Analysis of a Flying Capacitor Converter:
A Switched Systems Approach
Alex Ruderman, Boris Reznikov, and Michael Margaliot
Abstract
Flying capacitor converters (FCCs) attract considerable interest because of their inherent natural
voltage balancing property. Several researchers analyzed the voltage balance dynamics in FCCs using
frequency domain methods.
Recently, considerable research attention has been devoted to switched systems, i.e. systems composed of several subsystems, and a switching law that determines which subsystem is active at every
time instant.
In this paper, we propose a new approach to the analysis of an FCC. The analysis is performed in
the time–domain, treating the FCC as a switched system. The subsystems are the various configurations
obtained for each state of the circuit switches, and the switching law is determined by the modulation.
We demonstrate this new approach by using it to analyze a single–phase single–leg three–level
FCC. The switched systems approach provides simple closed–form expressions for the system behavior.
We show that the natural balancing property is equivalent to the asymptotic stability of a certain matrix.
We also show that it is possible to rigorously analyze properties such as the capacitor time constant
and relate them to the parameter values of the load, carrier frequency, and duty ratio.
Index Terms
Multilevel converters; flying capacitor; affine switched systems; periodic switching law.
I. I NTRODUCTION
Multilevel converters (MCs) are recently attracting considerable interest. When compared to
conventional converters, MCs allow higher power ratings, higher efficiency, and lower harmonic
distortion. The topology, modulation strategy, and performance of MCs have been extensively
studied over the last two decades (see, e.g. [1], [2], [3]). Although initially suggested for
high–voltage power applications, MCs may also be interesting for low– and medium–power
converters [4], [5].
Three basic MC architectures are: multiple point clamped (MPC) (or diode clamped), flying
capacitor, and cascaded H–bridge with separate DC sources. A shortcoming of the MPC architecture is that it operates in a limited (modulation index/load power factor) envelope [6]. The
maximal possible modulation index is achieved only for a pure reactive load. As the load power
factor approaches one, the modulation index is reduced due to limitations imposed by the need
to actively balance the capacitor voltages (see also [7]).
An abridged version of this paper was presented at the 13th International Power Electronics and Motion Control Conference
(EPE–PEMC’08), Poland, 2008.
AR (aruderman@elmomc.com) is with Elmo Motion Control Ltd, Israel; BR (reznikovb@spb.gs.ru) is with the
General Satellite Corporation, Russia; MM (michaelm@eng.tau.ac.il) is with the School of Electrical Engineering–
Systems, Tel Aviv University, Israel.
2
Several strategies for addressing the balancing problem in diode clamped DC/AC MCs with
passive front ends have been proposed (see [8] and the references therein). More generally,
voltage balancing is regarded as the most important problem in the field of high–voltage MCs [9].
The flying capacitor converter (FCC) is a multilevel pulse–width modulated (PWM) converter
whose internal architecture automatically guarantees the voltage balancing property for passive
loads. It thus provides an attractive alternative to the MPC. Furthermore, a single–leg FCC
may be used for both DC/DC and DC/AC conversion, whereas an MPC converter cannot
serve as a DC/DC converter. This is so because a DC current at a clamping point permanently
charges/discharges the DC bus capacitors, so no voltage balance is possible in principle. Even
for the case of DC/AC conversion, significant low frequency clamping point voltage oscillations
may appear for some operating conditions [6].
Several authors studied the dynamic behavior of FCCs (see, e.g., [9], [10], [11], [12], [13]).
The analysis is usually based on representing the switching strategy using a piecewise constant
function and then applying frequency domain methods. This is probably due to the important
role of the load current high–order harmonics in the voltage balancing process. Indeed, consider
an ideally smooth, ripple–free load current, and an appropriate phase–shifted voltage modulation
strategy with relatively high switching frequency. Then the capacitor will be charged and discharged by the same amount of load current on time intervals of equal durations. The capacitor
voltage will then oscillate around some average value, determined by the capacitor initial voltage,
and no voltage balancing will take place in this case.
However, the application of frequency domain methods in this context is non trivial. It
typically requires a double Fourier transform, lengthy computations, and many simplifications
and approximations. Analysis in the frequency domain also makes it more difficult to gain an
intuitive understanding of the physical mechanisms underlying the self balancing property.
Recently, considerable attention is devoted to the analysis of switched systems [14]. Consider m
continuous–time systems described by:
ẋ(t) = f i (x(t)),
i = 1, . . . , m.
(1)
Here x(·) ∈ Rn is the state vector, and f i (·) : Rn → Rn describes the dynamics of system i. A
switched system is a mathematical model in the form:
ẋ(t) = f σ(t) (x(t)),
(2)
where σ(·) ∈ {1, . . . , m} is the switching–law. This models a system that can switch between
the m subsystems (1). The switching–law determines which subsystem is active at which time
instant. For example, if σ(t) = 1 for t ∈ [0, 10), and σ(t) = 2 for t ∈ (10, 15), then the solution
of (2), starting at time 0, follows the dynamics ẋ = f 1 (x) for the first 10 seconds, and then
follows ẋ = f 2 (x) for the next 5 seconds.
Switched systems are useful for modeling the combination of continuous–time dynamics with
discrete switchings. The underlying philosophy is to explicitly take into account the coexistence
of both the continuous– and discrete–time dynamics. Numerous examples and applications can
be found in [14], [15], [16]. In particular, switched systems provide suitable models for electric
circuits that contain on/off switches. Here typical state-variables are the capacitor voltages and
inductor currents. Each possible configuration of the set of switches induces a continuous–time
dynamics of the state variables. The dynamics of the system thus changes every time a switch
opens or closes. In other words, the modulation of the switches determines the switching–law.
February 5, 2009
DRAFT
3
It is important to note that even when all the subsystems (1) are linear, i.e. f i (x) = Ai x,
with Ai ∈ Rn×n , the switched system may demonstrate a highly nonlinear behavior [15]. This
suggests that frequency–domain approaches may not be appropriate in the analysis of switched
systems.
In this paper, we demonstrate the switched systems perspective by applying it to analyze a
single–capacitor single–leg three–level FCC. Here each subsystem, corresponding to a possible
configuration of the switches, is a second-order affine linear system. Our analysis is based on
“stitching” together the analytical solutions of the subsystem trajectories, corresponding to each
modulation phase of the FCC.
This approach provides considerable insight into the FCC dynamics. In particular, we show
that the natural balancing property is equivalent to the asymptotic stability of a certain matrix.
Furthermore, we obtain explicit formulas for the capacitor time constant.
In an FCC, the switching is performed at a frequency that is much higher than the natural
frequency of the equivalent RLC–circuit. This is needed in order to guarantee that the current
and voltage ripples are relatively low (which is always true for a practical converter). Using this
fact, it is possible to use a small parameter approximation that further simplifies the analytic
expressions. This reveals how the capacitor charge rate depends on the inductive load parameters,
the carrier frequency, and the duty ratio. To the best of our knowledge, this is the first time that
such formulas are derived. Numerical simulations of the FCC demonstrate excellent agreement
with the analytic results.
The remainder of this paper is organized as follows. Section II reviews the FCC topology and
modulation strategy. Section III derives the mathematical model describing the FCC behavior in
the case of DC–modulation. The asymptotic and transient behavior of this model is rigorously
analyzed in Sections IV and V, respectively. Section VI demonstrates how the analytic results
obtained for the DC–modulated case can be used to analyze the case of AC–modulation. The
final section concludes and describes some possible directions for future research.
II. TOPOLOGY AND MODULATION
We consider a single–phase single–leg three–level FCC with an RL load (see Fig. 1). The
converter is composed of two equal voltage sources, four (ideal) switches, and a capacitor C.
The load is modeled by an inductor connected in series with a resistor. Wilkinson et al. [11]
refer to this topology as a two–cell inverter.
The four switches operate in two complementary pairs: when switch S i is on (off), switch S i
is off (on). The FCC can thus be in one of four possible configurations.
In the DC–modulated case, the modulation strategy is based on comparing a constant command
signal VCOM (t) ≡ VCOM to a triangular wave s(t). The state of the switches is determined
according to:
(
on, if s(t) < VCOM (t),
S1 =
off, otherwise,
(
on, if s(t) > −VCOM (t),
(3)
S2 =
off, otherwise.
The resulting modulation is periodic. In each period the FCC switches between four possible
phases. In phases P1 and P3 the switch configuration is S1 S2 (that is, both S1 and S2 are
February 5, 2009
DRAFT
4
S2
VDC / 2
+
-
S1
C
L
+
-
VDC / 2
vC
S1
R
iL
+
-
Fig. 1.
S2
Single–phase three–level FCC with an RL load.
s(t)
M
VCOM
0
−VCOM
t
4t1
4t2
4t3
4t4
TP W M
Fig. 2.
S¯1 S2
S1 S2
S1 S¯2
P4
P1
P2
S1 S2 S¯1 S2
P3
P4
DC modulation and corresponding switch states.
on); in phase P2 the configuration is S1 S 2 ; and in phase P4 it is S 1 S2 . The transition order
is P1 → P2 → P3 → P4 (see Fig. 2).
Let TP W M denote the time of one period, and let 4ti denote the time spent in phase Pi during
one period, so that
4
X
4ti = TP W M .
i=1
February 5, 2009
DRAFT
5
iL
L
R
vc
− +
iL
L
R
−
+
+
−
VDC /2
+
−
VDC /2
iL
L
R
VDC /2
(b)
(a)
Fig. 3.
vc
+ −
(c)
FCC topology: (a) in phases P1 and P3 ; (b) in phase P2 ; (c) in phase P4 .
A simple calculation shows that for the modulation strategy defined above:
4t1 = 4t3 = DTP W M /2,
4t2 = 4t4 = (1 − D)TP W M /2,
(4)
where D, the PWM duty ratio, is given by
D := M/(VCOM + M )
with M := maxt s(t) − VCOM .
The converter topology in each phase is depicted in Fig. 3. Phases P 1 and P3 correspond to
the same topology. Here the capacitor C is disconnected, so its voltage, v c (t), remains constant
during these phases. The configurations during phases P2 and P4 are similar, except for the fact
that voltages across the source and capacitor have opposite polarities.
It is desirable to regulate the converter such that the capacitor is charged to a value V DC /2.
With this principle in mind, it is possible to calculate the voltage on the load in each possible
configuration (see Fig. 3), and to use this to determine the duty ratio yielding the intended
functionality of the converter.
The main advantage of the FCC is that the average value of the voltage across the capacitor
converges to VDC /2 for a large range of initial conditions and duty ratios. The next example
demonstrates this.
Example 1 Consider the FCC depicted in Fig. 1 with: VDC = 100V , R = 1Ω, L = 0.25∗10−3 H,
C = 100 ∗ 10−6 F and TP W M = 300 ∗ 10−6 sec. The initial conditions (capacitor voltage and
inductor current) are zero. Fig. 4 depicts the behavior of the inductor current (i L ) and capacitor
voltage (vc ) as a function of time, for various values of the duty ratio D.1
Let īL (v̄c ) denote the average value of iL (vc ). It may be seen that in each of the four cases:
īL → VDC D/(2R),
v̄c → VDC /2.
Here are a few more observations concerning the behavior of the FCC that can be deduced
1
All the simulations in this paper were performed using MATLAB’s ODE45 numerical integration procedure.
February 5, 2009
DRAFT
6
60
60
50
50
v
c
40
30
30
20
20
10
L
i
L
0
−10
−20
c
10
i
0
v
40
−10
0
0.005
0.01
0.015
Time, s
0.02
0.025
−20
0
0.005
0.01
(a)
60
50
50
40
40
v
c
30
i
L
20
i
10
0
0
−10
−10
0.005
0.01
0.015
Time, s
0.02
vc
10
L
0
0.025
30
20
0.025
−20
0
0.005
(c)
Fig. 4.
0.02
(b)
60
−20
0.015
Time, s
0.01
0.015
Time, s
0.02
0.025
(d)
Load current and capacitor voltage for different duty ratios: (a) D = 0; (b) D = 0.2; (c) D = 0.4; and (d) D = 0.8.
from Fig. 4. A good approximation of the average current through the inductor is given by:
where the time constant is
īL (t) ≈ (1 − exp(−t/TL ))VDC D/(2R),
(5)
TL := L/R.
To demonstrate this, Fig. 5 depicts iL (t) for the case D = 0.8 and the exponential curve given
by (5).
Note that this implies that we can regulate the current through the load by changing the duty
ratio D. Note also that (5) is independent of the capacitance C.
A careful inspection of vc (t) in Fig. 4(a) suggests that this signal is composed of two
February 5, 2009
DRAFT
7
45
40
35
30
25
20
15
10
5
0
Fig. 5.
0
0.002
0.004
0.006
0.008
Time, s
0.01
0.012
0.014
0.016
iL (t) vs. the exponential curve given by Eq. (5) (solid line).
exponentials. A fast exponent that appears as spikes in the initial part of the response, and
a slower exponent that determines the overall rise of vc (t) toward its asymptotic value. The time
constant T2 of this slower exponent depends on D. As D is increased, the rise time increases,
i.e. T2 increases. For D ∈ [0, 1/2], T2 changes slowly with D. The change is more dramatic for
values D > 0.6.
The increase in the capacitor time constant has a clear physical interpretation. Indeed, as D
increases to one, the time intervals 4t2 and 4t4 go to zero (see (4)). This means that the
capacitor is never connected, so its time constant must go to infinity.
In the AC–modulated case, the state of the switches is again determined by (3), but the
command signal is sinusoidal: VCOM (t) = M sin(ηt). Note that in this case, a new phase,
corresponding to the state S̄1 S̄2 , appears. The corresponding topology is similar to that of
phase P1 , but the polarity of the voltage source is reversed.
The frequency of the command signal is usually much lower than that of the triangular wave, so
that in every cycle [kTP W M , (k + 1)TP W M ) of the triangular wave, VCOM (t) ≈ VCOM (kTP W M ).
Example 2 Consider the FCC with the parameter values as in Example 1. Fig. 6 depicts
the behavior of the inductor current (iL ) and capacitor voltage (vc ) as a function of time,
for VCOM (t) = 0.5 sin(1000t). It may be seen that v̄c again converges to VDC /2 = 50. The
inductor current iL is sinusoidal with the same frequency as the command signal. The amplitude
of iL can be controlled by changing the amplitude M of VCOM (t).
Summarizing, the average value of vc (t) converges exponentially to the desired value VDC /2,
and then the FCC operates as desired. This is true for various initial conditions and for both
DC– and AC–modulation. The time constant of this exponential behavior is of great importance.
The reason for this is that it provides a measure on how quickly the FCC returns to normal
operation after some perturbation of its state-variables.
The remainder of this paper is devoted to developing a suitable time–domain approach for
analyzing the FCC. We first analyze the DC-modulated case and then use the results to analyze
February 5, 2009
DRAFT
8
60
v
C
50
40
30
i
L
20
10
0
−10
−20
−30
Fig. 6.
0
0.005
0.01
0.015
Time, s
0.02
0.025
Load current and capacitor voltage for AC modulation.
also the AC–modulated case. As we will see below, this approach provides an analytic explanation
for many of the features demonstrated in the simulation results.
III. M ATHEMATICAL MODEL
Denote the state variables of the converter by x(t) = (x1 (t), x2 (t))0 , where x1 (t) = iL (t) (the
inductor current) and x2 (t) = vc (t) (the capacitor voltage).
Applying Kirchhoff’s laws yields a differential equation for each of the four modulation phases
in the form:
ẋ(t) = Ai x(t) + bi VDC /2,
i = 1, . . . , 4,
(6)
−1/TL 0
−1/TL 1/L
−1/TL −1/L
with A1 = A3 =
, A2 =
, A4 =
, b1 =
0
0
−1/C
0
1/C
0
1/L
−1/L
b3 = b4 =
, and b2 =
.
0
0
Note that (6) and the PWM implies that the FCC dynamics is described by a second–order
affine switched system with a periodic switching law.
We derive an expression for the solution of (6) using the standard variation of constants
method. Let y(t) := exp(−Ai t)x(t). Then (6) yields
ẏ(t) = −Ai exp(−Ai t)x(t)
+ exp(−Ai t)(Ai x(t) + bi VDC /2)
= exp(−Ai t)bi VDC /2,
February 5, 2009
DRAFT
9
so for any two times t and τ , with t ≥ τ :
y(t) = y(τ ) + (VDC /2)
Z
t
exp(−Ai s)bi ds
Z t
exp(−Ai s)bi ds.
= exp(−Ai τ )x(τ ) + (VDC /2)
τ
τ
Thus,
x(t) = exp(Ai t)y(t)
= exp(Ai (t − τ ))x(τ ) + ci (t − τ ),
(7)
where ci (t − τ ) := (VDC /2) τ exp(A
√ i (t − s))bi ds. p
Denote α := 1/(2TL ), w0 := 1/ LC and w := w02 − α2 . We consider from here on the
case where the capacitance is sufficiently small so that:
p
R < 2 L/C.
(8)
Rt
Note that this implies that w is real. p
This assumption is made for concreteness only.
√ All the
results below hold for the case R < 2 L/C once w is replaced by jw, with j = −1.
A calculation yields
exp(−t/TL ) 0
,
exp(A1 t) = exp(A3 t) =
0
1
1
sin(wt)
s− (wt)
wL
,
exp(A2 t) = exp(−αt)
1
− wC
sin(wt)
s+ (wt)
1
sin(wt)
s− (wt)
− wL
,
exp(A4 t) = exp(−αt)
1
sin(wt)
s+ (wt)
wC
where s− (l) := cos(l) −
sin(l), s+ (l) := cos(l) + wα sin(l), and
VDC
1 − exp(−t/TL )
c1 (t) = c3 (t) =
,
0
2R
1
VDC
sin(wt) exp(−αt)
− wL
c2 (t) =
,
1 − s+ (wt) exp(−αt)
2
1
VDC
sin(wt)
exp(−αt)
wL
c4 (t) =
.
1 − s+ (wt) exp(−αt)
2
P
Let tk denote the switching times, i.e., tk := ki=1 4ti (so in particular t4 = TP W M ). Suppose
that our initial state is x(0) and that we repeatedly apply the phase sequence P 1 P2 P3 P4 . Then (7)
February 5, 2009
α
w
DRAFT
10
yields
x(t1 ) = exp(A1 4t1 )x(0) + c1 (4t1 ),
x(t2 ) = exp(A2 4t2 )x(t1 ) + c2 (4t2 ),
(9)
x(t3 ) = exp(A3 4t3 )x(t2 ) + c3 (4t3 ),
x(t4 ) = exp(A4 4t4 )x(t3 ) + c4 (4t4 ),
x(t5 ) = exp(A1 4t1 )x(t4 ) + c1 (4t1 ),
..
.
Combining the first four equations yields
(10)
x(t4 ) = Ax(0) + b,
where
A := exp(A4 4t4 ) exp(A3 4t3 )
(11)
× exp(A2 4t2 ) exp(A1 4t1 ),
b := exp(A4 4t4 ) exp(A3 4t3 ) exp(A2 4t2 )c1 (4t1 )
+ exp(A4 4t4 ) exp(A3 4t3 )c2 (4t2 )
A calculation yields
where
+ exp(A4 4t4 )c3 (4t3 ) + c4 (4t4 ).
A = q×
2 2
r s− (z) + w2rLC sin2 (z)
r sin(z)
(rs− (z) − s+ (z))
wC
sin(z)
(rs− (z) − s+ (z))
wL
1
sin2 (z)r + s2+ (z)
w 2 LC
,
r := exp(−αDTP W M ),
z := w(1 − D)TP W M /2,
q := exp(−α(1 − D)TP W M ).
(12)
Since the modulation is periodic, (10) yields
x(t4(k+1) ) = Ax(t4k ) + b,
k = 0, 1, . . . .
(13)
Let pA (λ) := det(λI − A) denote the characteristic polynomial of the matrix A. A calculation
yields
pA (λ) := λ2 − ((s+ (z) − rs− (z))2 + 2r)qλ + r 2 q 2 .
(14)
It is clear that the dynamic behavior of the FCC is determined by (13). In particular, for small
values of k, (13) can be used to yield the time constants of the exponential behavior of v c (t)
and iL (t). For k → ∞, (13) describes the asymptotic behavior of the FCC as t → ∞. The next
two sections analyze these two cases.
February 5, 2009
DRAFT
11
IV. A SYMPTOTIC BEHAVIOR
In this section, we consider the periodic behavior of x(t) as t → ∞. The switching frequency
is usually much higher than the natural frequency of the equivalent RLC–circuit, i.e., T P W M 1/w. Hence, we assume that
(15)
z = w(1 − D)TP W M /2 1.
Lemma 1 There exists a constant p > 0 such that for all z ∈ [0, p) the eigenvalues of the
matrix A satisfy |λi | < 1, i = 1, 2.
In other words, the matrix A is asymptotically stable for all sufficiently small z.
Proof. See the Appendix.
Theorem 1 Suppose that the matrix A is asymptotically stable. Define
(16)
x̄0 := (I − A)−1 b.
Then, as t → ∞ the state–vector x(t) converges to a periodic solution. Each period is described
by the solution of:

A1 x(t) + b1 VDC /2, t ∈ [0, t1 ),



A x(t) + b V /2, t ∈ [t , t ),
2
2 DC
1 2
(17)
ẋ(t) =

A
x(t)
+
b
V
/2,
t
∈
[t
3
3 DC
2 , t3 ),



A4 x(t) + b4 VDC /2, t ∈ [t3 , t4 ),
with x(0) = x̄0 .
Proof. Eq. (13) is a linear difference equation for the state at times t4k , whose solution is:
k
x(t4k ) = A x(0) +
k−1
X
Aj b.
(18)
j=0
Taking the limit as k → ∞, and using the assumption that A is an asymptotically stable matrix
yields limk→∞ x(t4k ) = (I − A)−1 b. Using (6) completes the proof.
Note that Eqs. (16) and (11) imply that x̄0 depends on the modulation strategy and the
parameter values of the converter and the load, but is independent of the initial condition x(0).
This implies that perturbations in the values of the state–variables will have no effect on the
asymptotic behavior of the FCC. In other words, the natural balancing property is completely
equivalent to the asymptotic stability of the matrix A.
The fact that x̄0 is independent of the initial condition x(0) has an additional implication.
Recall that we considered the result of repeatedly applying the phase sequence P 1 P2 P3 P4 .
Now suppose that starting from x(0), we repeatedly apply, say, the phase sequence P 2 P3 P4 P1 .
Let x̃ := exp(−A1 4t1 )(x(0)−c1 (4t1 )). Then this is equivalent to repeatedly applying the phase
sequence P1 P2 P3 P4 with x(0) = x̃. Since our results are independent of the initial condition,
this implies that we will obtain exactly the same asymptotic behavior.
February 5, 2009
DRAFT
12
60
VC
50
40
30
20
IL
10
0
Fig. 7.
0
0.5
Time, s
1
1.5
−3
x 10
Asymptotic behavior for D = 0.2. Here x(0) = x̄0 .
Example 3 Consider the FCC with the parameters given in Example 1 and D = 0.2. In this
case, a calculation yields:
0.4029 −0.1359
11.7314
A=
,
b=
.
−0.3014 0.8493
10.5030
The eigenvalues of A are λ1 = 0.3248, λ2 = 0.9274, so A is asymptotically stable. Eq. (16)
yields
6.9462
.
x̄0 =
55.7891
Fig. 7 depicts the periodic solution of (17) for this case. It may be seen that this is in agreement
with the behavior depicted in Fig. 4(b) for large values of t.
For the particular case of zero duty ratio, x̄0 admits a particularly simple form.
0
Proposition 1 If D = 0, then x̄0 =
.
VDC /2
Note that this agrees, of course, with the simulation results depicted in Fig. 4(a).
Proof. For D = 0, 4t1 = 4t3 = 0 and 4t2 = 4t4 = TP W M /2. Hence, (11) simplifies to
A = exp(A4 TP W M /2) exp(A2 TP W M /2),
b = exp(A4 TP W M /2)c2 (TP W M /2) + c4 (TP W M /2).
A calculation shows that I − A has the form:
1
∗ exp(−αTP W M ) wL
(s+ (z) − s− (z)) sin(z)
,
∗ 1 − exp(−αTP W M )(s2+ (z) + w21LC sin2 (z))
February 5, 2009
DRAFT
13
where ‘∗’ denotes values that are not important for the proof. Also,
b = (VDC /2)
1
exp(−αTP W M )(s+ (z) − s− (z)) sin(z)
wL
×
,
1 − exp(−αTP W M )(s2+ (z) + w21LC sin2 (z))
where z = wTP W M /2. Using these expressions, it is easy to verify that (I −A)
0
VDC /2
= b.
Using (16) completes the proof.
Theorem 1 provides considerable information on the behavior of the FCC as t → ∞. We
now consider the analysis of the transient behavior of x(t) and, in particular, the associated time
constants.
V. T RANSIENT BEHAVIOR
Our approach is based on relating the eigenvalues of the matrix A to the time constants
of the original,
continuous–time,
system. To do so, suppose for a moment that A is diagonal,
λ1 0
and b = 0. Then, the solution of (10) satisfies
i.e. A =
0 λ2
xi (TP W M ) = λi xi (0),
i = 1, 2.
(19)
The response of a continuous–time system with a diagonal matrix and time constants T i , i = 1, 2,
is xi (t) = exp(−t/Ti )xi (0), so
xi (TP W M ) = exp(−TP W M /Ti )xi (0),
i = 1, 2.
Comparing this with (19) yields λi = exp(−TP W M /Ti ). We thus define the time constants as:
Ti := −TP W M / ln(λi ),
(20)
where λi are the eigenvalues of the matrix A.
For the sake of simplicity, we first consider the case D = 0, and only then turn to the case
of an arbitrary duty cycle D ∈ [0, 1).
A. Zero duty ratio
Consider the case where D = 0. Then
4t1 = 4t3 = 0,
4t2 = 4t4 = TP W M /2.
In other words, the PWM period consists of only two time intervals corresponding to phases P 2
and P4 . In this case, r = 1 and the characteristic polynomial of A given in (14) simplifies to
pA (λ) = λ2 − 2(1 + 2δ 2 )qλ + q 2 ,
with
(21)
α
sin(z), z = wTP W M /2, q = exp(−αTP W M ).
(22)
w
Since we consider the case where 0 < z 1 (see (15)), δ > 0 and we may treat δ as a small
parameter.
δ :=
February 5, 2009
DRAFT
14
The roots of (21) are:
λi = qsi ,
with
√
s1 = 1 + 2δ 2 − 2δ 1 + δ 2 ,
i = 1, 2,
(23)
√
s2 = 1 + 2δ 2 + 2δ 1 + δ 2 .
(24)
Note that 0 < s1 < 1 < s2 . Since, by definition, q ∈ (0, 1), this implies that λ1 ∈ (0, 1).
Also, λ2 ∈ (0, 1) whenever qs2 < 1.
Using (24) yields:
ln(s1 ) = −2δ + δ 3 /3 + O(δ 5 ),
ln(s2 ) = 2δ − δ 3 /3 + O(δ 5 ).
Using (20), (22), (23) and the Taylor series sin(x) = x −
T1 ≈
L
,
R
T2 ≈ 48
x3
3!
+ O(x5 ) yields the time constants:
L LC
.
R TP2 W M
(25)
The first time constant T1 = TL is, as expected, the one associated with the load. This agrees,
of course, with the transient behavior of iL (t) (see (5) and Fig. 5).
The second time constant can be expressed as T2 = 48LCT1 /TP2 W M . Our simulations indicate
that an exponential with time constant T2 provides a very good approximation of the transient
behavior of the capacitor’s voltage vc (t). In other words,
v̄c (t) ≈ (1 − exp(−t/T2 ))(VDC /2).
(26)
The next example demonstrates this.
Example 4 Consider the FCC with the parameter values given in Example 1 and D = 0. In
this case T2 = 1/300. Fig. 8 depicts the behavior of vc (t) vs. the function given by (26). It
may be seen that after a short initial transient, v̄c (t) and the exponential curve are practically
indistinguishable.
Summarizing, the analysis yields simple and explicit expressions for the two time constants T 1 =
TL and T2 that determine the behavior of iL and vc , respectively. Note that the fact that the
time constant of iL is just TL , and is independent of the capacitance C, has a simple physical
interpretation. Indeed, the opposite symmetry of the two phases P 2 and P4 (see Fig. 3) suggests
that on average the capacitor has no effect on the current iL .
B. General case
We now consider the more general case where D ∈ [0, 1). The analysis runs along the same
lines as in Section V-A, but with somewhat lengthier calculations.
We begin by deriving approximations of the two eigenvalues of (14). Denote b := −q((s + (z)−
rs− (z))2 +2r). Note that r = exp(−αDTP W M ) = exp(−αµz/w), with µ := 2D/(1−D). Using
February 5, 2009
DRAFT
15
50
45
40
35
30
25
20
15
10
5
0
Fig. 8.
0
0.002
0.004
0.006
0.008
Time, s
0.01
0.012
0.014
0.016
vc (t) vs. the exponential curve given by Eq. (26) (solid line).
the geometric series for the functions sin, cos, and exp yields:
α
α2
b/q = −2 + 2µ z − 2(2 + µ(2 + µ)) 2 z 2
w
w
α3 3
4
+ µ(3 + µ(3 + µ)) 3 z
3
w
2
α (2 + µ)(2 + 3µ)w 2 − α4 µ2 (9 + 2µ(4 + µ)) 4
+
z
3w 4
+ O(z 5 ).
(27)
The discriminant of (14) is
4 : = b2 − 4r 2 q 2
and expending yields:
= q 2 (s+ (z) − rs− (z))2 ((s+ (z) − rs− (z))2 + 4r),
4 = 4q 2 (2 + µ)2
where
α2 2
z (1 + p(z)) + O(z 5 ),
w2
α
p(z) := −2µ z
w
2
α (6 + µ(9 + µ(18 + 7µ))) − (2 + 3µ)w 2 2
+
z .
3(2 + µ)w 2
Using the series
February 5, 2009
√
1 + x = 1 − x/2 + x2 /8 − . . . yields
p
α
4 ≈ 2q(2 + µ) z(1 − p/2 + p2 /8).
w
(28)
DRAFT
16
10
9
8
7
6
5
4
3
2
1
Fig. 9.
0
0.1
0.2
0.3
0.4
D
0.5
0.6
0.7
0.8
The function f (D) as a function of D.
The roots of (14) are λ1,2 = (−b ±
√
4)/2 and using (27) and (28) yields
λi = qsi ,
(29)
with:
s1 = 1 − 2(1 + µ)αz/w + O(z 2 ),
s2 = 1 + 2αz/w + 2α2 z 2 /w 2
− (α/6)(3α2 (µ − 2) + (2 + 3µ)w 2 )z 3 /w 3 + O(z 4 ).
Using the series ln(1 + x) = x − x2 /2 + x3 /3 + O(x4 ) yields
ln(s1 ) = −2(1 + µ)αz/w + O(z 2 ),
ln(s2 ) = 2αz/w − (α/6)(2 + 3µ)(a2 + w 2 )z 3 /w 3 + O(z 4 ).
Using (20), (12), and (29) yields the time constants:
T1 (D) ≈ L/R,
T2 (D) ≈
48
L LC
.
2
(1 − D) (1 + 2D) R TP2 W M
(30)
Note that T1 , i.e. the time constant associated with iL is just TL . The second time constant
satisfies T2 (D) = f (D)T2 (0), with f (D) := (1−D)21(1+2D) (see (25)). Fig. 9 depicts the function f (D). It may be seen that f (D) increases with D. The increase is relatively moderate for
the range D ∈ [0, 1/2], and becomes more dramatic for values D > 0.6. This agrees well with
the changes in the behavior of vc as a function of D (see Fig. 4). As D → 1, T2 (D) → ∞. This
is reasonable since for D = 1 the capacitor is always disconnected, so its time constant must
go to infinity.
Exponentials with time constants Ti (D), i = 1, 2, provide an excellent approximation of the
average behavior of the state–variables of the converter. The next example demonstrates this.
February 5, 2009
DRAFT
17
60
v
c
50
40
30
20
iL
10
0
−10
Fig. 10.
0
0.005
0.01
0.015
Time, s
0.02
0.025
Behavior of the FCC for D = 0.5 and exponentials with time constants Ti (solid lines).
Example 5 Consider the FCC with the parameters given in Example 1 and D = 0.5. Fig. 10
depicts the FCC state–variables with x1 (0) = x2 (0) = 0. Also depicted are the exponentials (5)
and (26) with T1 (D) and T2 (D) given in (30). It may be seen that the exponentials provide an
excellent approximation of the average behavior of the state–variables.
Note that for R → 0, T1 , T2 → ∞, suggesting that for a purely reactive load the natural
balancing property disappears. This agrees with the observations in [11].
VI. AC MODULATION
Consider now the case of AC modulation, i.e. VCOM (t) = M sin(ηt), with M ∈ [0, 1]. As
noted above, the frequency of the command signal is usually much lower than that of the
triangular wave, so that in every cycle [kTP W M , (k+1)TP W M ) of the triangular wave, VCOM (t) ≈
VCOM (kTP W M ). This suggests that we may be able to approximate the time constants in this
case by averaging the time constants of the DC case over one period of V COM (t).
It turns out that it is actually easier to average the reciprocal of the time constant, namely, g(D) :=
1/T2 (D) = (1 − D)2 (1 + 2D)/T2 (0). Note that in our analysis of the DC-modulated case, we
assumed that D ∈ [0, 1] (see (4)). Thus, we average g(D(t)) = g(M sin(ηt)) on [0, π/η] (rather
than [0, 2π/η]) as this is the interval for which D(t) ∈ [0, 1]. A calculation yields
Z π/η
1
16M 3 − 9M 2 π + 6π
.
ḡ :=
g(M sin(ηt))dt =
π/η 0
6πT2 (0)
Thus, the averaged time constant is
T̄2 (M ) :=
1
T2 (0)
=
,
ḡ
1 − 1.5M 2 + 8M 3 /(3π)
so our approximation in the AC–modulation case is
v̄c (t) ≈ (1 − exp(−t/T̄2 (M )))(VDC /2).
February 5, 2009
(31)
DRAFT
18
60
50
40
30
20
10
0
Fig. 11.
0
0.005
0.01
0.015
Time, s
0.02
0.025
0.03
Capacitor voltage vs. the exponent (31) (solid line) as a function of time.
Extensive simulations indicate that (31) provides an excellent approximation of the average
behavior of vc (t) and that this holds for a large range of values of M and η. The next example
demonstrates this.
Example 6 Consider the FCC with the parameter values as in Example 1. Fig. 11 depicts the
capacitor voltage vC (t), as a function of time, for VCOM (t) = M sin(1000t) with M = 0.5. In
this case, T̄2 (M ) = 1/219.331. Fig 11 also depicts the approximation (31) for this value of ḡ. It
may be seen that v̄c (t) indeed provides an excellent approximation of the average value of the
capacitor voltage.
VII. D ISCUSSION
Multilevel converters combine continuous–time elements and on/off switches. This makes their
analysis and design highly non trivial. In this paper, we suggested an analysis approach that is
based on treating such converters as a switched system.
We demonstrated this approach for the case of a simple FCC. We analyzed this FCC in
the time–domain by combining the effects of the subsystems that correspond to the various
switching configurations. The analysis provides considerable information on the circuit behavior
and in particular on its natural balancing property. The latter property is completely equivalent
to the asymptotic stability of a matrix A. By analyzing the eigenvalues of A, we obtained simple
and explicit expressions for the time constants that determine the FCC behavior for both DC–
and AC–modulation.
Topics for further research include extending the analysis to the case of more complex loads,
and the analysis of other, more complicated, MCs.
Finally, the PWM modulation considered here may be viewed as an open–loop control strategy
for the FCC. The design of closed–loop controllers is recently attracting considerable interest
(see [17], [18], [19], [20] and the references therein). The time–domain model developed here
may also be useful in this context.
February 5, 2009
DRAFT
19
ACKNOWLEDGMENTS
We thank the anonymous reviewers for their helpful comments. The first and second authors
are grateful to the management of Elmo Motion Control and General Satellite Corporation,
respectively, for their encouragement and support.
A PPENDIX : P ROOF OF L EMMA 1
For z = 0, the characteristic polynomial of the matrix A (see (14)) becomes
p0A (λ) := λ2 + a1 λ + a0 ,
(32)
with a1 := −((1 − r)2 + 2r)q and a0 = r 2 q 2 .
It is well known [21, Fact 11.18.2] that a necessary and sufficient condition for asymptotic
stability of the polynomial p0A is that
|a0 | < 1 and |a1 | < 1 + a0 .
The condition |a0 | < 1 clearly holds since, by definition, r, q < 1. The second condition is
equivalent to
q(1 − exp(−αDTP W M ))2 < (1 − exp(−αTP W M ))2 .
Since α > 0, q < 1 and D ∈ [0, 1), this inequality indeed holds.
We conclude that the polynomial p0A is asymptotically stable, and a continuity argument
completes the proof.
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