, Bpk , the working current density, J and the
frequency of operation.
WA = 2
VI
k f J B pk
W is the winding window area, A is the flux path area and k is the packing
factor for the winding. W amd A are orthogonal, so indicate the volume, where
their ratio is fixed by the core shapes available. Clearly a high frequency
reduces the volume and weight of the transformer.
With some circuits for example the flyback converter, the mean flux in the
transformer is not zero. This must be allowed for on the design, typically by
putting an air gap in the magnetic path. This results in poor utilisation of the
core. Some cores are manufactured with an airgap and design curves are
used to determine the winding details.
Lectures 9-12, Page 20
Engineering IIA, 3B3 Switch-Mode Electronics
11.1.2 Equivalent Circuit
The equivalent circuit reflects the operation with a magnetising inductance and
two ‘ideal’ windings. Stray inductances and wire resistances are also included.
The iron core loss is represented by a resistor, Rm. Using the correct ferrite for
the frequency of operation allows Rm to be neglected.
11.2 Bridge circuits
11.2.1 Half bridge circuit
As the isolating transformer can only pass ac, the half bridge circuit can be used.
For low output voltages a split secondary as shown is preferred as only one
diode is in series with the load at any given time. An inductor must be used
as the first smoothing element (or design in transformer leakage inductance).
Follow with capacitor smoothing.
Lectures 9-12, Page 21
Engineering IIA, 3B3 Switch-Mode Electronics
For high voltage outputs a bridge rectifier gives the smallest reverse voltage
across an individual diode and better use of the transformer winding. Again,
add capacitor smoothing.
11.2.2 The full bridge circuit
The full bridge is capable of supplying the transformer primary with twice
the voltage when compared to the half bridge, and the output voltage can be
controlled by unipolar switching, but needs four transistors. Care must be
taken to ensure that there is no dc current in the transformer. A dc current
would be wasted power and could also saturate the transformer.
Lectures 9-12, Page 22
Engineering IIA, 3B3 Switch-Mode Electronics
11.3 Flyback converters
11.3.1 Basic flyback converter
The principle of the flyback converter is related to the buck-boost converter
described in section 8.3
Note the dot convention, which indictes the voltage sense of the windings.
When the transistor is on, the secondary voltage is negative with respect to the
diode D2. When the transistor turns off, a current into the dot is required to
maintain the transformer flux, so diode D2 becomes forward biased and a
secondary current flows.
The use of a transformer allows either side of the output to be earthed,
enabling a supply of either polarity to be obtained as well as allowing the
switch to be at ground. It may also be used to reduce the voltage stress on
the transistor.
11.3.2Analysis
Clearly, the energy stored in the inductor is maintained at the switching instant.
For analysis, the transformer is better represented by the magnetising inductance
and an ideal transformer (with the magnetising branch on the primary side).
Lectures 9-12, Page 23
Engineering IIA, 3B3 Switch-Mode Electronics
Current build up: 0 < t < ρT
current decay : ρΤ < t < T
Waveforms for continuous current condition.
Lectures 9-12, Page 24
Engineering IIA, 3B3 Switch-Mode Electronics
The current in LO varies from IO1 as the transistor turns on to IO2 where the
transistor turns off. The current charge, ∆ IO = IO2 - IO1
In integral form during current build up:
The secondary voltage is Vo during demagnetisation
The current rise must equal the current fall
This is similar to the result obtained for the buck-boost converter, using an
analysis of the current, but the output voltage in the flyback converter is also
transformed by N2/N1 allowing better choices in the design.
The voltage across the switch when it is off is
The voltage across the switch here is lower than that for a given output
voltage with the buck-boost by virtue of the transformer action. This
overcomes the main drawback of the buck boost, making it the better circuit
for high ratios of step up conversion. Like the buck boost it is usually used
in the discontinuous current mode, under the control of a dedicated control
IC, so the simple equation does not then apply.
Lectures 9-12, Page 25
Engineering IIA, 3B3 Switch-Mode Electronics
11.4 Forward converters
11.4.1 Basic forward converter
In a practical forward converter a tertiary winding for demagnetising the core is
often necessary. The energy stored due to IO is returned to the supply via the
tertiary winding and D3.
In the diagram, the transformer is shown as an ‘ideal transformer’ with a
separate magnetising inductance. Note the dot convention.
Transistor on 0 < t < ρT:
The primary voltage is VDC. D1 conducts and supplies load current. The
transistor current will consist of two components, i1 where i1 = N1/N2 i2 , and iO
which is the magnetising current. Appropriate transformer design ensures that
iO < 10% i1.
Transistor off, ρΤ < t < T
The transformer flux decays, reversing the transformer voltage so that diode D3
becomes forward biased. A current flows in the tertiary winding. In this period
the primary voltage is determined by the tertiary winding and the turns ratio,
(-N1/N3)VDC . This period is used to demagnetise the transformer.
During this time, diode D1 will be turned off and the load current continues
through Df . The voltage on the transistor is (1+N1/N3)VDC.
Lectures 9-12, Page 26
Engineering IIA, 3B3 Switch-Mode Electronics
Waforms for forward converter:
For no net flux in the core, the increase in current when the transistor is on must
equal the decrease in the off period.
Hence
To ensure total demagnetisation ρO must be less than 1 - ρ .
The maximum value of ρ , ρmax is then
Again, the presence of a transformer allows the output to be either polarity
and allows various circuits to be used on the primary side. Often the
number of turns on the tertiary winding (if it exists) equals the number of
turns on the primary.
Lectures 9-12, Page 27
Engineering IIA, 3B3 Switch-Mode Electronics
11.4.2 Two-switch forward converter
A bridge with only two switches and two freewheel diodes. The switches are
operated simultaneously (bipolar switching).
The circuit eliminates
demagnetising windings, since D3 and D4 take over the magnetising current
when T1 and T2 are switched off, so the reverse applied voltage demagnetises
the core. Max duty ratio of 50%.
This is a rather tedious circuit as a high side gate drive is required, but
otherwise simple and efficient with excellent Mosfets available below 60V.
Lectures 9-12, Page 28
Engineering IIA, 3B3 Switch-Mode Electronics
11.5 The push-pull converter
The duty cycle of the switches is fixed at 0.5. Obviously the two transistors
must not conduct simultaneously. On turning off T1 , D2 takes the magnetising
current for a short period until the transformer current reverses.
It is difficult to avoid the transformer saturating, as an exactly 50% duty
cycle is hard to achieve.
The basic output voltage is related to the input voltage by
N
V O = N 2 V DC
1
This is an important circuit as it reuires only one high efficiency n-channel
MOSFET in series with each primary, and one diode in the secondary side
so it is fairly efficient. Also, the MOSFETs are both at the ground potential
making them easy to drive. The poor utilisation of the transformer windings
and the conservatiuve design necessary to avoid saturation is compensated
for by the high frequency used (small transformer). This circuit is the basic
element of the popular 12V dc to 240 V ac inverters, where is creates the
340V dc needed for the output circuit.
The merits of each converter circuit are further reviewed in ‘The power
transistor in its environment’, NC 102.
Lectures 9-12, Page 29
Engineering IIA, 3B3 Switch-Mode Electronics
12 Switch-mode dc to ac inverters
Many applications, notably ac motor drives, require a variabe voltage variable
frequency ac source which has a lower harmonic content than can be obtained
by the simple square wave operation of a bridge converter as described in
lecture 10. An effective solution is to use sinusoidal pulse width modulation.
12.1 Sinusoidal pulse width modulation
In section 9.2 it was shown that the average output voltage of a full bridge can
be made proportional to a control voltage, vc. The control voltage vc can be
made sinusoidal to give a sinusoidal output, within limits imposed by the
switching frequency of the bridge. Consider one leg of a bridge
Waveform generation:
Lectures 9-12, Page 30
Engineering IIA, 3B3 Switch-Mode Electronics
12.1.1 Naturally sampled pwm
The triangular waveform is maintained at a constant amplitude and its
frequency fs, is the switching frequency (or carrier frequency). vc modulates the
duty cycle of the switches and has a variable frequency f1 and a variable
magnitude. The fundamental frequency of the inverter output is therefore f1 .
The inverter output will also contain harmonics related to f1 and fs. The
amplitude modulation ratio is defined as
where V̂ c is the amplitude of vc. The frequency modulation ratio mf is defined
as
The inverter leg shown is controlled as in 9.2 according to the scheme:
vc > vt
vc < vt
For m < 1, the following apply
1. The fundamental component is given by
1. the higher harmonics in the output are sidebands centred around the
switching frequency its multiples
where n and k are integers.
For odd n the harmonics exist only for even k and vice versa.
Lectures 9-12, Page 31
Engineering IIA, 3B3 Switch-Mode Electronics
12.1.2 Factors influencing the choice of switching frequency
Further consideration of the mathematics leads to an odd integer ratio for mf
such that there is ¼ wave symmetry in the resulting pwm waveform (like a
sinewave) so that cosines and even harmonics disappear from the Fourier series.
(Look closely at the waveforms above).
Harmonic spectrum for m = 0.8 and mf = 15 as in the waveforms above.
For m < 1 , the amplitude of the fundamental output is proportional to the
amplitude of the control voltage and the only harmonics are up around the
switching frequency (and multiples of it).
In general, a higher switching frequency makes the filtering of harmonics
easier but increases switching losses in the inverter. It is very advantageous
to have a switching frequency above 20 kHz so that it is inaudible.
12.1.3 Over modulation
The magnitude of the fundamental output frequency is mVDC/2 implying that m
> 1 will give a high output voltage, but the simple relationship breaks down
when m >1.
Lectures 9-12, Page 32
Engineering IIA, 3B3 Switch-Mode Electronics
Known as over modulation or pulse dropping, the limit is square wave operation
as in section 9.3.
The output at the fundamental frequency no longer varies linearly with the
magnitude of vc .
12.1.4 Implementation
The features described above are most conveniently combined using
microcontrollers or FPGAs. The odd integer synchronised ratio of fS to f1 can
then be produced easily.
Various alternative methods for producing sinusoidal pwm have been
implemented, the most common is based on pre-calculated waveforms
which explicitly eliminate particular harmonics.
Lectures 9-12, Page 33
Engineering IIA, 3B3 Switch-Mode Electronics
12.2 Sinusoidal pwm bridge inverters
12.2.1 Single phase bridge (2 legs)
The sinusoidal pulse width modulation scheme can be realised in a full bridge
using either the bipolar or unipolar switching methods, as described in sections
9.2 and 9.3.
For bipolar switching, the output is given by
The output voltage for m < 1 is simply
For a unipolar scheme use two references:
The effective doubling of the switching frequency is apparant.
Lectures 9-12, Page 34
Engineering IIA, 3B3 Switch-Mode Electronics
The unipolar scheme shifts the harmonics up in frequency.
where n and k are again integers. Both approaches give the same amplitude of
the fundamental for a given modulation ratio m.
The effect is better than it appears in the waveforms: All the components at
and around the modulating frequency are eliminated.
An alternative unipolar scheme employs reference waveforms staggered by
120O.
Lectures 9-12, Page 35
Engineering IIA, 3B3 Switch-Mode Electronics
The effective doubling of the switching frequency is retained, but the output
voltage magnitude (voltage gain) is reduced to 0.866 at m = 1 . The harmonic
content of the output voltage can be reduced by synchronising fs and f1 , making
mf odd integer, as before and a multiple of 3.
All Third harmonics in the phase voltages cancel in the line voltage due to the
120O difference VA to VB
12.2.2 Three phase bridge inverters
Three bridge legs can be connected to generate a three phase output.
Lectures 9-12, Page 36
Engineering IIA, 3B3 Switch-Mode Electronics
The unipolar waveforms above, with 120O between the reference waveforms,
provide the three modulated outputs, A, B, C.
Thus three line voltages can be constructed
The comments on over modulation made above apply. Here, however, an
important version of overmodulation is for m >>1, which is 180O conduction in
each leg, remembering that all Thirds will cancel:
The harmonics have amplitudes
2 3
V DC ( 15 + 17 + 1 + 1 + ....)
11 13
This feature is widely used in motor drives.
P.R. Palmer
November 2008