# Frequency-Dependent, Time-Invariant DC-DC Converter Modeling without Averaging - PowerPoint PPT Presentation

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## Frequency-Dependent, Time-Invariant DC-DC Converter Modeling without Averaging

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Title: Frequency-Dependent, Time-Invariant DC-DC Converter Modeling without Averaging

1
Frequency-Dependent, Time-Invariant DC-DC
Converter Modeling without Averaging
• Patrick Chapman
• Asst. Prof. UIUC
• April 10, 2006

Grainger Center for Electric Machines and
Electromechanics
2
Summary
• Overview time-invariant (TI) converter modeling
• Propose alternative method of TI modeling,
avoiding formal averaging
• Explore simulation results
• Probe small-signal analysis improvements

3
Converter Modeling
• DC-DC converter difficulties
• Nonlinear
• Time-varying
• Switched
• Longstanding modeling desires
• Easy control design
• Rapid simulation
• Accuracy
• No hand-waving

4
Conventional Approach Averaging
• Method dates from 70s
• Needed way to apply linear control design
• Needed time-invariant model for this
• Stability was main priority (fast response, less
so)
• Can be done on schematic (circuit-based)
• Can be done on equations (state-space)
• Methods produce equivalent models

5
Conventional State-Space Avg.
• Often cast in the time-window approach
• T is the switching period
• x is any time-domain variable
• Sometimes cast as weighted average of state
matrices
• d is duty cycle in period T
• n is integer
• Methods produce same model

6
Synopsis Conventional Avg.
• Removes time-varying state matrices
• Removes switching ripple
• Models often linearized
• Enables small-signal, linear analysis
• Very widely used
• Calls into question bandwidth limitation
• What disturbance frequency is valid?
• Aliasing-like effects neglected
• Can improperly track the average!

7
Tracking Issues in Avg. Models
• Occurs in certain converters
• Boost and buck-boost in particular
• Occurs even under ideal circuits with modest
ripple
• Two main causes
• Neglecting ripple, ESR in steady-state
derivations
• Sampling effects in feedback control

8
Regarding Moving Time-Window
• Averaging integral
• Relies on continuous time history
• Cant be implemented directly in hardware
• Can complicate rigorous analysis
• Imposing is questionable
• q is switching signal
• d is duty cycle command

9
Comparing q, d, and
Ratio 201 Natural 2-sided PWM should preserve
phase Averaging integral known to produce a
stair-step, delayed signal
10
Improvements to Conv. Avg.
• Higher order abstractions, ripple estimates
• Krein, Bass, et al (KBM formality)
• Sanders, Verghese, Caliskan, Stankovic
(generalized or multi-frequency)
• Modification of output equation to catch ripple
• Lehman (2004)
• Stability and convergence analysis (Tadmor)
• These approaches, among others, did much to
strengthen and extend averaging
• Discontinuous mode, other converters, correction
factors, etc.

11
Multi-Frequency Averaging (MFA)
• Accounts for ripple and switching frequency
• Uses higher-order averages
• Has implicit assumption of slowly-varying states
(at least during time window)

Also Generalized averaging
12
Perhaps Redefine MFA
Make these definitions
Then
13
Multiple Reference Frames (MRF)
• Quite similar to MFA
• In context of motor drives (Sudhoff)
• Multi-phase systems
• E.g., PM motor with nonsinusoidal back EMF
• Involves a time-window average w.r.t. rotor
position, not time
• Makes a time-invariant model ? linear analysis of
motor drives

14
PM Motor MRF Analysis
MRF theory, then, has similar properties as MFA
15
Floquet Theory
• A linear transformation of states
• Applies to models
• Linear
• Periodic coefficients
• Produces exact time-invariant model
• Applied by Visser to open-loop buck, boost
• Difficult, maybe infeasible, for closed-loop
• Closed-loop not periodic
• Nonlinear theory recent, but doesnt address
aperiodicity

16
Questions
• Can averaging be avoided?
• If so,
• Are resulting models better?
• Is rigorous analysis (error bounds) easier?
• Can they be extended?
• Aliasing-like effects
• Digital control

17
Alternative Approach
• Decompose signal along the lines of MFA
• Quasi-Fourier-Series (QFS) components
• 2N1 variables, a ? cosine, b ? sine
• New variables not defined yet just notational

18
Signal Derivatives
(2nd order, e.g.)
19
Switching Function Model
• d is the duty cycle
• f is the initial phase shift (usually arbitrary)

Can let d be a function of time (so long as
between 0, 1)
20
Duty Cycle Ripple
• Usually neglected
• Usually small
• Ripple in voltage or current may be fed back
• Important in proportional control
• Some filtering may be applied
• Current-mode control known to exhibit
ripple-dependent behavior
• Usually suffices to model one harmonic
• Analysis here is general, but specific N 1
example given

21
Obtaining d and f
• Duty cycle command (d) and d not the same

Sawtooth
Triangle
22
Obtaining d and f
• Times th and tl would occur if QFS variables
remained the same until intersection
• Variables change between switching edges
• At first seems suspect
• Cancels out in the infinite sum
• Necessary step to put in terms of new states

23
Switching Function Summary
• QFS coefficients
• Solution to transcendental equation
• Inconvenient, but actually quite easy
• Can use noniterative approximations

24
Product Terms
• Commonly we have qx product of switching
function and continuous variable
• Generates higher order harmonics
• Approximation is the neglect of these
• Though, can include in extra state eqns if
desired

25
Product Terms, contd
• Brute-force calculation (preferred here)
• Discrete convolution
• Use complex exponential QFS for concise statement

2nd order example
26
Converter Model
• Boost converter example
• Tools in place ? apply to state equations

27
Synthesizing Model
• Thus far, weve not completely defined variables
• After making substitutions
• Can equate trigonometric coefficients to satisfy
• Only one of infinite choices, but yields TI model
• My states, I make them up

28
Equating Coefficients
• Example
• Arbitrary choice a(t) 1, b(t) 0
• Other choices dont satisfy constant-in-steady-st
ate desire, e.g.

29
Extracting the Equations
• Model is synthesized as
• vb is constant input, matrices are constants
• Will yield constant state variables in steady
state

30
Closed-loop Simulation (sawtooth)
• Current-mode control (parameters from lit.)

31
Closed-loop Simulation (triangle)
32
Toward Error Bounds
• Was tacitly assumed N was sufficient order
• Error will occur due to truncation of higher
order terms
• Usually, error will be small given
• Low-pass filters
• Low levels of excitation at higher frequencies
• Can bound the steady-state error, if not transient

33
Comparison with MFA
• Generates same state-space equations
• Slightly improved consideration of switching
function here
• Avoided averaging integral
• Avoided slow-varying assumption
• Both neglect aliasing effects

34
More Comparison
• Both generally capture the average value more
precisely
• Both rely on truncations of actual signals
• Allowed for sawtooth vs. triangular carrier
• Proposed method was not a unique formulation
(infinite of zero terms)
• Linearization (below) is analytical with proposed
technique

35
Small-Signal Analysis
• Linearize the Qx terms (op operating point)
• Calculate the partials
• Messy derivation, but not bad end result

(Sawtooh)
(Triangle)
36
Linearized Boost
• Includes 1 harmonic in duty cycle command,
otherwise retains generality

37
Closed-loop Frequency Response
• 0th order component only (sawtooth)

Closed-loop gain differs by about 6 Higher
frequency analysis questionable due to neglect of
aliasing effects
38
Disturbing Issues
• Switching and sampling not exactly the same
• Cant apply directly sampling theorems
• But
• Higher frequency disturbances are effectively
aliased
• Signals aliased to low frequencies can pass
through easily
• Absolute phase shift of input disturbance matters
• (in typical small-signal analysis, only relative
phase matters)
• How can we account for this in a TI model?

39
Sinusoidal Disturbances
• Can use ideas from sinusoidal PWM
• Once linearized and perturbed, we introduce
unmodeled harmonics
• Choose important components from spectrum
• Let
• Use Bessel-function approach to get specific
components, e.g.

40
Example Disturbances
• 1 kHz switching, 400 Hz disturbance, dm 0.1

Actual freq 200 Hz
41
Example Disturbances
• 1 kHz switching, 500 Hz disturbance, dm 0.1

Actual freq 500 Hz
42
Example Disturbances
• 1 kHz switching, 900 Hz disturbance, dm 0.1

Actual freq 90 Hz
43
Example Disturbances
• 1 kHz switching, 1 kHz disturbance, dm 0.1

Actual freq 1 kHz
44
Other Work in Progress
• In most applications, the 2nd order effects dont
matter that much
• Sometimes, we design so they wont matter but
how can we be sure? How can we push the limit?
• Aliasing effects
• Digital sampling techniques
• Variable switching frequency
• Can we use MFA modeling techniques to investigate

45
Dominant Aliased Harmonic
• Looking between 0 and wsw only
• Harmonics of wsw reduced by factor of
• Model with an additional harmonic for fixed
frequency disturbance analysis

Relative sizes depend on fd
46
Digital Sampling Effects
• Voltage or current may be sampled
• Analog to digital converter
• Working with continuous time model
• Sampling can be modeled as 0th or 1st order hold
• Quantization error
• These create additional harmonics in small-signal
analysis
• Effects of (intentional?) over- or undersampling

47
Variable Switching Frequency
• Useful during transients
• Speed up switching to increase dynamic
performance
• Slow down during steady-state for lower switch
losses
• Let wsw vary with time must actually work in
terms of qsw the switching angle

48
Conclusion
• Multi-frequency techniques can capture
sometimes-important 2nd order effects
• Showed one technique that avoids formal averaging
• Seems to have easy instantiation (suggests
tool)
• Perhaps easier analysis
• Do other problems crop up?
• These methods have potential to tackle subtleties
near and above fsw/2