Design-Oriented Stability of Outer Voltage Loop in Capacitor Current Controlled Buck Converters

Abstract

Due to the inherent feedforward of load current, capacitor current (CC) control shows a fast transient response that makes it suitable for the power supplies used in various portable electronic devices. However, considering the effect of the outer voltage loop, the stable range of the duty-cycle is significantly diminished in CC controlled buck converters. To investigate the stability effect of the outer voltage loop on buck converters, a CC controlled buck converter with a proportion-integral (PI) compensator is taken as an example, and its second-order discrete-time model is established. Based on this model, the instability caused by the duty-cycle is discussed with consideration of the outer voltage loop. Then the dynamical effects of the feedback gain of the PI compensator and the equivalent series resistance (ESR) of the output capacitor on the CC controlled buck converter with a PI compensator are studied. Furthermore, the design-oriented closed-loop stability criterion is derived. Finally, PSIM simulations and experimental results are supplied to verify the theoretical analyses.

I. INTRODUCTION

Recently, the transient performance of power supplies has come to be regarded as an important index for powering microprocessors or portable electronics devices [1]-[4]. As a result, V² control featuring a fast transient response has been proposed and has attracted a lot of attention [5]-[9]. However, the stability of V² control switching dc-dc converters is greatly affected by the equivalent series resistance (ESR) of the output capacitor [10]. When the ESR of the output capacitor is large, V² control switching dc-dc converters operate in the stable period-1 state and show better control performance. On the other hand, when the ESR of the output capacitor is small, the converter operates in the unstable state, which results in subharmonic and chaotic oscillations [11], [12]. However, a larger ESR of the output capacitor can result in a larger output voltage ripple and a lower steady-state performance of the converter. To improve the stability of V² control with a small ESR output capacitor, the inductor current ripple was sensed to compensate the output voltage ripple in the control loop, which can deteriorate the load transient performance of the converter [3], [9], [13].

To solve the above problem, capacitor current (CC) control was proposed and widely used in switching dc-dc converters [14]-[19]. The capacitor current is equal to the difference between the inductor current and the output current. Thus, the control has inherent feedforward of the load current and shows a fast transient response [14]. In addition, CC control can eliminate the subharmonic and chaotic oscillations caused by the output capacitor voltage phase lagging behind the inductor current phase, and improve the stability of the converter [17]-[19].

To analyze and design a CC controlled switching dc-dc converter, the output impedance characteristic of the CC controlled buck converter was investigated in [14] and it was concluded that the CC control is equivalent to inductor current control combined with load current feedforward. Based on the describing function method, the stability of the capacitor current fixed off-time controlled buck converter was studied in [18]. In addition, the design optimization methodology for the capacitor current compensated constant on-time V² control was proposed in [19]. However, the stability effect of the outer voltage loop on the converter was not considered in these studies.

To study the stability effects of the outer voltage loop on converters, some research works were carried out [20]-[24]. Taking the compensator capacitor voltage in the outer voltage loop as a state variable, third-order discrete-time models for a peak-current-mode controlled switching dc-dc converter with a proportion-integral (PI) compensator were established. For these models, complex dynamic behaviors such as coexisting fast-scale and slow-scale instabilities [20], and the complex interaction between the tori and onset of three-frequency quasi-periodicity [21], were revealed. To solve the inaccuracy caused by the approximation used in [20], an accurate discrete-time model combined with the Floquet theory was proposed to analyze the subharmonic oscillation in a V²I_C controlled buck converter with a Type-I compensator in [22]. In addition, an improved discrete-time model was proposed to investigate the stability of a peak-current-mode controlled buck LED driver in [23]. However, the design-oriented closed-loop stability criterion has not been found in [20]-[23], which makes it inconvenient to choose the circuit parameters of converters.

By representing the compensator capacitor voltage in the outer voltage loop with a linear combination of the inductor current and the output capacitor, a second-order discrete-time model for a constant on-time current-mode controlled buck converter was proposed in [24], and used to obtains the design-oriented instability condition. In fact, constant on-time control is a kind of pulse frequency modulation control technique with a variable-frequency [24], whereas CC control is a kind of pulse width modulation control technique with a constant-frequency, where the stability can be affected by the duty-cycle. Specifically, CC control uses the capacitor current instead of the inductor current as the inner loop control signal, which can exhibit a fast load transient performance.

In this paper, a CC controlled buck converter with a proportion-integral (PI) compensator is taken as an example. The stability effect of the outer voltage loop on CC control is investigated and a design-oriented closed-loop stability criterion is derived. In Section II, a CC controlled buck converter with a PI compensator is briefly described and its second-order discrete-time model is established. In Section III, the dynamical behaviors of the converter with variations of the duty-cycle, feedback gain of the PI compensator, and ESR of the output capacitor are investigated by bifurcation diagrams, maximal Lyapunov exponents and loci of eigenvalues. In Section IV, by simplifying the second-order discrete-time model, a design-oriented closed-loop stability criterion for a CC controlled buck converter with a PI compensator is derived. Based on this, a stability interface to divide the stable space and unstable space in the space of the feedback gain, output capacitor ESR, and duty-cycle is given. In Section V, a PSIM simulation model and an experimental prototype are implemented to verify the theoretical results. Finally, some conclusions are made in Section VI.

II. SYSTEM DESCRIPTION AND ITS SECOND-ORDER DISCRETE-TIME MODEL

A. System Description

A schematic diagram of a CC controlled buck converter with a PI compensator is shown in Fig. 1. The capacitor current i_C = i_L ‒ i_o is sensed by sensing resistor R_s and taken as the inner control loop. Meanwhile, the proportional-integral (PI) compensator with the feedback gain g = R_a/R_in and the time constant τ_a = R_aC_a is used as the outer voltage loop. The CC controlled buck converter with a PI compensator has three state variables, consisting of the inductor current i_L(t), the output capacitor voltage v_C(t), and the compensation capacitor voltage v_a(t).

Fig. 1. Schematic diagram of a CC controlled buck converter with a PI compensator.

At the beginning of each switching cycle, the switch S is turned on and the sensed capacitor current R_si_C(t) increases. When R_si_C(t) increases to the control signal v_con(t), the switch S is turned off. Thus, the switching equation of the CC controlled buck converter with a PI compensator is written as

\(R_{\mathrm{s}} i_{C}(t)=v_{\mathrm{con}}(t)\)       (1)

In addition, the control signal can be expressed as

\(v_{\text {con }}(t)=(1+g) V_{\text {ref }}-g v_{\mathrm{o}}(t)-v_{a}(t)\)       (2)

where V_ref is the reference voltage, v_o = κ[ri_L(t) + v_C(t)] is the output voltage and κ = R/(R+r).

According to the states of the switch S and the diode D, the two switch states of the CC controlled buck converter with a PI compensator operating in the inductor current continuous conduction mode (CCM) can be identified as

Switch state 1: switch S is on and diode D is off.

Switch state 2: switch S is off and diode D is on.

In the m-th (m = 1, 2) switch state, the state equations of buck converter are described by

\(\dot{\mathbf{x}}(t)=\mathbf{A}_{m} \mathbf{x}(t)+\mathbf{B}_{m} V_{\text {in }}\left(t_{m-1} \leq t<t_{m}\right)\)       (3)

where x(t) = [i_L(t) v_C(t)]^T, and t_m–1 and t_m denote the time instants at the beginning and end of the m-th switch state, respectively. In addition, the matrices A_m and B_m are the state matrix and input matrix, whose expressions are written as

\(\mathbf{A}_{1}=\mathbf{A}_{2}=\left[\begin{array}{cc} -\frac{\kappa r}{L} & -\frac{\kappa}{L} \\ \frac{\kappa}{C} & -\frac{\kappa}{R C} \end{array}\right], \quad \mathbf{B}_{1}=\left[\begin{array}{l} \frac{1}{L} \\ 0 \end{array}\right], \quad \mathbf{B}_{2}=\left[\begin{array}{l} 0 \\ 0 \end{array}\right].\)

B. Second-Order Discrete-Time Model

Define the state variable x(t) at the time instant t_m–1 as x(t_m–1). By solving (3), the state variable x(t) at the time instant t_m can be obtained as [24]

\(\mathbf{x}\left(t_{m}\right)=\mathbf{P}_{m}\left(\tau_{m}\right) \mathbf{x}\left(t_{m-1}\right)+\mathbf{Q}_{m}\left(\tau_{m}\right) V_{\text {in }} \quad(m=1,2)\)       (4)

where τ_m = t_m – t_m–1 is the operation time interval of the m-th switch state, and the matrixes P_m and Q_m are expressed as

\(\mathbf{P}_{m}\left(\tau_{m}\right)=\left[\begin{array}{cc} a_{m}+\frac{\beta}{\omega} b_{m} & -\frac{\kappa}{\omega L} b_{m} \\ \frac{\kappa}{\omega C} b_{m} & a_{m}-\frac{\beta}{\omega} b_{m} \end{array}\right],\)

\(\mathbf{Q}_{1}\left(\tau_{1}\right)=\left[\begin{array}{c} \frac{1}{R}-\frac{1}{R} a_{1}+\frac{R-\alpha L}{\omega R L} b_{1} \\ 1-a_{1}-\frac{\alpha}{\omega} b_{1} \end{array}\right], \mathbf{Q}_{2}\left(\tau_{2}\right)=\left[\begin{array}{l} 0 \\ 0 \end{array}\right],\)

\(\alpha=\frac{\kappa}{2}\left(\frac{1}{R C}+\frac{r}{L}\right), \beta=\frac{\kappa}{2}\left(\frac{1}{R C}-\frac{r}{L}\right), \omega=\sqrt{\frac{\kappa}{L C}-\alpha^{2}},\)

\(a_{m}=\cos \omega \tau_{m} e^{-\alpha \tau_{m}}, \quad b_{m}=\sin \omega \tau_{m} e^{-\alpha \tau_{m}}.\)

Denote x_n = [i_L,n, v_C,n]^T and x_n+1 = [i_L,n+1, v_C,n+1]^T to be the sampling values of x = [i_L, v_C]^T at the beginning of the n-th switching cycle and the (n+1)-th switching cycle, respectively. In the n-th switching cycle, R_si_C increases to v_con at the time instant t = t₁, and the state variable x(t₁) is obtained as

\(\mathbf{x}\left(t_{1}\right)=\left[i_{L}\left(t_{1}\right) v_{C}\left(t_{1}\right)\right]^{\mathrm{T}}=\mathbf{P}_{1}\left(t_{\mathrm{on}}\right) \mathbf{x}_{n}+\mathbf{Q}_{1}\left(t_{\mathrm{on}}\right) V_{\mathrm{in}}\)       (5)

where t_on is the on-time of the n-th switching cycle, which can be calculated numerically by combining (1) and (5).

At t = (n+1)T, i.e. at the end of the n-th switching cycle, the state variable x_n+1 is written as

\(\mathbf{x}_{n+1}=\left[i_{L, n+1}, v_{C, n+1}\right]^{\mathrm{T}}=\mathbf{P}_{2}\left(T-t_{\mathrm{on}}\right) \mathbf{x}\left(t_{1}\right)+\mathbf{Q}_{2}\left(T-t_{\mathrm{on}}\right) V_{\mathrm{in}}\)       (6)

By referring to [24], the compensation capacitor voltage v_a(t) at t = t₁ and at t = (n+1)T is expressed as

\(v_{a}\left(t_{1}\right)=v_{a, n}+\frac{g}{\tau_{a}}\left(V_{\text {in }}-V_{\text {ref }}\right) t_{\text {on }}-\frac{g L}{\tau_{a}}\left[i_{L}\left(t_{1}\right)-i_{L, n}\right]\)       (7a)

and

\(v_{a, n+1}=v_{a}\left(t_{1}\right)-\frac{g}{\tau_{a}} V_{\mathrm{ref}}\left(T-t_{\mathrm{on}}\right)-\frac{g L}{\tau_{a}}\left[i_{L, n+1}-i_{L}\left(t_{1}\right)\right]\)       (7b)

where v_a,n and v_a,n+1 are sampling values of the compensation capacitor voltage v_a(t) at the beginning of the n-th and (n+1)-th switching cycles, respectively.

Substituting (7a) into (7b) yields

\(v_{a, n+1}=v_{a, n}+\frac{g}{\tau_{a}}\left(V_{\mathrm{in}} t_{\mathrm{on}}-V_{\mathrm{ref}} T\right)-\frac{g L}{\tau_{a}}\left(i_{L, n+1}-i_{L, n}\right)\)       (8)

which indicates that v_a,n+1 is a linear representation of i_L,n+1.

Therefore, the second-order discrete-time model of the CC controlled buck converter with a PI compensator in CCM can be unified as

\(\mathbf{x}_{n+1}=\mathbf{P}_{1}\left(t_{\mathrm{on}}\right) \mathbf{P}_{2}\left(T-t_{\mathrm{on}}\right) \mathbf{x}_{n}+\mathbf{Q}_{1}\left(t_{\mathrm{on}}\right) \mathbf{P}_{2}\left(T-t_{\mathrm{on}}\right) V_{\mathrm{in}}\)       (9)

It is marked that model (9) is only a second-order discrete-time model, unlike the existing third-order discrete-time model [20].

Based on model (9), the Jacobian of the CC controlled buck converter with a PI compensator for the given equilibrium state can be calculated by

\(\mathbf{J}_{n}=\left[\begin{array}{ll} J_{11} & J_{12} \\ J_{21} & J_{22} \end{array}\right]\)       (10)

where J₁₁, J₁₂, J₂₁ and J₂₂ are derived in the Appendix.

Correspondingly, the maximal Lyapunov exponents of the CC controlled buck converter with a PI compensator are calculated by

\(\left[\begin{array}{c} \lambda_{L 1} \\ \lambda_{L 2} \end{array}\right]=\lim _{n \rightarrow \infty} \frac{1}{n} \ln \left|\operatorname{eig}\left(\prod_{k=1}^{n} \mathbf{J}_{k}\right)\right|\)       (11)

\(\lambda_{L}=\max \left(\lambda_{L 1}, \lambda_{L 2}\right)\)       (12)

where J_k is the Jacobian of the converter evaluated along the trajectory, and eig(•) and max(•) are the eigenvalue function and maximal member function, respectively.

III. STABILITY EFFECTS WITH VARIATIONS OF SPECIFIC CIRCUIT PARAMETERS

To investigate the dynamic behaviors of the CC controlled buck converter with a PI compensator, the circuit parameters are chosen in Table I.

TABLE I TYPICAL CIRCUIT PARAMETERS FOR THE CC CONTROLLED BUCK CONVERTER WITH A PI COMPENSATOR

A. Instability Caused by Input Voltage

For a buck converter with a constant voltage output, the variation of the duty-cycle D = V_o/V_in (0 < D < 1) can be realized by varying the input voltage V_in. Thus, the stability effect of the duty-cycle on the CC controlled buck converter with a PI compensator can be explored by varying the input voltage. When the input voltage V_in is taken as the bifurcation parameter and the other circuit parameters are given in Table I, bifurcation diagrams and maximal Lyapunov exponents of model (9) are shown in Figs. 2(a) and 2(b), respectively.

Fig. 2. Stability effect with V_in increasing. (a) Bifurcation diagrams of i_L,n and v_o,n. (b) Maximal Lyapunov exponent.

In Fig. 2, with V_in decreasing, the first period-doubling bifurcation occurs at V_in = 14.43 V and the operation state of the CC controlled buck converter with a PI compensator is from period-1 to period-2, implying the occurrence of instability in the converter. With V_in decreasing further, the operation state changes from period-2, to period-4, to period-8, and then to chaos by successive period-doubling bifurcation and border-collision bifurcation routes.

At the first period-doubling bifurcation point, the duty-cycle is D = V_o/V_in ≈ 0.37, which is less than 0.5. This indicates that the stable range of the duty-cycle is significantly diminished with consideration of the outer voltage loop.

B. Stability Effects of the Feedback Gain and Output Capacitor ESR

According to (2), the control signal generated by the outer voltage loop is related to the feedback gain g and output capacitor ESR r. Thus, the stability effects of the feedback gain g and output capacitor ESR r on the CC controlled buck converter with a PI compensator are stressed.

Based on model (9) and the circuit parameters given in Table I, bifurcation diagrams with variations of g and r are shown in Figs. 3(a) and 3(b), respectively. From Fig. 3, the CC controlled buck converter with a PI compensator goes through a period-doubling bifurcation point at g = 6.41 or r = 87.2 mΩ, which leads to its operation state changing from period-1 to period-2. With g increasing or r decreasing further, the operation state of the converter changes from period-2, to period-4, to period-8, and finally to chaos.

Fig. 3. Bifurcation diagrams of i_L,n and v_o,n with increases in the (a) Feedback gain g; (b) Output capacitor ESR r.

The loci of two eigenvalues, λ₁ and λ₂, of the CC controlled buck converter with a PI compensator with g increasing or r decreasing are depicted in Fig. 4. The results imply that λ₁ and λ₂ are two nonzero real eigenvalues. When g increases from 6.0 to 7.0 or r decreases from 80 mΩ to 90 mΩ, λ₁ is always located in the unit circle, and λ₂ leaves the unit circle via ‒1 at g = 6.41 or r = 87.2 mΩ, resulting in the emergence of period-doubling bifurcation. Accordingly, typical eigenvalues and corresponding operation states are listed in Table II and Table III, respectively.

Fig. 4. Loci of the eigenvalues λ_i (i = 1, 2) when the (a) Feedback gain g is increasing; (b) Output capacitor ESR r is increasing.

TABLE II TYPICAL EIGENVALUES AND OPERATION STATES FOR DIFFERENT VALUES OF FEEDBACK GAIN

TABLE III TYPICAL EIGENVALUES AND OPERATION STATES FOR DIFFERENT VALUES OF OUTPUT CAPACITOR ESR

From Fig. 3 and Fig. 4, it can be found that the feedback gain of the PI compensator and the ESR of the output capacitor have significant influences on the stability of the CC controlled buck converter with a PI compensator.

C. Design-Oriented Closed-Loop Stability Criterion

Based on the condition that the rising slope m₁ = (V_in ‒ V_o)/L and falling slope ‒m₂ = ‒V_o/L are constant in each switching cycle, the second-order discrete-time model (9) can be approximated as

\(\left\{\begin{array}{l} i_{L, n+1}=i_{L, n}+m_{1} t_{\mathrm{on}}-m_{2}\left(T-t_{\mathrm{on}}\right) \\ v_{C, n+1}=v_{C, n}+\frac{T}{C}\left(i_{L, n}-I_{\mathrm{o}}\right)-\frac{\left(m_{1}+m_{2}\right) t_{\mathrm{on}}^{2}}{2 C}+\frac{\left(m_{1}+m_{2}\right) T t_{\mathrm{on}}}{C}-\frac{m_{2} T^{2}}{2 C} \end{array}\right.\)       (13)

where I_o = V_o/R is the averaged output current.

Combining (10) and (13), the Jacobian for the approximate second-order discrete-time model (13) is expressed as

\(\mathbf{J}_{n}=\left[\begin{array}{cc} J_{11} & J_{12} \\ J_{21} & J_{22} \end{array}\right]=\left[\begin{array}{cc} 1+\left(m_{1}+m_{2}\right) A & \left(m_{1}+m_{2}\right) B \\ \frac{T}{C}+\frac{m_{1} T}{C} A & 1+\frac{m_{1} T}{C} B \end{array}\right]\)       (14)

where

\(A=\frac{\partial t_{\mathrm{on}}}{\partial i_{L, n}}=\frac{-\left(g \kappa R-\kappa R_{\mathrm{s}}\right)\left(t_{\mathrm{on}}+r C\right)-R_{\mathrm{s}} R C}{\left(g \kappa R-\kappa R_{\mathrm{s}}\right)\left(i_{L, n}-I_{\mathrm{o}}+m_{1} t_{\mathrm{on}}+m_{1} r C\right)+m_{1} R_{\mathrm{s}} R C}\)       (15)

\(B=\frac{\partial t_{\mathrm{on}}}{\partial v_{C, n}}=\frac{-\left(g \kappa R-\kappa R_{\mathrm{s}}\right) C}{\left(g \kappa R-\kappa R_{\mathrm{s}}\right)\left(i_{L, n}-I_{\mathrm{o}}+m_{1} t_{\mathrm{on}}+m_{1} r C\right)+m_{1} R_{\mathrm{s}} R C}\)       (16)

In the neighborhood of the instability boundary, there exist the following approximate conditions

\(i_{L, n} \approx I_{\mathrm{o}}-0.5 \frac{m_{1} m_{2} T}{m_{1}+m_{2}}, t_{\mathrm{on}} \approx \frac{m_{2} T}{m_{1}+m_{2}}\)       (16)

Substituting (16) and κ = R/(R+r) into (15) yields

\(A=\frac{-\left(r C+\frac{m_{2} T}{m_{1}+m_{2}}\right)\left(g R-R_{\mathrm{s}}\right)-R_{\mathrm{s}}(R+r) C}{m_{1}\left(r C+\frac{0.5 m_{2} T}{m_{1}+m_{2}}\right)\left(g R-R_{\mathrm{s}}\right)+m_{1} R_{\mathrm{s}}(R+r) C}\)       (17a)

\(B=\frac{-\left(g R-R_{\mathrm{s}}\right) C}{m_{1}\left(r C+\frac{0.5 m_{2} T}{m_{1}+m_{2}}\right)\left(g R-R_{\mathrm{s}}\right)+m_{1} R_{\mathrm{s}}(R+r) C}\)       (17b)

The two nonzero eigenvalues of (14) are described as

\(\lambda_{1,2}=0.5\left(J_{11}+J_{22}\right) \pm 0.5 \sqrt{\left(J_{11}+J_{22}\right)^{2}-4\left(J_{11} J_{22}-J_{12} J_{21}\right)}\)       (18)

To ensure normal stable operation, λ_1,2 must locate within −1 and 1 [25]. From Section III.B, it is known that the stability of the CC controlled buck converter with a PI compensator is lost by the first period-doubling bifurcation, i.e., λ₁ = ‒1 or λ₂ = ‒1 is the critical stable condition, which indicates that

\(0.5\left(J_{11}+J_{22}\right) \pm 0.5 \sqrt{\left(J_{11}+J_{22}\right)^{2}-4\left(J_{11} J_{22}-J_{12} J_{21}\right)}=-1\)

or

\(1+\left(J_{11}+J_{22}\right)+\left(J_{11} J_{22}-J_{12} J_{21}\right)=0\)       (19)

Using (17) and putting the Jacobian elements of (14) into (19), a simplified relation for the circuit parameters of the CC controlled buck converter with a PI compensator is derived as

\(\left(g R-R_{\mathrm{s}}\right)\left[2 r C-\frac{m_{1}^{2}+m_{2}^{2}}{m_{1}^{2}-m_{2}^{2}} T\right]+2 R_{\mathrm{s}}(R+r) C=0\)       (20)

Furthermore, substituting the rising slope m₁ = (V_in ‒ V_o)/L and falling slope ‒m₂ = ‒V_o/L into (20) and defining g_c as a critical feedback gain, the design-oriented closed-loop stability criterion can be given as

\(g_{\mathrm{c}}=\frac{R_{\mathrm{s}}(R+r) C}{\left(0.5+\frac{D^{2}}{\Delta}\right) R T-R r C}+\frac{R_{\mathrm{s}}}{R}\)       (21)

i.e.

\(g_{\mathrm{c}}=\frac{\left(D^{2}+0.5 \Delta\right) R_{\mathrm{s}} T+R_{\mathrm{s}} R C \Delta}{\left(D^{2}+0.5 \Delta\right) R T-R r C \Delta}\)       (22)

where D = V_o/V_in and ∆ = 1 ‒ 2D.

Consequently, for the condition \(\left(0.5+\frac{D^{2}}{\Delta}\right) R T-R r C>0\), i.e.

\(0.5+\frac{D^{2}}{\Delta}>\frac{r C}{T}\)       (23)

when

\(g<g_{\mathrm{c}}=\frac{\left(D^{2}+0.5 \Delta\right) R_{\mathrm{s}} T+R_{\mathrm{s}} R C \Delta}{\left(D^{2}+0.5 \Delta\right) R T-R r C \Delta}\)       (24)

the CC controlled buck converter with a PI compensator is stable, otherwise it is unstable. It is necessary to note that the feedback gain g of the outer voltage loop as well as the capacitance C and ESR r of the output capacitor can affect the stability of the buck converter when the outer voltage loop is closed, which has not been previously considered [25].

Based on (23), (24) and the circuit parameters given in Table I, an interface to divide the stable space and unstable space in the g-r-D space is obtained by using MATLAB software, as shown in Fig. 5. It is indicated that the stable range of the duty-cycle is significantly diminished with the feedback gain increasing or the output capacitor ESR decreasing.

Fig. 5. Stability interface used to divide the stable space and unstable space in the g-r-D space.

IV. CIRCUIT SIMULATIONS AND EXPERIMENTAL VERIFICATIONS

A. PSIM Circuit Simulations

According to Fig. 5, six kinds of operation conditions with different values of the feedback gain g, output capacitor ESR r and duty-cycle D are chosen in Table IV. Using PSIM software and the circuit parameters given in Table I, simulation results for the six kinds of operation conditions given in Table IV are given in Figs. 6-8, where (a1) and (b1) are time-domain waveforms, and (a2) and (b2) are phase plane trajectories.

TABLE IV SIX KINDS OF OPERATION CONDITIONS FOR THE CC CONTROLLED BUCK CONVERTER WITH A PI COMPENSATOR

In Figs. 6-8, when (g, r, D) are located in the stable space, the CC controlled buck converter with a PI compensator operates in the period-1 oscillation state, as shown in Figs. 6(a)‒8(a). On the other hand, when (g, r, D) are located in the unstable space, the converter operates in the subharmonic oscillation state, as shown in Figs. 6(b)‒8(b).

Fig. 6. PSIM simulations under different values of g. (a) (g, r, D) = (6, 10 mΩ, 0.33). (b) (g, r, D) = (7, 10 mΩ, 0.33)

Fig. 7. PSIM simulations under different values of r. (a) (g, r, D) = (13, 90 mΩ, 0.33). (b) (g, r, D) = (13, 40 mΩ, 0.33)

Fig. 8. PSIM simulations under different values of D. (a) (g, r, D) = (1, 10 mΩ, 0.33). (b) (g, r, D) = (1, 10 mΩ, 0.49)

To validate the accuracy of the stability interface in the g-r-D space shown in Fig. 5, two sets of circuit parameters near the point (g, r, D) = (4.29, 15mΩ, 0.4) on the stability interface are chosen as (g, r, D) = (4.2, 15mΩ, 0.4) and (g, r, D) = (4.4, 15mΩ, 0.4). According to Fig. 5, it is known that (g, r, D) = (4.2, 15mΩ, 0.4) are located in the stable space, whereas (g, r, D) = (4.4, 15mΩ, 0.4) are located in the unstable space. For the two sets of circuit parameters, PSIM simulation results are given in Fig. 9(a) and 9(b). As shown in Fig. 9(a), the CC controlled buck converter with a PI compensator for (g, r, D) = (4.2, 15 mΩ, 0.4) operates in the stable periodic oscillation state. However, as shown in Fig. 9(b), for (g, r, D) = (4.4, 15 mΩ, 0.4), it operates in the unstable subharmonic oscillation state.

Fig. 9. PSIM simulations under two sets of circuit parameters near the stability interface. (a) (g, r, D) = (4.2, 15 mΩ, 0.4). (b) (g, r, D) = (4.4, 15 mΩ, 0.4)

It can be concluded from Figs. 6-9 that when the outer voltage loop is considered, some of the circuit parameters including the feedback gain, output capacitor ESR, and duty cycle have great dynamic effects on the stability of the CC controlled buck converter with a PI compensator. A small feedback gain, a large output capacitor ESR, and a small duty cycle are better for the normal stable operation of the CC controlled buck converter with a PI compensator. Therefore, the PSIM simulation results validate the theoretical analyses.

B. Circuit-Implemented Experimental Verifications

To verify the validity of the PSIM simulation results, an experimental prototype of the CC controlled buck converter is built in the lab, as shown in Fig. 10. In the prototype, a IRF540 MOSFET, a MBR2045CT, and a LM319 are used as the switch, the diode, and the comparator. Meanwhile, an LT1357 is used as the operational amplifier to construct the PI compensator.

Fig. 10. Photograph of the experimental prototype

Based on the circuit parameters in Table I, experimental results for the six sets of circuit parameters given in Table IV are shown in Figs. 11-14, where (a1) and (b1) are time-domain waveforms, and (a2) and (b2) are phase plane trajectories. When the chosen circuit parameters are located in the stable space, the CC controlled buck converter with a PI compensator operates in the stable period-1 oscillation state, as shown in Figs. 11(a)-14(a). However, when the chosen circuit parameters are located in the unstable space, the converter operates in the unstable subharmonic oscillation state, as shown in Figs. 11(b)-14(b). Consequently, the experimental results given in Figs. 11-14 validate the PSIM simulations given in Figs. 6-9.

Fig. 11. Experimental results under different values of g. (a) (g, r, D) = (6, 10 mΩ, 0.33). (b) (g, r, D) = (7, 10 mΩ, 0.33).

Fig. 12. Experimental results under different values of r. (a) (g, r, D) = (13, 90 mΩ, 0.33). (b) (g, r, D) = (13, 40 mΩ, 0.33).

Fig. 13. Experimental results under different values of D. (a) (g, r, D) = (1, 10 mΩ, 0.33). (b) (g, r, D) = (1, 10 mΩ, 0.49).

Fig. 14. Experimental results under two sets of circuit parameters near the stability interface. (a) (g, r, D) = (4.2, 15 mΩ, 0.4). (b) (g, r, D) = (4.4, 15 mΩ, 0.4).

V. CONCLUSIONS

In this paper, a second-order discrete-time model of the CC controlled buck converter with a PI compensator is established, upon which the instability of the converter caused by the duty-cycle is discussed and the stability effects of the feedback gain of the PI compensator and the ESR of the output capacitor on the converter are studied. By simplifying the second-order discrete-time model, a design-oriented closed-loop stability criterion is derived, upon which a stability interface used to divide the stable space and the unstable space in the space of the feedback gain, output capacitor ESR, and duty-cycle is given. Theoretical analyses indicated that the stable range of the duty-cycle of the CC controlled buck converter with a PI compensator is significantly diminished with the feedback gain of the PI compensator increasing or the ESR of the output capacitor decreasing, which is verified by PSIM simulations and circuit-implemented experimental results. The results in this paper provide a guideline for choosing the cicuit parameters of CC controlled buck converters.