power electronics - university of illinois · 2012-08-26 · • pf is often divided into a phase...
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Power ElectronicsDay 4 – Equivalent Sources, “Power Filtering” Analysis, Dc Conversion
P. T. KreinDepartment of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
© 2011 Philip T. Krein.
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Equivalent Sources
When a switch matrix operates to satisfy KVL and KCL, many of the waveforms become well defined.
Example: Matrix 2x2 ac voltage to dc current converter. The output must be +vin, -vin, or zero.
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Equivalent Sources
2,2
1,21,1
2,1
vOUT
vIN
vOUT
on2,22,1on,211,1
0
on2,1on2,21,1
in
in
vv
,21
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Equivalent Sources
• If switch action is specified, the output waveform becomes fully determined.
• We can treat the waveform as an ideal source (with an unusual shape).
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Sample Cases• Full-wave rectifier (Fig. 2.33)• Phase-delayed rectifier (Fig. 2.17)• Inverter into an ac current source (Fig. 3.5)• 60 Hz 3 to 60 Hz 1 conversion• Fig. 2.19, 60 Hz to 180 Hz
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Sample Cases
vOUT
vIN
delay60
delay0
vOUT
vIN
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Sample Cases
vOUT
vIN
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Sample Cases
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Equivalent SourcesAny of those waveforms can be a source.
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Equivalent Sources• Equivalent sources can be a powerful tool:
– Many converters act like an equivalent source in a linear circuit
– We can represent a source as a combination of Fourier components
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Equivalent Sources• With a source in a linear circuit,
analysis, filter design, etc. can proceed along familiar lines.
• This is a common way to design interfaces for rectifiers and inverters.
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Equivalent SourcesExample:
vIN
Outputterminals
I
VLOAD
VOUT
R
L
Ignore capacitorfor a moment:
We know VOUTR
L
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Equivalent Sources
We can represent the periodic waveform with a Fourier series.
Periodicwaveform
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Equivalent Sources
i is the sum of the contributions from each of the sources. We can break up the circuit.
i
Linear circuit
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Equivalent Sourcesi0
DC
i1
1st harmonic
i2
2nd harmonic
0n
nii
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Equivalent SourcesDC term:
RVI dc
dc
I
DC
VDC
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Equivalent SourcesAC terms, based on phasor analysis.
low~wante.g.,
Want low ripple
1I
~~
1
11 LjwR
VI
V1~
I1~
Usually, Fourier terms decrease in amplitude as 1/n. The fundamental is the largest.
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DAY 4 START Power Filtering• Filters (or interfaces) for converters
have needs distinct from those in signal applications.
• Filters must be lossless, and impedances of sources and loads are unknown.
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Power Filtering
• Two common methods of analysis– Equivalent sources– “Ideal action” assumption
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Filter Examples
vIN
VOUT
i
VOUT
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Filter ExamplesvOUT = Vin sq(t)
2
+1
-1
sq(t)
f = 100HZ
sq(t) = 2sin4
)cos(nωtn
n
1n
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Filter Examples
I1~
Fundamentalω
3ω
5ω
3rd term
vout(t) =
4Vin 2
sin)cos(nωt
n
n
1n
ω = 2100 rad/s
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Filter Examples
Look at examples based on the equivalent source method (such as Example 3.6.1).
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Ideal Action Assumption• In a power converter, we know what a
filter is trying to achieve.• Examples: low-ripple dc, ideal ac sine
wave, etc.• In general: give a large wanted
component and small unwantedcomponents.
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Ideal Action Assumption
• If the filter is well-designed, it ought to work.
• If it works, we know its output.• Now, use the “known” output with the
known input to compute values.
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Filter Examples
VIN
large L I
R
If L is large, then I is just dc.
Substitute
vIN
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Filter Examples
i
Ideal actionassumption
10 V
00 100s
vD
50 %Duty
L
10 V
f = 10 kHz
10 vD
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Filter Examples
5 V
10
0.5 A
Harmonics
Choose L to make < Limit.Too much work!
1~I
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Filter ExamplesIf L is large and the circuit works, the inductor current is almost constant and so is the voltage across the load resistor. This voltage can be represented by a constant voltage source. v
5 V
L
L
10 V
0Switch on: VL = 5 VSwitch off: VL = -5 V
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Filter Examples
L+5
-5
VL = L di/dt
If VL = 5V = L di/dt
LV5 = di/dt
= ti
If VL = -5V = L di/dt
LV-5
= di/dt
= ti
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Filter Examples
i
100s50s 150s0
<iL> = 0.5 A
Choose L to make = 0.005 AiLV5
= ti
= si
iLV5= sx
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Filter Examples
LV5 x0.005 A = x
L = 5 x 10-3250 x 10-6
L = 0.005 H
iL > 5 mH makes < 0.005 A
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Results and Comments
• Since we know the objective of our filters, it is reasonable to design them based on the assumption that the objective is met!
• This simple expedient is a very effective simplifying step.
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Results and Comments
• The ideal action assumption works better than one might expect.
• We will analyze this as we build up converter designs.
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Summary So Far• We can analyze the quality of a
converter output.• Equivalent sources give us a way to
deal with the interface problem.• The ideal action assumption helps
considerably with design.
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Filter Example• Consider a converter, shown, with
switch #1 duty ratio at 3/4.
VIN
IIN L iOUT
#2
VOUT
C R#1
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Filter Example• Let the switching frequency be 200 kHz,
L = 1 mH, C = 10 µF, R = 10 , Vin = 5 V.• By KVL and KCL, the switches
need to alternate.• We can determine the device types.
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Filter Example
VIN
iL L #2
VOUT#1
VIN
VOUT
VOUT
VIN
VOUTVOUT
ON statecurrents
OFFstatevoltages
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Filter Example
VIN
L #2
#1Diode
FCFB
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Load Current#2iL L
#1
0
5
-15
-5
-10
vL
0Lv
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Energy Balance
• With switch #1 on, the input energy to the inductor is (Vin)(iL)(3T/4). With switch #2 on, the input is (Vin - Vout)(iL)(T/4).
• The total must be zero. This requiresVout = 4 Vin = 20 V.
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Load Current• The load current is 2 A, and the load
power is 40 W.• The average input current must be
(40 W)/(5 V) = 8 A. This is iL.
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Current Ripple• If the inductor and capacitor are large
(we will check this), then iL and Voutare nearly constant.
• The inductor sees 5 V when #1 is on, so its current increases for 3.75 us.
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Current Ripple• The inductor sees 5 V - 20 V = -15 V
when switch #1 is off, and the current falls for 1.25 us.
• During the rise, vL = 5 V = L di/dt, but the rise is linear over 3.75 us, so (5 V)/L = i/t , t = 3.75 us.
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Current ripple
With a 1 mH inductor, this means i = (5 V)(3.75 us)/(1 mH), i = 0.0188 A.This is less than 0.25% of iL.Check the current fall. Does it match? Why?
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Current ripple
8
iL (A)
8.01
7.99
0 3.75 5 t (s)
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Voltage Ripple
• We can do the same thing to find ripple on the output capacitor.
• The capacitor current is known: With switch #2 off, the resistor draws out 2 A. With switch #2 on, the current is 8 A - 2 A = 6 A.
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Voltage Ripple
• iC is fully determined.• #2 off : iC = -2 A vC decreases• #2 on : iC = iL - 2 = 8 - 2 = 6 A vC increases
iOUT#2
VOUT#1
ILOAD
iC
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Voltage Ripple• Thus iC = 6 A for 1.25 us, and
-2A for 3.75 us.• Since iC = C dv/dt gives linear voltage
ramps, the voltage rises when iC = 6 A: (6 A)/C = v/t.
• The time involved is 1.25 us.
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Voltage Ripple• (6 A)(1.25 us)/(10 uF) = v = 0.75 V.• This is 3.75% of the 20 V dc level.• Not perfect, but still very nearly constant.• Thus with switching frequency of 200
kHz, L = 1 mH, C = 10 µF, R = 10 , Vin= 5 V, we get 20 V out and 3.75% peak-to-peak output ripple.
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Power Factor• A conventional measure in utility
systems is power factor -- the fraction of energy flow that does useful work.
• Recall that cross-frequency termsdo not contribute <P>.
• But, the cross terms do require current and voltage.
• The extra current means extra I2R loss, and should be avoided is possible.
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Power FactorCapture fraction of energy flow that performs useful work.
R
12 V
i2 R
10 Apeak60 Hz
<POUT >=0 pf=0
LOSS =49RP
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405
Power Factor• Power factor is defined by
• Ideally, this is 1. When harmonics or phase shifts are present, it is less than 1.
• pf can be less than 1 even in a linear circuit, but it is never greater than 1.
1RMSRMS IV
Ppf
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Engineering at IllinoisEngineering at Illinois
406
Power Factor Example
Two contributions to the pf : “Distortion power” and “Displacement power.” The “displacement factor:”
00 pfP
111
cos RMSRMS IV
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Power Factor Issues• pf is often divided into a phase effect at
the wanted frequency (displacementpower, with a displacement factor), and a distortion effect at unwanted frequencies.
• pf < 1 causes extra loss, and limits flow capabilities.
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Power Factor IssuesWhy do we want pf = 1 ?
1) Minimizes system loss. Maximizes “device utilization.”2) Gives more available power.
120 V, 12 Apf = 1 1440 Wpf = 0.5 720 W
3) ExamplesRectifiers can have pf ~ 0.3
![Page 57: Power Electronics - University Of Illinois · 2012-08-26 · • pf is often divided into a phase effect at the wanted frequency (displacement power, with a displacement factor),](https://reader036.vdocuments.net/reader036/viewer/2022080717/5f78291a4f323d6f1101ad3b/html5/thumbnails/57.jpg)
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Dc-Dc Converters
• We would like to have a dc transformer -- a device with Pin=Pout and Vout/Vin = a.
• Magnetic transformers cannot handle dc, but the dc transformer is still a valid concept.
• Our objective in dc-dc converter design is to approach a dc transformer as best we can.
![Page 58: Power Electronics - University Of Illinois · 2012-08-26 · • pf is often divided into a phase effect at the wanted frequency (displacement power, with a displacement factor),](https://reader036.vdocuments.net/reader036/viewer/2022080717/5f78291a4f323d6f1101ad3b/html5/thumbnails/58.jpg)
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Dc Transformers• We would like to have a box like this, for DC.
IIN IOUTVIN VOUT
aIIa
VV
IVPIVP
out
in
in
out
outoutoutininin
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Dividers?• We might try a voltage divider.• Two problems:
– No regulation– Losses within the “converter”
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Dividers?
R2
R1 VOUT
VIN
0then ,0PIf out
:loadNo21
1
inout V
RRRV
in
out
PP
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Dividers?
Load regulation issue.R2
R1 VOUT
VIN
RLOAD
We want Vout to be almost constant.
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Dividers?• The load regulation problem can be
addressed through excess loading:• Make the divider input draw so much
power that the load power causes no change.
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Divider Efficiency• Instead, if somehow all output power
is delivered to the load (best possible case), the efficiency is Vout/Vin.
• This occurs only at a single load value, if designed in advance. The design has no load regulation.
• Reality is always worse.
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Dividers -- Conclusion• Voltage dividers are useful for sensing
applications when the load power is intended to be zero.
• A voltage divider is not useful for dc-dc conversion.
• It is not a power electronic circuit, since the efficiency cannot be 100%.
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Sensing application
Can keep load change small, if is low
1K
1A
12 V 7
5 5 V
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Sensing application
5/12
IVP outoutout IVP ininin
IoutI in
PP
out Iout
IinVout
Vin
Vout
Vin
1 A
12 V 7
5
5 V
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Dc Regulators• Since a divider has no regulation, it
motivates new types of circuits.• In these types of “converters,” the
output is independent (within limits) of the input and of the load.
• They perform a regulation functionrather than energy conversion.
• We call them “dc regulators.”
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Amplifiers
• It is also possible to use amplifier methods for dc-dc conversion.
• These are common, because they have excellent regulation properties.
• In general, efficiency is poor.
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Shunt Regulator
Voltage divider, 12 V to 5 V, 1 W.– With exact values, best efficiency is 5/12.– To provide regulation, the divider current path
must carry much more than the load current.– Problems: line regulation, load regulation,
loss even if Pout = 0, low η.
Shunt regulator.– Zener diode in place of low-side resistor.– Requires IZ > 0.– For 12 V to 5 V, 1 W, R1 < 35 Ω.– Solves the line and load regulation challenges,
but not the others.
VOUT
IOUT
IZ
RS
VZ
VIN
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Example
12 V to 5 V regulation at up to 0.2 A.At 0.2 A load, the input current must be at least 0.2 A to ensure IZ > 0.This current flows through a drop of 7 V, so Rs < 35 .Try it . . .
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Example• Test a load of 0.1 A. The input current,
if the regulator works, is (12 V - 5 V)/(35 ) = 0.2 A. The load current is 0.1 A, so the zener current must be 0.1 A.
• This is wasteful, but it works.• Useful for generating low-power
reference voltages.
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ExamplePOUT = (0.1 A)(5 V)
= 0.5 W
PIN = (12 V)(0.2 A) = 2.4 W
20.8%
in
out
PP
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Series Regulator
• Instead find a series device that can provide an output that is approximately independent of the input.
• A bipolar transistor can do the job – in its linear operating region.
• With proper bias, the output depends on the base voltage.
• Not a switching method.
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Series Pass Arrangement
The emitter voltage follows the (low-power) base voltage.
IOUTIIN
VIN
VCONTROL
RLOAD
VOUT
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Series Pass ArrangementSuppose a 6 V output is needed.
IOUTIIN
precisepotential
rom shuntregulator)
6 V
(6.7 V)
VC
IB
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Example
Here, a shunt regulator provides the reference voltage for a series regulator.
6 V
IBIZ
1 K
6.7 V
12 V
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Series Pass Arrangement
• In the bipolar case, if there is high gain, the base current is very low.
• The emitter voltage will be roughly 0.7 V below the base voltage.
• This works provided the collector input is high enough.
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Series Pass ArrangementIf IB is small (high gain), then•
in
out
inin
outout
in
out
VV
IVIV
PP
IIN=IC
IOUT=IE=IB+IC
IC = IEIIN= IOUT.
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Series Pass Comments
• Common for local dc power, e.g., 12 V in, 5 V out, but extremely inefficient unless voltages are nearly the same.
• Notice that Iin Iout.• Best-case efficiency is Vout/Vin since
current is conserved.• Requires Vin > Vout + ~2 V
![Page 80: Power Electronics - University Of Illinois · 2012-08-26 · • pf is often divided into a phase effect at the wanted frequency (displacement power, with a displacement factor),](https://reader036.vdocuments.net/reader036/viewer/2022080717/5f78291a4f323d6f1101ad3b/html5/thumbnails/80.jpg)
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More Comments• Although this is common, it is only
acceptable when voltages are close.• Useful example: 14 V to 12 V regulator
for automotive application. Efficiency could be 86%.
• Poor example: 48 V to 5 V regulator for telephone application. Efficiency is only 10%.
![Page 81: Power Electronics - University Of Illinois · 2012-08-26 · • pf is often divided into a phase effect at the wanted frequency (displacement power, with a displacement factor),](https://reader036.vdocuments.net/reader036/viewer/2022080717/5f78291a4f323d6f1101ad3b/html5/thumbnails/81.jpg)
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Key Advantage
• Vout = Vcontrol - Vbe --- entirely independent of input, load, etc.
• This is a “linear regulator,” since Voutis a linear function of a control potential.
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Parting Comments
Series linear regulators make good filters -- if we can keep the input and output close together.Shunt regulators provide fine fixed reference voltages but are not so useful for power.
![Page 83: Power Electronics - University Of Illinois · 2012-08-26 · • pf is often divided into a phase effect at the wanted frequency (displacement power, with a displacement factor),](https://reader036.vdocuments.net/reader036/viewer/2022080717/5f78291a4f323d6f1101ad3b/html5/thumbnails/83.jpg)
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Now, Switching
• The circuits so far cannot provide 100% efficiency. We need switching.
• Two possibilities of general dc-dc conversion:– 2 x 2 matrix, voltage in, current out– 2 x 2 matrix, current in, voltage out.
• These are the direct dc-dc converters.
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Direct DC-DC Converters
2,2
1,21,1
2,1Source
Source
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Direct DC-DC ConvertersTwo direct converters for DC-DC:
2,2
1,21,1
2,1 2,2
1,21,1
2,1
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Voltage to Current
Output voltage is +V 0 or V
2,2
1,21,1
2,1
IIN
VIN
VOUT
IOUT
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Switch Relations• Output is +Vin if 1,1 and 2,2 are
on together, etc.• A switching function representation
is vout(t) = q11 q22 Vin - q12 q21 Vin
• But KVL, KCL require q11+q21=1,q12+q22=1.
![Page 88: Power Electronics - University Of Illinois · 2012-08-26 · • pf is often divided into a phase effect at the wanted frequency (displacement power, with a displacement factor),](https://reader036.vdocuments.net/reader036/viewer/2022080717/5f78291a4f323d6f1101ad3b/html5/thumbnails/88.jpg)
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Switch RelationsIn switching function form:
outoutin IqqIqqti 12212211
inout Vqqtv 2211 inVqq 1221
KVL+KCL:
11
2212
2111
qqqq
ininout VqqVqqtv 22112211 11
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Switch Relations
In this dc application, we are interested in <vout(t)>. The switching function averages are the duty ratios, and
inout Vqqtv 12211
inout VDDtv 12211 We can choose duty ratios D11 and D22 to provide a desired <vOUT>.
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Switch Relations
“Buck Converter” or“Step-Down Converter”
inoutinoutin VvVvV ii DDD 2010 2211
outin IDDi 12211
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Current to Voltage
Output current is +Ii 0 or -Ii
2,2
1,21,1
2,1IIN
VOUT
VIN
IOUT
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Switch Relations
12211
DDv
V inout
12211 VDDv outin
12211 IDDi inout
Boost Converter
2
inout
ii
Vv
DDD
010 2211
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Summary• The dc transformer is an important practical
function.• Non-switching methods, such as voltage
dividers and dc regulators, are not really suitable for power conversion.
• We considered two switching circuits that accomplish buck and boost dc-dc conversion functions – types of dc transformers.
![Page 94: Power Electronics - University Of Illinois · 2012-08-26 · • pf is often divided into a phase effect at the wanted frequency (displacement power, with a displacement factor),](https://reader036.vdocuments.net/reader036/viewer/2022080717/5f78291a4f323d6f1101ad3b/html5/thumbnails/94.jpg)
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Simplifications• In many applications, it is desirable to
share a common input-output node (ground reference).
• This requires one switch always on and one always off.
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Common-Ground Dc-DcExample: 2x2 switch matrix, with common
input-output ground#1
#2
![Page 96: Power Electronics - University Of Illinois · 2012-08-26 · • pf is often divided into a phase effect at the wanted frequency (displacement power, with a displacement factor),](https://reader036.vdocuments.net/reader036/viewer/2022080717/5f78291a4f323d6f1101ad3b/html5/thumbnails/96.jpg)
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Common-Ground Dc-Dc#2 ON
#1 ON
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Common-Ground Dc-Dc
With two switches left, label them #1 and #2.One becomes and one
This can be checked by testing current (on) polarity and voltage (off) polarity.
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Switching Functions
With ideal, or near-ideal, current and voltage sources, KVL and KCL require q1 + q2 = 1.The buck converter:
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Buck Converter
• The voltage vout is the “switch matrix output.”
• The load voltage is <vout> since <vL> = 0.
VIN
iIN
vOUT
L IOUT
<vOUT>#2
#1
RLOAD
![Page 100: Power Electronics - University Of Illinois · 2012-08-26 · • pf is often divided into a phase effect at the wanted frequency (displacement power, with a displacement factor),](https://reader036.vdocuments.net/reader036/viewer/2022080717/5f78291a4f323d6f1101ad3b/html5/thumbnails/100.jpg)
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Relationships
Average power: <pout>=<pin>
vout = q1 Vin
iin = q1 Iout
<vout> = D1 Vin
<iin> = D1 Iout
There is no loss.
Instantaneous power: pin(t) = q1 Vin Iout
= pout(t)
=D1VinIout
![Page 101: Power Electronics - University Of Illinois · 2012-08-26 · • pf is often divided into a phase effect at the wanted frequency (displacement power, with a displacement factor),](https://reader036.vdocuments.net/reader036/viewer/2022080717/5f78291a4f323d6f1101ad3b/html5/thumbnails/101.jpg)
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Relationshipsvout is the switching matrix output.
outID1
inVqoutv 1
in qV 1
inout Vqv 1
outin Iqi 1inVDoutV 1load voltage
outv outV load voltage
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Relationships
vOUT
L
VLOAD
C R
0Lv
vv loadoutoutV
constantVbigL load
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Relationships
outin IVD1outin pp
outin tptp
outin IVq1 outoutout Ivtp
outin IqV 1 ininin tiVtp
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The RMS “output”The voltage vout has an RMS value of
Is this relevant?
10
221
1 DVdtVtqT in
T
in
Notice that
DqRMS
tqtq 2
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A Design• A 24 V to 5 V converter, switching at
100 kHz. The nominal load is 25 W, and the ripple is to be less than 1% peak-to-peak.
• This could be met with a buck converter, since Vout < Vin.
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A Design• The duty ratio will need to be
Vout/Vin = (5 V)/(24 V) = 0.208• The output current is (25 W)/(5 V) = 5 A.• When switch #1 is on, the inductor sees
24 V - 5 V = 19 V.
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A Design• With #1 off, the inductor sees -5V• So, since vL = L di/dt, with #1 on,
19 V = L di/dt = L i/t
• The time involved is 0.208 T, or 2.08 us. We want i < 0.01(5 A).
• Thus (19 V)(2.08 us)/L < 0.05 A, and L > 0.792 mH
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A Design• We expect that D1 = 0.208,
fswitch = 100 kHz, L = 0.8 mH,and R = 1 will meet the need.
• Practice: What is the peak-to-peak ripple if L = 8 uH? it will be 100x as big
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Boost Converter
A boost converter is a buck converter flipped horizontally.
#2
#1
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Boost Converter
With common ground, the matrix reduces to two switches.Iin is formed as a voltage in series with L.
VIN
IIN
vIN
L iOUT
#1
#2
VOUT
C R
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Relationships• The input voltage to the switch matrix
is vin, the voltage across the transistor.• Since <vL> = 0, the average transistor
voltage matches Vin.
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Relationships
VOUT
0
0
C
inin
L
i
vV
v
VIN
IIN
vin
L iOUT
C R
ILOAD
iC vL
VOUT
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Relationships• By KVL and KCL, sources require
q1 + q2 = 1.• Then vin = q2 Vout
= (1 - q1) Vout, iout = q2 Iin
= (1 - q1) Iin.• The averages require <vin> = Vin, and
Vout = Vin/(1 - D1) = V /D
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Relationships
outIloadI
inout Iqi 2
inout DIi 11
inIq 11
inin Vv
1 outin DVV 1
outVq 11
outin Vqv 11
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Relationships
1DIf 1
V inin V
D
V1 1
inout D
V
:1DIf 1
0outV
VOUT
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Example
2 V to 5 V boost (input might be one Li-ion cell, for instance, with 2 V as its lowest value).Switching: 80 kHz. Load: 5 W. Input ripple: + 10 mA. Output ripple: + 1%.This gives a period of 12.5 us.
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Boost Example
With 2 V input and 5 V output, the load current at 5 W is 1 A, but the input current must be (5 W)/(2 V) = 2.5 A.With + 10 mA input ripple, the peak-to-peak value is 20 mA.
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Boost Example
• When switch #1 is on, the inductor voltage is 2 V, and current increases.
• The duty ratios: D2 = Vin/Vout = 0.40, and D1 = 1 - D2 = 0.60
• Switch #1 is on 0.60 T = 7.5 us.
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Boost Example
vL = L di/dt = 2 V with #1 on.Thus (2 V)/L = i/t,
t = 7.5 us.To get i < 0.02 A, we need
L > (2 V)(7.5 us)/(0.02 A), or
L > 0.75 mH.
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Boost Example
• What about Vout?• The capacitor current is
Iin - Iload = 2.5 A - 1 Awhen switch #2 is on, and -1 A when switch #1 is on.
• We want + 1% of 5 V, or a peak-to-peak change below 0.1 V.
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Boost Example#1 ON: iC= -1 A
1 A
iC
1 A
iC
2.5 A#2 ON: iC= 1.5 A
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Boost Example• With switch #2 on (duty ratio was
found to be 0.4, so time is 5 us), iC = 1.5 A
= C dv/dt = C v/t.
• (1.5 A)(5 us)/C = v < 0.1 V.• This requires C > 75 uF.
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Boost Example
Practice: What if fs is changed to 40 kHz? average values are the same ripple 2x
2 V
0.8 mH
60%Duty 5
10 F
2 to 5 V, 80 kHz boost converter:
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Comments• With a few practice examples, you should
be able to design a common-ground buck or boost converter.
• Challenge: Think about effects of nonideal switching.
• It is not so difficult to include some basic nonideal effects, such as switching device voltage drops and resistances.
• Consider an example with switch and diode voltage drop.
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Nonideal boost
VIN
IIN
(Still) vIN
L iOUT
VOUTC R
0.5 V
Voltage drop
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Nonideal boost
VIN0.5 V
#1 ON
vIN
#2 ON
1150
15.0 21
outin
VDVDvVVqVqv
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Nonideal boost• Switching function expressions still apply.• Boost: vin = q1(0.5 V)+ q2(Vout+ 1 V).• On average,
<vin> = Vin= D1(0.5V) + (1-D1)(Vout + 1 V), and
Vout = (Vin + 0.5D1 -1)/(1 - D1)• For current, iout = q2 IL, <iout> = D2 IL.• Since <iout> is the load current Iload, we
have IL = Iload/D2 = Iload/(1 – D1).
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Nonideal boost• The efficiency: Pin = Vin IL, Pout = VoutIload.• So Pin = Vin Iload/(1 – D1) and
Pout = (Vin + 0.5D1 -1)Iload/(1 - D1)• The efficiency ratio = (Vin + D1/2 -1)/Vin,
and = 1 – (1 – D1/2)/Vin.• This is less than 100%, reflecting the losses in
the switch forward drops.• Switching functions support analysis of
converters even with these extra parts.
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Indirect Dc-Dc Converters• The buck is a dc transformer with Vout < Vin.• The boost gives Vout > Vin.• How can we give full range? Use a buck as
the input for a boost.• That is, use the current source output of a
buck to provide the input source for a boost.• Remove redundant or unnecessary switches.
Result is the polarity reverser: buck-boost.
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Buck-Boost Development
Can be thesame one
Buck Boost
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Final Simplification
The switch across the current source is not necessary for KCL.Try removing it.The current source is a transfer source.
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Buck-Boost Converter
Left switch is FCFB. Right switch is FCRB.
Buck-Boost
#2#1
iIN iOUT
VIN VOUTIS
vt
Transfer source
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Relationships• To meet KVL and KCL, q1+q2 = 1.• There are really two matrices now. Let us
consider the transfer source, which is manipulated by both matrices.
• Transfer voltage is subject to control.• Transfer voltage vt = q1Vin- q2Vout.• Transfer source power is vt Is = q1 Vin Is- q2 Vout Is.• We want the average power in the transfer
source to be zero -- no loss.
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Relationships
<vtIs> must be zero, not to have losses in the transfer source.
outinstsst
outint
soutsinst
outint
VDVDIvIIv
VDVDvIVqIVqIv
VqVqvqq
KCLKVL
21
21
21
21
21 1:
vt
vt
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Relationships
This can be done if D1Vin= D2Vout.Since D1 + D2 = 1, we have D1Vin = (1 - D1)Vout.This becomes Vout = D1Vin/(1-D1).The polarity reversal comes from the cascade process.
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Buck-Boost
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Relationships
The buck-boost allows outputs both higher and lower than the input, but a polarity shift is present.The transfer source can be an inductor alone to avoid loss.
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Relationships
Consumes no average power.Maintains fixed I.
Can be approximated by an inductor.
This will be our transfercurrent source.
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What About Currents?
The input current: iin = q1 Is,The output current: iout = q2 Is,Average input: Iin = D1 Is,Average output: Iout = D2 Is.We do not really know Is. Add the above: Iin + Iout = (D1 + D2)Is = Is.
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Currents and Stresses• The transfer source sees a current equal to
the sum of input and output average currents.
• Each switch must carry Is, and each must block Vin + Vout.
• All device ratings are higher than either the input or output needs.