Simple Analysis of a Flying Capacitor Converter
Voltage Balance Dynamics for DC Modulation
A. Ruderman
(1)
(1)
, B. Reznikov
( 2)
, and M. Margaliot
( 3)
( 3)
Elmo Motion Control Ltd.,
General Satellite Corporation,
Tel Aviv University
aruderman@elmomc.com, reznikovb@spb.gs.ru, michaelm@eng.tau.ac.il
Abstract–Flying capacitor multilevel PWM converter with a
natural voltage balance is an attractive multilevel converter
choice because it requires no voltage balance control effort. Flying
capacitor converter practically does not suffer from voltage
balance imposed performance limitations as opposed to multiple
point clamped converter. Voltage balance dynamics analytical
research methods reported to date deal mostly with an AC
modulation case and are essentially based on a frequency domain
analysis using double Fourier transform. Therefore, these
methods require high mathematical skills, are not truly analytical
and rather difficult to use in an everyday practice by electrical
engineer. In this paper, we consider a DC modulation case to
demonstrate that a straightforward time domain approach based
on switching intervals piece-wise analytical solutions makes it easy
to obtain time-averaged discrete and continuous models for
voltage balance dynamics simulation. A primitive single-phase
single-leg three-level converter analytical investigation yields a
surprisingly simple accurate expression for capacitor charge /
discharge related time constant revealing its dependence on
inductive load parameters, carrier frequency, and duty ratio.
I.
( 2)
INTRODUCTION
Multilevel converters are being progressively used for
medium and high voltage / power applications. Multilevel
converter topologies, modulation strategies, and performances
have been extensively studied over the past two decades [1, 2].
Flying Capacitor (FC) multilevel PWM converter is an
attractive choice due to the natural voltage balance property.
In Multiple Point Clamped (MPC) converter used for front
end applications, it is possible to achieve capacitors voltage
balance only for a limited operation envelope in terms of
modulation index – load displacement angle. In such a MPC
converter, maximal possible modulation index M=1 is
achieved for pure reactive (inductive) load only (zero power
factor). With load power factor approaching unity, maximal
modulation index is theoretically compromised to about
M=0.55 for a three-phase and M=0.63 for a single-phase
converter because of the performance limitations that originate
from capacitor voltage balance [3].
Suppose ideally smooth, ripple free load current, relatively
high switching frequency, and appropriate phase shifted
voltage modulation strategy. Under the above assumptions, a
flying capacitor is charged and discharged by the same amount
of load current on time intervals of equal durations. It is clear
that the capacitor voltage is thus oscillating around some
average value that is defined by the capacitor initial voltage
and no any voltage balance conclusion may be drawn from this
simplistic model. Therefore, it is recognized that FC converter
voltage balance process is actually driven by the load current
high order harmonics.
Reported FC converter analytical voltage balance research
methods mostly deal with an AC modulation [4-8]. However,
from a methodology perspective it would be correct to get
started with a more simple DC modulation case.
The reported FC converter voltage balance analysis methods
are essentially based on frequency domain transformations.
This selection of the analysis tools probably comes from the
recognition of the important role of load current high order
harmonics in the whole voltage balance process. However,
intuitively it seems somewhat artificial to use "intermediate"
frequency domain methods for derivation of linear time
invariant models to be analyzed in time domain.
As a result, the reported FC converter voltage balance
methods are not "truly analytical" and not easy to understand.
The usage of the frequency domain methods for building linear
time invariant voltage balance models makes it difficult to gain
a thorough insight into FC converter capacitor voltage balance
physical mechanisms and apply this approach in an everyday
engineering practice.
In this paper, we apply a straightforward time domain
approach based on “sewing” analytical transient solutions of
consecutive PWM period switching subintervals to derive DC
modulated FC converter voltage balance dynamics models both discrete and continuous. This approach seems most
adequate for linear switched (variable structure) systems that
are naturally formed by idealized FC converters along with
their linear active-inductive loads.
By means of this technique, we study a primitive single
capacitor single leg three-level FC converter. We obtain
surprisingly simple accurate expression for a capacitor charge /
discharge related time constant by applying small parameter
technique. The small parameter naturally arises due to the fact
that converter switching is performed at a frequency that is
much higher than equivalent RLC-circuit natural one. In other
words, this means that current and voltage ripples are relatively
low that is always true for a good practical converter. This
approach allows revealing the capacitor charge rate
dependence on inductive load parameters, carrier frequency,
and duty ratio.
The suggested analytical approach may be easily adopted for
a DC modulated FC converter with an arbitrary number of
voltage levels (cells). We have to admit that a generalization to
an AC modulation case is not trivial and requires an adequate
technique to perform time averaging on a fundamental period.
L
S2
VDC / 2
VDC / 2
iL
S1
L
C
v
S1
R
Fig. 3. FC converter topology on intervals 1 and 3 ( S1S 2 ) (the capacitor
is disconnected)
i
VDC / 2
Fig. 2,b shows converter switching states and output voltage
waveform assuming equal voltage sources, ideal switches, and
the capacitor voltage v = VDC / 2 .
As readily seen from Fig.2, a switching period is comprised
of four intervals. Assuming ideal switches, intervals 1 and 3
generate the same converter topology shown on Fig. 3 – note
that the capacitor is disconnected and keeps its initial voltage
unchanged. Both intervals have the same duration
S2
Fig. 1. Single-phase three-level FC converter with RL-load
However, an insight into voltage balance mechanism gained
for DC modulation may be useful for AC modulation as well
and we solidify this statement with corresponding examples.
II. SINGLE-PHASE SINGLE-LEG THREE-LEVEL FC CONVERTER
TOPOLOGY AND MODULATION STRATEGY
A single-phase single-leg three-level FC converter with
active-inductive load is given in Fig. 1.
Fig.1 converter voltage modulation strategy is demonstrated
in Fig. 2,a. Instantaneous voltage command VCOM is scanned
by two opposite phase triangular wave carrier signals to define
converter switching instants. Carrier wave s1 is responsible
for switching a complementary switch pair
∆t1 = ∆t 3 =
where
D
TPWM
2
(normalized DC voltage command).
FC converter topologies on the intervals 2 and 4 are shown
in Fig. 4,a, b. The difference between the topologies is the
opposite polarity of both voltage source and capacitor.
The duration of both intervals is
(1 − D)
TPWM ,
2
∆t1 + ∆t 2 + ∆t 3 + ∆t 4 = TPWM .
∆t 2 = ∆t 4 =
vC
S1 − S1 ; s 2 - for
VDC / 2
S1
S2
S1
S2
S1
VT
S2
t
VDC / 2
VL
0
S 1 S2
S1 S 2
S1 S 2
4
1
2
S1 S 2 S 1 S 2
3
t
L
(2)
R
iL
vC
VDC / 2
(1)
TPWM - PWM period; D = VCOM / VT - PWM duty ratio
S2 − S2 .
VCOM
R
L
R
iL
4
Fig. 2. Voltage modulation process (a) and output voltage waveform with
switching states (b)
Fig. 4. FC converter topologies on intervals 2 (a - S1S 2 ) and 4 (b -
S1S 2 )
III. FC CONVERTER DYNAMICS MODELING
ω02 = 1 /( LC ) ; R < 2 L / C .
On each switching interval, FC converter may be modeled as
a linear time-invariant system. The FC converter under
consideration is a second order switched linear system with
load inductor current and capacitor voltage being state
variables. Therefore, assuming constant DC voltages, on each
interval the converter behavior is described by the following
state equation
X (t ) = A j (t ) X (0) + B j (t )
 i (t ) 
VDC
; X (t ) =  ,
2
v(t )
(3)
 i(0) 
X (0) = 
 - the interval initial conditions; j = 1,2,3,4.
v(0)
The state space equations’ matrices in (3) are obtained by
analytically solving ordinary linear time-invariant differential
equations on different switching intervals.
On interval 1 (Fig.3),
exp( −t / TL ) 0
A1 (t ) = 
;
0
1

(1 − exp(−t / TL ) ) / R 
B1 (t ) = 
,
0


(4)
(5)
where TL = L / R - load time constant.
On interval 1, the capacitor is disconnected and keeps its
initial voltage. The converter dynamic behavior on interval 3 is
identical to that of interval 1 (Fig.3):
A3 (t ) = A1 (t );
(6)
B3 (t ) = B1 (t ).
On interval 4,
A4 (t ) = exp( −αt ) A4' (t );
α
1


−
sin(ωt ) 
cos(ωt ) − ω sin(ωt )
ωL
A (t ) = 
;
α
1

sin(ωt )
cos(ωt ) + sin(ωt ) 
ωC
ω


1


exp(−αt ) sin(ωt )


L
ω
B 4 (t ) = 
,
α
 − exp(−αt ) sin(ωt ) + cos(ωt )  + 1
ω
 

VDC
;
2
V
X (t 2 ) = A2 (∆t 2 ) X (t1 ) + B2 ( ∆t 2 ) DC ;
2
V
X (t 3 ) = A3 (∆t 3 ) X (t 2 ) + B3 (∆t 3 ) DC ;
2
V
X (t 4 ) = A4 (∆t 4 ) X (t 3 ) + B4 ( ∆t 4 ) DC ;
2
V
X (t 5 ) = A1 ( ∆t1 ) X (t 4 ) + B1 ( ∆t1 ) DC ;
2
..........................................................
X (t1 ) = A1 ( ∆t1 ) X (0) + B1 ( ∆t1 )
where
A2 (t ) = exp( −αt ) A (t );
t 2 = ∆t1 + ∆t 2 ;
1
α


cos(ωt ) − sin(ωt )
sin(ωt )


'
L
ω
ω
A2 (t ) = 
;
1
α
 −
sin(ωt )
cos(ωt ) + sin(ωt ) 
ω
ωC


(7)
(10)
Recalling (1), (2), we now have a discrete FC converter
model for calculating state variables at the switching instants:
Now, suppose oscillating step response of the equivalent
LCR-circuit Fig.4 ("small" resistance). Then on interval 2
'
2
(9)
'
4
(11)
t1 = ∆t1 ;
t 3 = ∆t1 + ∆t 2 + ∆t 3 ;
t 4 = ∆t1 + ∆t 2 + ∆t 3 + ∆t 4 = TPWM ;
(12)
t 5 = TPWM + ∆t1
..........................................................
1


−
exp(−αt ) sin(ωt )


ωL
B 2 (t ) = 
,
α
 − exp(−αt ) sin(ωt ) + cos(ωt )  + 1
ω
 

where
α = 0.5 R / L = 1 /( 2TL ) ;
ω = ω 02 − α 2 ;
(8)
Example 1. Consider single-phase single capacitor three-level
FC converter with the following converter and load
parameters:
VDC = 100V ;
R = 1Ohm ; L = 0.25mH ; C = 100uF ;
TPWM = 300us ( f PWM = 3.33kHz ).
(13)
Load current (A) and
capacitor voltage (V)
FC Converter Simulated Transient (D=0)
60
50
40
30
20
10
0
-10 0
-20
VCOM
VC
S2
0.005
0.01
0.015
0.02
0.025
S2
t
VL
t
0
VDC/ 2−VC
FC Converter Simulated Transient (D=0.2)
Load current (A) and
capacitor voltage (V)
VT
VDC / 2
VC −VDC/ 2
a
S 1 S2
60
50
40
30
20
10
0
-10 0
-20
IL
0.005
0.01
0.015
0.02
0.025
b
FC Converter Simulated Transient (D=0.4)
60
50
40
30
20
10
0
-10 0
-20
VC
IL
0.005
0.01
0.015
0.02
0.025
Time, s
c
FC Converter Simulated Transient (D=0.8)
60
50
40
30
20
10
0
-10 0
-20
VC
0.01
0.015
0.02
0.025
Time, s
d
Fig. 5. FC converter load current and capacitor voltage simulation for zero
initial conditions and different duty ratios: a – D=0; b – D=0.2; c – D=0.4;
d – D=0.8
Converter dynamics simulation results obtained using Excel
by programming formulas (11), (12) are presented in Fig. 5.
Though demonstrated are results for zero initial conditions,
average load current and capacitor voltage always converge to
VDC D
;
2R
V
v(∞) = DC
2
S1 S 2
S1 S 2 S 1 S 2
for any set of initial conditions.
Here are more observations from FC converter dynamics
modeling experience along with their physical interpretation.
Intuitively, we expect the FC converter time averaged model to
behave like a 2nd order linear time invariant system. However,
time averaged load current transient behavior is practically
characterized by a single load time constant
TL = L / R
(15)
without any dependence on operating conditions (duty ratio).
This is because the average load voltage (instantaneous load
voltage example for ideally balanced capacitor voltage VDC / 2
is shown in Fig. 2) actually does not depend on capacitor
voltage as illustrated by Fig. 6.
An average capacitor voltage curve contains two exponential
terms – a small fast exponent with the load time constant and a
large slow exponent. The large dominating capacitor charge
time constant
IL
0.005
S1 S 2
Fig. 6. Average load voltage does not depend on capacitor voltage (equal
shaded areas shown)
VC
Time, s
Load current (A) and
capacitor voltage (V)
S1
IL
Time, s
Load current (A) and
capacitor voltage (V)
S1
S2
S1
i (∞ ) =
(14)
TC increases with a duty ratio increase – slowly
for duty ratios 0 < D < 0.5 and dramatically for D > 0.6 . The
explanation is that, for the duty ratio approaching unity,
capacitor time constant strives to infinity, because for D = 1
the capacitor is totally disconnected.
The FC convertor modeling approach based on analytical
solutions of corresponding linear time-invariant differential
equations on different switching intervals is not limited to a
single-phase single-leg three-level (single capacitor) converter
with DC modulation.
If the FC converter level count is more than three (two and
more flying capacitors), still there is no any problem to obtain
simple analytical solutions for individual switching interval
differential equations. The reason is that, in multiple flying
capacitors case, as different capacitors are connected in series,
the overall system of equations still has the 2nd order because
the capacitor voltages are linearly dependent.
For AC modulation, switching intervals durations (1), (2) are
not constant - they vary with sinusoidal voltage command. For
negative commanded voltages there is additional switching
topology (Fig.7) instead of that of Fig.3.
L
VDC / 2
As expected for inductive load, voltage (duty ratio) AC
component leads the current one. Capacitor voltage balance
dynamics still shows aperiodic behavior with a dominant slow
exponent. With average duty ratio increase, capacitor charge
rate slightly slows down (Fig. 8, a, b).
For AC frequency increase, the capacitor charge rate remains
unchanged (Fig. 8, b, c). This likely holds as long as the
capacitor charge time constant is larger than an AC period.
R
iL
+
IV. FC CONVERTER DYNAMICS ANALYSIS
Fig. 7. FC converter topology for negative output voltage ( S1S 2 )
(the capacitor is disconnected)
The FC converter simulation results for combined DC-AC
unipolar ( D ≥ 0 ) excitation
D(t ) = D0 − D0 cos(ω f t )
(16)
are presented in Fig. 8.
This section is devoted to analytical solutions for DC PWM
modulation. Suppose we take initial conditions of the interval 1
and substitute them, along with the interval 1 duration, into
interval 1 dynamic solution (3)-(5). This will give us initial
conditions for the interval 2. By substituting them with the
interval 2 duration into interval 2 dynamic solution (3), (7), (8),
we obtain initial conditions for the interval 3. Carrying out the
same procedure for the intervals 3, 4, we complete a PWM
period and obtain a linear difference vector equation in the
form
Load current (A) and
capacitor voltage (V)
FC Converter Simulated Transient - D=0.2-0.2COS(1000*t)
60
50
40
30
20
10
0
-10 0
-20
X (t + TPWM ) = A( D) X (t ) + B( D)
VC
(17)
where
D, %
A( D ) = A4 A3 A2 A1 ;
IL
0.005
VDC
,
2
0.01
0.015
0.02
0.025
B ( D ) = A4 ( A3 ( A2 B1 + B2 ) + B3 ) + B4 .
(18)
Time, s
a
Load current (A) and
capacitor voltage (V)
FC Converter Simulated Transient - D=0.4-0.4COS(1000*t)
90
80 D, %
70
60
50
40
30
20
10
0
-10 0
-20
VC
IL
0.005
0.01
0.015
0.02
0.025
X (∞ ) = (I − A( D ) ) B ( D )
−1
Time, s
b
Load current (A) and
capacitor voltage (V)
FC Converter Simulated Transient - D=0.4-0.4COS(2000*t)
90
80
70
60
50
40
30
20
10
0
-10 0
-20
VC
IL
0.01
0.015
0.02
VDC
,
2
(19)
I - unity matrix.
D, %
0.005
This procedure may be considered as a kind of averaging on
a PWM period (in “fast” time) to obtain averaged equations in
"slow" time.
While calculating the matrices A and B in (17), we started
with the switching interval 1. However, there are three more
possibilities to get started with the intervals 2, 3, and 4 that
will, generally speaking, generate different system matrices.
Assuming system (17), (18) stability, a steady state solution
may be found as
0.025
Time, s
c
Fig. 8. FC converter load current and capacitor voltage simulation for
combined DC-AC excitation: a - D(t ) = 0.2 − 0.2COS(1000⋅ t ) ; b D(t ) = 0.4 − 0.4COS(1000⋅ t ) ; c - D(t ) = 0.4 − 0.4COS(2000⋅ t )
Not only the solution (19) is a function of the duty ratio, it is
also the intervals order (1234, 2341, 3412, 4123) dependent.
For good practical converter, current and voltage ripples
should be low and the solution (19) is supposed to be close to
(14) for any intervals order.
Using "on average" derivatives approximations in “slow”
time
di i (t + TPWM ) − i (t )
≈
;
dt
TPWM
dv v(t + TPWM ) − v(t )
≈
,
dt
TPWM
(20)
we obtain the FC converter “averaged” differential equations in
the form
dX
1
( A( D ) − I )X + 1 B VDC .
=
dt TPWM
TPWM
2
(21)
An accurate solution and switched simulation will show
twice transistor switching frequency ripples in both load
current and capacitor voltage. These ripples are filtered out in
the averaged model (21).
Again, four discrete (and matching continuous) models are
possible dependent on the initial switching interval selection.
However, it may be shown that the averaged switched system
characteristic polynomial and eigenvalues (natural frequencies)
don't depend on specific initial interval selection.
A. Time Constants for Zero Duty Ratio DC PWM
Consider first zero duty ratio D = 0 . Then the PWM period
is comprised of the two intervals - 2 and 4 - with durations
∆t 2 = ∆t 4 = TPWM / 2 (unity matrices A1 , A3 in (18)). The
resulting matrix A characteristic polynomial does not depend
on the intervals 2 and 4 order -

 λ + (22)

The roots of (22) may be presented as
λ1 = exp(− αTPWM )λ1' ;
(23)
λ2 = exp(− αTPWM )λ'2 ,
T2 = −TPWM / ln(λ2 )
.

 λ + 1,

(24)
(25)
A small parameter
δ = (α / ω ) sin (0.5ωTPWM )
and logarithm
1 2 1 3
x + x + o( x 3 )
2
3
series expansions, we approximate the logarithms of (23) as
ln(1 + x ) = x −
ln λ1 = −αTPWM − 2δ − (1 / 3)δ 3 + o(δ 4 );
ln λ2 = −αTPWM + 2δ + (1 / 3)δ 3 + o(δ 4 ).
(27)
Expanding sine function for the small parameter (26)
sin x = x −
1 3
x + o( x 4 ) ,
6
we eventually obtain the approximate expressions for the two
time constants (25) for D = 0 :
T1 (0) =
L
,
R
T2 (0) = 48
(28)
L LC
.
2
R TPWM
(29)
The “small” time constant (28), as we expected based on
simulation experience and instantaneous load voltage analysis,
is actually the load associated one and does not depend on the
capacitance.
The “large” capacitor charge time constant (29) may be
interpreted as
LC
T1 (0)
2
TPWM
or capacitor equivalent charge resistance my be viewed as
Once we get the discrete system (17) eigenvalues (23), we
calculate equivalent continuous system time constants as
T1 = −TPWM / ln(λ1 );
1 2
x + o( x 3 )
2
T2 (0) = 48
0 < λ1 < λ2 < 1 , with λ1' , 2 being the roots of

α2
 ωT
P ' (λ ) = λ2 − 21 + 2 2 sin 2  PWM
ω
 2

0 < λ1' < 1 < λ'2 .
Using square root
1 + x2 = 1 +
Note that in (21) we use an actual PWM period and there is
no need to perform a limit transition TPWM → 0 .

α2
 ωT
P (λ ) = λ2 − 2 exp(− αTPWM )1 + 2 2 sin 2  PWM
ω
 2

+ exp(− 2αTPWM ).
naturally arises in (22), (24) because the switching frequency is
much higher than the natural frequency of the equivalent RLCcircuit ωTPWM << 1 .
(26)
RC = 48R
T12
48 L2
.
=
2
2
TPWM
R TPWM
According to our numerical calculations, for reasonable
converter parameters that provide low ripples of load current
and capacitor voltage, large time constant expression (29)
holds with an excellent accuracy.
Example 2. Consider a FC converter with the parameters (12).
Its transient for D = 0 , zero DC bus voltage and zero initial
load current is shown in Fig. 9.
Capacitor charge / discharge time constant according to (29)
amounts to
T2 (0) = 3.33ms .
This is what can be observed from capacitor transient graphs
- charge (Fig. 5, a) and discharge (Fig. 9).
C. Time Constants Estimation for AC PWM
Load current (A) and
capacitor voltage (V)
FC Converter Simulated Transient - D=0; Vdc=0
60
50
40
30
20
10
0
-10 0
-20
VC
IL
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Time, s
Fig. 9. Capacitor discharge for
VDC = 0V ; D = 0 ; i (0) = 0 A
Starting from section III, we assumed throughout this section
oscillating step response of the equivalent LCR-circuit (Fig.4).
For aperiodic step response ("large" resistance R > 2 L / C ),
α = 0.5 R / L ; α1 = α 2 − ω02 ; ω02 = 1 /( LC )
and characteristic polynomial is different

α2
α T
P (λ ) = λ2 − 2 exp(− αTPWM )1 + 2 2 sh 2  1 PWM
α1
 2

+ exp(− 2αTPWM ).
 
 λ + (30)

However, the small parameter analysis of (30), (23) shows
that approximate time constants expressions (28), (29) hold.
B. Time Constants for an Arbitrary Duty Ratio DC PWM
Now consider a general case of an arbitrary DC PWM duty
ratio 0 ≤ D ≤ 1 . First, it can be shown that the small load
associated time constant does not practically depend on duty
ratio meaning that (28) holds for any duty ratio
T1 ( D ) = T1 (0) =
L
, 0 ≤ D ≤ 1.
R
α 2 ( D) = α 2 (0)(1 − 3D 2 + 2 D 3 );
(31)
2
RTPWM
48L2 C
exp (− α (t)t ) ,
where α (t ) period equals
amounts to
(33)
T , an averaged decay factor
T
1
α 0 = ∫ α (t )dt
T0
(34)
and equivalent time constant is T0 ≈ 1 / α 0 given that T0 >> T .
Now consider sinusoidal PWM with modulation index M ,
0 ≤ M ≤ 1 , and relatively high fundamental frequency (that
is still essentially lower than a switching one)
(35)
D (t ) = M sin(ω f t )
for the sine arguments
Second, we found a small parameter approximation of the
slow exponent decay factor (inverse time constant) in the
following form:
α 2 ( 0 ) = 1 / T2 ( 0 ) =
For DC PWM, FC converter models (17), (22) obtained by
averaging on PWM period are linear time invariant. For AC
PWM, FC converter models (17), (22) are linear time-variable
and, unfortunately, we can’t directly apply the power of linear
time-invariant systems analysis.
However, if the load fundamental frequency is relatively
high, then, assuming a quasi-static behavior, the FC converter
linear time-invariant model for AC modulation may hopefully
be obtained by appropriate fast dynamics time averaging on a
fundamental period. We have to acknowledge that, in general,
it may be not trivial because an adequate dynamics averaging
mathematical tool is required.
For the single-capacitor FC converter (Fig.1), we obtained
analytical time constant (decay factor) expressions (30)-(32)
for DC modulation. For AC PWM, it is reasonable to assume
that the expression (30) for the small duty ratio independent
time constant holds.
For an exponent with a fast periodical decay factor variation
(observe α (1) = 0 and α (1) = 0 ).
Therefore, the equivalent capacitor charge time constant for
an arbitrary duty ratio is given by
'
0 ≤ωft ≤π
(36)
(not to enter into negative duty ratio problem).
Then, to obtain the large time constant for AC PWM, we
must average the DC PWM decay factor (31) on the half
period (36) (the quarter period will also do). This way, for AC
PWM anticipated slow exponent decay factor is
2
RTPWM
8
 3 2

α 2 (M ) =
M3
1 − M +
2
3π
48L C  2

(37)
and the equivalent capacitor charge time constant
T2 ( D ) =
T 2 ( 0)
.
1 − 3D 2 + 2 D 3
(32)
The capacitor charge time constant (32) increases with duty
ratio and, for D → 1 , T2 ( D) → ∞ because for unity duty ratio
the capacitor is totally disconnected.
T2 ( M ) =
48L2 C
.
8
 3 2
2
3
RTPWM 1 − M +
M 
3π
 2

(38)
The comparison of accurate DC PWM and anticipated AC
PWM capacitor charge decay factors and time constants is
given in Fig.10 and Fig.11 respectively.
Normalized Decay Constant (Inverse Time Constant)
1
Decay Constant, p.u.
0.9
AC (M)
0.8
0.7
0.6
DC (D)
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Duty Ratio D, p.u.
Modulation Index M, p.u.
Fig. 10. Normalized capacitor charge exponential decay constant for DC and
AC modulation
Normalized Capacitor Time Constant
Capacitor Charge Time Constant, p.u.
5
4.5
4
3.5
DC (D)
AC (M)
3
2.5
A general time domain approach to construct a family of FC
converter models – both discrete and continuous – by sewing
analytical solutions for consecutive switching intervals is
applicable to modeling a multilevel multiphase converter and is
not limited to DC PWM. However, for DC PWM the model is
linear time-invariant that makes voltage balance dynamics
analysis straightforward.
For a primitive single capacitor three-level FC converter, we
obtained extremely simple and accurate expression for a
capacitor related time constant by applying small parameter
technique. The small parameter naturally arises for practical
converters with low current and voltage ripples. The capacitor
time constant formula reveals the capacitor charge rate
dependence on inductive load parameters, carrier frequency,
and duty ratio and does not depend on the equivalent LCRcircuit transient behavior. This time constant increases with
DC PWM duty ratio and strives to infinity with the duty ratio
approaching unity (disconnected capacitor).
We have to admit that a generalization to an AC modulation
case is not trivial and an adequate mathematical technique to
perform FC converter dynamics averaging across the AC
trajectories is required. However, an insight into the voltage
balance mechanism gained from a DC PWM consideration
may definitely be useful for an AC PWM as well. For a single
capacitor single leg three-level FC converter, we estimated AC
PWM associated capacitor charge time constant dependence on
modulation index by averaging that obtained for DC PWM on
the AC fundamental period.
2
ACKNOWLEDGMENT
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
The first and second authors gratefully acknowledge Elmo
Motion Control and General Satellite Corporation management
respectively for on-going support to advanced applied power
electronics research and development.
Duty Ratio D, p.u.
Modulation Index M, p.u.
REFERENCES
[1]
Fig. 11. Normalized capacitor charge time constant for DC and AC modulation
Note twice DC PWM time constant increase for 50% duty
ratio and about 3 times AC PWM time constant increase for
100% modulation index (overmodulation is not considered).
One can also easily calculate the equivalent capacitor charge
time constant for combined DC-AC modulation (16) by
averaging on AC period. The comparison of Fig. 8, b and c
confirms the assumption that the fundamental frequency has no
impact on the capacitor charge rate as long as the capacitor
charge time constant is larger than an AC period.
[2]
[3]
[4]
[5]
[6]
V. CONCLUSION
While the reported FC converter voltage balance dynamics
research mostly deals with an AC modulation and is based on
heavy frequency domain transformations, DC modulation
analysis seems to be a missing intermediate link in physical
understanding of FC converter transient behavior.
[7]
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