Analysis of a Flying Capacitor Converter: A Switched Systems Approach

Transcription

1 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. INTRODUCTION 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 8), Poland, 28. 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 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], [1], [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( ) R n is the state vector, and f i ( ) : R n R n 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 [, 1), and σ(t) = 2 for t (1, 15), then the solution of (2), starting at time, follows the dynamics ẋ = f 1 (x) for the first 1 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.

3 3 It is important to note that even when all the subsystems (1) are linear, i.e. f i (x) = A i x, with A i R n 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 V COM (t) V COM to a triangular wave s(t). The state of the switches is determined according to: { on, if s(t) < V COM (t), S 1 = off, otherwise, { on, if s(t) > V COM (t), S 2 = (3) off, otherwise. The resulting modulation is periodic. In each period the FCC switches between four possible phases. In phases P 1 and P 3 the switch configuration is S 1 S 2 (that is, both S 1 and S 2 are

4 4 V DC /2 V DC / S2 S1 C L R + - v C S 1 S2 i L Fig. 1. Single phase three level FCC with an RL load. V COM V COM s(t) M t t 1 t 2 t 3 t 4 T P W M S 1 S 2 S 1 S 2 S S2 1 S 1 S 2 S1 S 2 P 4 P 1 P 2 P 3 P 4 Fig. 2. DC modulation and corresponding switch states. on); in phase P 2 the configuration is S 1 S 2 ; and in phase P 4 it is S 1 S 2. The transition order is P 1 P 2 P 3 P 4 (see Fig. 2). Let T P W M denote the time of one period, and let t i denote the time spent in phase P i during one period, so that 4 t i = T P W M. i=1

5 5 i L + L V DC /2 R v c i L + L + V DC /2 R v c+ + V DC /2 i L L R (a) (b) (c) Fig. 3. FCC topology: (a) in phases P 1 and P 3; (b) in phase P 2; (c) in phase P 4. A simple calculation shows that for the modulation strategy defined above: where D, the PWM duty ratio, is given by t 1 = t 3 = DT P W M /2, t 2 = t 4 = (1 D)T P W M /2, (4) D := M/(V COM + M) with M := max t s(t) V COM. The converter topology in each phase is depicted in Fig. 3. Phases P 1 and P 3 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 P 2 and P 4 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 V DC /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: V DC = 1V, R = 1Ω, L = H, C = F and T P W M = 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 (v c ) as a function of time, for various values of the duty ratio D. 1 Let ī L ( v c ) denote the average value of i L (v c ). It may be seen that in each of the four cases: ī L V DC D/(2R), v c V DC /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.

6 v c 4 4 v c i L i L Time, s Time, s (a) (b) i L 4 4 v c i L 1 v c Time, s Time, s (c) (d) Fig. 4. Load current and capacitor voltage for different duty ratios: (a) D = ; (b) D =.2; (c) D =.4; and (d) D =.8. from Fig. 4. A good approximation of the average current through the inductor is given by: where the time constant is ī L (t) (1 exp( t/t L ))V DC D/(2R), (5) T L := L/R. To demonstrate this, Fig. 5 depicts i L (t) for the case D =.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 v c (t) in Fig. 4(a) suggests that this signal is composed of two

7 Time, s Fig. 5. i L(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 v c (t) toward its asymptotic value. The time constant T 2 of this slower exponent depends on D. As D is increased, the rise time increases, i.e. T 2 increases. For D [, 1/2], T 2 changes slowly with D. The change is more dramatic for values D >.6. The increase in the capacitor time constant has a clear physical interpretation. Indeed, as D increases to one, the time intervals t 2 and t 4 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: V COM (t) = M sin(ηt). Note that in this case, a new phase, corresponding to the state S 1 S2, appears. The corresponding topology is similar to that of phase P 1, 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 [kt P W M, (k + 1)T P W M ) of the triangular wave, V COM (t) V COM (kt P W M ). Example 2 Consider the FCC with the parameter values as in Example 1. Fig. 6 depicts the behavior of the inductor current (i L ) and capacitor voltage (v c ) as a function of time, for V COM (t) =.5 sin(1t). It may be seen that v c again converges to V DC /2 = 5. The inductor current i L is sinusoidal with the same frequency as the command signal. The amplitude of i L can be controlled by changing the amplitude M of V COM (t). Summarizing, the average value of v c (t) converges exponentially to the desired value V DC /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

8 8 6 5 v C i L Time, s Fig. 6. 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. MATHEMATICAL MODEL Denote the state variables of the converter by x(t) = (x 1 (t), x 2 (t)), where x 1 (t) = i L (t) (the inductor current) and x 2 (t) = v c (t) (the capacitor voltage). Applying Kirchhoff s laws yields a differential equation for each of the four modulation phases in the form: ẋ(t) = A i x(t) + b i V DC /2, i = 1,..., 4, (6) ( ) ( ) ( ) 1/TL 1/TL 1/L 1/TL 1/L with A 1 = A 3 =, A 2 =, A 1/C 4 =, b 1/C 1 = ( ) ( ) 1/L 1/L b 3 = b 4 =, and b 2 =. 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( A i t)x(t). Then (6) yields ẏ(t) = A i exp( A i t)x(t) + exp( A i t)(a i x(t) + b i V DC /2) = exp( A i t)b i V DC /2,

9 9 so for any two times t and τ, with t τ: Thus, y(t) = y(τ) + (V DC /2) t = exp( A i τ)x(τ) + (V DC /2) x(t) = exp(a i t)y(t) τ exp( A i s)b i ds t τ exp( A i s)b i ds. = exp(a i (t τ))x(τ) + c i (t τ), (7) where c i (t τ) := (V DC /2) t τ exp(a i(t s))b i ds. Denote α := 1/(2T L ), w := 1/ LC and w := w 2 α 2. We consider from here on the case where the capacitance is sufficiently small so that: R < 2 L/C. (8) Note that this implies that w is real. 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(a 1 t) = exp(a 3 t) = ( exp(a 2 t) = exp( αt) ( exp( t/tl ) 1 ), 1 s (wt) sin(wt) wl 1 sin(wt) s wc +(wt) ( s (wt) exp(a 4 t) = exp( αt) 1 sin(wt) wl 1 sin(wt) s wc +(wt) where s (l) := cos(l) α sin(l), s w +(l) := cos(l) + α sin(l), and w c 1 (t) = c 3 (t) = V ( DC 1 exp( t/tl ) 2R c 2 (t) = V ( DC 1 sin(wt) exp( αt) wl 2 1 s + (wt) exp( αt) c 4 (t) = V DC 2 ( 1 wl sin(wt) exp( αt) 1 s + (wt) exp( αt) Let t k denote the switching times, i.e., t k := k i=1 t i (so in particular t 4 = T P W M ). Suppose that our initial state is x() and that we repeatedly apply the phase sequence P 1 P 2 P 3 P 4. Then (7) ), ), ). ), ),

10 1 yields Combining the first four equations yields where A calculation yields where x(t 1 ) = exp(a 1 t 1 )x() + c 1 ( t 1 ), x(t 2 ) = exp(a 2 t 2 )x(t 1 ) + c 2 ( t 2 ), x(t 3 ) = exp(a 3 t 3 )x(t 2 ) + c 3 ( t 3 ), (9) x(t 4 ) = exp(a 4 t 4 )x(t 3 ) + c 4 ( t 4 ), x(t 5 ) = exp(a 1 t 1 )x(t 4 ) + c 1 ( t 1 ),. A := exp(a 4 t 4 ) exp(a 3 t 3 ) x(t 4 ) = Ax() + b, (1) exp(a 2 t 2 ) exp(a 1 t 1 ), (11) b := exp(a 4 t 4 ) exp(a 3 t 3 ) exp(a 2 t 2 )c 1 ( t 1 ) + exp(a 4 t 4 ) exp(a 3 t 3 )c 2 ( t 2 ) + exp(a 4 t 4 )c 3 ( t 3 ) + c 4 ( t 4 ). A = q ( r 2 s 2 (z) + r w 2 LC sin2 sin(z) (z) (rs wl (z) s + (z)) r sin(z) (rs 1 wc (z) s + (z)) w 2 LC sin2 (z)r + s 2 + (z) r := exp( αdt P W M ), z := w(1 D)T P W M /2, Since the modulation is periodic, (1) yields ), q := exp( α(1 D)T P W M ). (12) x(t 4(k+1) ) = Ax(t 4k ) + b, k =, 1,.... (13) Let p A (λ) := det(λi A) denote the characteristic polynomial of the matrix A. A calculation yields p A (λ) := λ 2 ((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 i L (t). For k, (13) describes the asymptotic behavior of the FCC as t. The next two sections analyze these two cases.

11 11 IV. ASYMPTOTIC 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 z = w(1 D)T P W M /2 1. (15) Lemma 1 There exists a constant p > such that for all z [, p) the eigenvalues of the matrix A satisfy λ i < 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 x := (I A) 1 b. (16) Then, as t the state vector x(t) converges to a periodic solution. Each period is described by the solution of: A 1 x(t) + b 1 V DC /2, t [, t 1 ), A 2 x(t) + b 2 V DC /2, t [t 1, t 2 ), ẋ(t) = (17) A 3 x(t) + b 3 V DC /2, t [t 2, t 3 ), A 4 x(t) + b 4 V DC /2, t [t 3, t 4 ), with x() = x. Proof. Eq. (13) is a linear difference equation for the state at times t 4k, whose solution is: k 1 x(t 4k ) = A k x() + A j b. (18) Taking the limit as k, and using the assumption that A is an asymptotically stable matrix yields lim k x(t 4k ) = (I A) 1 b. Using (6) completes the proof. Note that Eqs. (16) and (11) imply that x depends on the modulation strategy and the parameter values of the converter and the load, but is independent of the initial condition x(). 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 x is independent of the initial condition x() has an additional implication. Recall that we considered the result of repeatedly applying the phase sequence P 1 P 2 P 3 P 4. Now suppose that starting from x(), we repeatedly apply, say, the phase sequence P 2 P 3 P 4 P 1. Let x := exp( A 1 t 1 )(x() c 1 ( t 1 )). Then this is equivalent to repeatedly applying the phase sequence P 1 P 2 P 3 P 4 with x() = x. Since our results are independent of the initial condition, this implies that we will obtain exactly the same asymptotic behavior. j=

12 12 6 V C I L Time, s x 1 3 Fig. 7. Asymptotic behavior for D =.2. Here x() = x. Example 3 Consider the FCC with the parameters given in Example 1 and D =.2. In this case, a calculation yields: ( ) ( ) A =, b = The eigenvalues of A are λ 1 =.3248, λ 2 =.9274, so A is asymptotically stable. Eq. (16) yields ( ) x = 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, x admits a particularly simple form. ( ) Proposition 1 If D =, then x =. V DC /2 Note that this agrees, of course, with the simulation results depicted in Fig. 4(a). Proof. For D =, t 1 = t 3 = and t 2 = t 4 = T P W M /2. Hence, (11) simplifies to A = exp(a 4 T P W M /2) exp(a 2 T P W M /2), b = exp(a 4 T P W M /2)c 2 (T P W M /2) + c 4 (T P W M /2). A calculation shows that I A has the form: ( exp( αtp W M ) 1 (s wl +(z) s (z)) sin(z) 1 exp( αt P W M )(s 2 + (z) + 1 w 2 LC sin2 (z)) ),

13 13 where denotes values that are not important for the proof. Also, b = (V DC /2) ( 1 wl exp( αt P W M)(s + (z) s (z)) sin(z) 1 exp( αt P W M )(s 2 + (z) + 1 w 2 LC sin2 (z)) ( where z = wt P W M /2. Using these expressions, it is easy to verify that (I A) ), V DC /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. TRANSIENT 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 i.e. A = and b =. Then, the solution of (1) satisfies λ 2 x i (T P W M ) = λ i x i (), i = 1, 2. (19) The response of a continuous time system with a diagonal matrix and time constants T i, i = 1, 2, is x i (t) = exp( t/t i )x i (), so x i (T P W M ) = exp( T P W M /T i )x i (), i = 1, 2. Comparing this with (19) yields λ i = exp( T P W M /T i ). We thus define the time constants as: T i := T P W M / ln(λ i ), (2) where λ i are the eigenvalues of the matrix A. For the sake of simplicity, we first consider the case D =, and only then turn to the case of an arbitrary duty cycle D [, 1). A. Zero duty ratio Consider the case where D =. Then t 1 = t 3 =, t 2 = t 4 = T P W M /2. In other words, the PWM period consists of only two time intervals corresponding to phases P 2 and P 4. In this case, r = 1 and the characteristic polynomial of A given in (14) simplifies to p A (λ) = λ 2 2(1 + 2δ 2 )qλ + q 2, (21) with δ := α w sin(z), z = wt P W M/2, q = exp( αt P W M ). (22) Since we consider the case where < z 1 (see (15)), δ > and we may treat δ as a small parameter.

14 14 with The roots of (21) are: λ i = qs i, i = 1, 2, (23) s 1 = 1 + 2δ 2 2δ 1 + δ 2, s 2 = 1 + 2δ 2 + 2δ 1 + δ 2. (24) Note that < s 1 < 1 < s 2. Since, by definition, q (, 1), this implies that λ 1 (, 1). Also, λ 2 (, 1) whenever qs 2 < 1. Using (24) yields: ln(s 1 ) = 2δ + δ 3 /3 + O(δ 5 ), ln(s 2 ) = 2δ δ 3 /3 + O(δ 5 ). Using (2), (22), (23) and the Taylor series sin(x) = x x3 3! + O(x 5 ) yields the time constants: T 1 L R, T 2 48 L R LC T 2 P W M. (25) The first time constant T 1 = T L is, as expected, the one associated with the load. This agrees, of course, with the transient behavior of i L (t) (see (5) and Fig. 5). The second time constant can be expressed as T 2 = 48LCT 1 /TP 2 W M. Our simulations indicate that an exponential with time constant T 2 provides a very good approximation of the transient behavior of the capacitor s voltage v c (t). In other words, The next example demonstrates this. v c (t) (1 exp( t/t 2 ))(V DC /2). (26) Example 4 Consider the FCC with the parameter values given in Example 1 and D =. In this case T 2 = 1/3. Fig. 8 depicts the behavior of v c (t) vs. the function given by (26). It may be seen that after a short initial transient, v c (t) and the exponential curve are practically indistinguishable. Summarizing, the analysis yields simple and explicit expressions for the two time constants T 1 = T L and T 2 that determine the behavior of i L and v c, respectively. Note that the fact that the time constant of i L is just T L, and is independent of the capacitance C, has a simple physical interpretation. Indeed, the opposite symmetry of the two phases P 2 and P 4 (see Fig. 3) suggests that on average the capacitor has no effect on the current i L. B. General case We now consider the more general case where D [, 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( αdt P W M ) = exp( αµz/w), with µ := 2D/(1 D). Using

15 Time, s Fig. 8. v c(t) vs. the exponential curve given by Eq. (26) (solid line). the geometric series for the functions sin, cos, and exp yields: The discriminant of (14) is and expending yields: b/q = 2 + 2µ α α2 z 2(2 + µ(2 + µ)) w w 2 z2 + 4 α3 µ(3 + µ(3 + µ)) 3 w 3 z3 (27) + α2 (2 + µ)(2 + 3µ)w 2 α 4 µ 2 (9 + 2µ(4 + µ)) z 4 3w 4 + O(z 5 ). : = b 2 4r 2 q 2 = q 2 (s + (z) rs (z)) 2 ((s + (z) rs (z)) 2 + 4r), where = 4q 2 (2 + µ) 2 α2 w 2 z2 (1 + p(z)) + O(z 5 ), p(z) := 2µ α w z + α2 (6 + µ(9 + µ(18 + 7µ))) (2 + 3µ)w 2 3(2 + µ)w 2 z 2. Using the series 1 + x = 1 x/2 + x 2 /8... yields 2q(2 + µ) α w z(1 p/2 + p2 /8). (28)

16 D Fig. 9. The function f(d) as a function of D. The roots of (14) are λ 1,2 = ( b ± )/2 and using (27) and (28) yields with: s 1 = 1 2(1 + µ)αz/w + O(z 2 ), s 2 = 1 + 2αz/w + 2α 2 z 2 /w 2 λ i = qs i, (29) (α/6)(3α 2 (µ 2) + (2 + 3µ)w 2 )z 3 /w 3 + O(z 4 ). Using the series ln(1 + x) = x x 2 /2 + x 3 /3 + O(x 4 ) yields ln(s 1 ) = 2(1 + µ)αz/w + O(z 2 ), ln(s 2 ) = 2αz/w (α/6)(2 + 3µ)(a 2 + w 2 )z 3 /w 3 + O(z 4 ). Using (2), (12), and (29) yields the time constants: T 1 (D) L/R, T 2 (D) 48 L (1 D) 2 (1 + 2D) R LC T 2 P W M. (3) Note that T 1, i.e. the time constant associated with i L is just T L. The second time constant 1 satisfies T 2 (D) = f(d)t 2 (), with f(d) := (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 (1 D) 2 (1+2D) the range D [, 1/2], and becomes more dramatic for values D >.6. This agrees well with the changes in the behavior of v c as a function of D (see Fig. 4). As D 1, T 2 (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 T i (D), i = 1, 2, provide an excellent approximation of the average behavior of the state variables of the converter. The next example demonstrates this.

17 17 6 v c i L Time, s Fig. 1. Behavior of the FCC for D =.5 and exponentials with time constants T i (solid lines). Example 5 Consider the FCC with the parameters given in Example 1 and D =.5. Fig. 1 depicts the FCC state variables with x 1 () = x 2 () =. Also depicted are the exponentials (5) and (26) with T 1 (D) and T 2 (D) given in (3). It may be seen that the exponentials provide an excellent approximation of the average behavior of the state variables. Note that for R, T 1, T 2, 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. V COM (t) = M sin(ηt), with M [, 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 [kt P W M, (k+1)t P W M ) of the triangular wave, V COM (t) V COM (kt P 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/T 2 (D) = (1 D) 2 (1 + 2D)/T 2 (). Note that in our analysis of the DC-modulated case, we assumed that D [, 1] (see (4)). Thus, we average g(d(t)) = g(m sin(ηt)) on [, π/η] (rather than [, 2π/η]) as this is the interval for which D(t) [, 1]. A calculation yields ḡ := 1 π/η π/η Thus, the averaged time constant is g(m sin(ηt))dt = 16M 3 9M 2 π + 6π. 6πT 2 () T 2 (M) := 1 ḡ = T 2 () 1 1.5M 2 + 8M 3 /(3π), so our approximation in the AC modulation case is v c (t) (1 exp( t/ T 2 (M)))(V DC /2). (31)

18 Time, s Fig. 11. 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 v c (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 v C (t), as a function of time, for V COM (t) = M sin(1t) with M =.5. In this case, T2 (M) = 1/ Fig 11 also depicts the approximation (31) for this value of ḡ. It may be seen that v c (t) indeed provides an excellent approximation of the average value of the capacitor voltage. VII. DISCUSSION 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], [2] and the references therein). The time domain model developed here may also be useful in this context.

19 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. APPENDIX: PROOF OF LEMMA 1 For z =, the characteristic polynomial of the matrix A (see (14)) becomes p A (λ) := λ2 + a 1 λ + a, (32) with a 1 := ((1 r) 2 + 2r)q and a = r 2 q 2. It is well known [21, Fact ] that a necessary and sufficient condition for asymptotic stability of the polynomial p A is that a < 1 and a 1 < 1 + a. The condition a < 1 clearly holds since, by definition, r, q < 1. The second condition is equivalent to q(1 exp( αdt P W M )) 2 < (1 exp( αt P W M )) 2. Since α >, q < 1 and D [, 1), this inequality indeed holds. We conclude that the polynomial p A is asymptotically stable, and a continuity argument completes the proof. REFERENCES [1] J.-S. Lai and F. Z. Peng, Multilevel converters a new breed of power converters, IEEE Trans. Industry Applications, vol. 32, pp , [2] D. G. Holmes and T. A. Lipo, Pulse Width Modulation for Power Converters: Principles and Practice. Wiley-IEEE Press, 23. [3] T. Meynard, H. Foch, P. Thomas, J. Courault, R. Jakob, and M. Nahrstaedt, Multicell converters: basic concepts and industry applications, IEEE Trans. Industrial Electronics, vol. 49, pp , 22. [4] B. A. Welchko, M. B. de R. Correa, and T. A. Lipo, A three-level MOSFET inverter for low-power drives, IEEE Trans. Industrial Electronics, vol. 51, pp , 24. [5] R. Teichmann and S. Bernet, A comparison of three-level converters versus two-level converters for low-voltage drives, traction, and utility applications, IEEE Trans. Industry Applications, vol. 41, pp , 25. [6] J. Pou, R. Pindado, and D. Boroyevich, Voltage-balance limits in four-level diode-clamped converters with passive front ends, IEEE Trans. Industrial Electronics, vol. 52, pp , 25. [7] S. Busquets-Monge, S. Alepuz, J. Rocabert, and J. Bordonau, Pulsewidth modulations for the comprehensive capacitor voltage balance of n-level diode-clamped converters, in Proc. IEEE Power Electronics Specialists Conference (PESC), Rhodes, Greece, 28, pp [8] S. Busquets-Monge, S. Alepuz, J. Bordonau, and J. Peracaula, Voltage balancing control of diode-clamped multilevel converters with passive front-ends, IEEE Trans. Power Electronics, vol. 23, pp , 28. [9] T. Meynard, M. Fadel, and N. Aouda, Modeling of multilevel converters, IEEE Trans. Industrial Electronics, vol. 44, pp , [1] X. Yuang, H. Stemmler, and I. Barbi, Self-balancing of the clamping-capacitor-voltages in the multilevel capacitorclamping-inverter under sub-harmonic PWM modulation, IEEE Trans. Power Electronics, vol. 16, pp , 21. [11] R. H. Wilkinson, H. de Mouton, and T. Meynard, Natural balance of multicell converters: the two-cell case, IEEE Trans. Power Electronics, vol. 21, pp , 26. [12], Natural balance of multicell converters: the general case, IEEE Trans. Power Electronics, vol. 21, pp , 26. [13] B. P. McGrath and D. Holmes, Analytical modeling of voltage balance dynamics for a flying capacitor multilevel converter, IEEE Trans. Power Electronics, vol. 23, pp , 28. [14] D. Liberzon, Switching in Systems and Control. Boston: Birkhäuser, 23.

20 2 [15] M. Margaliot, Stability analysis of switched systems using variational principles: an introduction, Automatica, vol. 42, pp , 26. [16] M. S. Branicky, Introduction to hybrid systems, in Handbook of Networked and Embedded Control Systems, D. Hristu-Varsakelis and W. Levine, Eds. Birkhauser, 25. [Online]. Available: [17] E. I. Silva, B. P. McGrath, D. E. Quevedo, and G. C. Goodwin, Predictive control of a flying capacitor converter, in Proc. 26th American Control Conference (ACC7), New York, USA, 27, pp [18] C. Feng, J. Liang, and V. G. Agelidis, Modified phase-shifted PWM control for flying capacitor multilevel converters, IEEE Trans. Power Electronics, vol. 22, pp , 27. [19] B. McGrath, T. Meynard, G. Gateau, and D. Holmes, Optimal modulation of flying capacitor and stacked multicell converters using a state machine decoder, IEEE Trans. Power Electronics, vol. 22, pp , 27. [2] A. Shukla, A. Ghosh, and A. Joshi, Improved multilevel hysteresis current regulation and capacitor voltage balancing schemes for flying capacitor multilevel inverter, IEEE Trans. Power Electronics, vol. 23, pp , 28. [21] D. S. Bernstein, Matrix Mathematics. Princeton University Press, 25.