Loading...

PPT – Frequency-Dependent, Time-Invariant DC-DC Converter Modeling without Averaging PowerPoint presentation | free to download - id: 439164-YzFhZ

The Adobe Flash plugin is needed to view this content

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

Summary

- Overview time-invariant (TI) converter modeling
- Propose alternative method of TI modeling,

avoiding formal averaging - Explore simulation results
- Probe small-signal analysis improvements

Converter Modeling

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

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

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

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!

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

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

Comparing q, d, and

Ratio 201 Natural 2-sided PWM should preserve

phase Averaging integral known to produce a

stair-step, delayed signal

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.

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

Perhaps Redefine MFA

Make these definitions

Then

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

PM Motor MRF Analysis

MRF theory, then, has similar properties as MFA

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

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

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

Signal Derivatives

(2nd order, e.g.)

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)

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

Obtaining d and f

- Duty cycle command (d) and d not the same

Sawtooth

Triangle

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

Switching Function Summary

- QFS coefficients
- Solution to transcendental equation
- Inconvenient, but actually quite easy
- Can use noniterative approximations

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

Product Terms, contd

- Brute-force calculation (preferred here)
- Discrete convolution
- Use complex exponential QFS for concise statement

2nd order example

Converter Model

- Boost converter example
- Tools in place ? apply to state equations

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

Equating Coefficients

- Example
- Arbitrary choice a(t) 1, b(t) 0
- Other choices dont satisfy constant-in-steady-st

ate desire, e.g.

Extracting the Equations

- Model is synthesized as
- vb is constant input, matrices are constants
- Will yield constant state variables in steady

state

Closed-loop Simulation (sawtooth)

- Current-mode control (parameters from lit.)

Closed-loop Simulation (triangle)

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

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

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

Small-Signal Analysis

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

(Sawtooh)

(Triangle)

Linearized Boost

- Includes 1 harmonic in duty cycle command,

otherwise retains generality

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

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?

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.

Example Disturbances

- 1 kHz switching, 400 Hz disturbance, dm 0.1

Actual freq 200 Hz

Example Disturbances

- 1 kHz switching, 500 Hz disturbance, dm 0.1

Actual freq 500 Hz

Example Disturbances

- 1 kHz switching, 900 Hz disturbance, dm 0.1

Actual freq 90 Hz

Example Disturbances

- 1 kHz switching, 1 kHz disturbance, dm 0.1

Actual freq 1 kHz

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?

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

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

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

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