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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... Time-Invariant DC-DC Converter Modeling without Averaging Summary Converter Modeling Conventional Approach: Averaging Conventional State-Space Avg. Synopsis: ... – PowerPoint PPT presentation

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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
  • Many other improvements and adaptations made
  • 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
    advanced controllers?

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
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