IEEE APE

Generalized Averaging
Method for Power
Conversion Circuits
IEEE PAPER PRESENTATION
In the subject of
APE
(Advanced Power Electronics)
Prepared by
UTSAV YAGNIK (150430707017)
M.E. Electrical
SSEC, BHAVNAGAR

About the Paper
• IEEE Transactions on Power Electronics.
• Dated 2nd April 1991
• Innovator(s)
1. Seth R. Sanders
2. J. Mark Noworolski
3. Xiaojun Z. liu
4. George C. Verghese
2

INDEX
Sr.
No.
Topic Slide
No.
1 About the paper 2
2 Brief Abstract 4
3 Introduction 5
4 Previously Developed Methods 8
5 The Generalized Averaging Technique 10
6 Key Properties of Fourier Coefficients 13
7 The Utility of the Method 20
8 Example 23
9 Conclusion 32
3

Brief Abstract
• The method of state space averaging has its
limitations with switched circuits that do not
satisfy a small ripple condition.
• Here, a more general averaging procedure has
been put forward based on state space averaging
which can be applied to broader class of circuits
and systems including resonant type converters.
4

Introduction
• State space averaging has been effective method
for analysis and control design in PWM.
• But it can be applied to a limited class of converters
due to
1. Small ripple condition
2. A small parameter which is related to the
switching period and the system time constants.
5

Introduction
• Here, the small parameter is typically small for
fast switching but not for resonant type
converters.
• The resonant converters have state variables
that exhibit predominantly oscillatory behaviour.
• This paper represents a method that can
accommodate arbitrary types of waveforms.
6

Introduction
• The method is based on a time-dependent
Fourier series representation for a sliding
window of a given waveform.
• For instance, to recover the traditional state
averaged model, one would retain only the dc
coefficient in this averaging scheme.
7

Previously developed methods
Sampled-data modelling:
• Creates a small signal model for the underlying
resonant converter with the perturbation in
switching frequency as the input.
Difficulty with this approach:
• Requirement of obtaining a nominal periodic
solution as a first step in the analysis.
8

Previously developed methods
Phase plane techniques:
• A basic approach to obtaining a steady state
solution for a resonant converter.
Limitation:
• Its restricted to second order systems.
• So, one can not incorporate additional state
variables that are associated with load or source
dynamics.
9

The Generalized Averaging
technique
• It is based on the fact that the waveform 𝑥 ∙ can
be approximated in the interval (𝑡 − 𝑇, 𝑡) to
arbitrary accuracy with a Fourier series
representation of the form
• 𝑥 𝑡 − 𝑇 + 𝑠 = 𝑘 𝑥 𝑘(𝑡)𝑒 𝑗𝑘𝜔 𝑠(𝑡−𝑇+𝑠)
• k = all integers, 𝜔𝑠 = 2𝜋 𝑇 , 𝑠 𝜖 (0, 𝑇] , and 𝑥 𝑘(𝑡)
are complex Fourier coefficients which are
functions of time since considered interval slides
as a function of time.
10

technique
• The analysis computes the time evaluation of these
Fourier series coefficients as the window of length T
slides over the actual waveform.
• The 𝑘 𝑡ℎ coefficient, which is index-k coefficient, is
determined by
• 𝑥 𝑘 𝑡 =
1
𝑇 0
𝑇
𝑥(𝑡 − 𝑇 + 𝑠)𝑒−𝑗𝜔(𝑡−𝑇+𝑠) 𝑑𝑥 … (2)
11

technique
• Here the approach is to determine an
appropriate state space model in which the
coefficients in earlier equation are the state
variables.
12

Key properties of Fourier
coefficients
• 1. Differentiation with respect to time:
•
𝑑
𝑑𝑡
𝑥 𝑘 𝑡 =
𝑑
𝑑𝑡
𝑥
𝑘
𝑡 − 𝑗𝑘𝜔𝑠 𝑥 𝑘 𝑡
• When𝜔𝑠(𝑡) is slowly varying the above equation
will be a good approximation.
• This approximation is useful in analysis of system
where drive frequency is not constant.
13

coefficients
• 2. Transforms of Functions of Variables:
• 𝑓(𝑥1, 𝑥2, … , 𝑥 𝑛) 𝑘 is the general scalar function
of arguments which will be computed.
• A procedure for exactly computing the above
function is available where function is
polynomial which is based on following
relationship…
(contd.)
14

coefficients
• 𝑥𝑦 𝑘 = 𝑖 𝑥 𝑘−𝑖 𝑦 𝑖
• Here sum is taken over all integers i.
• The computation is done by considering each
homogeneous term separately. The constant and
linear term are trivial to transform. The
transforms of quadratic terms can be computed
using above equation.
15

coefficients
• Higher order homogeneous terms are dealt with
factoring such term into product of two lower
order terms. Then the procedure can be applied
to each term separately.
• This process is guaranteed to terminate since
factors with only linear term will eventually arise.
16

coefficients
• 3.Application to State-Space Model of Power
electronics circuits:
• Here the method is applied to a state-space
model that has some periodic time dependence.
• Model equation:
•
𝑑
𝑑𝑡
𝑥 𝑡 = 𝑓{𝑥 𝑡 , 𝑢(𝑡)}
17

coefficients
• Where u(t) = some periodic function with time
period T.
• To apply the generalized averaging scheme to
above equation, one simply needs to compute
the relevant Fourier coefficients of both sides of
that equation. i.e.
•
𝑑
𝑑𝑡
𝑥
𝑘
= 𝑓(𝑥, 𝑢) 𝑘
18

coefficients
• The first step is to compute derivative of the kth
coefficient as below…
•
𝑑
𝑑𝑥
𝑥 𝑘 = −𝑗𝑘𝜔𝑠 𝑥 𝑘 + 𝑓(𝑥, 𝑢) 𝑘
• As previously stated, only dc coefficients(index
zero) will be retained for a fast switching PWM
circuit to capture the low frequency behaviour.
• The result would be exact average model.
19

The utility of the method
• It is straight forward to obtain a steady state
solution for a model by setting its variables
derivative to zero.
• This approach goes one step further in extending
the analysis to transient behaviour as well.
• After obtaining a steady state solution, the model
may be linearized about the steady state to obtain
small signal transfer functions from inputs such as
switching frequency or source voltage to variables
such as v or i. 20

• The essence in modelling is to retain only the
relatively large Fourier coefficients to capture
the interesting behaviour of the system.
• So, only the index-zero(dc) coefficients for a fast
switching PWM circuit to capture the low
frequency behaviour is retained.
• The result would be a precise state-space model.
21

• For a resonant dc-dc converter which has some
of its states exhibiting predominantly dc(or
slowly varying) waveforms, the index-one(and
minus one) coefficients for those states
exhibiting sinusoidal-type behaviour and the
index-zero coefficients for those states exhibiting
slowly varying behaviour.
22

Example
• 1. Series resonant converter with voltage
source load
23

Example
• The state-space model for this circuit is of the
form below:
• The tank resonant frequency is 36KHz.
24

Example
• And the waveforms are as below:
•
25

Example
• These waveforms were generated by stepping
the drive frequency between 38KHz and 40KHz.
• Following each step change in driving
frequency, the waveforms appear to be
amplitude modulated sinusoids.
• The waveforms settle down to an approximately
sinusoid steady state.
26

Example
• So, the waveforms can be approximated with the
fundamental frequency terms in the Fourier
series coefficients as below…
27

Example
• The term 𝑠𝑔𝑛(𝑖) 𝑙 can be evaluated using the
describing function approach by assuming i(t) is
approximated as a sinusoid over each interval of
length T.
28

Example
• So, the model approximated with time invariant
model as below:
29

Example
• Waveforms obtained by this approach:
30

Example
• From the comparison of waveforms it is evident
that both waveforms are quite the same and the
given analysis technique is a step ahead than
steady state computation where slow transient
behaviour is also included.
31

Conclusion
• A new approach to averaging power electronics
circuits is introduced.
• It has been effectively tested with the resonant
type converters analysis.
• This approach refines the state space averaging
technique of analysis providing framework for
design and study of small ripple conditions.
32

IEEE APE

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