Power Electronics
Will McLennan
Based on lectures by Dr Ewen Macpherson
Short Description
This course aims to equip students to enter the power electronics industry by providing an
understanding of the fundamental principles of power semiconductor devices and circuits,
and the knowledge and skills required to analyse and design such circuits. Students will
also be introduced to the central issues involved in the specification and design of power
electronic systems in various applications, including in switched mode power supplies,
HVDC links, flexible ac transmission systems and renewable energy systems.
Summary of Intended Learning Outcomes
• Explain the operation of buck, buck-boost and boost converters;
• Design SMPS circuits with isolation and explain SMPS control techniques;
• Explain the operation of resonant power supplies, and carry out basic design;
• Design high frequency wound components;
• Explain the operation of power semiconductors;
• Calculate the harmonics produced by inverter circuits;
• Understand the operation of inverter circuits in solar photovoltaic systems;
• Understand the operation of various power electronic FACTS devices;
• Explain the application of D.C. drives;
• Explain the operation of induction motor and doubly fed induction generators.
Contents
1 Switched mode power supplies
1.1 Basic converter circuits . . . . . . . . . . .
1.1.1 Buck . . . . . . . . . . . . . . . . . .
1.1.2 Buck-Boost . . . . . . . . . . . . . .
1.1.3 Boost . . . . . . . . . . . . . . . . .
1.1.4 Summary of basic converter circuits
1.2 SMPS with input/output isolation . . . . .
1.2.1 Flyback converter . . . . . . . . . .
1.2.2 Forward converter . . . . . . . . . .
1.2.3 Bridge converter (buck) . . . . . . .
1.2.4 SMPS with multiple outputs . . . .
1.3 SMPS: Circuit design considerations . . . .
1.3.1 Resonant converters . . . . . . . . .
1.3.2 Control requirements and techniques
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1
2
2
7
8
10
12
13
18
21
22
22
24
29
2 Design of magnetic components
2.1 Laws of electromagnetism . . .
2.2 Inductors . . . . . . . . . . . .
2.3 Transformers . . . . . . . . . .
2.4 Wound component production .
2.4.1 Core shapes . . . . . . .
2.4.2 Materials . . . . . . . .
2.4.3 Windings . . . . . . . .
2.4.4 Inductors (chokes) . . .
2.5 Transformer design . . . . . . .
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36
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45
45
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49
49
51
52
55
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3 Power semiconductor devices
3.1 PN junction diode . . . . . . . .
3.2 Schottky diode . . . . . . . . . .
3.3 Power MOSFETs . . . . . . . . .
3.4 Insulated gate bipolar transistor
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4 Gate drive circuits
57
4.1 Snubbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Inverters
5.1 Output voltage waveshape . . . . . . .
5.1.1 Pulse width modulation . . . .
5.2 Inverter applications . . . . . . . . . .
5.2.1 Uninterruptible power supplies
5.2.2 Solar photovoltaic systems . . .
5.3 Inverters connected to the grid . . . .
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62
62
67
69
69
71
72
6 Power Electronics in Power Systems
6.1 High voltage DC links . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Flexible AC transmission systems . . . . . . . . . . . . . . . . . . . . . . .
6.3 Unified power flow controller . . . . . . . . . . . . . . . . . . . . . . . . .
75
75
77
79
7 DC
7.1
7.2
7.3
7.4
7.5
machine drives
Two quadrant control
Four quadrant control
Power factor . . . . . .
Current control . . . .
Motor drives . . . . .
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82
83
83
85
85
86
8 Induction motor drives
8.1 Speed control . . . . . . . . . . . . . . . . . .
8.1.1 Adjusting the supply voltage . . . . .
8.1.2 Adjusting the supply frequency . . . .
8.1.3 Adjusting the effective rotor resistance
8.2 Doubly fed induction generator (DFIG) . . .
8.2.1 DFIG wind generation system . . . . .
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89
90
90
92
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98
100
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Power Electronics
1
1
Switched mode power supplies
All electronic circuits need a supply of power, yet the power supply is normally the
most neglected part of the complete system. It is often hastily designed (or bought
in) after the rest of the system is complete, and placed in the leftover space, which
is often too small and has inadequate ventilation for cooling purposes. For low power
consumption units, a battery, or a solar cell, will suffice. Other units will run from the
mains supply. Whatever the primary energy source, almost always an electronic power
supply is required to process the power to produce a constant dc output voltage.
Over the past 20 years, there have been significant changes in the design of power
supplies. The most important of these has been the widespread change from linear
power supplies to those which operate on a switching basis, so called Switched Mode
Power Supplies (SMPS). The principal reason for the move to SMPS is their much greater
efficiency - typically 80-90% as opposed to 30-40% for linear units. This greatly reduces
the cooling requirements, and allows a much higher power density.
The concept of high frequency switching of transistors to provide a controllable dc output
has been around for some time. What has allowed the widespread adoption of SMPS
technology has been the availability of a range of suitable active and passive components.
The advent of MOSFETs with high power rating has been a particularly important
advance, together with the availability of high- speed diodes and improved magnetic
materials. Now circuits can be designed to operate at switching frequencies into the
MHz range, with consequent reductions in cost and in volume of the power supply.
System designers naturally wish to avoid any system faults occurring due to the power
supply, so the specifications for the power supply normally include large safety margins.
As the power supply will have its own safety margins, it is often grossly over-specified,
and as a result is considerably larger, heavier and more expensive than is necessary.
When the systems designer is specifying the power supply, he will initially consider the
required voltage, together with the maximum current. However, voltage and current
ratings are only the beginning of a long list of parameters which need to be specified
such as:
• Number of outputs (voltage of each)
• Output current rating
• Input voltage range
• Is electrical isolation required?
• Ripple voltage
• Output voltage regulation (affected by variations in voltage/current/temperature)
• Transient response due to sudden changes in load and input voltage
Power Electronics
2
• Efficiency
• Protection
• Electromagnetic Interference (EMI)
• Hold-Up Time
• Temperature range
• Physical dimensions
• Certification
• Economics
1.1
Basic converter circuits
Many different SMPS circuit topologies have been developed, most of which are derivatives of the buck, the buck-boost or the boost converters.
1.1.1
Buck
In the basic buck circuit, transistor T1 is switched at high frequency (20 kHz to 1 MHz)
to produce a chopped output voltage V2 . This is then filtered by the L-C circuit to
produce a smooth load voltage. The output voltage can be controlled by varying the
mark:space ratio of V2 .
T1
I0
L
D
Vi
V2
C
V0
Load
R
V2
t on
T
t
Figure 1: Buck converter circuit
For an inductor in steady state, the average voltage across an inductor over one complete
cycle is zero because the current through the inductor at the start of the cycle must equal
Power Electronics
3
the current at the end of the cycle:
Z
VL .dt = 0
(1.1)
If the average voltage across the inductor is zero, then the average of V2 = Vo (V0 is
assumed to be kept constant by the large capacitor C). Thus the green area above =
the yellow area.
V2
V0
t
Figure 2: Buck converter circuit output
Vo
ton
= D = Transistor duty ratio
=
Vi
T
(1.2)
Thus in the ideal case the output voltage is independent of load. However, in a realistic
circuit there will be losses associated with:
1. Transistor ‘on’ state voltage drop
2. Transistor switching losses
3. Diode forward voltage drop
4. Inductor effective resistance,
All of which increase the dependency of Vo on load current Io . The voltage across the
inductor VL = V2 − Vo .
When the transistor is on, V2 = Vi , so the voltage across the inductor vL = Vi − Vo .
Thus iL increases linearly. When the transistor is off, the current through the inductor
L cannot instantaneously fall to zero, so the ‘freewheeling’ diode is included to provide a
return path for the current to circulate through the load. When the diode is conducting,
V2 = 0. Thus during the off period the voltage across the inductor VL = −Vo . So iL
decreases linearly.
T1 on:
VL = L
diL
= Vi − Vo
dt
(1.3)
diL
= −Vo
dt
(1.4)
T1 off:
VL = L
Power Electronics
4
V2
V0
V0
t
iL
I0
ΔI
t on
t
Figure 3: Buck converter circuit voltage and current
This extend of this so called ‘ripple current’ can be calculated as such:
diL
∆I
=L
dt
ton
Vi − Vo
∆I = ton
L
Vo T
∆I =
(1 − D)
L
Vl = L
(1.5)
(1.6)
(1.7)
Where the maximum current is given by:
Imax = I0 +
∆I
2
(1.8)
Note that the transistor must be rated for the peak inductor current. The previous section analysed the buck converter for the continuous current mode, ie. when the inductor
current never falls to zero in the cycle. However, many switched mode power supplies
are designed to operate in the discontinuous current mode, where for a proportion of
each cycle the inductor current is zero.
If the load on a buck regulator is slowly reduced, eventually the minimum of iL just
touches zero. This is the boundary condition between continuous and discontinuous
modes of operation.
If the load is reduced still further, the inductor current is zero for part of the cycle, and
the circuit is said to be operating in the discontinuous mode.
When the inductor current is zero, the voltage across the inductor is zero, therefore
v2 = Vo . This increases the average value of v2 , so the output voltage Vo rises. Thus
in the discontinuous mode the output voltage is no longer independent of load. Let the
diode duty ratio = D1 . For the period the diode is conducting (D1 .T ), the inductor
Power Electronics
5
iL
Vi
Io
V2
Vo
V2
1
2
3
Vo
t
iL
Io
t
Figure 4: Discontinuous mode
voltage vL = −Vo and dince the average voltage across an inductor over one cycle is
zero:
(Vi Vo ).D.T = Vo .D1 .T
(1.9)
Therefore:
Vo
D
=
Vi
D + D1
The ripple current (looking at the ‘off’ period) is:
∆I =
(1.10)
Vo .D1 .T
L
(1.11)
The output current, iL , averaged over one cycle is:
Io =
∆I
1
× (D.T + D1 .T ) ×
2
T
(1.12)
Combining these equations gives:
Vo
2D
q
=
Vi
D + D2 +
(1.13)
8LIo
T Vo
In the continuous mode the output voltage Vo does not depend on the load current Io:
i.e. this circuit has very good load regulation and the equation simplifies to Vo /Vi = D.
Power Electronics
6
In the discontinuous mode the output voltage Vo depends on the load current Io: i.e.
this circuit has poor load regulation. The inductor size controls the current slope (but
doesnt affect the average current). A smaller inductor means that the circuit is closer
to discontinuous mode operation, as the minimum current is nearer to zero.
At the boundary condition,
Vo (1 − D)
2Io f
Thus if the circuit is designed to operate in continuous mode,
L=
L>
(1.14)
Vo (1 − D)
2Io f
(1.15)
for all possible values of D and Io . This is what usually determines the value of L chosen.
The capacitor is chosen to keep the ripple on the output voltage Vo to an acceptable
value.
C charging
iL
ΔI
C discharging
Io
t
Figure 5: The capacitor is chosen to keep the ripple on the output voltage Vo to an
acceptable value
In the above diagram in figure 5, the capacitor is charging when the inductor current iL
is greater than the load current Io . It is discharging when iL is less than Io .
∆Q = C∆Vo
1 ∆I T
=
2 2 2
(1.16)
(1.17)
∆Vo is the peak-to-peak output ripple voltage. Typically this is limited to about 5%
of the nominal output voltage Vo. To keep the ripple voltage less than the maximum
allowed value ∆Vo(max) ,
Vo (1 − D)
C>
(1.18)
8∆Vo(max) Lf 2
for all possible values of D. In practice, however, the output ripple voltage is usually
much more affected by the ripple current through the capacitor’s ESR (equivalent series
resistance) than by the charge/discharge of its capacitance. It is, therefore, important
to choose a filter capacitor with a low ESR value. ∆Vo = ∆I.Rc
Power Electronics
1.1.2
7
Buck-Boost
In the basic buck-boost regulator above, energy is stored in inductor L during the ontime of transistor T1 . When T1 is switched off, the voltage across L reverses as the
inductor transfers the stored energy (= 1/2Li2 ) to the smoothing capacitor and to the
load. Note that the output voltage is the opposite polarity to the input (i.e. Vo is
negative).
T1
T 1 on:
iL
Vi
T1
T 1 off:
Vo
Load
Vo
Load
iL
Vi
Figure 6: Buck-boost converter
When T1 is on, iL increases and C supplies load, when T1 is off, iL flows up through
load and diode and C is recharged. For an inductor in steady state the average votlage
is zero, therefore
Vi ton = Vo tof f = 0
(1.19)
Therefore:
Vo
=−
Vi
ton
tof f
=−
DT
(1 − D)T
=−
D
1−D
(1.20)
Losses which have been neglected are resistance of the inductor winding, the transistor on-state voltage drop, the forward volt drop across the diode, and the transistor
switching losses. However, the formulae as presented are usually adequate as a first
approximation.
If the load is reduced, the inductor current will fall, but as long as it is in continuous
mode the shape of iL will stay the same, and the inductor voltage vL (and hence Vo )
will be unaffected.
If the load current is reduced beyond that at the boundary condition, the circuit goes
into discontinuous mode operation. During the period the inductor current iL is zero, the
inductor voltage vL is also zero. Thus the green shaded area is now narrower, therefore
if it is to have the same area as the yellow shaded area it must be deeper; thus Vo
Power Electronics
8
is increased (but still negative). Thus in the discontinuous mode the output voltage
depends on the value of load current. If the diode duty ratio is D1 , then the diode is
conducting for D1 .T . Thus:
D
Vo
=−
(1.21)
Vi
D1
Hence it can be shown that:
Vo
= −D
Vi
r
Vo T
2Io L
(1.22)
Thus in discontinuous mode the voltage is dependent of load.
The peak-to-peak current ripple ∆I is obtained from:
Vi=L
∆I
∆I
=L
ton
DT
(1.23)
The inductor current only goes to the load during tof f , therefore the average load current is the average inductor current during the transistor off-period. At the boundary
condition between continuous and discontinuous mode of operation, the average load
current is:
∆I tof f
DT Vi (1 − D)T
Io =
=
(1.24)
2 T
2L
T
Thus,
Vo (1 − D)2
L=−
(1.25)
2Io f
When choosing the filter capacitor, during Ton the load current supplied by C is given
by”
∆Vo
Io = −C
(1.26)
Ton
Therefore:
∆Vo =
DIo
fC
(1.27)
The above is the minimum value of capacitance required to keep the ripple voltage
≤ ∆Vo . However, in practice the value of capacitance chosen will be much higher, as
the capacitor ESR is almost always the more important parameter in limiting ripple
voltage. Unlike the continuous mode buck converter, the capacitor in the buck-boost
converter supplies all the load current during the period the transistor is on. Therefore the capacitor in the buck-boost sees a much higher ripple current, placing a more
stringent ESR requirement on it.
1.1.3
Boost
During the transistor on-time the current builds up (linearly) in the inductor due to
vL = Vi . During this period, no current flows through the diode and the load is supplied
Power Electronics
9
vL
T1 on:
Io
Vi
vL
T1 off:
iL
iL
Vi
Vo
Load
Vo
Load
Io
Figure 7: Boost converter
entirely by the capacitor. When the transistor is turned off, the inductor current has to
keep flowing. The only path now is through the diode and load.
If the average inductor voltage is zero, for this to be true, Vo > Vi . Therefore:
Vo
1
=
Vi
1−D
(1.28)
Theoretically, as D tends to 1.0, Vo tends to infinity. In reality, Vo will fall to zero. This
is because a high value of D means that the diode is on for a very short period, yet during
this period it has to conduct all the energy going to the load during the whole cycle.
Thus large currents will flow, and the i2 R losses will become very high. [If D = 1.0, then
the diode would never conduct, therefore the capacitor would never be recharged so the
output voltage would fall to zero.] However, the formula above is reasonably accurate
up to D = 0.8.
If the load current is reduced beyond that at the boundary condition, the circuit goes
into discontinuous mode operation. During the period the inductor current iL is zero, the
inductor voltage vL is also zero. Thus the green shaded area is now narrower, therefore
if it is to have the same area as the yellow shaded area it must be deeper; thus Vi − Vo
is increased (but still negative). Thus in the discontinuous mode the output voltage
depends on the value of load current. If the diode duty ratio is D1 , then the diode is
conducting for D1 .T . Thus: Vi .D.T = (Vo Vi ).D1 .T . Hence:
Vo
D + D1
=
Vi
D1
(1.29)
Power Electronics
12
10
Vo /V i
ideal
10
8
6
4
real
2
D
0.2
0.4
0.6
0.8
1.0
Figure 8: Curve of D for a boost converter against Vo /Vi
It can be shown that:
1
Vo
=
Vi
2
s
1+
2D2 Vo
1+
f Io L
!
(1.30)
The boundary condition is given by:
L=
Vo D(1 − D)2
2Io f
(1.31)
If L is less than this value, it will be operating in discontinuous mode. As with the
Buck-Boost converter, the load is supplied entirely by the filter capacitor during the
transistor ‘on’ period. Hence the formula for output ripple is exactly the same as for the
Buck-Boost converter, i.e.:
DIo
∆Vo =
(1.32)
fC
1.1.4
Summary of basic converter circuits
Buck
Buck-Boost
Boost
Same polarity
Inverse polarity
Same polarity
Step-down
Step up or down
Step-up
Table 1: Circuit summary
In the continuous mode, the inductor current, iL , never falls to zero during any part
of the switching cycle. To maintain the current, the inductance has to be considerably
larger than is required in the discontinuous mode. Although the inductor current is
Power Electronics
11
continuous, in the buck-boost and boost converters the current into the converter output
stage (ie. the diode current) is discontinuous. However, in the buck converter the
inductor current is the current into the output stage, which is continuous and has a
relatively small ripple value. The buck converter is, therefore, easier to filter, and is the
most popular switching configuration, particularly at high power levels.
Buck
Buck-Boost
Boost
Vo
=D
Vi
(1.33)
Vo
D
=−
Vi
1−D
(1.34)
1
Vo
=
Vi
1−D
(1.35)
Open Loop Regulation:
The basic equations for continuous mode operation derived earlier are shown above. It
can be seen that the output voltage does not depend on the load current Io (taking a
first approximation and ignoring parameters such as the inductor resistance). Hence the
Open Loop Load Regulation is very good. However, Vo does depend on Vi, so the Open
Loop Line Regulation is poor.
Closed Loop Regulation:
The large inductor required for continuous mode operation, together with the filter
capacitor, constitutes a 2nd order delay in the feedback control loop, making it more
difficult to stabilise and resulting in poor closed loop response.
Also there is a small ripple current, smaller peak transistor current, the capacitor ESR
less critical and more difficult to stabilise control loop.
In the discontinuous mode the inductor current falls to zero each cycle. This results in
high inductor current peaks, placing an arduous duty on the switching transistor and
the filter capacitor, as well as the inductor itself.
Buck
Vo
2D
q
=
Vi
D + D2 +
Buck-Boost
Vo
= −D
Vi
Boost
Vo
1
=
Vi
2
r
s
1+
(1.36)
8LIo
T Vo
Vo T
2Io L
2D2 Vo
1+
f Io L
(1.37)
!
(1.38)
Power Electronics
12
Open Loop Regulation
As shown earlier in this section, when a buck, buck-boost or boost converter operates
in the discontinuous mode the output voltage depends on the load current Io. Hence
the open loop load regulation is poor. As in the continuous mode, the output voltage
depends on the input voltage, so the open loop line regulation in the discontinuous mode
is also poor.
Closed Loop Response
As the inductor starts each cycle with zero stored energy, it is possible for the control circuit to obtain any energy level and hence output current on a cycle-by- cycle basis. The
inductor, therefore, has no effect on the small signal closed loop characteristic, leaving
only the capacitor as the delay element in the loop. Converters operating in the discontinuous mode, therefore, are very stable and have a good closed loop response.
Also there is a larger ripple current, larger peak transistor current, a low capacitor ESR
required (bigger C) and it is easier to stabilise control loop.
It can be seen from the above that these converter have very different open loop and
closed loop characteristics, depending on whether operation is in the continuous or discontinuous current mode. It is important that a converter designed for one mode of
operation is not used in the other, as the different feedback characteristic may to lead to
instability. If the load current in a continuous mode converter is reduced below a minimum value discontinuous mode operation will result. Thus a continuous mode regulator
should not be run on very light loading.
1.2
SMPS with input/output isolation
Most SMPS are required to have transformer-coupling between the input and the output(s). Transformer coupling provides the following advantages over the basic regulators
so far described:
1. The output is electrically isolated from the input. This is usually a requirement
when operating from a 230 volt or 115 volt mains supply, to keep the mains voltages
well apart from a low voltage load. It also allows the output to be earthed, if
required.
2. The transformer turns ratio can be chosen to give an output voltage widely different
from the input voltage. The basic buck, buck-boost and boost circuits have the
output voltage limited to within approximately a factor of 10 of the input voltage.
3. With transformer coupling the polarity and step-up/step-down restrictions of the
basic circuits no longer apply.
4. By having more than one transformer secondary, multiple outputs at different
voltage levels can be obtained.
Power Electronics
13
However, the introduction of a transformer adds considerably to the size and weight of
the SMPS, and introduces further losses into the circuit. In addition, the transformer
leakage inductances may lead to severe voltage spikes in the circuit. A typical arrangement for an off-line (eg. connected to the 230 V, 50 Hz mains) power supply is shown
below in figure 9.
dc
output
ac
input
Inverter
Rectifier
Rectifier
Filter
Figure 9: Typical ‘off-line’ SMPS
The inverter operates at high frequency (usually somewhere between 50kHz and 500kHz),
resulting in a much smaller transformer than if it is at 50Hz. The transformer should
operate in the linear region of the B-H curve, between −Bmax and +Bmax . In the positive
1/2 cycle B increases to just below +Bmax and in the negative 1/2 cycle it returns to its
starting point just above −Bmax . According to Faradays Law:
v=n
dφ
dB
= nA
dt
dt
(1.39)
Therefore, over one cycle:
Z
1
B2 − B1 =
v1 dt
(1.40)
N1 A
In the steady state, B at the start of the cycle (B1 ) equals B at the end of the cycle
(B2 ). Therefore over one cycle:
Z
v1 dt = 0
(1.41)
i.e. the average voltage across the primary over one complete cycle equals zero. If the
average voltage across the primary does not equal zero, the transformer will ‘walk’ into
saturation (i.e. two steps up, one step down)
Figure 10 above shows the ‘full’ transformer equivalent circuit (although even this ignores parameters such as the inter-winding capacitance and the inter-turn capacitance).
The lower figure shows a simplified transformer equivalent circuit. The magnetising
inductance cannot be neglected in power supply circuits, as its presence significantly
affects the circuit design.
I1 = Iµ + nI2
(1.42)
1.2.1
Flyback converter
This circuit is a derivation of and operates in a very similar manner to the buck-boost
converter, but here the inductor has a secondary winding (or windings). Note that the
Power Electronics
I1
R1
14
L1
N1 N2
Iμ
Lμ
V1
I1
V1
Iμ
R2
L2
V2
Rc
N1 N2
nI2
I2
Lμ
V2
Figure 10: Transformer equivalent circuit
wound component is an inductor with a secondary winding, combining the functions of
both an inductor and a transformer. An inductor is an energy storage device, and as
such requires an air gap in the magnetic circuit (it is not possible to store significant
amounts of energy in the ferromagnetic part of the core). An ideal transformer directly
couples energy between the primary circuit and the secondary, and does not store energy.
In this circuit an air gap is required, as energy is stored in the device.
D
1:n
V1
N1
N2
C
Vo
Load
T1
Figure 11: The flyback converter
The equivalent circuit of the coupled inductor is the same as for a transformer: the
difference is in the value of the magnetising inductance Lµ . In a transformer Lµ is
designed to be as large as possible (no air gap, large number of primary turns), to keep
Iµ as small as possible and minimise losses. In the coupled inductor Lµ is the primary
inductance of the inductor, and hence will be designed to be much smaller.
During the transistor on-time, current builds up in a linear manner in the primary
Power Electronics
15
circuit.
di1
(1.43)
dt
storing energy (= 1/2 L1 i21 ) in the inductor. During this period, the diode prevents
any current flowing in the secondary: meanwhile, the load is supplied by the capacitor.
During the transistor off-time, the energy stored in the inductor (= 1/2 L2 i22 ) is released
to the load, also recharging the capacitor.
Vi = L1
T on:
V1
T off:
V1
Vo
Load
Vo
Load
Figure 12: Current flow in the flyback converter with T on and off
It can be shown that for continuous mode operation:
Vo
nD
=
Vi
1−D
(1.44)
where n is the turn ratio. The boundary condition for continuous mode operation,
is:
Vo (1 − D)2
L1 >
(1.45)
2Io f n2
for all possible values of D and Io .
Unlike the buck family of converters which are almost always operated in the continuous
current mode, the flyback converter is more commonly used in the discontinuous mode,
due to an inherent closed loop instability when operated in the continuous mode. The
term (1 − D) in the denominator has a dramatic effect on the response of the circuits to
sudden load changes. If load is suddenly increased, the output voltage will drop. The
feedback circuit will compensate by increasing D to bring Vo back to its rated value. The
inductor has a large value in continuous mode circuits, so iL cannot quickly increase to
the new load current, so the immediate effect of increasing D is only to reduce the time
current is flowing through the diode in the secondary, so reducing the energy transferred,
causing the output voltage to reduce still further.
Power Electronics
16
This makes continuous mode flyback converters very difficult to stabilise with a voltage
feedback loop, and for this reason they are almost always used in the discontinuous
current mode. [In control theory terminology, the term (1 − D) in the denominator leads
to a right half plane zero in the open loop transfer function. The effect of an RHP zero
is, like a normal LHP zero, to reduce the gain slope by -20 dB/decade, but it increases
the phase lag by 90◦ .]
The above argument holds also for the buck-boost and the boost regulators, which
also have a term (1 − D) in the denominator when operated in the continuous mode.
For this reason, both are normally used in the discontinuous mode. In the discontinuous
mode a smaller inductor is required. The disadvantages of discontinuous mode operation
are:
1. High peak transistor current (typically twice that in continuous mode)
2. Large filter capacitor (due to high peak current)
During the transistor on-time (D.T ), current builds up in the primary of the inductor
to a peak value i1 = I1 . During this period energy is stored in the inductor. When the
transistor is switched off, no current can flow in the primary, but amp-turns is maintained
by current i2 flowing in the secondary instead of i1 in the primary. During this period
energy is released from the inductor to the secondary circuit. During toff = D1 .T , i2
falls to zero, demagnetising the core.
I2
I1
i1
i2
t
D1.T
D.T
(1-D)T
T
Figure 13: Flyback converter in discontinuous mode
For a given specification (Vi , Vo , Io (max), f , output ripple voltage), and assuming ideal
components, the following equations apply.
1. Amp-turns must be maintained at turn-off, therefore:
I1 N1 = I2 N2
(1.46)
Power Electronics
17
2. The output current Io is the average of i2 , i.e.
Io = 12 I2 D1
(1.47)
3. The slope of the primary current i1 is given by:
Vi = L1
I1
DT
(1.48)
4. The slope of the secondary current i2 is given by:
Vi = L1
I1
DT
5. Input power = output power (assuming no losses):
I1
Vi
D = Vo Io
2
(1.49)
(1.50)
At the boundary between continuous and discontinuous modes, D1 = 1D. Even then
there are 6 unknowns (D, I1 , I2 , L1 , L2 , n) and only 5 equations, so there is no single
analytical solution. A 6th equation is (energy in over one cycle) = (energy out over one
cycle). However, this is not an independent equation: it can be derived from the others.
The equations can easily be entered into a spreadsheet, so that the effect of varying
parameters can be tested.
The output voltage is controlled by the amount of energy transferred from the primary →
secondary → output, per cycle. In a lossless flyback converter, energy stored per cycle in
the primary inductor equals energy released per cycle by the secondary inductor, equals
the output energy. Hence:
Vo Io
2
2
1
1
(1.51)
2 L1 I1 = 2 L2 I2 = f
If the output voltage is too low, more energy needs to be passed from the primary to
the secondary to raise the output voltage, it could be corrected by increasing the duty
ratio, allowing time for I1 to build up to a higher value. However, this could push the
circuit into continuous mode operation (in the waveforms on the previous page it is very
close to the limit).
To prevent this, the coupled inductor could be redesigned such that the primary inductance L1 is reduced. This will cause I1 to increase, and as energy is proportional
to the current squared, the net effect is to increase the primary energy, for the same
duty ratio. Another way to take the circuit further from continuous mode operation is
to reduce the secondary inductance. This will increase the peak secondary current, so
for the same average secondary current, the period of conduction of diode D1 will be
shorter. Changing the secondary inductance has no effect on the output voltage, as the
circuit energy is set by the primary circuit.
In practice the isolated flyback converter has two very useful features:
Power Electronics
18
1. It is able to boost the input voltage (independent of the transformer turns ratio),
making it very attractive as a low power EHT supply (e.g. in CRT television sets,
CRT computer monitors etc.).
2. Because the wound component is the energy storage inductor, any secondary only
requires a diode and a capacitor to produce an independent dc supply. Additionally
it is possible to include an extra winding for the feedback signal, providing isolation
in the feedback circuit, yet without introducing errors due to the volt drop across
an output inductor. It is thus an attractive circuit when a number of independent
outputs is required.
It should be noted, however, that the peak voltage across the transistor during tof f
is:
Vo
VT 1 = Vi +
(1.52)
n
The flyback converter is very common for powers up to about 100 watts.
1.2.2
Forward converter
The forward converter is very similar to the basic buck converter, but the addition
of a transformer provides both isolation and the possibility of a very wide voltage
range.
D1
1:n
L
C
D2
V1
Vo
Load
T1
I1
V1
Iμ
Lμ
n.I2
N1
N2
I2
V2
Figure 14: Forward converter and equivalent circuit
During the transistor on-time, diode D1 is forward biased, and energy is transferred
from the input to the load. During the off-time, D1 is reverse biased and D2 is forward
biased to maintain continuous current through the inductor. In the forward converter
Power Electronics
19
shown above, a serious problem would occur when T1 is switched off. During the on-time
the magnetising current iµ will have been building up linearly (approximately), storing
energy in the transformer core. When the transistor is turned off, there must be a path
for iµ to flow, or very high voltages will result, which will destroy the transistor.
There are a number of schemes to solve this problem, some of which include extra
switching components. A common technique is to include a tertiary winding on the
transformer core, as shown above, which conducts during the off period, providing a
path for im u.
+
1:1:n
D1
L
D2
V1
T1
C
Vo
Load
C
Vo
Load
D3
+
1:1:n
D1
L
D2
V1
T1
D3
Figure 15: Forward converter with tertiary
In a forward converter with a tertiary winding, when the transistor is on current flows
through the primary via the transistor, and also in the secondary (out of dot, through
diode D1 ). Diode D3 is reverse biased. The magnetising current iµ increases. When
the transistor is off no current can flow in the primary. Amp-turns must be maintained
(you can’t interrupt current in an inductor - the magnetising inductance L? in this case).
Amperes Law states:
Z
N.I =
H.dl
(1.53)
i.e. magnetising current must be maintained round the ferrite core, but it doesn’t matter
which coil as long as (N.I) is maintained. During ton current was flowing into the dot
on the primary. With T1 off, magnetising current cannot flow in the primary, nor can
it flow into the dot on the secondary because of D1 , but it can flow into the dot on the
tertiary, up through D3 and back to the supply. This applies Vi to the tertiary, but in
the opposite direction (positive away from the dot), thus causing iµ to reduce.
Power Electronics
20
i1
ni2
iμ
+Vi
in primary
iμ
t
in tertiary
vprimary
t
-Vi
Figure 16: Current and voltage in a forward converter with tertiary
i1 = iµ + n.i2
(1.54)
If the number primary turns = number tertiary turns, i? will rise and fall at the same
rate, therefore the duty cycle D must not exceed 0.5 (50 %) to allow time for iµ to
fall to zero. This results in poor transformer utilisation as energy is being transferred
from the primary to the secondary only during the on period. During the transistor-off
period, the tertiary winding induces a voltage of −Vi across the primary, which adds to
the input voltage to put 2.Vi across the transistor (if the tertiary winding has the same
number of turns as the primary). The d.c. gain of the forward converter when primary
tertiary is 1:1 is:
Vo
= nD
(1.55)
Vi
If the turns ratio of the primary to the tertiary winding is increased (number primary
turns > number tertiary turns), the duty ratio may exceed 0.5, but the resulting voltage
across the transistor during the off period will then be greater than 2Vi . When operating
from a rectified mains voltage (typically 340 volt d.c.), this places a severe duty on the
transistor.
The forward converter is normally used in the continuous mode, where the low current
ripple does not place a heavy duty on the filter capacitor. However, in the continuous
mode the closed loop response is poor (similar to the buck regulator) and can be difficult
to stabilise due to the 2nd order response of the LC filter. At high powers ( > 500 watts)
Power Electronics
21
the poor transformer utilisation results in a bulky design, and other circuits such as the
Bridge Converter are likely to be preferred.
1.2.3
Bridge converter (buck)
In the bridge converter, transistors T1 and T4 conduct together, then transistors T2
and T3 , thus producing a square ac voltage waveform equal to ±Vi on the transformer
primary. The duty ratio D of each transistor must be limited to 0.5, to prevent both
transistors in one leg (T1 and T3 or T2 and T4 ) conducting simultaneously, thus causing
a short circuit across the supply voltage.
+
T1
D1 D2
1:n
T2
+
T1
D3 D4
D1 D2
1:n
T2
-
D3 D4
C
+
V
- o
C
+
V
- o
D6
T4
D5
L
v2
V1
T3
L
v2
V1
T3
D5
T4
D6
Figure 17: Bridge converter
Due to the full wave rectification of the centre-tapped transformer rectifier, the effective
switching frequency, and the effective duty ratio, of the secondary voltage v2 are double
that of v1 . This affects the design of the filter inductor and capacitor, as they are filtering
v2 .
Vo
= 2nD
Vi
(1.56)
As T1 and T3 (also T2 and T4 ) are in series, the gate drives cannot be controlled from a
common reference voltage. This may require the gate drives to be transformer coupled,
Power Electronics
22
+V i
-V i
nV i
v1
T1,T4
T1,T4
t
T2,T3
v2
D5
D6
D5
t
Figure 18: The effective doubly of frequency by a bridge converter due to full wave
rectification
which adds considerably to their complexity. It is important that the positive and negative half cycles applied to the transformer primary are identical, so that the average
voltage across the transformer primary is zero (thus preventing saturation of the transformer core). The bridge converter is generally used in high power applications (greater
than 500 watts) where the transformer in the forward converter becomes excessively
large, and the voltage/current stresses on the single transistor in the forward converter
make it too expensive.
1.2.4
SMPS with multiple outputs
In many applications, more than one output is required, with each output likely to have
different voltage and current specifications. The basic buck, buck-boost and boost regulators are not suitable for multiple output applications. However, multiple outputs can
be readily obtained using any of the converters which have an isolating transformer, by
employing a separate secondary winding for each output, as shown in the forward converter above. In this circuit, each output voltage will be determined by the corresponding
turns ratio n1 , n2 ,or n3 .
1.3
SMPS: Circuit design considerations
The choice of switching frequency depends on a compromise between size and efficiency.
The size of the transformer, inductor and filter capacitor can be greatly reduced by operating at high frequencies. However, semiconductor device switching losses and transformer core losses increase with frequency, reducing the overall circuit efficiency.
Power Electronics
+
23
1 : 1 : n1
Vo1
Vi
-
n2
Vo2
n3
Vo3
Figure 19: SMPS with multiple outputs
The basic design equation for a transformer with a square wave input is:
V1
4f BM
(1.57)
V1
4.44f BM
(1.58)
A.N1 =
For a sinusoidal waveform this becomes:
A.N1 =
The term (A.N ) is a (very) rough measure of the size of a transformer. Thus, it can
be seen from this equation that the size of the transformer is inversely dependent on
frequency; therefore operation at high frequency will reduce the transformer size. Transformer efficiency, however, is reduced as frequency increases, as the eddy current and
hysteresis losses both depend on frequency. In a filter inductor, the boundary condition for continuous mode is proportional to 1/f so as frequency increases, inductor size
decreases. Similarly for the filter capacitor as frequency increases the required capacitance decreases. The output ripple voltage depends on the value of filter capacitance,
the switching frequency and the load current. Thus, by increasing the frequency, the
capacitance can be reduced without increasing the ripple voltage.
More importantly, the output ripple voltage depends on the ESR of the filter capacitor.
As the frequency increases, a larger value of ESR is allowable for the same ripple voltage,
meaning that a smaller capacitor can be used. In the switching transistors the switching
losses increase as the frequency increases. Typical transistor switching waveforms are
shown below in figure 20, including the power dissipated in the device (energy lost is the
area under the power curve).
Clearly, the semiconductor switching losses will increase with frequency. Switching devices are continually improving, particularly with the use of MOSFETs instead of bipolar
Power Electronics
24
+Vcc
R
i
vds
vds
power
i
t
Figure 20: Typical transistor switching waveforms
transistors, so this limitation on switching frequency is much less important than it was a
few years ago. However, above about 150kHz these losses can become significant. Also as
frequency increases problems with electromagnetic interference increase. The reduced
efficiency of high frequency SMPS creates problems of cooling the switching devices.
This may require the use of larger heat sinks and/or the inclusion of fan cooling, which
will add to the size and weight of the power supply. The advantages of high frequency
are:
• Small transformer
• Small capacitor
• Small inductor
Disadvantages:
• Increased transistor switching losses
• Larger heat sinks
• Increased transformer core losses
• Increased EMI
Most power supplies are designed to operate with switching frequencies between 80 kHz
and 500 kHz, although some designers are experimenting into the Megahertz range.
1.3.1
Resonant converters
The use of resonant switching reduces the effects of losses and EMI. When the transistor
is off, the current through it is zero, therefore the power is zero. When the transistor is
on, the voltage across it is very low (although not zero), so the on-state losses are low.
However, when a transistor is switched from the off-state to the fully conducting state,
there is a period when the current through the device has risen before the voltage across
the device has dropped to zero, as shown before in figure 20 above. The power dissipated
Power Electronics
25
in the transistor during this interval can be appreciable, and at high switching frequencies
can lead to the transistor overheating. The exact shape of these curves depends on the
type of transistors used (eg. MOSFETs, IGBTs or bipolar transistors).
In quasi-resonant power supplies, the main switching waveforms are designed to be
sinusoidal rather than square, with transistor switching taking place at either natural
current zeros (zero-current switched converters), or natural voltage zeros (zero-voltage
switched converters).
i
L
E
vL
vC
C
2E
vC
E
i
t
π LC
Figure 21: A resonant L-C circuit and waveforms
In the circuit above, E = vL + vC
vL = L
i=C
di
dt
dvC
dvL
= −C
dt
dt
Therefore:
vL = −LC
d2 vL
dt2
(1.59)
(1.60)
(1.61)
i.e.,
d2 vL
1
vL = 0
+
2
dt
LC
(1.62)
vL = a sin ωt + B cos ωt
(1.63)
Which has a solution of the form:
Power Electronics
26
where
ω=√
1
LC
(1.64)
To solve this, we need to look at the boundary conditions. At t = 0, i = 0 and vL = 0,
therefore A = 0 and B = E.
vL = E(1 − cos ωt)
r
C
i=E
sin ωt
L
(1.65)
(1.66)
Note that the maximum voltage across the capacitor is 2E. The previous analysis assumes
that there is no resistance in the circuit. In reality there will always be some resistance,
if only due to the resistance of the inductor and the wires.
2E
vC
E
Small series
resistance
i
t
π LC
2E
vC
E
Large series
resistance
i
t
π LC
Figure 22: Effect of series resistance on a resonant LC circuit
The circuit shown below in figure 23 is a conventional buck converter, with a resonant
L-C circuit added, it is a zero-current switched converter.
Both iL and vC continue to vary sinusoidally at the resonant frequency until t5 , with
vC peaking at t4 when iL = Io . At time t5 , iL = 0, therefore Io is supplied from Cr ,
discharging Cr linearly. The transistor will cease conducting at t5 . The transistor gate
drive should be removed between t5 and and t6 , to prevent it conducting again at t6 .
Power Electronics
27
Dr
T1
Lr
Lc
iL
vc
Vi
Dc
Cr
Io
Cc
Vo
Load
i
R
L
E
C
t1
t2
vC
vC
iL
Vi
Io
vL
t3
t4
t5 t6
t7
t8
Figure 23: A zero current switched converter with waveforms
t
Power Electronics
28
At time t7 : Vc = 0, therefore Io flows through the freewheeling diode. At time t8 the
transistor is switched on again, and the cycle repeated. It can be seen from the above
that both switch-on and switch-off occur at natural current zeros, greatly reducing the
transistor switching losses. In addition, as there are no very fast edges, EMI is also
considerably reduced.
The above circuits have two main disadvantages:
1. The capacitor voltage rises to twice the input voltage, and the inductor current to
more than twice the output current, placing larger stresses on components than
with the standard buck converter.
2. Output voltage control is achieved by varying the transistor off-time (t7 → t8 ) only,
as the on- time is effectively fixed by the LC time constant of the resonant circuit. Thus changing the output voltage involves changing the switching frequency:
output control is by frequency modulation rather than by pulse width modulation
at constant switching frequency as with more conventional SMPS. This is considerably more difficult to achieve, and can cause other problems (eg. design of the
transformer to handle a wide range of frequencies). It can be shown that:
Vo
fs
=
Vi
fn
(1.67)
where fs is the switching frequency and Fn is the resonant frequency.
Also, as iL must rise to more than 2Io if iL is to reach zero at t5 , then:
r
C
> Io
E
L
(1.68)
Another circuit is the zero-voltage switched converter.
The zero voltage switched buck converter above operates as follows, starting with the
transistor in the ‘on’ state.
Time t1 : The transistor is switched off. The voltage across the capacitor vC cannot
change instantaneously, therefore the voltage across the transistor during turn-off is zero.
vC will increase linearly, due to the approximately constant load current Io .
Time t2 : vC = Vi therefore the freewheeling diode is no longer reverse biased and will
start to conduct load current. iL will therefore start to fall, with Cr and Lr resonating.
Time t5 : vC = 0, and cannot go negative due to the anti-parallel diode. Current iL
increases linearly. The gate drive to the transistor should be applied during this period
(as the anti-parallel diode is conducting, it will not conduct until t6 ).
Time t6 : The transistor starts to conduct again.
Time t7 : iL = Io , therefore the freewheel diode is reverse biased.
Time t8 : The transistor is switched off again, and the cycle repeated.
Power Electronics
29
Dr
Do
vc
iL
t1
t2
Lo
iL
T1
Cr
Vi
Lr
Io
Co
Vo
vC
t3
t4
t5 t6
t7
t8
Load
t
Figure 24: A zero voltage switched converter with waveforms
It can be seen from the above that both transistor turn-on and turn-off occur at natural
voltage zeros, therefore switching losses are greatly reduced. Output voltage control is
achieved by controlling the period (t8 t7 ), ie. by frequency modulation.
There are a large number of resonant switching circuits which include an isolation transformer, an example of which is shown below in figure 25. This is a conventional bridge
converter, with ZVS included. Operation is very similar to the ZVS buck converter
described above.
+
+
-
Figure 25: A conventional bridge converter, with ZVS included.
1.3.2
Control requirements and techniques
A power supply should be designed to:
Power Electronics
30
1. Have good line regulation, such that the output remains constant if the input
voltage varies;
2. Have good load regulation, such that the output remains constant if the load
changes;
3. Have good transient response to system disturbances, such as sudden changes to
the input voltage, or to the load;
4. Remain stable under all operating conditions.
The above requirements are met by designing a feedback control system, which will
control the duty ratio, D, of the transistor to keep the output constant at all times. A
sudden increase in load current will produce one of the effects on the output voltage Vo
listed below:
1. Critically damped (optimum)
2. Under-damped (oscillatory)
3. Unstable
4. Over-damped (sluggish response)
A good power supply will return to the nominal output voltage very quickly, with minimal oscillations. The control block diagram of a simple buck regulator with feedback
control is shown below in figure 26.
vi
Vref
vc
D
Error
PWM
Amplifer
Modulator
Power
Circuit
v2
Load
Filter
Vo
Figure 26: Control block diagram of a simple buck regulator with feedback control
In designing the feedback loop, the transfer function of each block must be determined,
and the loop transfer function optimised. The design of the Power Circuit and Filter has
usually been finalised before the control loop is designed, so in practice loop optimisation
is achieved by appropriate design of the error amplifier / PWM modulator.
The PWN controller can operate in 3 ways:
• Voltage mode control
• Voltage feedforward control
• Current mode control
Power Electronics
31
In most SMPS, the output voltage Vo is controlled by comparing the output voltage with
a reference voltage, and using the resulting error to adjust the duty ratio D.
Io
v2
Vi
v gate
Vo
Load
Error
Amplifier
Comparator
+
-
vc
+
Sawtooth
Generator
Vref
Vs
vc
t
Vgate
t
t on
T
Figure 27: Voltage mode control and waveforms
In voltage mode control, D is obtained by comparing the control voltage vc with a fixed
frequency sawtooth voltage, as illustrated in figure 27. It can be seen that if the control
voltage is increased, the duty ratio D is also increased. Thus:
D=
ton
vc
Vo
=
=
T
Vs
Vi
(1.69)
In this technique, the sawtooth waveform has a fixed amplitude as well as fixed frequency,
and the duty ratio D is adjusted by a change in the output voltage altering the control
(error) voltage vc . This technique used to be very popular as it is very simple, but it does
have drawbacks. The open loop line regulation is poor; i.e. if the feedback loop is broken,
a change in Vi will cause a significant change in the output voltage Vo . Voltage mode
control also has a poor closed loop transient response to changes in Vi , as a change in Vi
will result in a compensating change in D being delayed by the output L-C filter.
Voltage feedforward control is very similar to voltage mode control, except that the magnitude of the sawtooth waveform, Vs , is proportional to the input voltage Vi . A reduction
Power Electronics
32
of, say, 20% in the input voltage Vi will cause Vs to reduce by 20%. This will cause the
duty ratio D to increase by 20%. The transient response to input voltage changes is now
excellent, as the delay caused by the L-C filter does not affect the feedforward control
line. Thus the compensating change in D is instantaneous and precise.
Vs is proportional to Vi therefore,
D=K
vc
Vi
→
Vo = Kvc
(1.70)
Vo is now independent of Vi - excellent open loop line regulation. It is an ideal buck
regulator (continuous mode) as it is independent of load and line voltage. It has excellent
closed loop transient response to input voltage changes but it is not suitable for all
topologies.
Feedforward control is only suitable for some converters because it removes the dependance on Vi . It can be seen that Vi disappears from the dc gain in only the buck
(continuous mode) and the buck-boost (discontinuous mode) converters. In all other
converters voltage feedforward provides only partial compensation, and is rarely used.
However, as the large majority of SMPS are continuous mode buck or discontinuous
mode buck-boost (or derivatives of these circuits), voltage feedforward control is a very
important and easily implemented technique.
Current mode control can be implemented using the circuit below in figure 28.
iL
Vi
T1
r
Io
v2
Vo
vgate
Q
S
R-S
Flip-Flop
R
Clock
Comparator
+
Load
Clock
S
Comparator
inputs
Error
Amplifier
vc
vs
+
vref
t
vc
vs
t
Comparator
Output, R
vgate
t
t
Figure 28: Current mode control and waveforms
In the buck regulator shown above, the clock sends a pulse to the R-S flip-flop which sets
the Q output to logic ‘1’, thus switching on the main transistor. The inductor current
will then build up in a linear fashion:
Vi − Vo = L
diL
dt
(1.71)
Power Electronics
33
When vs (= iL .r) reaches the control voltage vc , the comparator output changes to a
logic ‘1’, resetting the flip-flop Q output to ‘0’, hence switching off T1 . vs immediately
goes to zero, and the comparator output reverts to logic ‘0’.
The advantages of current mode control are:
1. Because the inductor current is being controlled directly, the effect of the inductor
in the feedback loop is eliminated, making what (in continuous mode circuits) is
a second order control system into a first order system by removing the inductor
pole from the loop. This makes the resulting system much easier to stabilise.
2. Current limit protection is easily implemented, by limiting the maximum value
of vc and hence iL (a Zener diode across vc would give an effective, though not
adjustable, clamp).
3. As with voltage feed forward control, current mode control eliminates the dependence of Vo on input voltage Vi . If the input voltage Vi increases, then the inductor
voltage (Vi − Vo ) increases, and the gradient diL /dt increases. Thus vs reaches vc
more quickly, reducing ton and hence the duty ratio D. There are no delays in the
loop, so duty ratio compensation is instantaneous and precise.
Circuits such as the flyback, forward and bridge converters provide electrical isolation
between the input and the output, so that the output is floating. However, as soon as the
feedback loop is closed, there is another connection between the input and the output,
as shown below in figure 29. For the output to be floating, some form of isolation is
necessary in the feedback loop. This can be provided by magnetic coupling (using a
transformer), or opto-coupling.
Rectifier/
Filter
Switching
Circuit
dc
Driver
Circuit
Comparator
dc
Error
Amplifier
Figure 29: Isolation in the feedback loop
Transformers only work with ac, so a transformer cannot be inserted directly between the
output and the error amplifier, or between the error amplifier and the comparator. If a
transformer is to be inserted here, a high frequency ac carrier waveform would have to be
generated, and the dc signal used to amplitude modulate the carrier. This is sometimes
Power Electronics
34
done, but it adds considerably to the complexity of the circuit. Much more common is
to insert a pulse transformer between the driver circuit and the switching transistor(s).
However, this produces another problem, as the control and driver circuits are connected
to the secondary, and therefore must derive their power from the secondary. This can
create start-up problems, as there will be no secondary voltage until the transistors have
started switching.
Another alternative is to use an opto-isolator, either between the output and the error
amplifier, or between the error amplifier and the comparator. This means that the
control and driver circuits can be powered from the dc input (which may cause problems
if operating from a high d.c. voltage). However, opto-isolators tend to be very nonlinear, and extra biasing circuitry may be necessary to produce a reasonably linear
transfer function. This is not a major problem if the opto-isolator is placed after the
error amplifier, as any non-linearities are compensated for by the feedback loop.
Figure 30: An opto-isolator
In circuits where regulation is not particularly tightly specified, the feedback signal can
be taken from an extra winding on the transformer, as shown above. This is the simplest
but least accurate technique, as no feedback compensation will be made for any volt
drops on the secondary circuit, such as across the rectifying diodes or the filter inductor.
[Ideally, the feedback signal should be taken from as close to the load as possible.]
In many applications, there is more than one output, each fed from a separate transformer secondary. The output voltage feedback signal, which is fed into the control
circuit to set the transistor duty ratio D, can usually only be taken from one output.
Hence the value of D will be set at a level which will keep that particular output at the
required voltage, regardless of the voltage at the other outputs. A change in the input
voltage Vi would affect each output in the same proportion, so the feedback signal from
one output would produce an adjustment in the duty ratio D which would correct each
output. Hence the line regulation for the auxiliary outputs would be approximately the
same as for the main output.
In a practical converter where the filter inductors have some resistance, a change in load
current will produce a change in the output voltage. In the case of the main output, the
feedback signal will produce a compensating adjustment in D. However, a change in load
in any of the auxiliary outputs will not (significantly) affect the main output voltage;
Power Electronics
35
Rectifier/
Filter
Switching
Circuit
Driver
Circuit
Comparator
Error
Amplifier
Figure 31: Taking the feedback signal from an extra transformer winding
therefore the feedback circuit will make no compensating adjustment to D. Thus the
load regulation of the auxiliary outputs will not be as good as for the main output.
This will be accentuated if discontinuous mode operation is used, as the output voltage
inherently depends on the load.
I is possible to ‘share’ feedback from 2 outputs by winding inductors on same core which
improves dynamic response. Typically, a multiple output SMPS may have 0.5% load
regulation on the main output, but only 3% on each of the auxiliary outputs. If close
regulation on an auxiliary output is required, then a post-regulator must be used. This
is normally either a linear regulator or a basic buck converter (with its own feedback
circuit) placed at the appropriate output, which will add considerably to the size and
weight of the total power supply.
Power Electronics
2
2.1
36
Design of magnetic components
Laws of electromagnetism
Ampere’s law:
I
B.dl = µ0 I
(2.1)
Faraday’s law:
I
E.dl = −
∂
∂t
ZZ
B.dA
(2.2)
These can be simplified into more useful forms when regarding transformers:
I
mmf =
H.dl = N I
(2.3)
E=N
dφ
dB
= NA
dt
dt
(2.4)
B is the magnetic flux density, units Tesla, T.
H is the magnetic field intensity, units Amperes per metre, Am−1 .
B = µ0 µ r H
(2.5)
The units of B.H is energy/m3 . Thus the area under the B.H curve represents energy
density. The area enclosed within the hysteresis curve represents energy lost (per unit
volume) in the magnetic core during one complete cycle. If B goes above the saturation
flux density Bs , very high values of H, and hence I, result, which is likely to lead
to destruction of semiconductor components. Br is the remanent flux density. If the
magnetising current (and hence H) is reduced to zero, there will still be some remanent
flux left.
2.2
Inductors
The purpose of an inductor is to store energy. Energy storage depends directly on the
inductance of the device, which for a long straight coil is:
L=
Nφ
µ 0 µ r N 2 Ac
=
I
le
(2.6)
Where N is the number of turns, Ac is the cross sectional area of the inductor core,
le is the effective length of the magnetic path, µ0 is the permeability of free space
(4π × 10−7 N A−2 ) and µr is the relative permeability.
Power Electronics
37
B
Bs
Br
H
Figure 32: BH hysteresis curve with simplification (dotted line)
When there is an air gap the characteristic is still dominated by the air gap, despite
the ferrite path length = 100 x air gap (typical). The composite B-H characteristic of a
ferrite core with an air gap is shown below in figure 33.
The blue line on the left is the characteristic of the ferrite alone, with relative permeability of 3000 at 0.1 tesla flux density. This characteristic is highly non- linear, with
saturation occurring at 0.33 tesla at 100◦ C. The green line sloping linearly to the right
is the air gap characteristic, with an actual relative permeability of 1.0 but an effective
permeability of 100. [The effective permeability is the average value that would be obtained if the air gap were spread out over the entire length of the magnetic path through
the core. In this case, the total magnetic path length is 100 times greater than the gap
length, resulting in an effective permeability of 100.]
The pink line represents the overall characteristic which is the sum of the magnetic
fields in the ferrite and in the gap. Even though the distance through the ferrite is 100
times longer, its permeability is so large the composite characteristic is dominated by
the air gap. The effective permeability of the composite is approximately 100, averaging
the actual permeabilities of 1.0 in the gap and 3000 in the ferrite over the entire path
length.
The energy stored per unit volume (energy density) at any operating point is the area
to the left of the curve from the origin up to the operating point:
3
Energy/m =
Z
1
H.dB ≈ B.H
2
(2.7)
Power Electronics
38
ɸ
i
B (Tesla)
ferrite
μr = 1
0.3
N
air
μr = 1
0.2
0.1
H(A.m-1)
1000
ɸ
ferrite with
air gap
2000
3000
Figure 33: Effect of an air gap
The approximation applies if operating on the linear part of the curve (constant permeability). The curves also show that the area to the left of the ferrite characteristic alone
is very small, indicating that it is not possible to store significant energy in the ferrite
because its high permeability results in a very small magnetic field intensity. This is ideal
for a transformer, where energy storage is undesirable, but not for an inductor, whose
main function is to store energy. When it is necessary to store a significant amount of
energy, this can be accomplished by introducing an air gap in series with the magnetic
core.
Assuming minimal flux fringing in the gap, the flux density in the gap B will be the same
as in the core. For air µr = 1, and for the ferrite core µr = 3000 typically, therefore the
magnetic field strength in the gap will be much higher than in the ferrite. Therefore, to
a first approximation, the ferrite can be ignored and all the mmf can be considered to
exist in the gap. Thus, for the gapped inductor:
H=
NI
lg
(2.8)
It is usually assumed that all of the energy in the system is stored in the gap. The
energy stored in the field outside the gap causes a few percent error in the calculation
of stored energy and inductance; but, in practice, it is usually possible to use this
assumption.
Without an air gap the inductance is very large, which would seem to be good for energy
storage. However, only very small currents are possible without going into saturation:
thus, little energy can be stored.
E = 12 LI 2
(2.9)
Power Electronics
39
B (Tesla)
0.3
0.2
0.1
H (A.m-1)
1000
2000
3000
Figure 34: Energy storage in an inductor with an without an air gap
An alternative is to use a powder core which stores its energy in a series of microscopic ‘gaps’ in the binder which holds the magnetic particles together - the gap is thus
effectively distributed round the core.
2.3
Transformers
The function of a transformer is to transfer energy, not to store energy. Thus, it contrasts with the inductor, which is designed to store energy. In practice, the transformer
parasitics cause losses and the device is less than 100% efficient. When the primary
winding is connected to an ac voltage source, a small current flows to magnetise the
core, even if the secondaries are all open circuit. This magnetisation develops voltages
in the secondary windings. Since the same magnetic flux couples the primary and secondary windings, the open-circuit voltages induced in the secondaries are proportional
to the number of turns on each winding.
V2
N2
=
V1
N1
(2.10)
It is important that the sense of the different windings is clearly specified since this
dictates the polarity of the ends of the windings with respect to each other. It is conventional to identify polarity by a dot at the ends of the windings which are of the same
polarity. Thus, if at the instant considered in figure 35 below, i1 (t) is in the direction
shown, then the dotted end of the winding will be positive. This means that the dotted
ends of both secondaries are positive. Since these are essentially voltage sources, the
directions of the secondary currents will be as shown.
The physical size of a transformer depends on its volt-amp (VA) rating. Cores are often
rated on their volt-amp capacity, ie. the product of total rms current and the operating
Power Electronics
40
i1(t)
i2(t)
v2(t)
v1(t)
v3(t)
i3(t)
Figure 35: A basic transformer
rms voltage. If we assume an efficiency of η, then the primary volt-amp rating will
be:
v2 i2 + v3 i3
(V A)1 = v1 i1 =
(2.11)
η
There is imperfect coupling between primary and secondary coils, in reality:
v2
N2
=
N1
v1 .K
(2.12)
Where K is the coupling coefficient, K ≈ 0.98 → 0.999. The coupling coefficient and
efficiency are different. Efficiency is related to transformer losses (copper losses in the
windings, hysteresis and eddy current losses in the core). The coupling coefficient is the
proportion of the magnetic flux that is coupled by both primary and secondary windings,
and is related to leakage inductance.
2.4
Wound component production
Devices are usually created for operation between typically -10◦ C to +45◦ C for commercial applications and typically -50◦ C to +100◦ C for military applications. The most
commonly used material for windings is copper. When considering wound components,
the low resistivity of copper is important for two reasons;
• Losses (and hence efficiency) of a wound component depends on, among other
things, the resistance of the windings,
• Regulation of the output voltage depends on any voltage drop in the resistance of
the wound component winding.
The rms current density in a winding is limited because of the associated temperature
rise. An rms current density of 5 A mm−2 causes typically a 30◦ C temperature rise with
natural convection cooling for a transformer or inductor.
The skin effect results in a uniform current density for dc and a current density greater at
surface as currents induced in conductor by field for ac currents. In conductors carrying
Power Electronics
41
dc the current density will be uniform across the cross-section of the conductor. However,
in high frequency circuits the current density will be much higher at the surface. The
skin depth δ is the distance from the surface of the conductor to where the current
density is 1/e times the surface current density. Thus, when the skin depth is smaller
than the radius of the conductor the centre of the conductor is carrying virtually no
current: it is thus not efficient to increase the radius beyond the skin depth.
conductor
walls
approximate
current
density
current
density
δ
δ
δ
real
current
density
Figure 36: The skin effect
r
δ=
ρ
πµ0 µr f
(2.13)
For copper δ = 0.2 mm at 100kHz and δ = 0.1 mm at 400kHz. To overcome the skin
effect there are various solutions:
1. Use 2 strands of smaller diameter wire, rather than one thick wire. More than 2
strands becomes difficult to wind.
2. Use foil (tape), with a maximum thickness of 2δ. However, more than about 4
turns is not usually feasible to wind.
3. Use Litz wire. Here each strand (insulated from each other) spends part of its
length on the surface and part in the centre. Thus all wires carry the same current.
Litz wire is expensive, and care has to be taken that each strand is properly
connected at each end (adding further to the cost).
2.4.1
Core shapes
There are many different shapes of cores, all with their advantages and disadvantages.
Power Electronics
42
δ
Foil (tape)
2 strands
“Litz” wire
Figure 37: Solutions to the skin effect
Figure 38: E core transformer
Power Electronics
43
E-cores are generally used for higher power transformers (>130 watts). They are also
appropriate for inductors, as an air gap can easily be created either by grinding back one
of the central limbs, or by inserting shims between the two halves. They come with a
plastic bobbin which fits over the central limb, which eases the winding. The relatively
open construction allows good cooling of the core and windings.
Toroidal cores are available from small beads up to a maximum size of around 4 cm
outside diameter.
Advantages:
• The main advantage of the toroidal shape is that it is available in different aspect
ratios - deep with small diameter to shallow with large diameter - and can thus be
fitted into unusual spaces. Cores can also be stacked for additional power.
• With toroids good magnetic coupling can be achieved.
• Cores give minimum reluctance for a given core volume.
Disadvantages:
• Toroids are more difficult to wind than bobbins, especially for multi-layer construction.
• Difficulty is experienced in insulating between layers.
• Mechanical fitting to boards can be difficult.
• Achieving a suitable termination is more difficult than for the other core configurations.
Figure 39: Toroidal core transformer
In pot cores the windings are almost completely enclosed by the core, resulting in very
low leakage flux, and hence very low EMI. However, the enclosed construction means
that there is very poor cooling available for the windings. RM cores are a slightly more
Power Electronics
44
open version of the pot core. Not as low leakage and EMI, but better cooling. They’re
easy to wind and can be mounted on a PCB.
Screw core
Sleeve
Pot core
Coil Former, 1 section,
with 10 pin terminals
Pot core
Figure 40: Pot core transformer
2.4.2
Materials
For transformers operating at mains frequency, soft magnetic steel with laminations
(to reduce eddy current in the core) is used. At high frequency (¿10kHz) laminations
would have to be unmanageably thin to be effective, so ferrites with high resistivity
are normally used. These are ceramic materials made from iron oxides sintered with
(usually) manganese and zinc oxide (MnZn), although sometimes Nickel and Zinc is
used.
The peak operating flux density depends very much on the material used and the temperature at which the device will operate. Designs are usually restricted to approximately
80% of saturation level but where an increased safety factor is required, for example to
cope with any change in characteristic with increased temperature and to accommodate
the magnetic conditions during overload or short circuit, operating flux densities may
be reduced to as low as 50% of the saturation value. It should be noted that Bsaturation
reduces with increase in temperature. Typical values of Bsaturation are 150-350 mT.
2.4.3
Windings
Care must be taken in winding power transformers, to minimise leakage inductance and
capacitance, and also to ensure a symmetrical and predictable voltage in the output
windings.
Power Electronics
45
Windings on bobbins have a layer of insulation tape between each layer of the winding.
For insulation between windings (ie between the primary and the secondary), a suitable
polyester tape may be used for insulation since these have high dielectric and mechanical
strength.
In toroidal windings interlayer insulation may be with terylene tape. Interwinding insulation uses the same material. With high voltage devices, it is good practice to wind the
primary on one side of the toroid and the secondary on the other, although this reduces
coupling efficiency. This technique is also used to minimise capacitive effects at high
frequency.
2.4.4
Inductors (chokes)
If ferrites are used, a gap must be included on the centre limb (E, RM and pot cores) for
energy storage. As it is not usually possible to include an air gap with a toroidal core,
ferrite toroids are not normally used for inductors. Increasingly composite powdered
metal cores are used for high frequency inductors. Here high permeability metal particles
are bound together by a non- magnetic binding material. The non-magnetic binding
material acts as a distributed air gap, so no separate air gap is required. These materials
are not normally used for transformer applications.
The precise material chosen will largely depend on the operating frequency. Materials
with a high relative permeability generally have a low resistance, and therefore high core
losses at high frequencies.
Current densities of 3-10 A.mm−2 with 5 A.mm−2 as typical. With chokes, it is important to take into account the IR voltage drop in the choke (due to output direct current)
for output voltage regulation. A typical value of IR drop is 0.1 - 0.15 V for a 12 V
supply.
For ferrite cores typical values of Bsaturation are 300-400 mT, whereas for powdered metal
cores values of 800 mT are possible.
2.5
Transformer design
The design approach adopted is one of many possible, and it is not claimed that this
is the only, or indeed, the best, design methodology. The example uses components
from one specific manufacturer, but again, it should be emphasised that many other
manufacturers produce components that are equally suitable.
The transformer specification will contain information on which the engineer will base
their designs. This list is typical of the information which is required. If not supplied
by the client, the designer should request it before starting out on the design. If some
Power Electronics
46
details are not available, the designer should decide on suitable values, based on his
experience and on discussions with the end-user.
Other information needed include mechanical details and often it is mainly mechanical
restrictions which limit transformer design. The following information is required: Volume, i.e. maximum height, width, length of transformer. If the component is toroidal,
a minimum inner diameter is required. Method of fixing, e.g. printed-circuit-board
mounted or chassis mounted. Weight limitation. Terminations, e.g. flying leads or
former terminal pins.
The working environment must be described in order that component construction, materials etc., may be designed. The following details are used: Ambient temperature,
e.g. 0◦ C to +40◦ C. Component finish, e.g. dry or varnish impregnated or resin encapsulated. Vibration (frequency and amplitude). Screening, specify electromagnetic and
electrostatic screening requirements. Figure 41 below might well be the transformer used
in a typical bridge converter with a typical frequency of 100kHz.
primary
secondary A
vin
25A
vA1
0V
5A
vB1
+5V
+12V
0V
secondary B
10μs
4μs
Primary Voltage:
300V
0.1μs
Figure 41: Example transformer with primary voltage
Normally the input voltage, and therefore the duty ratio, would vary. Here, fixed values
for both are used for simplicity. The total VA is required to enable an appropriate
Power Electronics
47
core to be selected. The transformer output VA includes the output power from each
output, plus the losses, mainly due to the diodes. A 0.8 volt drop across the diode is
assumed (this is slightly high to take into account the resistance of the wires and the
inductor).
V Asecondary A = (25 × 5) + (25 × 0.8) = 145W
(2.14)
V Asecondary B = (12 × 5) + (5 × 0.8) = 64W
(2.15)
The core material depends mainly on the operating frequency and temperature. Each
manufacturer has a range of materials. E cores are the most common shape for medium/high
powers, as cooling is normally a main consideration. The core size is chosen to minimise
losses and reduce temperature rise. Losses are hysteresis losses and copper losses (eddy
current losses are negligible in ferrites).
2.7
Physteresis ∝ f 1.7 × Bmax
W/m3
(2.16)
The above formula is empirical. Increasing the cross sectional core area reduces flux
density, hence reduces losses: however, note that a bigger area means the volume goes up,
and that the hysteresis losses are watts per unit volume, so it is not simple to optimise.
Also, increasing the cross sectional core area increases the length of one complete turn,
hence a longer wire needed, which increases the i2 R losses.
There are many ways of choosing a core, with each manufacturers’ publicised data based
on their own preferred method. The simplest is where the cores are chosen purely on
power rating. Note that this does not always produce the optimum solution, and more
detailed methods are common. The windings must all fit into the ‘window’ area, L.H,
as shown in figure 42 below.
H
L
D
“window”
Figure 42: Window in which windings must fit
A flux density of Bmax = 0.2T is chosen as Bsat reduces with temperature, so lower
values of B are required if the core is likely to operate above about 100◦ C. Higher values
increase hysteresis losses and a lower value would be chosen if core temperatures were
> 100◦ C. The secondary voltage can be calculated using:
V0 = (vsec − 0.8) × 2D
(2.17)
Power Electronics
48
In the above formula: 0.8 takes account of the diode volt drop + an allowance for
inductor resistance. The secondary duty ratio is 2D as it is a bridge converter. For full
wave rectification:
N ABmax 2f
∆B
=
(2.18)
V = NA
∆t
D
During the positive ‘on’ period, the flux swings from −Bmax → +Bmax , i.e. a swing of
2Bmax . In the negative 1/2 cycle it goes back to −Bmax . From this the number of turns
on the secondary can be calculated to be 0.564, however this must be an integer. For low
output voltage transformers, it is usual to calculate the number of secondary turns first,
starting with the lowest output voltage. As there must be an integer number of turns,
errors are reduced. A ratio of 4:9:170 will suit this example, minimising rounding errors.
As the secondary is centre-tapped, each half of secondary A must have 4 turns.
When designing the windings aim for a current density in the wires of about 5A.mm−2 .
Higher than this gives unacceptably high losses (the wire will overheat). Much lower
than this means that the core window area has to be bigger than necessary to fit the
wires in (remember that hysteresis losses are proportional to volume, so there is a penalty
to pay for having a bigger window). If the wire diameter is > 2 times skin depth, the
centre of the wire is not being used (no current flow in the centre), reducing the effective
cross sectional area. Thus the current density in the outer part of the wire will increase,
probably to considerably higher than 5A.mm−2 .
√
In the secondary A winding, each 1/2 conducts 25A for 1/2 of the time. Irms = Idc × 0.5.
For a current density of 5A.mm−2 this gives a diameter of 2.1mm and a skin depth of
0.2mm. To ensure that all the wire is used need to use foil twice the skin depth. 4 turns
of foil is as many as you’d ever want to wind.
In the secondary B winding, each 1/2 conducts 5A for 1/2 of the time, using the same
procedure, the diameter required is 0.95mm with a skin depth 0.2mm, this would require
too many turns of foil, instead using two strands of 0.7mm diameter wire means that
there is some wastage in the centre, but not too much.
Knowing the diameters of the secondary windings, the primary winding can be calculated. The input power can be calculated from the output power and the efficiency and
knowing the input voltage the input current can be deduced, using this value and the
same current density as before the required diameter is 0.5mm. There will be a very
small area in the centre where no current will flow, but this is negligible.
All of these windings need to fit into the window area (L.H). Frequently the design has
to be re-calculated as it doesn’t fit! In that case, a larger core with a bigger window,
or thinner wires (which will then run hotter), will be required. Often several design
iterations are required before a final design is reached. The fill factor is the proportion
of the window filled with copper (primary and secondaries). It is unrealistic to expect to
achieve a fill factor of greater than about 0.6, as allowance has to be made for wire insulation, a layer of insulation tape between the primary and secondary windings, and small
spaces between wires due to round wires being used, and also imperfect winding.
Power Electronics
3
3.1
49
Power semiconductor devices
PN junction diode
The simple pn junction diode can handle up to 7,500A, 4,000V with junction temperatures up to200◦ C. It is very rugged. The maximum current that a diode can conduct is
normally determined by thermal considerations. During normal operation, the diode p-n
junction dissipates approximately 1 watt of power for each ampere of forward current.
For safe operation the temperature of the junction should not rise above 200◦ C. The
actual pellet of silicon which performs the rectification is small and has a very low thermal capacity. For this reason, the silicon pellet is mounted between heavy copper parts
in a symmetrical arrangement that results in a uniform distribution of thermal stresses,
and low thermal resistance. Figure 43 below shows a double-sided heatsink. In lower
power applications single-sided heatsinks are normally used. Note that in the figure, the
heatsink (made of aluminium) forms part of the electrical circuit, so is live.
heatsink
silicon
copper
case
Figure 43: A double-sided heatsink pn junction diode
The device and accompanying heatsink are chosen to ensure that the junction temperature does not rise to levels that will destroy the device. The junction temperature can
be calculated according to:
θj − θa = P.Rth
(3.1)
where θj is the junction temperature, θa is the ambient temperature, P is the power
dissipated in the junction and Rth is the total thermal resistance.
Rth can be split up into Rth-jc (thermal resistance from the junction to the case), Rth-ch
(thermal resistance from the case to the heatsink), and Rth-ha (thermal resistance from
the heatsink to ambient).
Rth = Rth-jc + Rth-ch + Rth-ha
(3.2)
Power Electronics
50
The thermal resistances will be specified by the device and heatsink manufacturers.
A larger heatsink will have a lower thermal resistance Rth-ha , thus a lower junction
temperature will result for the same current (and hence same power).
The major source of power loss in a silicon diode arises from the forward-conduction
voltage drop, usually around 0.5 to 0.8 volt. Although the characteristic of an ideal
diode is given by:
qv
I = I0 e kT − 1
(3.3)
In power diodes the forward characteristic is dominated by the device’s dynamic resistance. This resistance is largely due to the lightly doped regions and long depletion
layers required to achieve high reverse breakdown voltages. The reverse characteristics
ar eshown below in figure 44.
reverse voltage
1μA
reverse
current
150°C
25°C
Figure 44: Reverse characteristics of a pn diode
Initially, reverse current increases slightly as the reverse voltage increases, but then tends
to remain relatively constant, even though the reverse voltage is increased significantly.
The figure also indicates that an increase in operating temperature causes a substantial
increase in reverse current for a given reverse voltage. Reverse-blocking thermal runaway
may occur because of this characteristic if the reverse dissipation becomes so large that,
as the junction temperature rises, the losses increase faster than the rate of cooling.
However, generally the reverse leakage current can be considered as negligible.
If the reverse voltage is continuously increased, it eventually reaches a value at which a
very sharp increase in reverse current occurs. This voltage is called the breakdown or
avalanche voltage. If the diode is operated beyond this point, it may be destroyed as a
result of thermal runaway.
After a silicon diode has been operated under forward-bias conditions, some finite time
interval (in the order of a microsecond) must elapse before it can return to the reversebias condition. During this period, charge carriers in the device constitute a reverse
current known as the reverse-recovery current.
Power Electronics
51
if
t rr
vf
t
Figure 45: Reverse recovery time of a pn diode
The reverse-recovery time imposes an upper limit on the frequency at which a silicon
diode may be used. Any attempt to operate the diode at frequencies above this limit
results in a significant decrease in rectification efficiency and may also cause severe
overheating and resultant destruction of the diode because of power losses during the
recovery period.
Ratings for silicon diodes are determined by the manufacturer on the basis of extensive
testing. These ratings express the manufacturer’s judgement of the maximum stress
levels to which the diodes may be subjected without endangering the operating capability
of the device. Diode ratings include: peak reverse voltage, forward current, and i2 t (the
latter is a measure of energy that the device can safely absorb, used for choosing a
fuse).
3.2
Schottky diode
In low voltage applications (<15 volt), the on-state voltage of a pn junction diode can
represent a major circuit loss. For example, in a Buck regulator with a 5 volt output,
the diode volt drop of typically 0.8 volt means that the maximum efficiency is only 86%,
disregarding any other losses in the circuit. In such applications Schottky diodes are
increasingly used.
A Schottky diode is formed by placing a thin film of metal in direct contact with a
semiconductor. The metal-semiconductor structure forms a low-resistance ohmic contact to semiconductor materials of all types. The operation of the device depends on
quantum-mechanical effects which are beyond the scope of this course. The arrangement
Power Electronics
52
+5V
0V
Figure 46: Forward converter where a Schottky diode would be used
has a rectifying v-i characteristic very similar to that of a pn junction diode. The major
difference is that at any given forward current, the voltage across the Schottky diode is
typically 0.3 V less than that across a pn junction diode.
In the reverse direction, the Schottky diode has a reverse leakage current that is larger
than that of a comparable silicon pn junction diode. With present fabrication techniques,
the breakdown voltage of a Schottky diode cannot reliably be made much larger than
100 volts.
A Schottky diode turns on and off faster than a comparable pn junction diode. The basic
reason is that Schottky diodes are majority carrier devices and have no stored minority
carriers that must be injected into the device during turn-on and removed during turnoff. Thus, during turn-off, there will be no reverse current associated with removal of
stored charge. However, reverse current, associated with the growth of the depletion
layer charge in reverse bias, will flow.
3.3
Power MOSFETs
Ideally, a transistor used as a switch will have the following characteristics: A high
input (gate) impedance, a low on-resistance, a high off-resistance, an ability to withstand
high overvoltages, fast switching and ruggedness. A transistor is schematically drawn
as:
drain
gate
source
Figure 47: Schematic transistor
Note that power MOSFETs generally have a vertical structure, compared to the lateral structure of low power MOSFETs. This vertical structure allows many cells to be
Power Electronics
53
connected in parallel to form the complete transistor. Although p-channel MOSFETs
exist, the lower mobility of holes reduces their performance, so n-channel devices are
generally preferred in power applications. The power MOSFET is a device that evolved
from MOS integrated circuit technology. It has now largely replaced the power bipolar
transistor due to the latters large base drive current requirement and its limited switching speed capability (the bipolar device is current driven, while the MOSFET is voltage
driven).
gate
gate oxide
source
p
n
polysilicon
gate
source
n+
n+
current
path
p
n
n+
integral
“body”
diode
drain
Figure 48: Schematic transistor
Power MOSFETs are always used as switches; hence operation is in the linear region (to
the left of the pinch-off curve). The linear relationship between id and vds is described
by the drain-source resistance:
vds
(3.4)
Rds(on) =
id
MOSFETs generally have very low switching losses, and are commonly used in switching
circuits up to several hundred kHz. They are available in voltage ratings in excess of
1000 V (but with small current ratings), and with current ratings up to 100 A (but with
small voltage ratings). MOSFETs are easily paralleled because their on-state resistance
has a positive temperature coefficient. Thus if one device is conducting a higher current,
its higher temperature will cause Rds(on) to increase, hence reducing its current and thus
forcing it to equitably share its current with the other MOSFETs in parallel.
Due to the structure of the power MOSFET, there is an integral ‘body’ diode between
the source and drain, in anti-parallel with the transistor. This can be used as a feedback
diode in bridge converters, but its characteristics tend to be poor, so an external diode
is normally included.
The physical structure of a MOSFET results in capacitance between the terminals. The
Power Electronics
id
54
pinch-off
increasing
vgs
vds
Figure 49: MOSFET Characteristic
polysilicon gate / gate oxide structure determines the capacitance from gate to source
(Cgs ) and gate-to-drain (Cds ). The pn junction formed during fabrication results in a
junction capacitance from drain to source (Cds ).
MOSFETs require the continuous application of a gate-source voltage to be in the onstate. The oxide layer that separates the gate from the channel region gives an input
impedance of 109 to 1011 ohms, so no gate current flows except during turn-on and turnoff, when the gate capacitance is being charged or discharged. Sufficient drive current
must be sourced or sunk to change the gate voltage in the required time. The gatesource voltage should be greater than the gate-source threshold voltage, typically about
4 volt. A higher voltage (approximately 10 volt) is usually used to ensure that the device
is fully turned. However, values of vgs in excess of 20 volt are likely to break down the
gate oxide layer, destroying the device.
The switching times in a power MOSFET are very short, being in the range of a few
tens of nanoseconds to a few hundred nanoseconds depending on the device type. The
switching time is determined by the gate capacitance which must be charged and discharged, and the current available from the drive circuitry. The driving impedance and
the input capacitance affect the switching speed of the MOSFET: the lower the driving
resistance the faster the capacitance will charge and the faster the device will switch
on.
Claims of zero gate current for power MOSFETs are valid only at low frequency. As
the switching frequency rises, the gate drive circuit needs to be able to source and sink
the pulse currents required to charge and discharge the high input capacitance of these
devices. Because of the relatively low drive voltage requirement and its high input
impedance, the device is well suited for the control of high power directly from low-level
logic circuits such as CMOS, provided adequate charge is supplied to charge up the input
capacitance.
Power MOSFETs have been portrayed as being capable of fast switching speeds, but
Power Electronics
55
having a relatively large forward voltage drop in the on state due to the on-state resistance Rds(on) . Rds(on) increases rapidly with the device reverse voltage rating. Thus,
only devices with small voltage ratings have low on-state resistance.
3.4
Insulated gate bipolar transistor
The Insulated Gate Bipolar Transistor (IGBT) has appeared on the scene relatively
recently as a successful semiconductor device that combines the advantages of the power
MOSFET and the bipolar junction power transistor. Like the power MOSFET, it is
a voltage controlled switch, and its gate control requirements are practically the same
as for a power MOSFET. However, its ON state voltage drop is typically lower than
that of a power MOSFET: in this respect it is closer to a bipolar power transistor.
Unlike the power MOSFET, the IGBT has no integral reverse (body) diode. IGBTs are
manufactured in voltage and current ratings extending well beyond what are normally
available in power MOSFETs IGBTs up to 6,500V and up to 2,400A are commonly
available. However, they are not able to operate at switching frequencies as high as
power MOSFETs.
gate
gate oxide
emitter
J2
J1
polysilicon
gate
emitter
collector
gate
n+
p
n+
J3 p
current
n- path
n-
n+
p+
n+
p+
emitter
collector
gate
collector
emitter
Figure 50: Insulated gate bipolar transistor
Figure 50 shows the junction structure of a typical IGBT cell. It is very similar to
an n-channel power MOSFET; however, in the IGBT, there is an additional p+ layer
over the drain layer of the power MOSFET structure. This p+ region constitutes the
‘collector’ of the IGBT. The collector and the emitter are the power terminals of the
IGBT. Comparison with the power MOSFET structure shows that the emitter’s place
in the structure is identical to that of the source in the power MOSFET. The gate is the
Power Electronics
56
control terminal in both devices. The switching control voltage for the IGBT is applied
between the gate and the emitter. The circuit symbol generally used for the IGBT is
as shown. It is similar to that of an npn bipolar junction power transistor, but with an
insulated gate terminal in place of the base.
The additional p+ layer eliminates the integral body diode of the MOSFET, as junction
J1 is reverse biased when the IGBT is reverse biased. In ‘asymmetric’ (or ‘punchthrough’) IGBT’s the n region consists of an n+ and an n- region. The n+ region
serves to reduce the ON-state forward voltage drop, but also reduces the reverse voltage
blocking capability of the device to a few 10s of volts. In ‘symmetrical’ (or ‘non-punchthrough’) IGBT’s the n+ layer is not included, increasing the reverse voltage blocking
capability to the forward blocking voltage value, but at the cost of increased forward
voltage drop in the ON state.
With no gate voltage applied, there is no inversion layer beneath the gate, so the MOSFET junction J2 is reverse biased (when the device is forward biased), giving the IGBT
its forward blocking capability.
When a gate voltage is applied, an inversion layer forms beneath the gate. Current flows
through the inversion layer from collector, injecting holes from the p+ region into the
n- region, reducing the ON state voltage of the n- region compared to the MOSFET.
These hole move across the n- region to the p region, which acts as the collector of a
pnp transistor. The p region is connected to the IGBT emitter metal contacts. Thus
there are two parallel current paths:
• Through the pnp transistor, from the IGBT collector to the IBGT emitter.
• Through the p+/n+ diode at the IGBT collector and the MOSFET inversion layer
to the IGBT emitter. Most of the IGBT current takes this path.
The gate drive circuits for IGBTs are similar to those for power MOSFETs. The switching performance at turn ON is very similar to that of the power MOSFET and the time
specifications are about the same. The IGBT is different to the MOSFET as regards
turn OFF switching behaviour. During turn OFF, the initial fall in current (to about
25% of the ON state current) is steep, similar to that of the power MOSFET. But this is
followed by a longer ‘tail’ during which the decay takes place relatively slowly. The tail in
the current decay waveform is because of the time needed for the excess minority carriers
(injected holes) in the base region of the pnp transistor to disappear by recombination.
Since there is no external terminal in contact with this n zone through which these excess carriers can be ‘sucked out’, the carrier lifetime by and large determines this tail
duration. The overall turn OFF time is thus longer than in the power MOSFET.
Power Electronics
4
57
Gate drive circuits
The purpose of the gate drive circuit of a power semiconductor is to switch it on and
switch it off. This section will concentrate on the gate drive requirements and circuits
of power MOSFETs. The requirements for IGBTs are very similar, as they have a very
similar high impedance gate.
When a transistor is switched from the OFF-state to the ON-state, there is a period when
the current through the device has risen before the voltage across the device has dropped
to zero, as shown in figure 51 below. The converse is true at turn- off. The instantaneous
power dissipated in the transistor during this interval (vdci) can be appreciable, and, at
high frequencies, can lead to the transistor overheating. Fast turn-on and turn-off times
are necessary to reduce these switching losses.
+Vcc
R
i
Vds
Vds power
i
t
Figure 51: MOSFET switching
MOSFETs have the ability to switch large currents very quickly. Their very high gate
impedance means that the gate drive circuit requirements are far less onerous than for
bipolar transistors and thyristors. However, there are several requirements that must be
met if the MOSFET is to perform to its full capability. These include:
1. An ON state gate-source voltage vgs of between 4 and 20 volts. Less than 4 volts,
and the MOSFET will not be full turned on, while a vgs more than 20 volts is
likely to break down the gate oxide layer, destroying the device. Values of 10-15
volts are common.
2. An OFF state value of vgs of between 0 and -20 volts.
3. The ability to source/sink currents sufficiently rapidly to charge/discharge the
gate-source capacitance to ensure fast turn-on/turn-off.
4. Isolation between the control circuit and the MOSFET gate/source (not required
in all applications).
A MOSFET will begin to conduct when the threshold voltage is reached (= 2-4 V), and
will be fully on when vgs = 7-8 V. The waveforms in figure 52 below are idealised: in
practice, there is a short period when both vgs and ig are constant, just after vgs has
Power Electronics
58
reached the threshold voltage. The rate of rise and the magnitude of the gate current ig
determine the rise and fall times of the drain current.
vgs
t
ig
control
t
Figure 52: Gate drive control
Some integrated circuits specifically designed for switched mode power supply control
purposes can provide up to 100 mA of sink and source output capability, and when
directly driving a MOSFET can switch reasonably efficiently at 50-200 kHz. However,
to switch efficiently at higher frequencies, several amperes of drive may be required,
requiring a dedicated driver circuit - also available as an integrated circuit.
+15V
to load
control
circuit
power
MOSFET
Figure 53: Simple gate drive
When the npn bipolar transistor in the figure ?? is turned on, the gate source capacitance
of the MOSFET will rapidly discharge, giving fast turn-off. However, the disadvantage
of this circuit is that MOSFET turn-on will be slow, as the resistor will slow the charging
of the gate capacitance. An improved version is shown below in figure 54, with a totem
pole output ensuring rapid charge/discharge of the gate capacitance.
In some applications it is necessary to provide isolation between the control circuit and
the power circuit. In this case, an isolating pulse transformer is normally used: a typical
circuit is shown below in figure 55. When the input is high, the body diode of the
auxiliary MOSFET conducts to switch on the power MOSFET. When the transformer
saturates, this diode will prevent the power MOSFET gate capacitance discharging.
Power Electronics
59
+15V
to load
power
MOSFET
control
circuit
0V
Figure 54: Improved gate drive
When the input is negative, the auxiliary MOSFET will be switched on, allowing the
power MOSFET gate to discharge.
to load
1:1
power
MOSFET
0V
Figure 55: Gate drive with isolation
4.1
Snubbers
High frequency power conversion circuits subject power transistors to high instantaneous
power dissipation during switching, both at turn-on and at turn-off. Additionally, hardswitching power electronic circuits generally produce very high frequency noise, due to
resonance (‘ringing’) between circuit/device parasitics (eg. the MOSFET capacitance
resonating with transformer leakage inductance). The peaks of this ringing can add
significantly to the stresses imposed on a transistor.
Snubbers are (usually) passive circuits composed of diodes, resistors and impedances
(capacitors or inductors) which reduce the severity of electrical effects on the transistor
while switching. Of particular interest are:
• The rate of increase of voltage dv/dt while the transistor is being turned off.
• The rate of increase of current di/dt while the transistor is being turned on.
Power Electronics
60
v
t
Figure 56: Ringing in an L-C resonant circuit
The simplest snubber circuit is a capacitor snubber circuit. At turn off the capacitor
reduces the rate of rise (dv/dt) of the transistor voltage, hence reducing the turn-off
losses. While this circuit reduces the turn off-losses, there is a serious problem at turnon, as the transistor will place a short-circuit across the capacitor, causing a very high
current to flow (for a very short period of time). This may destroy the transistor. With
a snubber, the voltage will rise much more slowly at turn-off as the capacitor takes time
to charge up, greatly reducing the turn-off losses as shown below in figure 57.
+Vcc
R
vds power
i
t
Figure 57: Simple capacitor snubber
Other snubber circuits include the RC snubber, the resistor in the above circuit reduces
the current when the transistor turns on, but at the cost of reduced effectiveness of the
snubber at turn-off. A high value of R makes the snubber ineffective at turn-off, whereas
a low value of R causes a high current to flow at turn-on. The RC snubber is also effective
at reducing any ‘ringing’ that occurs at switching. There is also the RCD snubber which
avoids these problems by using a diode to allow effectively zero resistance at turn-off,
while the capacitance is required to discharge through the resistor at turn-on. This is the
most common snubber circuit. An inductor can be included to create a turn-on/turn-off
snubber which limits di/dt at turn-on, thus reducing the turn-on losses.
While snubbers are effective in reducing the transistor switching losses, there is usually
no reduction in overall circuit losses - the losses are merely transferred from the transistor
Power Electronics
R
61
D
C
R
C
R
L
D
C
Figure 58: From left to right: RC, RCD and turn-on/turn-off snubbers
to the resistor. However, the reduced stress on the transistor will improve the circuit
reliability considerably.
Design of the snubber capacitor is a compromise between reducing transistor losses and
reducing snubber circuit losses (and costs). A larger value of capacitor C increases the
effectiveness of the snubber, but, as the energy stored during turn-off (energy = 1/2
CV2 ) is all dissipated in the resistor, overall losses will rise. A loss of 1% of total circuit
power might be considered acceptable. During the transistor on-time the capacitor must
discharge to zero. Hence the RC time constant is typically set such that:
RC =
ton
2
(4.1)
Power Electronics
5
5.1
62
Inverters
Output voltage waveshape
In a single phase inverter, diagonal transistors conduct together: T1 and T4 in the
positive 1/2 cycle, and T2 and T3 in the negative 1/2 cycle. The output voltage is a
square of ±Vdc .
+
Vdc
T1
D1 v D2
ac
T2
T3
i ac
D3
T4
D4
+Vdc
vac
T1 T4
γ
-Vdc
T1 T4
γ
π
T2 T3
2π
ωt
T2 T3
Figure 59: A single phase inverter
With sinusoidal waveforms:
Vpeak =
√
2.Vrms
(5.1)
But this is not a sine wave, therefore to calculate the rms voltage we need to look at the
true definition of root mean square.
s
Z γ
1
2 d(ωt)
Vrms =
.2
Vdc
(5.2)
2π
0
2
When the current is non-zero, it is either +Vdc or −Vdc . The current squared is +Vdc
2,
for γ radians due to the ‘+Vdc contribution’ and the ‘−Vdc contribution’ is also +Vdc
2
hence in total +2Vdc . To get the ‘mean’ we need to average it over one complete cycle,
hence this is all divided by 2π. So, the rms of the output waveform is:
r
γ
Vac-rms = Vdc
(5.3)
π
Power Electronics
63
However, the rectangular waveform consists of a fundamental and a series of harmonic
frequencies. Normally what is wanted is only the fundamental sine wave: the harmonics
are a nuisance. Analysis of the output voltage waveform can by achieved by Fourier
techniques in which any periodic wave can be broken down into a series of sine waves,
consisting of a fundamental and its harmonics.
Simplification can be achieved if the zero time axis is moved as shown (this removes
the cosine terms) and the amplitude of the waveform is normalised to unity. For a
rectangular waveform, only odd harmonics are found. We get that:
f (ωt) = a1 sin(ωt) + a3 sin(3ωt) + a5 sin(5ωt) + . . .
(5.4)
where
2
an =
π
Z
2
=
π
Z
=
π
f (ωt). sin(nωt) d(ωt)
(5.5)
0
π
− γ2
2
π
+ γ2
2
sin(nωt) d(ωt)
γ
4
sin n
nπ
2
(5.6)
(5.7)
The rms harmonics with input Vdc are given by:
√
2 2
nγ
V1 =
Vdc sin
nπ
2
(5.8)
The above graph in figure 60 shows the peak value of each harmonic against pulse width
γ. Vdc has been chosen as 78.5 volts to produce a fundamental of 100 volts for γ = 180◦ .
A good area to work in is around γ = 130◦ : the fundamental is high, yet the harmonics
are generally fairly low.
The Total Harmonic Distortion (THD) is a measure of how ‘non-sinusoidal’ a waveform
is:
pP∞
2
n=2 Vn
THD =
× 100%
(5.9)
V1
where
2
Vac(rms)
= V12 +
∞
X
Vn2
(5.10)
n=2
A typical requirement is that the THD is less than 5%.
Filters can be used to remove harmonics. A LC filter in series has zero impedance at
the resonant frequency ω = ωr :
1
ωr = 2πfr = √
LC
(5.11)
Power Electronics
64
Inverter Harmonics: Vdc = 78.5V
110
100
90
Peak Volts
80
n=1
70
n=3
60
n=5
50
n=7
40
n=9
30
20
180
165
150
135
120
105
90
75
60
45
30
15
0
0
10
Pulse Width (degrees)
Figure 60: Inverter harmonics
It has a finite impedance to all other frequencies, given by:
1
Xtotal = j ωL −
ωC
(5.12)
A typical inverter filter will comprise different LC filters tuned to different frequencies.
such as the one shown below in figure 61.
L1
C1
L3
L5
C3
C5
Figure 61: An inverter filter
The fundamental (let us assume this is 50Hz) sees L1 C1 as a short circuit, so passes
through unaffected. The 3rd harmonic (150Hz) sees L3 C3 as a short circuit, so no 3rd
harmonic reaches the load. The 5th harmonic (250Hz) sees L5 C5 as a short circuit, so no
5th harmonic reaches the load. More filters tuned to other harmonics (eg. 7th , 9th ) could
be included, but as their magnitude is less anyway, they are usually omitted to save
Power Electronics
65
cost. These filters are very heavy and expensive, particularly for high powers (> 1kW
or so).
An improvement is the 3-phase bridge inverter which has an extra limb (extra pair of
transistors) as shown below in figure 62.
+
1/
2Vdc
T1
T2
T3
T4
T5
T6
va
vb
vc
0V
1/
2Vdc
-
+1/ 2Vdc
- 1/ 2 Vdc
+1/ 2Vdc
- 1/ 2 Vdc
+1/ 2Vdc
- 1/ 2 Vdc
va
T1
T1
T4
vb
T2
vc
T6
T5
T3
t
t
t
Figure 62: A 3 phase bridge inverter
The capacitors shown above provide a centre-point for the dc supply, to give a reference
point. They are not needed in a real inverter - they are included here only to help
understand how the circuit works. Look at the ‘a’ phase: When T1 is on, va = + 21 Vdc .
When T2 is on, va = − 12 Vdc . Note that T1 is on for 1/2 the cycle, and then T2 is on for
the other 1/2 cycle.
Now look at the other two phases - the ‘b’ and ‘c’ phases. The switching of the ‘b’ phase
transistors is delayed by 120◦ compared to the ‘a’ phase transistors. The switching of
the ‘c’ phase transistors is then delayed by 120◦ compared to the ‘b’ phase transistors.
The RMS of the fundamental phase voltage is given by:
√
2 2 Vdc
180
Vphase-1-rms =
sin
(5.13)
π 2
2
√
2
Vdc
(5.14)
=
π
Power Electronics
66
The line voltage vab is the difference between va and vb , i.e. vab = va − vb . The RMS of
the fundamental line voltage is given by:
√
2 2
120
(5.15)
Vline-1-rms =
Vdc sin
π
2
√
√
2 2
3
=
Vdc
(5.16)
2
√π
6
=
Vdc
(5.17)
√π
(5.18)
= 3Vphase-1-rms
Also Vline leads Vphase by 30◦ . The other line voltages can be produced in a similar
fashion.
+
1/
2Vdc
T1
T2
T3
T4
T5
T6
0V
+Vdc vab
-Vdc
+Vdc
vbc
-Vdc
+Vdc
vcd
1/
2Vdc
va
vb
vc
t
t
t
-Vdc
Figure 63: A 3 phase bridge inverter with line voltages
Unlike single phase inverters, it is not possible to adjust the pulse width with a 3-phase
bridge inverter to control the magnitude of the output voltage. The pulse width always
needs to be 180◦ . Thus the only way of controlling the output voltage is to control the
input voltage Vdc .
Power Electronics
5.1.1
67
Pulse width modulation
To reduce the difficulty introduced by the harmonic content in the inverter output, another technique is commonly used: pulse width modulation. Here, the inverter switches
are switched at a much higher frequency (the carrier frequency) than the required output, with the ON periods longer near the sinusoid peak value. For an output of 50Hz,
carrier frequencies in excess of 1kHz are normal.
vac
fundamental
sine wave
+Vdc
T2 , T 3 on
t
T1 , T 4 on
-Vdc
Figure 64: Pulse width modulation
A common technique to control the switching of the inverter transistors (or IGBT’s) is
to compare a reference sinusoid vref at the fundamental frequency with a triangular wave
vc at the carrier frequency. The intersections give the switching points. When vc > vref ,
then switches S2 and S3 are on, so vac = −Vdc , and when vc < vref , then switches S1
and S4 are on, so vac = +Vdc .
The frequency ratio is given by:
N=
fc
fref
(5.19)
Note that N should always be an odd integer, to ensure that there are no even harmonics
in the output. Also, the carrier waveform should be synchronised to the reference waveform to eliminate sub-harmonics (frequencies lower than the fundamental). The carrier
frequency is usually > 1 kHz. The modulation index is given by:
m=
V̂ref
V̂c
(5.20)
If N is large (> 15):
V̂1 ≈ m.Vdc
(5.21)
By controlling the reference voltage the peak output voltage can be controlled. If vref = 0
20
Power Electronics
68
reference
sine wave
carrier
t
Control
waveforms
fundamental
vac
Inverter output
voltage
waveforms
t
Figure 65: Pulse width modulation, control and output waveforms
Power Electronics
69
then Vac(average) = 0, but if vref = vc then Vac(average) = vc , i.e.,
vac(average) = Vdc ×
v̂ref
= Vdc × m
v̂c
(5.22)
If N is large then the peak output voltage can be approximated as before. It is thus
possible to control the magnitude of the output voltage by controlling m. Normally
the magnitude of the carrier is kept constant, and the output controlled by varying the
magnitude of Vref .
Normal operation with PWM is with m < 1.0. If m > 1.0, it is overmodulated and higher
output voltages can be obtained, but at the cost of increased harmonic content. The
harmonics in an inverter with PWM appear as sidebands around the carrier frequency
and its multiples, ie. at k.fc ± l.fref where k, l are integers. It can be shown that if k is
odd, there are harmonics only at even values of l, and if k is even, there are harmonics
only at odd values of l.
The Total Harmonic Distortion of an inverter with PWM is very similar to the THD
for an inverter with a single pulse 1/2 cycle. However, in PWM the harmonics are far
removed from the fundamental, and are thus easily filtered out. A disadvantage of PWM
is the increased switching losses due to more switching operations per cycle. However,
up to medium powers IGBT’s can be used, which are able to switch at several 10’s of
kHz without excessive losses. PWM is now used in the vast majority of inverters.
5.2
5.2.1
Inverter applications
Uninterruptible power supplies
Uninterruptible power supplies (UPS) are required for critical loads, where a failure
in the mains power supply would have severe consequences such as computer systems,
emergency lighting and other safety systems. The power supply can be interrupted for
a number of reasons:
• Outage (complete blackout)
• Overvoltage (sustained)
• Undervoltage (sustained) ‘brownout’
• Voltage spikes < 1 ms
• Under/Over frequency
• Excessive harmonics on supply
During mains healthy conditions, the rectifier supplies the inverter and ensures that the
battery is fully charged. The inverter supplies the load (even when the mains is healthy).
If there is a power cut, then the rectifier switches off, but the battery maintains the dc
Power Electronics
70
ac
supply
Rectifier
Battery
Inverter
Load
Figure 66: UPS - No Break Supply
power to the inverter, which continues to supply the load. As there is no interruption
in the dc voltage, the load will see no disturbance at all. The battery will continue to
supply the inverter (and hence the load) either until the battery is fully discharged (in
which case the supply to the load is lost), or until the mains supply returns. The battery
size will be selected to ensure that it can supply the load for the vast majority of power
interruptions. If the interruption is very long, however, a controlled shutdown must be
arranged.
In case of inverter failure, a bypass supply is usually included. The bypass supply
and inverter are switched using back-to-back thyristor pairs. These can be switched
exceedingly quickly (1-2 microseconds), so if the inverter output is synchronised to the
bypass supply (same voltage, same frequency, same phase), then the load will see a
continuous sine wave - not even a blip.
ac
supply
Rectifier
Battery
Inverter
Load
Figure 67: UPS - with bypass supply
The inverter must maintain a constant output ac voltage even as the input dc voltage
changes (e.g. as the battery discharges). This is achieved by controlling the PWM
modulation index m.
1
Vrms = √ .m.Vdc
(5.23)
2
Power Electronics
5.2.2
71
Solar photovoltaic systems
Solar photovoltaic (PV) systems directly convert solar energy to dc electricity using a
pn junction silicon cell. The dc output depends on the solar intensity: in poor sunlight,
the I-V curve above will retain the same shape, but with lower values of I and V.
I
short
circuit
max power
= V.I
decreasing
insolation
open
circuit
V
I
Variable
voltage, dc
V
Figure 68: Solar photovoltaic system
If the PV panel has an open circuit on the output, then the output voltage will be high,
but the output current will be zero. Hence the output power will be zero. If the PV
panel has a short circuit on the output, then the output voltage will be zero, but the
output current will be high. Hence the output power will be zero. The aim is to extract
maximum power from the device. Thus the output voltage/current need to be controlled
to operate in the circled region.
If the system is isolated (not connected to the grid), a battery is usually required to
provide power when the PV output is insufficient to meet demand (e.g. night time).
A dc-dc converter (sometimes called a d.c. chopper) converts the PV output voltage
to the battery voltage. Note that in this case the dc-dc converter duty ratio controls
the converter input voltage, as the converter output voltage is fixed by the battery.
Small perturbations are then made to D and the output power is monitored, to set the
operating point at the maximum power point. This is termed Maximum Power Point
Tracking (MPPT). An inverter converts the battery d.c. voltage to a.c., via a step-up
transformer.
The inverter required for an isolated PV system is very similar to that used in a UPS
system as shown in figure 66 above. Both are fed from a battery, and both need to
produce a constant 230 V rms output voltage with minimum harmonic content. PWM
is almost always used.
Some PV systems (e.g. on the Rankine Building at Kings Buildings) feed into the
Power Electronics
72
grid, and therefore dont need a battery. In this case a chopper is not required, and
the inverter will control the PV voltage Vdc (by controlling the modulation index m)
ensuring maximum power point tracking. Note that in this case m is controlling the
inverter input voltage.
The inverter would still normally employ PWM to minimise harmonics injected into
the grid. A series inductor is necessary to reduce any ripple on the current waveform
to within acceptable limits (usually very close to a sine wave). The phasor diagram of
an inverter (with output voltage E1) feeding power into the grid (voltage E2) is shown
below in figure 69.
+
L
I
Vdc
E1
E2
E1
δ
I.XL
E2
I
Figure 69: The phasor diagram of an inverter feeding power into the grid.
Note that this is very similar to the phasor diagram for a synchronous generator feeding
power into the grid system. The power equation is exactly the same as for a synchronous
generator.
E1 .E2 . sin δ
P =
(5.24)
XL
Unlike with a synchronous generator, it is possible to directly control the phase of the
inverter relative to the grid, by controlling precisely when the IGBTs are switched. An
example grid connected PV system is shown below in figure 70.
5.3
Inverters connected to the grid
With regards to the grid connected PV system described before, what happens if E1
lags E2 ? If the power is negative, that means power is flowing in the opposite direction,
Power Electronics
73
50 Hz
230 V
Variable
voltage, dc
Inverter
Figure 70: A grid connected PV system
ie. from the ac system to the dc system. On the dc side, Vdc cannot go negative (due to
the diodes), so for the dc power to be negative the dc current has to be negative.
I
E2
δ
E1
I.XL
Figure 71: The phasor diagram for an inverter connected to the grid, in this case the
converter is a rectifier.
Thus this converter can operate either as a rectifier or as an inverter, depending on
whether E1 lags or leads E2 . Also, the power factor of the converter system (cosine of
the angle between I and E2 ) can be controlled by varying the magnitude of E1 . This
is very similar to a synchronous machine connected to the grid. The main differences
are:
1. Changes in power, power factor, can be made very quickly, as there is no machine
inertia.
2. The phase angle δ can be controlled directly. In a synchronous machine the angle
δ is controlled indirectly by controlling the power flow according to the power
equation.
Compared to other rectifier circuits (diode bridge, thyristor bridge), this ‘active’ rectifier:
Power Electronics
1. Can operate at unity power factor
2. Draws near sinusoidal currents from the ac supply
3. Can control the output dc voltage.
Active rectifiers can be either single phase or three phase.
Figure 72: Single and 3 phase active PWM rectifiers
74
Power Electronics
6
75
Power Electronics in Power Systems
6.1
High voltage DC links
Conventional HVDC links are based on phase controlled thyristor converters. These
have been in existence since the 1970’s, and are well proven. The cross-channel EnglandFrance link (rated at 2000 MW and commissioned in 1986) uses this technology. However, these systems do have limitations:
1. They require a stiff grid at each end, as they require a firm sinusoidal grid voltage
to switch off (commutate) the thyristors. Thus they are not suitable to supply,
for instance, an island community which does not have its own power station.
Generally, the link rating should not be more than about 10% of the generating
capacity at each end.
2. They draw very non-sinusoidal currents from the grid, thus need large (and expensive) filters to prevent distortion of the mains voltage, even in stiff systems.
3. The power factor depends on the delay angle α, and may often be poor, requiring
synchronous condensers to be installed (these are synchronous machines running
at zero power factor, with no real power flow, simply to correct the power factor).
I
I
Figure 73: A High Voltage DC (HVDC) link
An alternative to the phase controlled (thyristor) bridge converter is the active rectifier /
inverter. Thyristor inverters have the advantage that they can handle considerably higher
powers that active converters. Active converters can operate at unity power factor, and
draw/supply near- sinusoidal currents if PWM is used. The switches (IGBT’s or GTO’s)
are self commutated (by removal of the gate drive (IGBT), or by applying a negative
gate voltage (GTO)), so can supply a passive island load. The active converter HVDC
system is termed an ‘HVDC Light’ system. The first HVDC Light systems was installed
in 1999 in Sweden, on the island of Gotland, rated at 50MW (±80kV). There are still
very few such schemes in existence. In this system Vdc is always positive, the current
can be both positive and negative and power can flow in both directions.
Power Electronics
76
Idc
Vdc
E 1a
E 1b
E 1c
I 1a
I1b
I1c
Idc
E 1x
E 1y
E 2x
E 2y
Vdc
E 1z
Converter
1
δ1
E 1a
E 1x
E 2z
Converter
2
I
I1aXL
Figure 74: A High Voltage DC (HVDC) Light link
I 2a
I 2b
I 2c
E 2a
E 2b
E 2c
Power Electronics
77
At the sending end, the power factor is normally set to unity. The power sent is controlled
by controlling the angle δ.
If PWM is used, then the converter line voltage E1xy(rms) is controlled by varying the
modulation index m. Due to the limited power rating capability of IGBT’s, GTO based
systems with square wave outputs are sometimes used rather than PWM. In this case, the
converter line voltage E1xy(rms) is controlled by varying dc link voltage Vdc (to increase
Vdc charge the dc capacitors by making the sending power greater than the receiving
power for a short period).
At the receiving end, the power factor is set depending on the load requirements. The
(real) power received is set by:
P =
3E2a E2x sin δ2
XL
(6.1)
To ensure efficient operation of the system, the dc link should be operated close to its
rated voltage at all times (thus minimising I 2 R losses). If the power sent by converter
1 = the power received by converter 2, the link voltage will be constant. If the power
sent by converter 1 > the power received by converter 2, current will flow into the link
capacitors, and the link voltage will rise. If the power sent by converter 1 < the power
received by converter 2, current will flow out of the link capacitors, and the link voltage
will fall.
Converter 2 will be controlled to supply the required load power. Converter 1 is controlled to keep the link voltage constant (which means that it is matching the converter
2 power).
An example of this system is the Eire-UK link. This uses the HVDC light technology to
connect Deeside (Wales) to Dublin. It is rated at 500MW ± 200kV. It covers 186km on
the ocean floor and 70km underground, at an expected cost of £390m to be completed
September 2012.
6.2
Flexible AC transmission systems
Power electronics is increasingly being used to improve performance in ac power systems,
in particular:
1. Improve power factor
2. Control load flow in ac lines
3. Damp out oscillations
Conventional power factor correction is carried out by installing power factor correction
capacitors, to bring the overall power factor to (close to) unity. However, as loads change,
Power Electronics
78
IL
I
IC
C
E
E
ɸ
IL
L
E 1a
R
E
I
ɸ
IL
IC
Figure 75: Power factor correction
the optimum amount of capacitance required is likely to vary. Normal installations only
provide a fixed capacitance.
Static VAR Controller (SVAR systems) have several (banks of) capacitors that can be
switched in/out using back-to-back thyristor pairs as an ac switch (note these are not
using phase control the thyristors are either continually on or continually off). To switch
off, the thyristor gate pulses are removed. The thyristor will remain on until the current
through it reaches zero (in a 50Hz system this will always occur within 10ms). As the
current through a capacitor leads the voltage by 90◦ , the voltage across the capacitor will
be a maximum (either positive or negative). The thyristors are turned on by applying
a gate pulse. It is essential that this occurs when the ac line voltage = the capacitor
voltage, or a very large current will flow. As the capacitor is switched off fully charged,
the thyristors therefore need to be turned on at a voltage peak.
Figure 76: A static VAR controller
Power Electronics
79
An alternative to the SVAR Controller is the STATCOM (static synchronous compensator). The converter terminal voltages Ex , Ey and Ez are controlled to be in phase
with Ea , Eb and Ec respectively. The resulting phasor diagram is shown below in figure
77, with Ix leading Ea by 90◦ (i.e. the circuit looks like a capacitor).
Ea
Eb
Ec
Iz Iy Ix
XL E
x
Ey
Vdc
Ez
Ea
I xXL
Ix
Ex
Figure 77: A STATCOM (static synchronous compensator)
As Ix leads Ea by 90◦ , no real power flows in the converter - only reactive power. Thus
the average dc link current is zero, so the capacitor does not charge/discharge. Thus
a relatively small capacitor can be used, to keep the voltage constant over a 1/2 cycle.
The amount of reactive power is controlled by controlling the magnitudes of Ex , Ey and
Ez . This can be achieved in two ways:
1. If PWM (with IGBT’s) is used, then Ex , Ey and Ez are controlled by varying the
converter modulation index m, with Vdc kept constant.
2. Due to the limited power rating capability of IGBT’s, GTO based systems with
square wave outputs are sometimes used rather than PWM. In this case, Ex , Ey
and Ez are controlled by varying the capacitor voltage Vdc . This is achieved by
making small, temporary, changes to the phase of Ex , Ey and Ez allowing a small
amount of real power to charge/discharge the capacitor.
6.3
Unified power flow controller
The most sophisticated FACTS device is the UPFC. The maximum amount of power
that can be transmitted across the system before synchronism is lost is limited by the
Power Electronics
80
line inductance L. If the line is very long, this can be a severe restriction considerably
more restrictive than the current (thermal) rating of the line.
It can be very difficult to control the power flow in an ac system. δ cannot be controlled
directly: if the systems are large, it can be very slow to change the phase angles due
to generator inertias. The voltages can be controlled with transformer tap-changers,
but only in discrete steps and again, not quickly. The situation is very different if an
active converter with controllable output voltage EC is put in series as shown below in
78.
Vdc
converter
IxX L
I
EC
E1
δ
L
Ex
Ex
E2
EC , I XL
E1, E 2
I
Figure 78: A power electronic controller
If E1 = E2 then I.XL = EC and I lags EC by 90 ◦ . No real power is drawn from the
converter and the capacitor voltage is constant. Consider the system comprising the
intermediate voltage Ex, L, E2 (the green box)
P =
3Ex E2 sin δ
XL
(6.2)
The magnitude of Ex and the phase angle δ is controlled by varying the magnitude
and phase of EC (the converter output voltage). This can be achieved simply and very
quickly.
Ex
EC
δ
E1
IXL
EC
E2
I
Figure 79: Power flow in AC system when E1 and E2 are not exactly in phase
Normally E1 will not have exactly the same magnitude and phase as E2 . The case above
in figure 79 is if E2 > E1 , but with them still in phase with each other. Here I is not
perpendicular to EC , so the converter is supplying real power, which has to come from
Power Electronics
81
somewhere - the capacitor. Hence the capacitor will quickly discharge. The solution to
this is the UPFC.
EC
I
IXL
L
E1
converter V
dc
1
converter
2
Ex
E2
Figure 80: A Unified Power Flow Controller (UPFC
Converter 1 acts as a STATCOM, controlling the reactive power in the system. However,
unlike in the STATCOM, if the converter voltage is not completely in phase with the
main line voltage, then some real power can flow, recharging the capacitor if discharged
by Converter 2 (or vice versa). Converter 2 controls the real power flow in the system,
as just previously described. Usually transformers are required between the converters
and the ac system, both to provide isolation and reduce the voltage to a level suitable
for the converters. For simplicity, the above is drawn for single phase. Any real such
system would always be three phase.
UPFC’s provide very fast control and the two converters are PWM (IGBT’s) for fairly
low power, low harmonics and square wave (GTO’s) for medium power, high harmonics.
Power Electronics
7
82
DC machine drives
The steady state equivalent circuit of a separately excited dc machine is shown below.
The armature winding is on the rotor, supplied through the mechanical commutator,
and the field winding is on the stator.
Ra
Ia
If
E
Va
Figure 81: Simple DC machine
Va = E + Ia .Ra
(7.1)
E ∝ speed × If
(7.2)
There are 4 modes of operation of the machine shown below in 82, the 4 quadrants of
operation shown on the right hand side (torque-speed) applies to any machine, not just
dc machines. The left hand graph (Ia − Va ) applies only to dc machines.
Torque
Ia
Reverse
Generator
(braking)
Forward
motor
Reverse
motor
Forward
Generator
(braking)
Va
Reverse
Generator
(braking)
Forward
motor
Reverse
motor
Forward
Generator
(braking)
Speed
Figure 82: The 4 quadrants of operation of a DC machine
Referring to 81:
• Forward motor: As shown,
• Reverse generator (braking): Va and E reversed,
Power Electronics
83
• Reverse motor: Va , E and Ia reversed,
• Forward generator (braking): Ia reversed.
7.1
Two quadrant control
The simplest dc drive is to control the armature voltage via a 3-phase bridge rectifier
(or 1-phase for small motors), with the field current kept constant. This circuit can
theoretically operate in quadrants 1 and 2. Quadrants 3 and 4 are not possible as
current Ia cannot reverse (current can’t go backwards through a thyristor).
Ia
Ra
Vdc
E
Figure 83: A 6 thyristor bridge rectifier
Vdc =
3
VM cos α
π
(7.3)
In quadrant 1 (α < 90◦ ) the drive is a rectifier, Vdc is positive). In quadrant 2 (α > 90◦ )
the drive is an inverter, Vdc is negative. As the drive cannot operate in quadrant 3, it
is impossible to drive it in the reverse direction, therefore quadrant 2 is useless: there
is little point to being able to slow down the machine in the reverse direction, if you
cannot get it going in the reverse direction in the first place.
7.2
Four quadrant control
Quadrant 4 operation can be achieved in the system below in figure 84, by moving the
switch to position B. The machine will operate as a dc generator with EMF E driving
current (in an anti-clockwise direction) through the resistor R. The energy dissipated in
R has to come from somewhere - the rotational kinetic energy of the machine ( 21 Iω 2 ).
Hence the machine slows down.
Full 4 quadrant operation is possible with two bridges in antiparallel. Note that now
the machine current Ia can reverse by using bridge 2.
Power Electronics
84
A
Ia
B
Ra
R
Vdc
E
Figure 84: Resistive braking
I dc1
Ra
A
B
C
V
dc1
Bridge 1
A
B
C
I a Vdc2
E
I
dc2
Bridge 1
Figure 85: Full 4 quadrant operation with two bridges in antiparallel
Power Electronics
85
Using two bridges in anti parallel operates in the four quadrants as:
• Quadrant 1 (Forward motor): Bridge 1 is a rectifier,
• Quadrant 2 (Reverse generator): Bridge 1 is an inverter,
• Quadrant 3 (Reverse motor): Bridge 2 is a rectifier,
• Quadrant 4 (Forward generator): Bridge 2 is an inverter.
7.3
Power factor
The power factor of a 3 phase bridge rectifier is given by:
P.F =
3
cos α
π
(7.4)
Where α is the thyristor firing delay angle. A disadvantage of using the 3-phase bridge
rectifier to control dc machines is that low speed operation requires a low armature
voltage, hence a low vale of cos α, hence a low power factor at low speeds. The power
factor can be improved by using a diode bridge (α = 0), with the dc voltage controlled
by a dc chopper.
IGBT
Ia
Ra
E
Figure 86: Improved power factor using a diode bridge
In a chopper drive, if regeneration is required, the circuit in figure 87 can be used, but
note that the dc source must be able to accept a reverse current (ie. absorb power), so
this cannot be fed directly from a diode bridge (current cant go backwards through a
diode).
7.4
Current control
Note that at rest E = 0, therefore if a high voltage is applied to the armature winding, a
very high current will result (limited only by Ra , the armature winding resistance).
Power Electronics
86
D2
iL
Vdc
S1
D1
S1
L
R
E
V1
Figure 87: Regenerative braking
Ia
Ra
Va
If
E
Figure 88: Current control
It is essential that a low armature voltage (large delay angle α) is applied when the
machine is at rest. As the machine accelerates, E will increase, therefore Va should be
increased at the same rate to keep Ia approximately constant, as
Ia =
7.5
Va − E
Ra
(7.5)
Motor drives
There are 3 options for controlling the speed of a machine (not just dc machines), fixed
speed and variable loop which cane be open and closed loop. The use of variable speed
control adds very considerably to the capital costs, but can produce very significant
energy savings by allowing operation at reduced speeds. In many applications variable speed operation is essential. For accurate speed control, closed loop control is
required.
Open loop control could be achieved using the circuit below in figure 89, although simple
the resulting speed will not be accurate. Any change in load will cause the speed to
change, with any speed correction having to be carried out manually.
With closed loop control comes accurate speed control, with any speed variation being
corrected automatically by the speed feedback loop. A fast response to changes in the
speed setting requires a large torque, resulting in large motor currents. Motors are
sufficiently rugged to be able to withstand a high current for a few seconds, but power
Power Electronics
Speed
Setting
87
Ra
Ia
Control
Circuit
Power
Converter
Va
E
Figure 89: Open loop control
electronic devices can be destroyed in milliseconds. The resulting very large currents
in this system will almost certainly destroy the power converter, so this system is not
used.
Speed
Setting
+
-
Ra
Ia
Error
Amplifier
Control
Circuit
Power
Converter
Va
E
Speed
Sensor
Figure 90: Closed loop control
Instead a closed loop control with current control is used. This is the same as the previous
circuit, but with the addition of a current feedback loop which provides a current limit
to protect the converter. This system is drawn for a dc machine drive, but the blocks
are identical for any closed loop drive (eg induction machine drive).
In steady state conditions, the driving (motor) torque is exactly balanced by the load
torque (including friction), thus there is no accelerating torque and the speed is constant.
In figure 91 above, the required speed setting is increased at t0 . The applied voltage Va
is increased, until Ia = (Va E)/Ra equals the current limit setting Icl . As the machine
accelerates, the back EMF E increases, therefore Va is increased to maintain Ia = Icl ,
thus maintaining maximum torque.
Power Electronics
Speed
Setting
+
-
88
Speed
Error
Amplifier
+
Current
Error
Amplifier
-
Ia Ra
Control
Circuit
Power
Converter
Va
E
Current feedback
Speed
Sensor
Speed feedback
Figure 91: Closed loop control with current control
When the machine reaches the new desired speed at t1 , no accelerating torque is required,
therefore the current Ia will come out of current limit, falling to a level that corresponds
with the motor torque exactly balancing the load torque.
The curves on the left in figure 92 are ideal and assume a very fast response and settling
down period around t1 . In practice they are as shown on the right. As the accelerating
torque reduces between t0 and t1 , speed, and hence E, will rise more slowly. Hence the
curved shape of E (and thus Va ).
volts
Va
Ia
E
t
current
I cl
Ia
t
torque
accelerating
torque
motor torque
load torque
t0
t
t1
t0
t1
Figure 92: Machine voltage, current and torque both ideally (left) and practical (right)
Power Electronics
8
89
Induction motor drives
The vast majority of motor drives used in industry today are induction motor drives.
They are cheap and rugged, particularly with the squirrel cage rotor. In an induction
motor a 3-phase supply is connected to the 3-phase stator winding, with the phase
windings distributed evenly around the circumference of the stator (in a machine with
one pole-pair). A magnetic field is produced, which has constant magnitude and rotates
at a speed ω0 radians/second (termed ‘synchronous speed’).
a
ω0
b’
c
ɸs
ωr
c’
b
a’
Figure 93: A 3 phase induction motor
The most common rotor is the squirrel cage rotor. This has a laminated steel core, with
aluminium or copper bars set in grooves on the circumference. At each end, the bars
are short-circuited by aluminium or copper end rings. The magnetic field rotates at a
speed governed by the supply frequency and the number of pole pairs.
ω0 =
2πf
rad.s−1
p
(8.1)
Slip is a measure of the speed of the rotor relative to the synchronous speed. In normal
(motor) operation the rotor speed is slightly lower than synchronous speed (i.e. a very
small slip)
ω0 − ωr
N0 − Nr
slip, s =
=
(8.2)
ω0
N0
where ωr is the rotor speed in rad.s−1 and Nr is the rotor speed in rpm.
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90
For a typical induction machine the torque-speed curve will look like figure 94 below.
Note that the torque is zero at synchronous speed. Normal operation is just below
synchronous speed, with a slip around s = 0.05.
torque
speed
s=1
ω0
s=0
Figure 94: Typical torque speed curve for an induction motor
The efficiency of an inductor motor can be considered to be ratio of the rotor output
power to the air gap power.
T ωr
η=
=1−s
(8.3)
T ω0
So for the highest efficiency, the motor needs to operated with minimum slip.
8.1
8.1.1
Speed control
Adjusting the supply voltage
The motor operating point is where the load torque-speed characteristic intercepts the
motor torque-speed characteristic. If the motor terminal voltage is reduced from V1 to
V2 , then the motor torque speed curve changes from the blue line to the green line in
figure 95 below. The operating point moves to the left and the speed reduces - but
not by much. Reducing the voltage does not change the speed by much, but it reduces
the available torque quite considerably. The efficiency is reduced, as the machine is
operating with a greater slip.
Voltage control is achieved by phase control of the thyristors, varying the thyristor firing
delay angle α as shown below in figure 95.
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91
torque
V1
V2
load
speed
ω0
Figure 95: Induction motor speed control by adjusting the voltage
Va
• Voltage increased by reducing α
Vb
• Harmonics produced
Vc
α
• Increased losses
• Suitable for soft starting
vac
π
2π
ωt
Figure 96: Voltage control is achieved by phase control of the thyristors
Power Electronics
8.1.2
92
Adjusting the supply frequency
A better way to control the speed of an induction motor is to control the supply frequency. This has the effect of changing the synchronous speed. Figure 97 below shows
the torque-speed curves for 3 different supply frequencies.
Torque
Load
ω02
ω03
Speed
ωr
ω01
Figure 97: Torque-speed curves for 3 different supply frequencies
Note that high torques are now available at low speed. Also, slip will be small (so
high efficiency operation), as it will always be operating close to synchronous speed.
The frequency can be controlled by a variable frequency inverter. The 6-diode bridge
rectifier converts the constant voltage, constant frequency ac supply to a constant dc
voltage as shown below in figure 98.
+
constant
voltage,
constant
frequency
-
variable voltage,
variable frequency
Figure 98: A 6 diode bridge rectifier to convert ac to dc
The PWM inverter then converts the fixed dc voltage to a variable voltage, variable
frequency supply. Voltage is controlled by controlling the PWM modulation index,
Power Electronics
93
m. Frequency is controlled by controlling the frequency of the IGBT gate pulses. For
maximum torque capability, the peak value of B (and hence Φ) should be just below the
knee point of the B-H curve - ie. just below saturation.
If the frequency ω is reduced while keeping the voltage magnitude constant, then the
flux Φ will rise, with the risk that the iron core will go into saturation. To ensure that
the flux magnitude is kept constant (and saturation avoided), the ratio Vs /f should
be kept constant. Thus if the frequency is reduced, the voltage should be reduced in
proportion.
dΦ
vs = n
(8.4)
dt
1
Φ=
n
=−
Z
Vs(max) sin(ωt) dt
(8.5)
1 Vs(max)
cos(ωt)
n ω
(8.6)
The direction of rotation of an induction motor may be reversed (3rd quadrant) by interchanging 2 of the phases (causing the magnetic field to rotate in the opposite direction).
This could be done by a mechanical switch. The same effect can also be achieved very
simply by changing the switching order of the transistors in the circuit above in figure
98. T1, T2 and T3 are the transistors on the top, with T4, T5 and T6 on the bottom.
Note that 2 transistors in the same leg (eg. T1 and T4) are never on simultaneously,
otherwise there is a short circuit of the dc link.
Change
To
T5
T6
T1
T1
T6
T5
T2
T3
T4
T4
T3
T2
T5
T6
T1
T1
T6
T5
Figure 99: Change of transistors for reverse rotation
Regenerative braking in the 2nd /4th quadrants is achieved by reducing the inverter frequency such that the rotor is rotating faster than the magnetic field. For regeneration,
the inverter frequency is reduced to produce the green torque speed curve below in figure
100. This can be achieved very quickly, but due to the inertia of the machine the speed
cannot change quickly, hence the operating point moves down into quadrant 4 with a
negative torque, which acts to slow down the machine.
For braking, dc link power (= Vdc Idc ) must be negative. As the inverter diodes prevent
Vdc becoming negative, the dc link current Idc must reverse. However, current cannot
Power Electronics
94
torque
ω02
speed
ω01
Figure 100: Torque-speed diagram for regeneration
reverse through the rectifier diodes, do the circuit shown earlier does not allow regenerative braking. If the diode bridge rectifier is replaced by an a 2nd PWM converter as
shown below in figure 101, then full 4 quadrant operation is possible. Vdc still cannot
go negative (due to the diodes), but the dc link current can now reverse through the
IGBT’s in the left hand converter.
The LHS PWM converter is operating as an inverter feeding power back into the grid.
The RHS PWM converter is operating as an active rectifier.
8.1.3
Adjusting the effective rotor resistance
The peak torque on the torque-speed curve occurs at a slip speed which depends on the
effective rotor resistance Rr in figure 102. Peak torque occurs when:
s=
Rr
ωLr
(8.7)
If the rotor resistance is varied, a set of curves as shown in figure 103 results.
Varying the rotor resistance is clearly not possible with the standard squirrel cage rotor
which is rotating. Varying rotor resistance is possible if the squirrel cage rotor is replaced
by a wound rotor, where the ends of the (3 phase) windings are brought out inside the
shaft to slip rings, which make a sliding contact with carbon brushes connected to
external resistors.
Power Electronics
95
+
-
Figure 101: If the diode bridge rectifier is replaced by an a 2nd PWM converter, then
full 4 quadrant operation is possible.
Rs
Rr
Ls
Lr
n.I r
Vs
Rc
s.n.Vs
Lμ
Ir
Figure 102: Slip control
torque
small
Rr
large
Rr
s=1
speed
ω0 ωr
s=0
Figure 103: Torque-speed curve with slip control
Power Electronics
96
slip rings
stator
rotor
stator
ωr
• Efficiency = (1 - slip)
• Poor efficiency at low speed
brushes
R
Figure 104: Slip control with a wound rotor
While variable speed control with this arrangement is perfectly possible, efficiency is
poor at low speeds, as the system will be operating with a high value of slip (energy is
being dissipated in the external resistors). An improvement on the previous arrangement
is the Static Kramer Drive shown below in figure 105. Here, the resistors are replaced
by a rectifier-inverter system, that takes the energy that previously was dissipated in
the resistors and feeds it back into the mains supply, usually via a step-up transformer
(as rotor voltages are usually fairly low).
The frequency of the rotor currents fr = s.fs , therefore the rotor current is rectified in
a diode bridge, and then converted to 3-phase, 50 Hz by a line-commutated inverter.
The diode bridge has an approximately unity power factor, therefore as far as the rotor
circuit is concerned it is equivalent to a resistor. The dc link voltage is set by the inverter
delay angle α:
3√
Vdc =
2Vrms cos α
(8.8)
π
The slip ring voltage is then given by:
Vdc =
3√
2Vslip-rms
π
(8.9)
Thus the magnitude of the voltage at the slip-rings is set by the rectifier-inverter link,
and controlled by the delay angle α of the line-commutated inverter.
Note that the curves are not the same as for simply increasing rotor resistance, as
the slip-ring voltage is now being set. If the slip ring voltage is equal to the induced
rotor voltage (= s.n.Vs ), then no currents will flow in the rotor, and no torque will be
produced. An attraction of this system is that the rectifier/inverter system is only rated
for the slip power, which is a lot less than the total system power. Thus, compared to the
variable frequency system the power electronic system is considerably cheaper. However,
Power Electronics
97
transformer
stator
rotor
stator
ωr
Vdc
Figure 105: A static Kramer drive
torque
increasing
slip-ring volts
slip-ring
volts = 0
speed
ωr
ω0
Figure 106: Torque-speed curve for a static Kramer drive
Power Electronics
98
the system will only work with an induction motor with a wound rotor and slip-rings,
which is considerably more expensive then the the normal squirrel cage machine (and
less mechanically robust).
8.2
Doubly fed induction generator (DFIG)
A development from the Static Kramer Induction Motor Drive is the Doubly Fed Induction Generator, which is becoming increasingly popular in wind generation systems. In
a static Kramer the rotor frequency is set by the speed of rotation and the diode rectifier
accepts any frequency. In an induction machine with wound rotor, φr and φs rotate at
the same speed, which is essential for constant torque. For example, in an induction
machine with 1 pole pair, fs = 50 Hz and slip = 0.1: φs rotates at 2,000rpm and the
rotor roatates at 2,700 rpm. fr = sfs = 5Hz. φr rotates at 300 rpm relative to the rotor,
therefore φr rotates at 2, 700 + 300 = 3000 rpm.
The DFIG is very similar to the static Kramer Drive, except that the diode bridge is
replaced with an active converter. The 6-thyristor bridge inverter is also replaced with
a 2nd active converter. There are two types of converter that can be used:
1. PWM converters using IGBT’s. These produce low harmonics, but the power is
limited by the IGBT ratings;
2. Square wave converters using GTO’s. These have much higher harmonics, but are
able to handle higher powers.
transformer
stator
rotor
stator
ωr
+
Figure 107: Doubly fed induction machine
In a DFIG:
Power Electronics
99
• Diode rectifier replaced by active converter
• Rotor frequency set by converter
• Power flow in either direction
• Must still ensure φr and φs rotate at the same speed
The frequency of the rotor currents is now set by the left hand converter. For a torque to
be produced, this frequency must be such that the rotor magnetic field φr is rotating at
exactly the same speed as the stator magnetic field φs . Note that the active converters
now allow power to flow through them in either direction: the polarity of the dc link
voltage will not change, but the dc link current can now flow in either direction.
Earlier we looked at the case where the stator field φs rotated anti-clockwise at 3,000
rpm, rotor field φr rotated anti-clockwise at 300 rpm relative to the rotor, and the rotor
itself rotated at 2,700 rpm. If the switching order of the converter IGBT is reversed, the
rotor field φr will now rotate at 300 rpm in the opposite direction (clockwise) relative
to the rotor. Thus for the φr and φs to rotate at the same speed, the rotor itself must
rotate super-synchronously at 3,300 rpm.
The rotor speed can be varied by controlling fr . By reversing the phase sequence of the
converter, the rotor can also operate at speeds greater than synchronous speed.
2πfs 2πfr
±
rad.s−1
p
p
60
=
(fs ± fr ) rpm
p
rotor speed =
(8.10)
(8.11)
In a conventional squirrel cage induction machine, the power dissipated in the rotor,
η = s.Pi (this neglects any losses in the stator). In the Static Kramer Drive, this power
is recovered and returned to the supply. When operating a static Kramer drive as a
motor, the slip is positive and it is sub-synchronous, however a DFIG, if the phase
rotation in the LHS converter is reversed, then:
1. The machine speed will be greater than the synchronous speed,
2. 2. Therefore the slip is negative,
3. 3. Therefore the power flow through the converters (=s.Pi) is reversed, flowing
into the rotor,
4. 4. Therefore the mechanical output power is the sum of the stator power and the
rotor power.
With the phase rotation still reversed (ie. slip s still negative, so super-synchronous
operation), if now the mechanical load is replaced with an energy source (eg a wind
turbine):
Power Electronics
100
1. Pi is negative (power flowing back into supply)
2. Power flow through the converters is from the rotor into the supply (s is negative,
Pi negative)
3. The majority of the wind energy flows from the stator direct to the grid, with only
a small portion going via the converters.
Still with an energy source (eg. wind turbine) connected to the rotor, if the rotor
converter phase rotation is now changed back to the original situation:
1. Slip s is positive, so the machine will rotate sub-synchronously,
2. If s positive and Pi is negative, the direction of converter power is into the rotor.
8.2.1
DFIG wind generation system
stator
Gearbox
rotor
stator
Converter
1
Converter
2
Figure 108: Doubly fed induction machine for a wind turbine
This system can run at variable speed, both sub-synchronously and super-synchronously.
The speed will be chosen to extract the maximum amount of power from the turbine as
the wind speed varies. The speed is controlled by the frequency and phase rotation of
converter 1. The converters are only rated for slip energy. System power factor can be
controlled to unity by controlling the magnitude and phase of the slip ring voltage (by
converter 1).
The advantages of using a DFIG in wind turbine are that
• Power electronics rated only for slip energy
• Optimise speed to maximise wind power
Power Electronics
101
• Can control power factor
An alternative to the DFIG is the permanent magnet (synchronous) generator. This machine is similar to a conventional synchronous machine, except that permanent magnets
are used on the rotor instead of a field winding.
Gearbox
Permanent
Magnet
(Synchronous)
Generator
Rectifier
Inverter
Figure 109: Permanent magnet (synchronous) generator for a wind turbine
This generator is allowed to rotate at the appropriate speed for maximum wind energy,
therefore producing a variable frequency, variable voltage output. This is then converted
to fixed (50Hz) frequency, fixed voltage via a rectifier- dc link-inverter system. This
system is very common in small systems, but is more expensive than the DFIG in larger
systems because the power electronics has to be rated for the full wind power (not just
the slip power) and the generator itself is more expensive than an induction machine.
However, many see this as the way forward in the future. Its major advantage is that
the part load efficiency is higher than for the DFIG. Over the lifetime of the system, it is
anticipated that the income from the extra energy generated will more than compensate
for the additional capital cost.
Older systems used a simple squirrel cage induction generator directly connected to the
grid. While this had the advantage of being cheap and simple, its disadvantages are that
it is effectively fixed speed, so maximum power is not extracted from the wind and the
power factor cannot be controlled. New systems rarely use this technique now.
Power Electronics
102
Gearbox
Squirrel
Cage
Induction
Generator
Figure 110: Squirrel cage generator for a wind turbine