Concept Kit:PWM Buck Converter Average Model

PWM Controller Power Switches Filter & Load
(Voltage Mode Control) U?
U?
PWM_CTRL BUCK_SW
L
VOUT
1 2 Vo
-

-
+

PWM C
+
D
1/Vp
Rload
REF ESR

VREF VP = 2.5
VREF = 1.23

Concept Kit:
PWM Buck Converter
Average Model
All Rights Reserved Copyright (C) Bee Technologies Corporation 2011 1

Contents
• Concept of Simulation
• Buck Converter Circuit
• Averaged Buck Switch Model
• Buck Regulator Design Workflow
1. Setting PWM Controller’s Parameters.
2. Programming Output Voltage: Rupper, Rlower
3. Inductor Selection: L
4. Capacitor Selection: C, ESR
5. Stabilizing the Converter (Example)
• Load Transient Response Simulation (Example)
Appendix
A. Type 2 Compensation Calculation using Excel
B. Feedback Loop Compensators
C. Simulation Index


Concept of Simulation
Block Diagram:

PWM Controller Power Switches Filter & Load
(Voltage Mode
Control) Averaged Buck Parameter: VOUT
Switch Model •L
-
+

Parameter: •C
• VP • ESR
VREF • VREF • Rload

Models:
U? U? L
PWM_CTRL BUCK_SW 1 2 Vo
-
C
PWM D
+ Rload
1/Vp ESR
REF

VP = 2.5
VREF = 1.23


Buck Converter Circuit
Power Switches Filter & Load
U2
BUCK_SW L
1 2 Vo

D C
Vin
Rload
ESR

0 PWM Controller
Type 2 Compensator
C2

R2 C1

U3 Comp Rupper
PWM_CTRL

- FB
d
PWM +
1/Vp Rlower
REF

VP = 2.5
VREF = 1.23 0


Averaged Buck Switch Model
iin U2 iout
BUCK_SW

+ D +

vin vout
D
- -

• The Averaged Buck Switch Model represents relation between input and output
of the switch that is controlled by duty cycle – d (value between 0 and 1).
• Transfer function of the model is

vout = d  vin
• The current flow into the switch is

iin = d  iout

Buck Regulator Design Workflow
1 Setting PWM Controller’s Parameters: VREF, VP

2 Setting Output Voltage: Rupper, Rlower

3 Inductor Selection: L

4 Capacitor Selection: C, ESR

5 Stabilizing the Converter: R2, C1, C2

• Step1: Open the loop with LoL=1kH and CoL=1kF then inject an AC signal to generate Bode plot. (always default)
• Step2: Set C1=1kF, C2=1fF, (always keep the default value) and R2= calculated value (Rupper//Rlower) as the initial values.
• Step3: Select a crossover frequency (about 10kHz or fc < fosc/4). Then complete the table.
• Step4: Read the Gain and Phase value at the crossover frequency (10kHz) from the Bode plot, Then put the values to
the table
• Step5: Select the phase margin at the fc ( > 45 ). Then change the K value until it gives the satisfied phase margin, for this
example K=6 is chosen for Phase margin = 46.
• Remark: If K-factor fail to gives the satisfied phase margin, Increase the output capacitor C then try Step1 to Step5
again.

6 Load Transient Response Simulation


Buck Regulator Design Workflow
U2
BUCK_SW 3 L
1 2 Vo

D C
Vin
Rload
ESR

4
0
5 Type 2 Compensator
C2

R2 C1
2
U3 Comp Rupper
PWM_CTRL

- FB
d
PWM +
1/Vp Rlower
REF

1 VP = 2.5
0
VREF = 1.23


1 Setting PWM Controller’s Parameters
U? vcomp
PWM_CTRL • VREF, feedback reference voltage, value
- FB is given by the datasheet
d PWM
• VP = (Error Amp. Gain  vFB ) / d
+
1/Vp
Error Amp.
REF • vFB = vFBH – vFBL
VP = 2.5 • d = dMAX – dMIN
VREF = 1.23
• Error Amp. Gain is 100 (approximated)
The PWM block is used to transfer the error voltage
(between FB and REF) to be the duty cycle. where
3.0V

2.0V
VP is the sawtooth peak voltage.
SEL>> VP vFBH is maximum FB voltage where d = 0
0V
V(osc) V(comp)

vFBL is minimum FB voltage where d =1(100%)
dMAX is maximum duty cycle, e.g. d = 0(0%)
V(PWM) Duty cycle (d) is a value from 0 to 1 dMIN is minimum duty cycle, e.g. d =1(100%)
Time

 If vFBH and vFBL are not provided, the default value, VP=2.5 could be used.

1 Setting PWM Controller’s Parameters (Example)
 If the VP ( sawtooth signal amplitude ) does not informed by the datasheet,
It can be approximated from the characteristics below.

from
vFBH
VP = (Error Amp. Gain  vFB )/d

vFB = •Error Amp. Gain = 100 (approximated)
25mV
•from the graph on the left, vFB = 25mV
(15m - (-10m))
vFBL
d = 1 (100%) • d = 1 – 0 = 1

VP ≈ ( 100  25mV )/1
dMIN dMAX
≈ 2.5V
LM2575: Feedback Voltage vs. Duty Cycle

 If vFBH and vFBL are not provided, the default value, VP=2.5 could be used.

2 Setting Output Voltage: Rupper, Rlower

• Use the following formula to select the resistor values.

 Rupper 
VOUT  VREF1   Type 2 Compensator Vo
 Rlower  C2

• Rlower can be between 1k and 5k. R2 C1

U3 Comp Rupper
Example PWM_CTRL
Given: VOUT = 5V - FB
d
VREF = 1.23 PWM
+
Rlower = 1k 1/Vp Rlower
Error Amp. REF
then: Rupper = 3.065k
VP = 2.5
VREF = 1.23 0


3 Inductor Selection: L

L
1 2 Vo Inductor Value
C
• The output inductor value is selected to set the
converter to work in CCM (Continuous Current
Rload
ESR
Mode) or DCM (Discontinuous Current Mode).
• Calculated by

LCCM 
VI , max VOUT  RL, min
2 fosc VI , max
Where
• LCCM is the inductor that make the converter to work in CCM.
• VI,max is input maximum voltage
• RL,min is load resistance at the minimum output current ( IOUT,min )
• fosc is switching frequency


3 Inductor Selection: L (Example)

L
1 2 Vo Inductor Value
C
from

ESR
Rload
LCCM 
VI , max VOUT  RL, min
2 foscVI , max
Given:
• VI,max = 40V, VOUT = 5V
• IOUT,min = 0.2A
• RL,min = (VOUT / IOUT,min ) = 25
• fosc = 52kHz

Then:
• LCCM  210(uH),
• L = 330(uH) is selected


4 Capacitor Selection: C, ESR

L
1 2 Vo Capacitor Value
• The minimum allowable output capacitor value should
C
be determined by
Rload
ESR

C  7,785 
VI , max
F
VOUT  L( H)
Where
• VI, max is the maximum input voltage.
• L (H) is the inductance calculated from previous step ( 3 ).

• In addition, the output ripple voltage due to the capacitor ESR must be considered as
the following equation.

VO , RIPPLE
ESR 
IL , RIPPLE

4 Capacitor Selection: C, ESR (Example)

L
1 2 Vo Capacitor Value
From
C
C  7,785 
VI , max
F
ESR
Rload
VOUT  L( H)
and
VO , RIPPLE
ESR 
IL , RIPPLE
Given:
• VI, max = 40 V
• VOUT = 5 V
• L (H) = 330
Then:
• C  188 (F)

In addition:
• ESR  100m


5 Stabilizing the Converter
U2
BUCK_SW L H(s)
1 2 Vo

D C
Vin
12Vdc Rload
ESR

0
Type 2 Compensator G(s)
• Loop gain for this configuration is C2

R2 C1

Rupper
GPWM
T ( s)  H ( s)  G( s)  GPWM
3.066k
U3 Comp
PWM_CTRL

- FB
d
PWM +
1/Vp Rlower
REF
1.0k

VP = 2.5
VREF = 1.23 0

• The purpose of the compensator G(s) is to tailor the converter loop gain
(frequency response) to make it stable when operated in closed-loop
conditions.


5 Stabilizing the Converter (Example)

U2 L
Specification: BUCK_SW
1
330uH
2 Vo
VOUT = 5V D C
330uF
VIN = 7 ~ 40V Vin Rload
12Vdc ESR 5
ILOAD = 0.2 ~ 1A 100m

PWM Controller: 0
G(s) Type 2 Compensator
VREF = 1.23V 1 C2

VP = 2.5V e.g. Given values
from National R2 C1
fOSC = 52kHz
Semiconductor Corp.
IC: LM2575 Rupper

Rlower = 1k, 2 U3
3.1k
PWM_CTRL
Rupper = 3.1k, LOL - FB
L = 330uH, 3 1kH
d
PWM +

C = 330uF (ESR = 100m) 4 COL
1/Vp
REF
Rlower
1.0k
1kF

VP = 2.5
0
Task: 1Vac
V3 VREF = 1.23

0Vdc
• to find out the element of the
Type 2 compensator ( R2, C1, 0

and C2 )


The element of the Type 2 compensator ( R2, C1, and C2 ), that stabilize the converter, can
be extracted by using Type 2 Compensator Calculator (Excel sheet) and open-loop
simulation with the Average Switch Models (ac models).
U2 L
BUCK_SW 330uH
1 2 Vo

D C
330uF
Vin Rload
12Vdc ESR 5
100m

0
Step2 Set C1=1kF, C2=1fF,
Type 2 Compensator
and R2=calculated value
Step1 Open the loop with C2
1f
(Rupper//Rlower) as the
LoL=1kH and CoL=1kF then inject R2 C1 initial values.
an AC signal to generate Bode 0.756k 1k

plot. Rupper
3.1k
U3
PWM_CTRL

LOL - FB
d
PWM
1kH +
1/Vp Rlower
COL 1k
1kF REF

VP = 2.5
V3 VREF = 1.23 0
1Vac
0Vdc
 C1=1kF is AC shorted, and C2 1fF is AC opened (or
Error-Amp without compensator).
0



Type 2 Compensator Calculator Step3 Select a crossover frequency
(about 10kHz or fc < fosc/4 ), for
Switching frequency, fosc : 52.00 kHz this example, 10kHz is selected.
Cross-over frequency, fc Then complete the table.
(<fosc/4) : 10.00 kHz
Rupper : 3.1 kOhm
values from 2
Rlower : 1 kOhm
R2 (Rupper//Rlower) : 0.756 kOhm (automatically calculated)
Calculated value of
PWM the Rupper//Rlower
Vref : 1.230 V
values from 1
Vp (Approximate) : 2.5 V



Gain: T(s) = H(s)GPWM
80

40
Step4 Read the Gain and Phase value
at the crossover frequency (10kHz)
0
(10.000K,-44.211) from the Bode plot, Then put the values
-40
to the table.
-80

DB(v(d))
180d

Phase  at fc
Parameter extracted from simulation
90d (10.000K,65.068)
Set: R2=R1, C1=1k, C2=1f
Gain (PWM) at foc ( - or + ) : -44.211
SEL>>
0d
Phase (PWM) at foc : 65.068
100Hz 1.0KHz 10KHz 100KHz
P(v(d))
Frequency

Tip: To bring cursor to the fc = 10kHz type “ sfxv(10k) ” in Search Command.

Cursor Search



Step5 Select the phase margin at fc
K-factor (Choose K and from the table) (> 45 ). Then change the K value
K 6 (start from K=2) until it gives the
 -199  (automatically calculated)phase margin, for this
satisfied
example K=6 is chosen for Phase
Phase margin : 46 (automatically calculated) 46.
margin =

R2 : 122.780 kOhm (automatically calculated)
As the result; R2,
C1 : 0.778 nF (automatically calculated)
C1, and C2 are
C2 : 21.600 pF (automatically calculated) calculated.

Remark: If K-factor fail to gives the satisfied phase margin, Increase the output
capacitor C then try Step1 to Step5 again.

 K Factor enable the circuit designer to choose a loop cross-over frequency and phase
margin, and then determine the necessary component values to achieve these results. A very
big K value (e.g. K > 100) acts like no compensator (C1 is shorted and C2 is opened).


The element of the Type 2 compensator ( R2, C1, and C2 ) extraction can be completed by Type 2
Compensator Calculator (Excel sheet) with the converter average models (ac models) and open-loop
simulation. U2 L
BUCK_SW 330uH
1 2 Vo

D C
330uF
Vin Rload
12Vdc ESR 5
100m

0
The calculated values of the
Type 2 Compensator
C2
type 2 elements are,
21.6p
R2=122.780k, C1=0.778nF,
R2
122.780k
C1
0.778n
and C2=21.6pF.
Rupper
3.1k
U3
PWM_CTRL

LOL - FB
d
PWM
1kH +
1/Vp Rlower
COL 1k
1kF REF

VP = 2.5
V3 VREF = 1.23 0
1Vac
0Vdc
*Analysis directives:
0 .AC DEC 100 0.1 10MEG


Gain and Phase responses after stabilizing
80

40
(9.778K,0.000)

0

-40

SEL>>
Gain: T(s) = H(s) G(s)GPWM
-100
DB(v(d))
180d

Phase  at fc
90d
(9.778K,45.930)

0d

100Hz 1.0KHz 10KHz 100KHz
P(v(d))
Frequency

• Phase margin = 45.930 at the cross-over frequency - fc = 9.778kHz.
Tip: To bring cursor to the cross-over point (gain = 0dB) type “ sfle(0) ” in Search Command.

Cursor Search


Load Transient Response Simulation (Example)
The converter, that have been stabilized, are connected with step-load to perform load transient
response simulation.

U2 L
Load
BUCK_SW 330uH
1 2 Vo

D C
330uF I1
Vin Rload I1 = 0
20Vdc ESR 25 I2 = 0.8
100m TD = 10m
TF = 25u
TR = 20u
PW = 0.43m
PER = 1
0 Type 2 Compensator
C2
21.6p 5V/2.5 = 0.2A step
R2 C1
to 0.2+0.8=1.0A load
122.780k 0.778n

Rupper
3.1k
U3
PWM_CTRL

- FB
d
PWM +
1/Vp Rlower
REF
1k

*Analysis directives: VP = 2.5
VREF = 1.23 0
.TRAN 0 20ms 0 1u


Load Transient Response Simulation (Example)

Simulation Measurement
5.2V 4.0A
1 2
Output Voltage Change
5.1V 3.5A

5.0V 3.0A

4.9V 2.5A

4.8V 2.0A

4.7V 1.5A

Load Current
4.6V 1.0A

4.5V 0.5A

>>
4.4V 0A
9.9ms 10.1ms 10.3ms 10.5ms 10.7ms 10.9ms
1 V(vo) 2 I(load)
Time

• The simulation results are compared with the measurement data (National
Semiconductor Corp. IC LM2575 datasheet).


A. Type 2 Compensation Calculation using Excel
Switching frequency, fosc : 52.00 kHz Given spec, datasheet
Cross-over frequency, fc
(<fosc/4) : 10.00 kHz Input the chosen value ( about 10kHz or < fosc/4 )
Rupper : 3.1 kOhm Given spec, datasheet, or calculated
Rlower : 1 kOhm Given spec, datasheet, or value: 1k-10k Ohm
R2 (Rupper//Rlower) : 0.756 kOhm (automatically calculated)

PWM
Vref : 1.230 V Given spec, datasheet
Vp (Approximate) : 2.5 V Given spec, or calculated, (or leave default 2.5V)

Parameter extracted from simulation
Set: R2=R2, C1=1k, C2=1f
Gain (PWM) at foc ( - or + )
: -44.211 dB Read from simulation result
Phase (PWM) at foc : 65.068  Read from simulation result

K-factor (Choos K and  from the table)
K 6 Input the chosen value (start from k=2)
 -199  (automatically calculated)

Phase margin : 46 (automatically calculated) Target value > 45

R2 : 122.780 kOhm (automatically calculated)
C1 : 0.778 nF (automatically calculated)
C2 : 21.60 pF (automatically calculated)


B. Feedback Loop Compensators
Type2 Compensator

C2
Type 1 Compensator Type2a Compensator
VOUT VOUT VOUT
C1 R2 C1 R2 C1

Rupper Rupper Rupper

PWM_CTRL PWM_CTRL PWM_CTRL

- FB - FB - FB
d d d
PWM PWM PWM
+ + +
1/Vp Rlower 1/Vp Rlower 1/Vp Rlower
REF REF REF

0 0 0

Type1 Compensator Type2 Compensator Type2a Compensator

Type2b Compensator Type3 Compensator

R2 C2

VOUT VOUT
C1 R2 C1

C3
Rupper
Rupper

PWM_CTRL PWM_CTRL R3

- FB - FB
d d
PWM PWM
+ +
Rlower 1/Vp Rlower
1/Vp
REF REF

0 0

Type2b Compensator Type3 Compensator


C. Simulation Index

Simulations Folder name
1. Stabilizing the Converter.................................................... ac
2. Load Transient Response.................................................. stepload

Libraries :
1. ..¥bucksw.lib
2. ..¥pwm_ctr.lib

Tool :
• Type 2 Compensator Calculator (Excel sheet)


Concept Kit:PWM Buck Converter Average Model

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