Chapter 10
Switching DC Power Supplies
• One of the most important applications of power
electronics
10-1
Linear Power Supplies
• Very poor efficiency and large weight and size
Chapter 10 Switching
DC Power Supplies
10-2
Switching DC Power Supply: Block
Diagram
• High efficiency and small weight and size
Chapter 10 Switching
DC Power Supplies
10-3
Switching DC Power Supply: Multiple
Outputs
• In most applications, several dc voltages are
required, possibly electrically isolated from each other
Chapter 10 Switching
DC Power Supplies
10-4
Transformer Analysis
• Needed to discuss high-frequency isolated supplies
Chapter 10 Switching
DC Power Supplies
10-5
Magnetic circuits
• Toroid with ferrite core
• Ampères law
A
n
 H  ds   i j
s
j 1
Hl  Ni  Fm
• Magnetomotive
force Fm
N
N turns
kierrost a

R
i
– Similar to voltage
source
10-6
Analogy with circuit theory
• Magnetic field
  BA   HA  
Ni
A  Ni  Fm
l
• Magnetic conductance, permeance

A
l
• Often easier to use the inverse, i.e. reluctance

Rm 
7
l
A

Fm  Rm
Fm
Rm
10-7
Flux and inductance
• Inductance of a winding is defined with the flux
so that
2
  N   Li
N
LN 
Rm
2

• Faraday’s law for induction
• Induced voltage tries to prevent the change in
the flux
– When resistive voltage drops are taken into account
d
di
u  Ri 
 Ri  L
dt
dt
8
10-8
Equivalent circuit of transformer
• Power systems
– Power flow in the system
– Load are reduced either to primary or secondary
• Electric machines
– Ofthen quantities are reduced to stator side
– Rotor quantities are often impossible to be measured
=> analysis done in stator side
• SMPS
– Interested both on primary and secondary quantities,
reduced values are not used
– Quantities are not reduced and equivalent circuit is a
bit different
9
10-9
Ideal transformer
• Independent on load or frequency
• No losses
p1  u1i1  u2i2  N1i1  N 2i2
• Load resistor R in primary side
2
u1 N1

u2 N 2
2
 u1 
 N1 
p2  u2i2 

 R´   R  
 R
R
R´
 u2 
 N2 
u22
u12
• Permeability  of the core material is infinite
– Reluctance is zero and transformer is not needing
any magnetizing current
10-10
Direction of the magnetization (1/2)
• Winding direction is marked with a dot
• Ampère’s and Faraday’s laws:
–
–
–
–
Voltage induced by flux changes has same the polarity of the dot
When current flows in from the dotted side
Fluxes are adding, i.e. currents are magnetiing in the same direction
If other winding is open ja the current of the oth increases in the
direction of the dot, a voltage induces in the second winding tries to
cause a current cancelling the flux created by the first current
i1
v1
N1 :N2
i2
v2
10-11
Direction of the magnetization (1/2)
• Positive current below is defined to flow
outwards from the transformer
• Rule to remember:
– From dot in and dot out
i1
v1
N1 :N2
i2
v2
10-12
Real transformer
• Two winding toroid is used as an example
A
m
i1
R
N1
v1
 1
i2
N2
 2
v2
10-13
Equivalent circuit
• Primary winding flux
 N i  N 2i2 N1i1 
1  N1   m   1   N1  1 1


R
R 1 
m

A
m
i1
R
N1
v1
 1
i2
N2
 2
v2
 N2 N2 
N12
N
1
1
 i1 

i2  i1  Lm  L 1   Lm 2 i2
  N1 N 2 2
 Rm R 1 
N1
N1 Rm


 m +  1  m
•
•
Magnetizing inductance
Leakage in primary
F 1 = N1 i1
Rm
 m –  2
 1
 2
R 1
R 2
F 2 = N2 i2
N12
N12
Lm 
, L 1 
Rm
R 1
10-14
Primary flux and inductances
• Equivalent circuit for primary flux
– Ideal transformer used
L 1
1
i1
N2
i2
N1
N2
1  i1  Lm  L 1   Lm
i2
N1
N1 :N2
i2
Lm
Ideal
ideaalinen
10-15
Secondary
 N i  N 2i2 N 2i2 
 2  N 2   m   2   N 2  1 1


R
R
m
2 

A
m
i1
 N2 N2 
N 22
N1
2
2


i


N
N
i


i
L

L

L
i1




i
2
1 2 2
2
2 m2
2
m2

R
R
N
N 2 Rm
2 
2
 m
N
2
R
N1
v1
 1
2
 2
v2
 m +  1  m
•
•
Magnetizing inductance seen
from the secondary
Leakage inductance
F 1 = N1 i1
Rm
 m –  2
 1
 2
R 1
R 2
F 2 = N2 i2
N 22 N 22
N 22
Lm 2 

Lm , L 2 
2
Rm N1
R 2
10-16
Secondary flux and inductances
N1
 2  i2  Lm 2  L 2   Lm 2
i1
N2
i1
N1 :N2
N1
i1
N2
i2
Lm 2
L 2
2
ideaalinen
Ideal
10-17
Equivalent circuit of transformer
• Previous equations for primary and secondary
are connected
– Magnetizing inductance in primary as the switch is
normally in primary too
L 1
1
i1
N2
i2
N1
N1 :N2
i1
i2
Lm
L 2
i2
2
Lm 2
ideaalinen
Ideal
ideaalinen
Ideal
L 1
1
N1 :N2
N1
i1
N2
i1
N1 :N2
i2
L 2
2
2
N12
N
N
Lm 
Lm 2  2  2 Lm
Rm
Rm N12
2
Lm
ideaalinen
Ideal
L'm 2

N12
N 22
Lm 2  Lm
10-18
Magnetizing curve
• Bm saturation
• Br remanence flux density
• Depending on the converter topology, core is
magnetized
– 1) in one direction (quadrant 1)
– 2) two directions (quadrants 1 and 3)
10-19
Transformer as part of SMPS
• Magnetization in one quadrant
– Changing previous dc-dc topologies and adding
transformer galvanically isolated version are
obtained
• Flyback (derived from buck-boost)
• Forward (derived from buck)
• Magnetization in two quadrants
– Single-phase dc-ac inverter produces high
frequency ac (square wave) and it is fed through
high frequency transformer and rectified
• Push-pull
• Half-bridge
• Full-bridge
10-20
Transformer inductances (1/2)
• Lm is magnetizing inductance seen in primary
• Should be as high as possible so that effect of in
switches small
• L1, L2, leakage inductances
– Should be small, are adding voltage stresses of
switches in hard switching
– Need to be considered when selecting switches and
dimensioning snubber circuits
10-21
Transformer inductances (2/2)
• Flyback
• Magnetic circuit acts
– Isolation
– and as energy storage (inductance)
• Lm cannot be large
– Resonant converters
– Leakage inductances can be used for ZCS/ZVS
10-22
PWM to Regulate Output
• Basic principle is the same as discussed in Chapter 8
Chapter 10 Switching DC Power Supplies
10-23
Flyback derived from buck-boost
• Galvanic isolation
– Second winding added to buck-boost inductance
id
id
+
Vd
–
+
vL
-
iL
L
+
–
Vo
C
io
R
+
–
C
Vd
Vo
io
–
+
N1 : N2
Chapter 10 Switching DC Power Supplies
10-24
+
V
–
Flyback Converter
• Derived from buck-boost; very popular at small
power (< 50 W ) power levels
Chapter 10 Switching DC Power Supplies
10-25
Flyback Converter
• Switch on and off states (assuming incomplete core
demagnetization)
Chapter 10 Switching DC Power Supplies
10-26
Operation
• Switch is turned on
– Primary current increases but in secondary no current
because of the diode
• Switch is turned off and primary current stops
– Induced secondary voltage tries to keep flux constant
– Secondary current flows and discharges energy
stored in the magnetic circuit => flyback
• Flyback in Finnish ” epäsuoraan tehonsiirtoon
perustuva tasasähkönmuuttaja” + i
d
–
C
Vd
Vo
io
–
+
N1 : N2
Chapter 10 Switching DC Power Supplies
10-27
Flyback waveforms
•Switch turned of
 t =  0 + Vd t
N1
0 < t < ton
•Peak flux
 =  ton =  0 + Vd ton
N1
• Switching
waveforms
(assuming
incomplete core
demagnetization)
Chapter 10 Switching DC Power Supplies
10-28
Voltage ratio
• After tun-off
– Energy stored in air gap keeps secondary current
flowing and energy is transferred
– Secondary voltage is –Uo and flux decreases
 t =  – Vo t – ton
ton < t < Ts
N2
• In steady state flux change (voltage integral)
must be zero  T =  – Vo T – t
s
s
on
N2
V
V
= 0 + d ton – o T s – ton =  0
N1
N2
=>
Vo N2 D
=
Vd N1 1 – D
Chapter 10 Switching DC Power Supplies
10-29
Currents
• During on-time current increases
im t = isw t = Im 0 +
Vd
t
Lm
0 < t < ton
Im = Isw = Im 0 +
Vd
ton
Lm
• After turn-off output voltage –Uo is reflected into
the primary and magnetizing current decreases
Vo N1 N2
im t = Im –
t – ton
Lm
ton < t < T s
• Setting im Ts = im 0 the same voltage equations as
in the previous page is obtained
Chapter 10 Switching DC Power Supplies
10-30
Voltage and current ratings
• Diode current
iD t =
V N N
N1
N
im t = 1 Im – o 1 2 t – ton ton < t < T s
N2
N2
Lm
• Average of diode current is equal to Io
– Peak value of switch and magnetizing current
Im = Isw =
N2 1
N 1 – D Ts
Io + 1
Vo
N1 1 – D
N2
2L m
• Voltage over switch when not conducting
vsw = Vd +
N1
Vd
Vo =
N2
1–D
Chapter 10 Switching DC Power Supplies
10-31
Demagnetizing area
• At low loads magnetizing current goes to zero
before next cycle
• It can be shown that in this area
Vo
R
D
Vd
2 Lm f s
– R = equvalent load resistance, Uo/Io
– Lm = magnetizing inductance
• Output voltage is not only dependent on D,
equal to DCM in dc-dc converters
Chapter 10 Switching DC Power Supplies
10-32
Other Flyback Converter Topologies
• Not commonly used
Chapter 10 Switching DC Power Supplies
10-33
Comparison
• Two-transistor flyback
– Transistor turned on and off simultaneously
– Voltage rating half
– No snubber needed in primary
• Energy of leakage inductances discharges through diodes
• Parallel connected
– More reliable, redundancy
– Efective switching frequency increased when 180 °
phase-shift between PWM’s
– Standard modules, several parallel connections, load
sharing with current control
Chapter 10 Switching DC Power Supplies
10-34
R1
C1
Blocking oscillator or ringing
choke converter
iB
Ud
Q1
iC
UCE
R2 uB
N3
–
• For low powers <
50 W, cheap, no
extra components
ipeak
iC
iS
iB
iS
D1
Io
iL
uB
+
N2
R1
Ud
C1
iB
Q1
iC
Uo
RL
2 Ud
UCE
UCE
iL
R2 uB
Io
N3
–
DTs
ipeak
iC
Ts
iS
Chapter 10 Switching DC Power Supplies
iB
10-35
Turn-on
• Transistor Q1 receives base current through R1
and starts to conduct
• Peak value of primary current
Ipp = Isw =
Vd
V
DmaxTs = d ton
Lm
Lm
• Voltage induced over N3 tries to prevent flux
increase and together with R2 creates base
V
N V
current to Q1
IB = B = 3 d
R2
N1 R2
– At some point base current is not enough and
transistor turns off
• Switching frequency changes with load
Chapter 10 Switching DC Power Supplies
10-36
Forward Converter, Buck derived
• Derived from Buck; idealized to assume that the
transformer is ideal (not possible in practice)
Chapter 10 Switching DC Power Supplies
10-37
Operation
• After turn-on primary current inceases
– Secondary induced voltage tries to cancel increase in
flux => D2 blocking and D1 conducting
N2
Vd – Vo
N1
0 < t < ton
• Voltage over inductance L
• After turn-off diode D2 conducts vL = – Vo ton < t < Ts
• Voltage integrals
vL =
N2
Vd – Vo DTs = Vo 1 – D Ts
N1
Chapter 10 Switching DC Power Supplies
=>
Vo N2
=
D
Vd N1
10-38
Forward Converter:
in Practice
• In practice magnetizing
current needs to be taken
into account
•Demagnetizing winding N3
•Magnetizing energy is
discharged to supply
•Switching waveforms
(assuming incomplete core
demagnetization)
Chapter 10 Switching DC Power Supplies
10-39
Forward, waveforms
• During turn-on
v1 = Vd
v1
0 < t < ton
• And magnetizing
current
increaseslinearly
• When switched off
i1 = – im and it
discharges
magnetic energy
N1i1  N3i3  N 2i2  0

Vd
Ts
DTs
tm
–
N1
N3
Vd
isw
i1
im
i1 = – im
iL
i3 
N1
im
N3
Chapter 10 Switching DC Power Supplies
10-40
Demagnetization
• During demagnetization primary voltage
N
u1   1 U d
N3
ton  t  ton  tm
• Demagnetization time tm can be calculated from voltage
integral over Lm
t
N
N
U d ton 
1U
N3
m  3D
t

0

d m
Ts N1
• Time to demagnetize is (1 – D)Ts and therefore
maximum duty cycle
N
1  Dmax 
3
N1
Dmax  Dmax 
1
1  N3 N1
• Often N3 = N1 because of bifilar winding, then Dmax = 0,5
Chapter 10 Switching DC Power Supplies
10-41
Forward Converter: Other Possible Topologies
• Two-switch Forward converter is very commonly used
Chapter 10 Switching DC Power Supplies
10-42
Comparison
• Two-transistor foward
– Voltage rating half
– At turn-off magnetizing energy flows through diodes
– No demagnetizing winding needed
• Parallel connected
– Same advantages as in flyback
Chapter 10 Switching DC Power Supplies
10-43
Push-Pull Inverter
• Derived from
Buck
(2*Forward)
•Square-wave
ac produced in
primary
•Leakage
inductances
become a
problem
Chapter 10 Switching DC Power Supplies
10-44
Voltages
• D1 conducts as T1 conducts
– Voltage over output inductor
uL 
N2
Ud Uo
N1
0  t  ton
• During D no switch conducting, current of L is
split equally between secondaries
• During second half-cycle T2 conducts
u L  U o
T
ton  t  ton  D  s
2
• Voltage integral
 N2

U

U

d
o  DTs  U o D
N
 1


U o N 2 DTs
DTs
N

 2
Ud
N1 DTs  D N1 DTs  Ts 2  DTs
Uo
N
2 2 D
Ud
N1
0  D  0,5
Chapter 10 Switching DC Power Supplies
10-45
Half-Bridge Converter
• Derived from Buck
Chapter 10 Switching DC Power Supplies
10-46
Voltages
• During turn-on
N2 U d
uL 
Uo
N1 2
0  t  ton
• When no switch conducts
u L  U o
• Voltage integral
 N2 U d

 U o  DTs  U o D

 N1 2


Ts
ton  t  ton  D 
2
U o 1 N 2 DTs
DTs
1 N2


U d 2 N1 DTs  D 2 N1 DTs  Ts 2  DTs
Uo N2

D
Ud
N1
0  D  0,5
Chapter 10 Switching DC Power Supplies
10-47
Full-Bridge Converter
N2
uL 
Ud Uo
N1
uL  U o

Uo
N
2 2 D
Ud
N1
0  t  ton
ton  t  ton  D
0  D  0,5
• Used at higher power levels (> 0.5 kW )
Chapter 10 Switching DC Power Supplies
10-48
Full bridge vs. Half
• Ratio of windings in Full Bridge (FB) and Halfbridge (HB)
 N2 
 N2 
 2



N
N
 1  HB
 1  FB
• If output current is assumed ideal and
magnetizing current zero
 I sw  HB  2  I sw  FB
• FB should be used in high power applications
Chapter 10 Switching DC Power Supplies
10-49
Current-Source Converter
• More rugged (no shoot-through) but both switches
must not be open simultaneously
Chapter 10 Switching DC Power Supplies
10-50
Operation
• Push-Pull with added input inductor
– Current sourced
• When both switches are conduction current in
inductor increases
– Energy is stored and when only one switch conducts
energy is transferred to secondary
• Equivalent to boost
Uo N2
1

Ud
N1 2 1  D 
D  0,5
• Power/weigth ratio lower than in voltage sourced
converters
Chapter 10 Switching DC Power Supplies
10-51
Compensation of nonsymmetric voltage
• Topologies with two quadrant magnetization
– Positive and negative voltage pulses in primary must
be equal
– Otherwise every cycle increases voltage integral and
transformed flux => so called flux walking
– Transformer saturates
• One solution is to add a series capacitor
– Removes dc-component from supply
• Position is same as in resonant converter but here C is larger
– ESR, equivalent series resistance must be small as
all current flows through it
Chapter 10 Switching DC Power Supplies
10-52
Series dc capacitor
• Capacitor C3 added in series with primary
• D5 and D6 to discharge leakage inductances
D1
L
D2
R1
C1
230V
D5
O1
115V
C4
V1
R2
C2
D1
RL
C3
V2
O2
D4
D3
Chapter 10 Switching DC Power Supplies
T1
D2
10-53
Waveforms
Q
• Unsymmetry e.q.
Gate driving
due to different
Q
delays in gate,
Q1 slower
Voltage V1
• a) voltage
A >A
without capacitor
• b) capacitor
removes dc
Primary voltage V2
component
1
2
1
A1
2
A1 = A2
A2
DC voltage over C3
A1
A2
Chapter 10 Switching DC Power Supplies
10-54
Resonance?
• Capacitor and output inductance, resonance
frequency
2
 N1 
fR 
, jossa LR  
 L
2π LR C
 N2 
1
• LR output inductance in primary
• Capacitor charging should be linear
• Resonance frequency must be low enough
when compared to ƒs
• Suitable value e.q. 0,25·ƒs
Chapter 10 Switching DC Power Supplies
10-55
Example
• E.q. ƒs = 20 kHz, output inductance 20 mH and
turns ration 10
1
C
2πf R2  N1
N2  L
2
 0,5 μF
• Capacitor’s dc-component has an effect on
primary voltage
– Positive cycle, it is removed
– Negative cycle, it is added (or vice versa)
– If capacitor is too small dc component is ”too” large
Chapter 10 Switching DC Power Supplies
10-56
Linear charging
• Capacitor voltage is
I
I Ts
U C  dt 
Dmax
C
C 2
– I is average of primary current
• And it needs to be small enough when
compared to supply voltage, 10-20 %
Chapter 10 Switching DC Power Supplies
10-57
Example, cont.
•
•
•
•
•
Continuing the previous example
Supply voltage 320 Vdc
Average primary current in worst case 2,3 A
Maximum duty cycle 80 %
Capacitor voltage
I Ts
2,3
0,8
UC 
Dmax 
V  92 V

6
3
C 2
0,5 10 2  20 10
Chapter 10 Switching DC Power Supplies
10-58
Size of capacitor
• It can be concluded that 92 V is too high when
compared to input 320/2 = 160
– Deteriates SMPS operation especially at lower limit of
input
• E.q. if 30 V is upper limit
I Ts
2,3
0,8
C
Dmax 
F  1,53 μF
UC 2
30 2  20 103
– In principle 30 V capacitor is enough but using e.q.
200V component is safer
– At higher power current rating of the capacitor
becomes a problem
Chapter 10 Switching DC Power Supplies
10-59
Ferrite Core Material
• Several materials to choose from based on applications
Chapter 10 Switching DC Power Supplies
10-60
Units
• Often non-standard units are used with magnetic
circuits
• In SI-system
– 1 gauss = 0,1 mT = 0,1·10–3 Vs/m2
– 1 örstedt = (1/4p)·103 A/m
Chapter 10 Switching DC Power Supplies
10-61
Core Utilization in Various Converter
Topologies
• At high switching frequencies, core losses limit
excursion of flux density
Chapter 10 Switching
DC Power Supplies
10-62
Maximum flux density
• When N1 = N3
Ud
DBmax 
4 N1 Ac f s
kun D  0,5
– Ac = Area of the cross section of transformer
• In Forward converter magnetization in one
quadrant
Bm  Br
DBmax 
2
• Half- and Full-bridge, magnetization in two
quadrants
DBmax  Bm
Chapter 10 Switching DC Power Supplies
10-63
Flux density (1/2)
• High saturation flux density Bm
– => high DBmax
– smaller cross section and transformer
• Small switching frequency
– Bm limits DBmax
– Increasing switching frequency reduces transformer
size
– However at high frequencies Bm must be reduced to
reduce losses or it is necessary to use better ferrite
material with lower losses
Chapter 10 Switching DC Power Supplies
10-64
Flux density (2/2)
• One quadrant magnetization
– Remanence must be small
– In practice air gap is used
• Two quadrant magnetization
– Air gap prevents saturation at start up and transients
– Unsymmetry of primary voltage in steady state needs
to be removed
Chapter 10 Switching DC Power Supplies
10-65
Flyback
• Magnetic circuit is used as energy storage
– It has to have an air gap
– Stores energy
– Prevents saturation
Chapter 10 Switching DC Power Supplies
10-66
Transformer losses (1/2)
• General equation for core losses
Pcore
m3
 kf a  DB 
b
– DB = maximum flux swing during ƒs
– Constants k, a, b depend on core material
2 ,𝑅 = 𝐹 𝑅
𝑃𝑤 = 𝑅𝑎𝑐 𝐼𝑟𝑚𝑠
𝑎𝑐
𝑟 𝑑𝑐 = winding ac resistance
• Copper losses are due to winding resistance
• Fr accounts for temperature and eddy currents
10-67
Transformer losses (2/2)
• It follows that
– Pcore increases as core size and DB increase
– Pw reduces as core size and DB increase
• Previous changes are opposite
• When designing a transformer
– Target is to minimize losses when output power P0
is given
– It can be shown that optimum case is when winding
and core loss densities are equal
10-68
Transformer dimension for
different topologies
• Next pages, summary of transformer
dimensioning, output power
– Reference is /Pressman, 1998 pages. 277-294/
• In the following equations
– Filling factor of winding is assumed to be 0,4
– When calculating primary quantities efficiency is
assumed to be 0,8
10-69
Quantities
• In the equations of following page units are
• Bmax = flux density in Gauss
• Acore = core area in cm2
• AW = winding area in cm2
• Dcma = inverse of current density when area is expressed in
circular mills
𝐷𝑐𝑚𝑎 =
Area of wire in circular mills
𝐼𝑅𝑀𝑆
• circular mill = area of a circle when diameter is 1 mm.
circular mill  5,07  10-6  cm2
10-70
Topologies
• Forward
0, 0005 Bmax fAcore Aw
Po 
Dcma
• Push-pull
0, 001Bmax fAcore Aw
Po 
Dcma
• Half-bridge
0, 0014 Bmax fAcore Aw
Po 
Dcma
• Full-bridge, same as half-bridge
• Output power increases from the same core as
topology is changed toward bridge converters
– At the same time more power semiconductor
devices are needed
10-71
Summary
•
•
http://www.ferroxcube.co
m/
Data Handbook Soft
Ferrites and Accessories
10-72
Feedback Control
10-73
Content
•
•
•
•
•
Linearisation of the converter and modulation
PWM-modulator
Optimizing the controller
Voltage feed forward
Current control
10-74
Control to Regulate Voltage Output
• Linearized representation of the feedback control
system
Chapter 10 Switching DC Power Supplies
10-75
Methods
• Middlebrook, Cúk
– state-space averaging
– Linear model in some operating point
• Vorperian
– average PWM switch
– Newer method and used often, a bit easier
– Only the non-linear part is averaged, i.e. switch and
diode
10-76
State-space averaging
• R. D. Middlebrook , S. Cúk A; A General Unified
Approach to modelling Switching-Converter
Power Stages, IEEE PESC 1976
• State-Space Averaging, SSA
• Target is transfer function
In CCM
10-77
Step 1
• State variables are x (matrix)
– Current of inductance
– Capactor voltage
– E.g. in Cúk more variables are needed
• Also losses of components taken into account
• Two set of equations, on and off
State equations
and
output voltage
uo  C1 x
uo  C 2 x
dTs
1- d  Ts
10-78
STEP 2 Averaging
• Previous equations
uo  C1 x
uo  C 2 x
dTs
1- d  Ts
are time averaged
10-79
STEP 3 Linearisation, ac-component in
variables
• Supply voltage is assumed constant
• Inserted in the averaged equations and further
Terms containing multiplications of two ac components, small and neglected
A  A1D  A2 1  D 
B = B1D  B2 1  D 
10-80
Steady state solution
• In steady state derivatives and ac-components
AX + BU d  0
are zero
• From the averaged equation remains
• Similarly, output voltage is
• Steady state output voltage is
U o  CX
• Steady state voltage ratio
Uo
 CA1B
Ud
10-81
Step 4 Laplace transformation
• Laplace of
• Output voltage equation
Laplace transformed and inserted => transfer
function is
10-82
Forward Converter: An Example
• The switch and the diode are assumed to be ideal
Chapter 10 Switching DC Power Supplies
10-83
Switch conducts
• In matrix form
Chapter 10 Switching DC Power Supplies
10-84
Switch not conducting
• Only difference is that Ud = 0
• Matrixes are then
A2  A1 ja B2  0
• Output voltage in both cases
Chapter 10 Switching DC Power Supplies
10-85
Transfer function
• After some steps
• Characteristic polynomial
s 2  2 o s   o2
o 
1
LC
r r
1
 C L
L
  RC
2 o
1
z 
rC C
Chapter 10 Switching DC Power Supplies
10-86
Forward Converter:Transfer Function Plots
• Example
considered earlier
Chapter 10 Switching DC Power Supplies
10-87
Flyback Converter:Transfer Function Plots
Chapter 10 Switching DC Power Supplies
10-88
Rigth half plane zero
• Flyback has zero in right half plane
– This is seen in the phase of the transfer function
• In steady state duty cycle is increased
– Time (1 – d)Ts is reduced
– Time dTs increases and output capacitor is
discharged longer
– Immediately after the increase of d output voltage
decreases and it doesn’t increase
• In steady state increase of d increases also
output voltage
Chapter 10 Switching DC Power Supplies
10-89
Linearizing the PWM Block
• The transfer function is essentially a constant with
zero phase shift
Chapter 10 Switching
DC Power Supplies
10-90
Gain of the PWM IC
• It is slope of the characteristic
Chapter 10 Switching
DC Power Supplies
10-91
Typical Gain and Phase Plots of the OpenLoop Transfer Function
• Definitions of the crossover frequency, phase and
gain margins
Chapter 10 Switching
DC Power Supplies
10-92
A General Amplifier for Error Compensation
• Can be implemented using a single op-amp
Chapter 10 Switching
DC Power Supplies
10-93
Type-2 Error Amplifier
• Shows phase boost at the crossover frequency
Chapter 10 Switching
DC Power Supplies
10-94
Coefficents of the controller
• Simple and often in SMPS used method is so called K-factor
• H. Dean Venable; The K-Factor: A New Mathematical Tool for
Stability Analysis and Synthesis, Proceedings of Powercon
10, March 22-24, 1983.
• Select cross and with K-factor
z 
•
 cross
 p  K  cross
K
It can be shown the the required phase Boost give K as
 o Boost 
K  tan  45 

2


Chapter 10 Switching DC Power Supplies
10-95
Boost and phase margin
• Phase margin
PM  180o  1  c
– c is phase of controllerTc at cross
– 1 is sum of phases of power stage Tp and modulator
Tm
• Required phase boost is
PM  180o  1  c  180o  1  90o  Boost
 Boost  PM  1  90o
• PM is normally selected to be 45° - 60°,
– => Boost => K
Chapter 10 Switching DC Power Supplies
10-96
Gain
• Amplifier gain is selected so that the open loop
gain of TOL (G1Gc) at cross over frequency is one
Gc   cross  
1
1
1


G1   cross  T1   cross  T p   cross  Tm   cross 
• This gives
Gc 
1
KR1C2 cross

1
G1
• Resistor R1 can be selected freely and other
components are
C2 
G1
KR1 cross

2

C1  C2 K  1
Chapter 10 Switching DC Power Supplies
R2 
K
C1cross
10-97
Phase boost
• With the previous controller structure phase can
be boosted only 90 degrees
• E.g. flyback has zero in rigth half plane
– Controller with two zeros and poles can boost the
phase 180 degrees
Chapter 10 Switching DC Power Supplies
10-98
Voltage Feed-Forward
• Makes converter more immune for input voltage variations, duty cycle
is changed immediately
• Overcompensation should be avoided
Chapter 10 Switching DC Power Supplies
10-99
Voltage versus Current Mode Control
• Regulating the output voltage is the objective in both
modes of control
Chapter 10 Switching DC Power Supplies
10100
Various Types of Current Mode Control
• Constant frequency, peakcurrent mode control is used
most frequently
Chapter 10 Switching DC Power Supplies
10101
Peak Current Mode Control
• Slope compensation is needed
Chapter 10 Switching DC Power Supplies
10102
Instability in current control
• E.g. noise in measurement and current is
according to the dashed line, slope still m1
• The disturbance DIo increases after every cycle
when duty cycle is larger than 0,5
Control voltage is constant
vc
iL
–m2
DI0
DI2
m1
D1 Ts
DT s
D1' Ts
DI1
t
D 'Ts
10103
Change in current
• Current increase with slope m1 and decrease
with slope m2. In steady state
m2 D
D


m1 D´ 1  D
• Based on the previous figure
DI o   D  D1  Ts m1




DI1  D´ D1´ Ts m2  1  D  1  D1  Ts m2   D  D1 Ts m2
• After first pulse
m2
D
DI1  
DI o  
DI o
m1
1 D
10104
Geometric series
• With recursion, after the n:th pulse
n
D 

DI n   
 DI o
 1 D 
• If duty cycle D is larger than 0,5
– Geometric series where constant is larger than one
– => disturbance is amplified as shown in the figure =>
unstable
10105
Slope compensation
• Instability is removed if vc is not constant
• Compensating voltage used in vc or added in
the measured current
iL
ohjausjännite
vc
–m
DI0
–m 2
m1
D1Ts
DT s
D 1' T s
DI1
DI2
t
D 'Ts
• Slope compensation guarantees stability
10106
Disturbance with slope compensation
• It can be shown that after n:n cycles
n
 m2  m 
DI n   
 DI o
 m2  m 
• When m is selected properly disturbance
decreases even if D larger than 0,5
• Fastest response when m = m2.
10107
Optimal compensation
• Disturbance is removes already after the first
cycle
iL
–m 2
–m 2
DI0
m1
DI1 = 0
DI2 = 0
t
• Is usefull even if duty cycle is less than0,5
• Optimal slope changes with operating point,
normally slope kept constant
10108
A Typical PWM Control IC
• Many safety control
functions are built in
Chapter 10 Switching DC Power Supplies
10109
Digitalization
•
Digitalization has several levels, often can be understood as digital control
–
•
Motor drives, UPS and high power supplies have already been for long
totally digital
–
•
Digital power, also the inner control loop should be digital
In low power supplies still inner loop is most often analog and microcontroller is used for
interfacing
Digital Control Advantages:
– Flexibility, Ease of upgrade, Man to Machine Interface (MMI)
– Sophisticated control techniques
• possibility to change control laws, switching frequency and improve efficiency and/or
dynamics
– Reduced number of components, Unsensitivity to components’ ageing
•
Digital Control Disadvantages:
– Design complexity, Cost, Dynamic performance (sampling frequency,
quantization...)
10110
Digital Control in Power
Electronics
• Typical organization of a digital current controller
10111
Analog vs. Digital Pulse Width
Modulation
Analog implementation of a PWM modulator. Analog
comparator determines the state of the switches by
comparing the carrier signal c(t) and the modulating
signal m(t)
Simplified organization of a digital PWM. At the beginning of the
counting period, the gate signal is set to high and goes low at the
match condition occurrence, i.e., any time the binary counter value
is equal to the programmed duty-cycle. Binary counter triggers an
interrupt request for the microprocessor at the beginning of each
modulation period.
10112
Sampling delay
•
•
•
Because of the sample and hold effect, the response of the modulator to
any disturbance, e.g. to one requiring a rapid change in the programmed
duty-cycle value, can take place only during the modulation period
following the one where the disturbance actually takes place
Double update or multi-sampling are ways to reduce the delay
Multi-sampling presents some limitations as well
– need for proper filtering of the switching noise
– need for non conventional hardware
– generation of dead bands
Quantization error eq Sample and hold delay effect
Multi-sampled PWM
10113
Synchronized sampling and
switching processes
•
Sampling takes place always at the
beginning (or in the middle) of the
modulation period, the average current
value is automatically obtained and this
is what we want to control
–
•
If the sampling frequency is lower than the
switching one, an aliased, low frequency
component appears on the reconstructed signal
Shannon’s theorem conditions will
always be violated
–
The typically recommended high ratio between
sampling frequency and sampled signal
bandwidth will “never” be possible in power
electronics
10114
Limit Cycle Oscillations, LCO
• The desired set-point for the control variable d is not anyone of its
possible values => system oscillates with period TLCO
• Converter output current
– Is proportional to the integral of the inverter average output voltage,
that is in turn, proportional to the duty-cycle
– Commutations between the two states are determined by the current
controller, that reacts to the current error build-up by changing the
duty-cycle
– Integrator in current control
reduces the effect of LCO
10115
(Model) Predictive or dead beat
current control
• At any given control iteration, we want to find
the average inverter output voltage, that can
make the average inductor current to reach its
reference by the end of the modulation period
– Requires detailed model of the system
– Fast responses
• Hot topic in control theory and in power
electronics control
10116
(Model) Predictive or dead beat
current control
10117
•
•
•
Custom designed digital control
chips
Software programmable devices, like microcontrollers or digital signal
processors are the most commonly adopted approach to digital control in
power electronics
BUT flexibility is costly especially if perfomance requirements are high
Custom designed digital control chips
– Optimize the cost of the controller, tailoring the hardware to the exact
functionality it is called to provide
– Maximizing the performance level of a given power converter topology, by
compressing computation times to the minimum and pushing switching
frequency, small-signal control bandwidth and large-signal speed of response as
high as possible.
•
Some leading integrated circuit manufacturers are nowadays offering
application specific digital control chips, designed and optimized for
particular DC-DC converter topologies and applications
10118
Example: Distributed Energy Source
(DER) / Energy Storage (ES)
10119
Example: STLUX family from ST
Microelectronics
•
•
Especially designed for lighting applications
SMED (state machine event driven) technology which allows the device to
pilot six independently configurable PWM clocks with a maximum resolution
of 1.3 ns
–
•
•
autonomous state machine, which is programmed to react to both external and internal
events and may evolve without any software intervention
Each SMED is configured via the STLUX internal microcontroller
A set of dedicated peripherals complete the STLUX:
–
–
–
–
–
4 analog comparators with configurable references and 50 ns max. propagation delay
It is ideal to implement zero current detection algorithms or detect current peaks
10-bit ADC with configurable op amp and 8-channel sequencer
DALI: hardware interface that provides full IEC 60929 and IEC 62386 slave interface
96 MHz PLL for high output signal resolution
10120
Example:
STLUX
10121
Example: 100 W LED street lighting
application using STLUX385A
• The application consists of a PFC regulator followed by a
zero voltage switching (ZVS) LC resonant stage
• LED current is adjusted using a primary side regulation
(PSR) control technique
10122
References
• Simone Buso, Paolo Mattavelli, Digital Control in Power Electronics,
Second Edition, Morgan& Claypool Publishers, 2015
• Tobias Geyer, Model Predictive Control of High Power Converters
and Industrial Drives, Wiley, 2016.
• C. Buccella, C. Cecati, and H. Latafat, Digital Control of Power
Converters A Survey, IEEE Transactions on Industrial Informatics,
Vol. 8, No. 3, August 2012, pp. 437–447.
• StMicroelectronics, AN4461, Application note, 100 W LED street
lighting application using STLUX385A
10123
Current Limiting
• Two options are shown
Chapter 10 Switching DC Power Supplies
10124
Implementing
Electrical
Isolation in the
Feedback Loop
• Two ways are shown
Chapter 10 Switching
DC Power Supplies
10125
Implementing Electrical Isolation in the
Feedback Loop
• A dedicated IC for this application is available
Chapter 10 Switching
DC Power Supplies
10126
Input Filter
• Needed to comply with the EMI and harmonic limits
Chapter 10 Switching
DC Power Supplies
10127
ESR of the Output Capacitor
• ESR often dictates the peak-peak voltage ripple
Chapter 10 Switching
DC Power Supplies
10128