1.
Everything you ever wanted to know about Buck Converter
Design for LED Lighting - unless you are writing a book
Woody Smith
-Design Geek
LINE-IN
PART 1: ANALOG
CONTROLLERS
1

2.
2
Index
Agenda and goals pages 5,6,7
LED drivers first thoughts and spec’s pages 8-14
Basic Buck Theory pages 15-19
a) current mode page 20-158
1) First order models pages 21-36
2) Averaged models pages 37-46
3) more accurate models pages 47-158
a)intro pages48-57
b)block diagram CPM transfer functions pages 58-64
c) Tymerski model and transfer func’s pages 65-67
d) Effect of current feedback on Q pages 68
e) Low Q approx and transfer functions pages 69-77
f)example design pages 78-81

3.
3

4.
4

5.
AGENDA AND GOALS
1.)Develop a complete design procedure for advanced controller buck converters for
LED Drivers Systems
•Focusing on:
•Overall power efficiency of the lighting assembly
•LED operating efficacy (lumen output per watt of input power)
•Thermal management of the LEDs and their driver circuit
•AC power factor correction (PFC) for the driver circuitry
•AC harmonics generation (distortion)
•Meeting EMI restrictions
•Whether or not dimming is required
•LED driver reliability and service life (to match that of the LEDs)
•Circuit protection devices needed
•Electronics space efficiency (assembly size)
•Cost/competitive position
5

6.
AGENDA AND GOALS
Design Approach
1.)Matlab/LTpice macro simulation to handle the majority of the design tasks
--Combined with an Excel spreadsheet for spec and equations
---Showing the strengths of each controller
Current Mode
1.)Peak, Average, OCPM
V2 mode
Sliding Mode
COT
1.)Fixed on time
2.)Fixed off time
Sigma Delta
2.) Comparison of the various architectures
Decision based on the overall system spec
3.) final design in Spectre/Hspice 6

7.
AGENDA and GOALS
Design Criteria
•LED operating efficacy (lumen output per watt of input power)
•Overall power efficiency of the lighting assembly
•Thermal management of the LEDs and their driver circuit
• only 10-25% of the power is converted to Lumens!!
•AC power factor correction (PFC) for the driver circuitry
•AC harmonics generation (distortion)
•Meeting EMI restrictions
•Whether or not dimming is required
•LED driver reliability and service life (to match that of the LEDs)
•Circuit protection devices needed
•Electronics space efficiency (assembly size)
•Cost/competitive position
-From an EDN article by
Jim Young, ON Semiconductor, and
Usha Patel, Littelfuse, Inc. - October 8, 2012
7

8.
First some thoughts on a possible new
direction for LED Lighting
8

9.
AC dimmer
Optional LED
Microcontroller
AC
Control
(I2C, PWM, Analog)
Voltage,
current,
frequency,
Harmonics, etc.
Monitor
Temperature,
Brightness,
Monitor
GWSnet
Interface
BACnet Bus
Smart Light - Rethink
Customized for building
communication medium
Gateway
Power supply built into the socket
RF
Interface
9

10.
AC dimmer
Optional
LED
Microcontroller
AC
Control
(I2C, PWM, Analog)
Voltage,
current,
frequency,
Harmonics,
etc.
Monitor
Temperature,
Brightness,
Monitor
BACnet
Interface
BACnet Bus
Smart Light – Rethink 2
Customized for building
communication medium
Gateway
Power supply built into the socket
RF
Interface
SMPSPFC
Optional
10

11.
The following is a primer for a top-level down design
methodology. It emphasizes behavioral modeling and design
oriented analysis(to borrow a phrase from Middlebrook).
Our goal is to shorten the
design cycle and
increase physical insight.
-Chip and board
11

12.
First Steps
• What’s your key specs?
• Rank and review them in comparison to existing designs
• Build an Excel spreadsheet and project timeline
• Review this with your team
• Revise and review again
• Build a complete system level behavioral model-Matlab and spice
• Check all spec’s!
• Revise your Excel spreadsheet and timeline
• Verify your resource needs
• List all assumptions
• Make sure your Timeline has “guard bands.”
TORA, TORA, TORA!
12

13.
Strawman Spec
• 90% efficiency
• 50,000 hr. lifetime with greater than 70% of the original luminosity
• 100W equivalent
• Assume 100 lm/W for Leds and 15lm/W for Incandescent -6’ish X advantage
• Assume Iout=700mA
• Vled(700mA) ~ 3.2v (phosphorous coated blue for white’ish light )
• 100W/6*700mA÷3.2≤8 leds required
• Thermal temp≤ 125°C
• 10-100% dimming Triac based
• Triac bleeder ~20mA
• Class B EMI spec
13

14.
1. Tmax =125C
2. Tmin= -25C
3. PFC= .9
4. Power eff=.85% min
5. 10-100% dimming
6. Front edge, triac dimmers
7. 20mA bleeder current
8. Vin =100-140V
9. Iout=500mA ±10ma
10.Vout= 30-32V
11.Inductor ripple=20-40%
12.vout-ripple=?
13.50k hr lifetime
14.Driver dimensions less than
2x3x5mm-working on it
15. driver cost ≤ $1.50 in 100k lots
16. internal OTP, OVP OCP
1. OTP at 125±5
2. OVP at input (MOV fuse-
typically at 200V)
3. OCP-TBD
17. class C EN6001
18.UL approved
Specification list
14

15.
Ok, let’s get down to it. Let’s start with peak mode Buck theory
………let’s start with some solid insights from Abraham Pressman and follow it
with a few(hundred) slides from:
U. of Colorado, ECEN 5807
Great stuff with a solid balance between theory,
design and ……math.
15

16.
From A. Pressman, Switching Mode
Power Supply , pp 177-178
Fig.
5.5a
Fig. 5.5b
Fig. 5.5c
16

17.
From A. Pressman, Switching Mode Power Supply , p 178
KEY
POINTS!!
17

18.
From A. Pressman, Switching Mode Power Supply , p 179
18

19.
Time to start the work
of understanding the
theory and math of
Buck Converters
19

20.
A pretty standard CPM buck
20

21.
21

22.
Look at
Ltspice
sims
22

23.
What’s the
optimal ramp
Size?
23

24.
Pertubation is
dying, but what
are you losing?
24

25.
25

26.
1st Order models work
For small bandwidth systems
Download the Fairchild:
fan4810_http://www.fairchildsemi.com/products/analo
g/fan4810designtools.xls
A wonderful tool for it’s purpose. Use it to explore the
design space.
Read the appnotes and enjoy the wonderful design process, but with your
eyes wide open for it’s limitations
26

27.
Read the appnotes; look at pfc setup, check the EMI filtering
There is a ton of things to be learned here-dissect it!!
27

28.
This a useful large signal, quasi- linearized model w/o averaging or
small signal considerations.
ic
iL
28

29.
Play with the ramps, the compensation. Observe the simplicity of
the switch structure
29

30.
1st order model’s limitations foster a need for a
more advanced approach.
• Start with the large signal models and average
them over the duty cycle phases
• Good, but not yet high frequency aware or
sensitive to sampling effects
• or linear
• Now let’s add State Space matrix formulation and
add small signal perturbations and linearize
• Excellent, but still not complete
• Time to include sampling-Ridley and gang to the
rescue. But which model?
30

31.
31

32.
Ah, the small
signal
perturbation
32

33.
33

34.
Positive effects on noise
and Vin sensitivity too
34

35.
35

36.
LOOP
Even if D is always less than .5(for any setup) use a ramp. CPM
means faster response, lower noise, max-current control on
startup
36

37.
37

38.
38

39.
39

40.
40

41.
41

42.
Erickson Textbook
42

43.
Tan model
Study this
closely
43

44.
44

45.
45

46.
Starting with either state space equations or the Tymerski-Vorperian models
you will arrive at this schematic. The only remaining question.: how to model
sampling effects?
46

47.
47

48.
48

49.
49

50.
50

51.
51

52.
52

53.
53

54.
54

55.
55

56.
∞
Or
At least greater than one
56

57.
57

58.
58

59.
59

60.
Ignore this for the moment
How cool is this equation. Look how Fm (ie, comp ramp)influences
both loops. Later we will add loop compensation, loop delay, and
sampling effects and it gets really interesting. 60

61.
61

62.
62

63.
63

64.
64

65.
65

66.
66

67.
67

68.
Study this
carefully
68

69.
Erickson
Book
69

70.
You’ll want to
derive this
70

71.
Compare to first-order approximation of the
sampled-data control-to-current model
hfs
s
sT
sT
c
L
sssT
e
esi
si s
s















1
1
)/(1
1
1
11
1
1
)(ˆ
)(ˆ
)/(
1
)/(
1


s
ssT
s
s
e s




 s
a
s
hf
f
m
m
DD
f
f
2
221
1
1
1





Control-to-inductor current response behaves
approximately as a single-pole transfer function
with a high-frequency pole at
Model (4) is consistent with the sampled-data small-signal model
71

72.
A quick aside on
2nd order
transfer function
roots
72

73.
73

74.
Wow, Q< 1 gain is pretty nicely behaved
and the phase variation is getting smooth
74

75.
At low Q the poles are separate but interact depending
on their frequency separation(a decade of frequeny to
really separate them phase-wise. Gain-wise decades of
separation removes most interaction)
75

76.
76

77.
77

78.
Approx iL= ic
78

79.
Compare to first-order approximation of the
high-frequency sampled-data control-to-current
model
hfs
s
sT
sT
c
L
sssT
e
esi
si s
s















1
1
)/(1
1
1
11
1
1
)(ˆ
)(ˆ
)/(
1
)/(
1


s
ssT
s
s
e s



kHz32
221
1
1
1
2







 ss
a
s
hf
ff
m
m
DD
f
f
Control-to-inductor current response behaves
approximately as a single-pole transfer function
with a high-frequency pole at 32k vs.39kHz~20%
with Ma=M2. play with
Ma. I grew up using
Ma=( .5-.75) x M2, but
is that opt?
79

80.
2nd-order approximation in the small-signal averaged model
Except Fm
80

81.
Example
• CPM buck converter:
Vg = 10V, L = 5 mH, C = 75 mF, D = 0.5, V = 5 V,
I = 20 A, R = V/I = 0.25 W, fs = 100 kHz
• Inductor current slopes:
m1 = (Vg – V)/L = 1 A/ms
m2 = V/L = 1 A/ms
A/V25.0
2
'

L
TDD
F s
g
D = 0.5: CPM controller is stable for any compensation ramp, ma/m2 > 0
Select: ma/m2 = Ma/M2 = 1, Ma = 1 A/ms
1/A1.0
1
2
1
21




s
a
m
TMM
M
F
81

82.
Example (cont.)
kHz2.8
1
2
1

LC
fo

1
L
C
RQ
47.047.0
1
1



 Q
L
VRCF
R
VF
QQ
gm
gm
ckHz3.1851  o
gm
oc f
R
VF
ff
kHz4.81  ccp fQf
kHz39/2  cchfp Qfff
Duty-cycle control
Peak current-mode control (CPM)
82

83.
83

84.
84

85.
85

86.
Repeating the obvious about these models
• Averaged models depend on small variations on all variables
• Particularly duty cycle
• It works well for DCM too, but only if key control variables are well
controlled
• Peak current mode wants small ∆IL to minimize switching uncertainty;
minimum peak to average error(always check this at low current levels)
• Low peak/average means low distortion but big inductors
• Leading to shallow current ramps and more noise sensitivity at low
vin
• Why? Low vin shallow ramps and the same level of
switching noise (comparator switching spikes(~.1-.2v?)
• Serious layout worries
86

87.
2nd-order approximation in the small-signal averaged model
87

88.
(Tan)
88

89.
][ˆ)1(]1[ˆ][ˆ ninini cLL  
Discrete-time dynamics: )(ˆ)(ˆ zizi Lc 
Z-transform: )(ˆ)1()(ˆ)(ˆ 1
zizzizi cLL   
1
1
1
)(ˆ
)(ˆ




zzi
zi
c
L

Discrete-time (z-domain) control-to-
inductor current transfer function:
ss TjsT
ee 










1
1
1
1
Difference equation:
• Pole at z = 
• Stability condition: pole inside the unit circle, || < 1
• Frequency response (note that z1 corresponds to a delay of Ts in
time domain):
89

90.
Equivalent hold: )(ˆ)(ˆ),(ˆ][ˆ sizitini LLLL 
ic[n]
m1
m2
ic + ic
iL[n]
d[n]Ts
iL[n-1]
ma(t)
iL(t)
iL[n]
Ts
90

91.
Equivalent hold
• The response from the samples iL[n] of the
inductor current to the inductor current
perturbation iL(t) is a pulse of amplitude iL[n] and
length Ts
• Hence, in frequency domain, the equivalent hold
has the transfer function previously derived for
the zero-order hold:
s
e ssT
1
£[u(t)-u(t)𝛿(t+Ts)]=
91

92.
Complete sampled-data “transfer
function”
s
sT
sT
c
L
sT
e
esi
si s
s






1
1
1
)(ˆ
)(ˆ


2
2
1
2
'
1
m
m
D
D
m
m
mm
mm
a
a
a
a






Control-to-inductor current small-signal response:
Ridley, Tan,
Middlebrook-
whoever. The
central issue is
this equation.
How do we
approximate it?
92

93.
Example
• CPM buck converter:
Vg = 10V, L = 5 mH, C = 75 mF, D = 0.5, V = 5 V,
I = 20 A, R = V/I = 0.25 W, fs = 100 kHz
• Inductor current slopes:
m1 = (Vg – V)/L = 1 A/ms
m2 = V/L = 1 A/ms
2
2
2
2
1
2
1
1
'
1
m
m
m
m
m
m
D
D
m
m
mm
mm
a
a
a
a
a
a









s
sT
sT
c
L
sT
e
esi
si s
s






1
1
1
)(ˆ
)(ˆ


D = 0.5: CPM controller is stable for any compensation ramp, ma/m2 > 0
93

94.
Control-to-inductor current responses
for several compensation ramps (ma/m2 is a
parameter)
10
2
10
3
10
4
10
5
-40
-30
-20
-10
0
10
20
magnitude[db]
iL/ic magnitude and phase responses
10
2
10
3
10
4
10
5
-150
-100
-50
0
frequency [Hz]
phase[deg]
ma/m2=0.1
ma/m2=0.5
ma/m2=1
ma/m2=5
5
1
0.5
0.1
MATLAB file: CPMfr.m
Look how the comp ramp kills
the Ti loop gain and BW
Large Fm approx
94

95.
First-order approximation
hfs
s
sT
sT
c
L
sssT
e
esi
si s
s















1
1
)/(1
1
1
11
1
1
)(ˆ
)(ˆ
)/(
1
)/(
1


s
ssT
s
s
e s



Control-to-inductor current response behaves
approximately as a single-pole transfer function
with a high-frequency pole at
Same prediction as HF pole in basic model (4) (Tan)

 s
a
s
hf
f
m
m
DD
f
f
2
221
1
1
1





PADE 1st order
Delay of
the control
signal
95

96.
Control-to-inductor current responses
for several compensation ramps (ma/m2 = 0.1, 0.5, 1,
5)
10
2
10
3
10
4
10
5
-40
-30
-20
-10
0
10
20
magnitude[db]
iL/ic magnitude and phase responses
10
2
10
3
10
4
10
5
-150
-100
-50
0
frequency [Hz]
phase[deg]
1st-order transfer-function approximation
96

97.
Second-order approximation
2
2/)2/(1
1
2
1
11
1
1
)(ˆ
)(ˆ






















ss
s
sT
sT
c
L
sssT
e
esi
si s
s



2
2
2/)2/(2
1
2/)2/(2
1















ss
sssT
ss
ss
e s




2
221
12
1
12
m
m
DD
Q
a








Control-to-inductor current response
behaves approximately as a second-
order transfer function with corner
frequency fs/2 and Q-factor given by
Pade 2nd order
approximate
2nd order polynomial
97

98.
10
2
10
3
10
4
10
5
-40
-30
-20
-10
0
10
20
magnitude[db] iL/ic magnitude and phase responses
10
2
10
3
10
4
10
5
-150
-100
-50
0
frequency [Hz]
phase[deg]
Control-to-inductor current responses
for several compensation ramps (ma/m2 = 0.1, 0.5, 1,
5)
2nd-order transfer-function approximation
Vs
Ideal transfer function
98

99.
99

100.
100

101.
Example
• CPM buck converter:
Vg = 10V, L = 5 mH, C = 75
mF, D = 0.5, V = 5 V,
I = 20 A, R = V/I = 0.25 W, fs
= 100 kHz
• Inductor current slopes:
m1 = (Vg – V)/L = 1 A/ms
m2 = V/L = 1 A/ms
A/V25.0
2
'

L
TDD
F s
g
D = 0.5: CPM controller is stable for any compensation ramp, ma/m2 > 0
Select: ma/m2 = Ma/M2 = 1, Ma = 1 A/ms
1/A1.0
1
2
1
21




s
a
m
TMM
M
F
101

102.
Example (cont.)
kHz2.8
1
2
1

LC
fo

1
L
C
RQ
47.047.0
1
1



 Q
L
VRCF
R
VF
QQ
gm
gm
ckHz3.1851  o
gm
oc f
R
VF
ff
kHz4.81  ccp fQf
kHz39/2  cchfp Qfff
Duty-cycle control
Peak current-mode control (CPM)
102

103.
103

104.
Fig. 9
These
is the
key
point
104

105.
Here’s the
another key
point, look at the
phase at fs/10
105

106.
Placeholder until
I do my sense
section
106

107.
107

108.
108

109.
Fig. 6
109

110.
Fig. 6
110

111.
111

112.
How Do I Model Current Mode Converters?
I follow a dual Pantheon of “Gods”
Middlebrook
Erickson
Ridley
Tymerski/Vorperian
Basso
Dixon
Lehman Brooks
Versus
• My goal to get a solid physical insight into the operation and modeli