Part II"
Converter Dynamics and Control!
7. !AC equivalent circuit modeling!
8. !Converter transfer functions!
9. Controller design!
10. Input filter design!
11. !AC and DC equivalent circuit modeling of the
discontinuous conduction mode!
12. !Current programmed control!
Fundamentals of Power Electronics!
1!
Chapter 7: AC equivalent circuit modeling!
Chapter 7. AC Equivalent Circuit Modeling!
7.1 !Introduction!
7.2 !The basic AC modeling approach!
7.3 !State-space averaging!
7.4 !Circuit averaging and averaged switch modeling!
7.5 !The canonical circuit model!
7.6 !Modeling the pulse-width modulator!
7.7 !Summary of key points!
Fundamentals of Power Electronics!
2!
Chapter 7: AC equivalent circuit modeling!
7.1. !Introduction!
Objective: maintain v(t)
equal to an accurate,
constant value V.!
A simple dc-dc regulator system, employing a
buck converter!
Power
input
Switching converter
Load
There are disturbances:!
• in vg(t)!
• in R!
+
vg(t) +
–
v(t)
R
Feedback
connection
–
• in element
values!
• in Vg!
Transistor
gate driver
c(t)
Compensator
v
Pulse-width c
Gc(s)
modulator
c(t)
–+
There are uncertainties:!
v
Voltage
reference vref
vc(t)
• in R
dTs Ts
t
t
Controller
Fundamentals of Power Electronics!
3!
Chapter 7: AC equivalent circuit modeling!
Applications of control in power electronics!
DC-DC converters!
Regulate dc output voltage.!
Control the duty cycle d(t) such that v(t) accurately follows a reference
signal vref.!
DC-AC inverters!
Regulate an ac output voltage. !
Control the duty cycle d(t) such that v(t) accurately follows a reference
signal vref (t).!
AC-DC rectifiers!
Regulate the dc output voltage.!
Regulate the ac input current waveform.!
Control the duty cycle d(t) such that ig (t) accurately follows a reference
signal iref (t), and v(t) accurately follows a reference signal vref.!
Fundamentals of Power Electronics!
4!
Chapter 7: AC equivalent circuit modeling!
Converter Modeling!
Applications!
Aerospace worst-case analysis!
Commercial high-volume production: design for reliability and yield!
High quality design!
Ensure that the converter works well under worst-case conditions!
– Steady state (losses, efficiency, voltage regulation)!
– Small-signal ac (controller stability and transient response)!
Engineering methodology!
Simulate model during preliminary design (design verification)!
Construct laboratory prototype converter system and make it work under
nominal conditions!
Develop a converter model. Refine model until it predicts behavior of
nominal laboratory prototype!
Use model to predict behavior under worst-case conditions!
Improve design until worst-case behavior meets specifications (or until
reliability and production yield are acceptable)!
!
Fundamentals
of Power Electronics!
5!
Chapter 7: AC equivalent circuit modeling!
Objective of Part II!
Develop tools for modeling, analysis, and design of converter control
systems!
Need dynamic models of converters:!
How do ac variations in vg(t), R, or d(t) affect the output voltage v(t)?!
What are the small-signal transfer functions of the converter?!
• Extend the steady-state converter models of Chapters 2 and 3, to
include CCM converter dynamics (Chapter 7)!
• Construct converter small-signal transfer functions (Chapter 8)!
• Design converter control systems (Chapter 9)!
• Design input EMI filters that do not disrupt control system operation
(Chapter 10)!
• Model converters operating in DCM (Chapter 11)!
• Current-programmed control of converters (Chapter 12)!
Fundamentals of Power Electronics!
6!
Chapter 7: AC equivalent circuit modeling!
Modeling!
• Representation of physical behavior by mathematical means!
• Model dominant behavior of system, ignore other insignificant
phenomena!
• Simplified model yields physical insight, allowing engineer to
design system to operate in specified manner!
• Approximations neglect small but complicating phenomena!
• After basic insight has been gained, model can be refined (if
it is judged worthwhile to expend the engineering effort to do
so), to account for some of the previously neglected
phenomena!
Fundamentals of Power Electronics!
7!
Chapter 7: AC equivalent circuit modeling!
nd what is most important. Once this basic insight is gained, it may be
efine the model, by accounting for some of the previously ignored phelife that real, physical systems are complex, and their detailed analysis
ntractable and useless mathematical mess. Approximate models are an
Finding
the response to ac variations"
ng understanding and
physical insight.
Neglecting
the switching
ripple! operatapter 2, the switching ripple is small
in a well-designed
converter
uction mode (CCM). Hence, we should ignore the switching ripple, and
ing ac variations in the converter waveforms. For
example, suppose that
Gate drive!
1
(a)!
the control
roduced intoSuppose
the control
signal vc (t),
such that
signal varies sinusoidally:!
vc (t) = Vc + Vcm cos !m t
0.5
vc (t)
(7.1)
0
t
20 Ts
0
5
10
15
causes the duty
onstants, |VThis
cm | ⌧ Vc , and the modulation frequency !m is much smaller
cycle to be modulated (b)!
6
iL (t)
ching frequency
!
=
2⇡
f
.
This
control
signal
is
fed
into
pulse-width
S
s
hiL (t)ia
sinusoidally:!
T
4
generates a gate drive signal having switching
frequency f s = 1/T s
d(t) = D + period
Dm cost
during each switching
depends
on the2 control signal vc (t) applied
mt
t
e resulting transistor
drive
signal is illustrated
in Fig. 7.2(a), and a
0
Assume Dgate
and D
are
m
0
5
10
15
20 Ts
tor current and
output⏐D
voltage
waveforms iL (t) and v(t) are illustrated in
constants,
m⏐ < D,
−13
(c)!
the modulation
m of v(t) isand
illustrated
in Fig. 7.3. This
spectrum
contains
components at
v(t)
−14
ωm is and
much
hv(t)iT
y as well asfrequency
its harmonics
sidebands; these
components
are small in
−15
the converter
hing ripple smaller
is small.than
In addition,
the spectrum
−16 contains a low-frequency
switching frequency ωs =
t
ulation frequency
!
.
The
magnitude
and
phase
of
this
component
de−17
m
2πfs.!
0
5
10
15
20 Ts
ntrol signal and duty cycle variation, but also on the frequency response
8!
of Power Electronics!
Chapter 7: AC equivalent circuit modeling!
neglect Fundamentals
the switching
ripple, then this low-frequency component
remains
s
s
Output voltage spectrum"
with sinusoidal modulation of duty cycle!
Spectrum
of v(t)
tm
ts
Contains frequency components at:!
• Modulation frequency and its
harmonics!
• Switching frequency and its
harmonics!
• Sidebands of switching frequency!
Fundamentals of Power Electronics!
9!
Switching
harmonics
{
{
Switching
frequency and
sidebands
{
Modulation
frequency and its
harmonics
t
With small switching ripple, highfrequency components (switching
harmonics and sidebands) are small.!
If ripple is neglected, then only lowfrequency components (modulation
frequency and harmonics) remain.!
Chapter 7: AC equivalent circuit modeling!
Objective of ac converter modeling!
• Predict how low-frequency variations in duty cycle induce
low-frequency variations in the converter voltages and
currents!
• Ignore the switching ripple!
• Ignore complicated switching harmonics and sidebands!
Approach:!
• Remove switching harmonics by averaging all waveforms
over one switching period!
Fundamentals of Power Electronics!
10!
Chapter 7: AC equivalent circuit modeling!
Averaging to remove switching ripple!
Average over one switching
period to remove switching
quivalent Circuit Modeling
ripple:!
ddhi
i LL(t)
(t)iTTss
L
L (t)iT s
L
== hv
vL(t)
Ts
dt
dt
C (t)iT s
d dhv
vC(t)
T s = hiC (t)iT
C
C
= i C(t) T s
dt
dt
s
enotes the average
of x(t) over an interval of length T s :
where!
1
hx(t)iT s =
Ts
Z t+T s /2
t T s /2
x(⌧)d⌧
Note that, in steady-state,!
vL(t) T = 0
s
iC(t) T = 0
s
(7.2)
by inductor volt-second
balance and capacitor charge
balance.!
(7.3)
loy the basic approximation of removing the high-frequency switching ripple
ver one switching period. Yet the average value is allowed to vary from one
d to the next, such that low-frequency variations are modeled. In e↵ect, the
11!
Fundamentals
of Power Electronics!
7: AC
circuit modeling!
e” of Eq.
(7.3) constitutes
low-pass filtering of the
waveform.Chapter
A few
of equivalent
the
nces on averaged modeling of switching converters are listed at the end of this
Nonlinear averaged equations!
The averaged voltages and currents are, in general, nonlinear
functions of the converter duty cycle, voltages, and currents. Hence,
the averaged equations!
L
C
d i L(t) T
dt
d vC(t) T
dt
s
s
= vL(t) T
s
= i C(t) T
s
constitute a system of nonlinear differential equations.!
Hence, must linearize by constructing a small-signal converter model.!
Fundamentals of Power Electronics!
12!
Chapter 7: AC equivalent circuit modeling!
Small-signal modeling of the diode!
Nonlinear
diode, driven
by current
source having
a DC and small
AC component!
Linearization of the diode i-v
characteristic about a quiescent
operating point!
+
i = I+i
v = V+v
–
i
Actual
nonlinear
characteristic
5A
4A
Quiescent
operating
point
3A
Small-signal
AC model!
i
rD
+
2A
v
1A
0
–
Linearized
function
0
i(t)
I
0.5 V
1V
v
v(t)
V
Fundamentals of Power Electronics!
13!
Chapter 7: AC equivalent circuit modeling!
Buck-boost converter:"
nonlinear static control-to-output characteristic!
0
0
0.5
1
D
V = Vg D/(1 – D)!
Quiescent
operating
point
–Vg
V
Linearized
function
D = 0.5
Actual
nonlinear
characteristic
Fundamentals of Power Electronics!
Example: linearization
at the quiescent
operating point!
14!
Chapter 7: AC equivalent circuit modeling!
Result of averaged small-signal ac modeling!
Small-signal ac equivalent circuit model!
Vg – V d (t)
+
–
1:D
L
Dv : 1
+
vg(t) +
–
I d (t)
I d (t)
C
v(t)
R
–
buck-boost example!
Fundamentals of Power Electronics!
15!
Chapter 7: AC equivalent circuit modeling!
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