Frequency Response Analysis for
DC-DC Converters
Without Small-Signal Linearization
Kasemsan Siri
Electrical and Electronic Systems Department
The Aerospace Corporation
El Segundo, CA 90245
(310) 336-2931
kasemsan.siri@aero.org
Copyright 2003 The Aerospace Corporation. All Rights Reserved
Introduction
n Familiar Issues:
o Stubborn design flaws
o Schedule slips
o Cost overruns
n Powerful analysis approach for assisting design and
development of power conversion products:
o Converge in timely manner
o Significantly mitigate the above shortcomings
n Necessities
n Choices of analysis approach
n Benefits
2
Convergence of Product Design
n Conventional design:
o Heavily dependent on prototyping experimental results
o Verify by frequency response measurements
o Time-consuming to identify or fix design flaws
n Modernized design:
o Apply modeling & simulation to uncover design
performance prior to prototyping
– verify typical steady-state & transient performance
– not always uncovering frequency response
n Advanced design:
o Virtual prototyping through modeling & simulation
o Capable of uncovering frequency response
o Better chance of uncovering flaws before prototyping
3
Necessities
n Frequency response analysis tools are necessary
for advanced design:
o
o
o
o
Assist in identifying and resolving design flaws
Validate the design prior to hardware prototyping
Cost effective
Timely complete product development
n Dilemma of design & test engineers
o Avoid analytical modeling
o Rely on prototype testing (so getting used to)
o Over simplified testing due to equipment
limitations
4
Analysis Choices
n Conventional method with small-signal linearization:
o Limited to “linearizable averaged” product models
o Discard parasitic and non-linear effects
o Quick to get results
n “Virtual network analyzer” method without small-signal
linearization (FFT, Fourier Series)
o Also applied to highly non-linear switching models
o Retain “as is” models as designer’s schematics
o Significant analysis time but worthwhile
5
Applicability of Analysis Choices
Analysis Choices
Conventional
Linearized AC
Analysis
Fast Fourier
Transformation
(Fixed time-step)
Fundamental
Extraction from
Fourier Series
6
Linearizable
Averaged
Model
Non-linear
Switching
Model
a
r
a
a
a
a
Analysis, Modeling, and Simulators
Analysis
Approaches
Modeling
Schemes
Simulator
Platforms
7
Linearized AC
FFT
Fourier Series
Linearizable
Averaged Models
Non-Linear
Switching Models
Circuit-Oriented
Control -Oriented
FFT ANALYSIS on
Control-Oriented Simulator
n FFT
frequency response analysis
without small-signal linearization
g No need to use approximated models
g Extract frequency response directly from
simulated time-domain signals
n Control-oriented
simulators offer more flexibility
gUse of programmable SCRIPT files, allowing
n repetitive simulations & data acquisition,
n repetitive & programmable FFT processing .
8
Converter System with Signal-Injection for
Loop-Gain Response Analysis
Vin
+
Ve
Vin /Ve = Frequency Response
of Array-Voltage
Regulation Loop-Gain
9
Control-Oriented Large-Signal Modeling
n Modeling
in control-oriented simulators requires
overhead analytical effort,
g to convert circuits into control blocks.
n
The control blocks derived from circuits can be
g interconnected transfer functions and/or
g linked sets of state and output equations
¤ which are formatted in vectors and matrices.
n Achieve
control-oriented power system model by
g combining all derived control-oriented models and
g connecting them together.
n Simulate
& verify the system model in time-domain
before frequency response analysis.
10
SIMULINK Model of the Whole Power System
Vin
11
V5
Find Vin/V5 = ?
Non-Linear Switching Model of
the Power Converter & Control System
Vin
V5
Find Vin/V5 = ?
12
FFT Analysis Flow Charts
Update
Small Signal
Frequency
Simulate until
Steady State
Done
?
Time-Domain
Data
Acquisition
Data
Pre-Filtering
13
Plot
Results
Record
Results
FFT on
Filtered
Data
Get |Mag|
& ∠θ
Well Correlated Results from Both Large-Signal
Averaged and Non-Linear Switching Models
FFT works well for both averaged and non-linear switching
models developed in a control-oriented simulator
14
Circuit-Oriented Non-Linear Switching Model
for Loop Gain Extraction (SIMPLORER)
ICA:
EQU
FREQ:=INT(10^FREQ_LOGAR)
NPERIO:=NDELAY+1.1
TEND:=NPERIO/FREQ
FUND_MAG:=FFT1.AMPL[1]
FUND_PHS:=FFT1.PHIDEG[1]
FUND_MAG2:=FFT2.AMPL[1]
AMPL:=0.1
FREQ
GAIN_MAG
GAIN_PHS
4.999k
-0.704895
70.3515
FUND_PHS2:=FFT2.PHIDEG[1]
PHASE:=0
GAIN_MAG:=20*log((FUND_MAG)/((FUND_MAG2)+1p)+1p,10)
GAIN_PHS:=ASIN(SIN((FUND_PHS-FUND_PHS2)*pi/180))*180/PI
NDELAY:=INT(0.004*FREQ)+1
TDELAY:=(1/FREQ)*NDELAY-0.5/SCAN_FREQ*0
FREQ_LOGAR:=4.69897*0+3*0+3.69897
INPUTDELAY:=(FRAC(TDELAY*FREQ)*(1/FREQ))*0
SCAN_FREQ:=8*FREQ*0+200k
FUND_PHS
FUND_PHS2
0.121886k
51.5343
DELAY:=(TEND-1m)*0
AM1
VIP
VA
L1
NL_ARRAY_SRC
R3
C1
VM1
C := 0.75u
+
R := 2.5
D
L := 15u
S
L := 13u
R4
L2_CURRENT
C := 10u
R := 1
C2
V
NL
VLOAD
L2
MOS1
A
RL
R5
D1
R := 2.5
R := 2
I1
gndA
XY
C3
C4
C := 10u
C := 30u
C0
C5
gndB
40
RL.V [V]
30
25
C := 100u
20
C := 44u
15
10
ARRAY_SRC_VI_VX_IY
0
60
0
Power Stage Switch Model
LINE-FILTER MODEL
VM1.V [V]
AM1.I [A]
50
0.5m
1m
1.5m
2m
2.5m
3m
3.5m
4.4m t [s]
35
30
L2.I [A]
25
20
40
15
30
10
20
5
0
0
0.5m
1m
1.5m
2m
2.5m
3m
3.5m
4m4.4m t [s]
10
0
2.8
2.5
-10
0
0.5m
1m
1.5m
2m
2.5m
3m
3.5m
4m 4.4m t [s]
4
GAIN
R1
GAIN1
KP := 0.1
R := 20k
R2
EMF := 4
-
R := 20k
Rsignal.V [V]
3
2.5
2
1.5
1
0.5
C := 4700p
VSPT
Vsetpoint
Rsignal.V [V]
SUM2.VAL
R7.V [V]
-0.5
0 0.5m
+
1.5m
2.5m
1.5
0.6
1
0.4
CLOCK.VAL
0.2
0.5
3.5m4.4m t [s]
Verr
SAWTOOTH1.VAL
CLOCK.VAL
0.8
-27m
53m
2.9m 2.9m
Vin
1
2
3m 3m 3m 3m 3m 3m
3m 3m
3m t [s]
3m
3m
3m t [s]
0 1m 2m 3m 4.4m t [s]
C := 560p
CLOCK
SUM1 VFB
R := 1Meg
Vsine
AMPL := AMPL
Ve
Find Vin/Ve = ?
5
FFT1
FFT2
FFT
FFT
3
2
NAND2
OmniCaster3
BS=>Q
nand21
SAWTOOTH1.VAL
R7
0.4
0.2
E2
-0.1
0 1m 2m 3m 4.4m t [s]
SAWTOOTH1
4
Ve
Vin
Q=>BS
-
0.7
SUM1.VAL
GAIN1.VAL
0 1m 2m 3m 4.4m t [s]
OmniCaster2
+
Array-Voltage Error Amplifier
5.5
OmniCaster2.val
-0.2
Comparator
EMF := Rsignal.V
SINE1
1.2
E4
Rsignal
L2_CURRENT
1meg
NAND2
SUM2
nand22
Ri
1.2
KP := -0.099
1.2
OmniCaster1.val
GAIN
4
3.5
SUM2.VAL
-0.2
0 1m 2m 3m 4.4m t [s]
Peak Current Control PWM
2.5
2
1
1.5
0
0.5m
1m
1.5m
2m
2.5m
3m
3.5m
4m 4.4m t [s]
1k
OmniCaster1
3
0.5
R8
E3
Q=>BS
1
0
0
1m
2m
3m
4.4m t [s]
R8.V [V]
0.6
0.2
-0.2
0 1m 2m 3m 4.4m t [s]
Excellent Correlation Between
SIMPLORER Simulation and PSPICE Test Data
Test-Mag(dB)
Mag vs. Frequency
SIMPLORER-Mag(dB)
50.00
40.00
Mag(dB)
30.00
20.00
10.00
0.00
1.00E+02
-10.00
1.00E+03
1.00E+04
1.00E+05
-20.00
-30.00
Frequency(Hz)
Phase vs. Frequency
Test-Phase(degree)
SIMPLORER-Phase(degree)
Phase(degree)
100.00
50.00
0.00
1.00E+02
1.00E+03
1.00E+04
-50.00
-100.00
Frequency)Hz)
1.00E+05
FOURIER ANALYSIS APPROACH USING
CIRCUIT-ORIENTED SIMULATION
17
FOURIER EXTRACTION ALGORITHM
Re(Vout)
A sinωt
Im(Vout)
Re(VFB)
Im(VFB)
18
Loop-Gain Response from Two Different
Circuit-Oriented Analysis Approaches
19
Benefits of Linearization-Free Analysis
n Extended accuracy including:
oNon-linear effects
oParasitic components
oretaining “as is” circuit behavior like “virtual prototype”
n Modeling simplicity:
oNo need for overhead analytical effort in deriving linearizable
averaged model
oDirect construction of “as is” models from schematic circuit
diagrams
n Applicable to larger classes of product design
oParticularly, DC-DC converter systems of which linearizable
models are not available.
20
Conclusion
n FFT or Fourier-series analysis is applicable to converter
system models being developed in both controloriented and circuit-oriented simulators, offerring
o Repetitive simulation runs & programmable data processing
o Flexible choices of large-signal modeling & simulation platforms:
• Pulse-by-pulse non-linear switching model
• Large-signal averaged model
o Retention of the system non-linearities and parasitic effects
n ‘Virtual network analyzer’ is necessary for switch-mode
DC-DC converter systems of which the linearizable
averaged models are not available.
n The powerful analysis tool offers:
o Modeling simplicity,
o Direct application to the time-domain response, and
o Extended accuracy of the extracted frequency response
21
Acknowledgement to Contributors
n John Kooker: University of California at San Diego
oProduction of the test converter model in
SIMPLORER
n Jason Ly: The Aerospace Corporation
oApplication of FFT and data plotting using
MATLAB Script files
oCo-simulation between MATLAB & SIMULINK
22