Advanced Power Electronic Converters and Power Quality Conditioning
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YuJun Mao, ChiSeng Lam, SaiWeng Sin, ManChung Wong, Rui Paulo Martins, "Review and Selection Strategy for HighAccuracy Modeling of PWM Converters in DCM", Journal of Electrical and Computer Engineering, vol. 2018, Article ID 3901693, 16 pages, 2018. https://doi.org/10.1155/2018/3901693
Review and Selection Strategy for HighAccuracy Modeling of PWM Converters in DCM
Abstract
Among various modeling methods for DCDC converters introduced in the past two decades, the statespace averaging (SSA) and the circuit averaging (CA) are the most general and popular exhibiting high accuracy. However, their deduction approaches are not entirely equivalent since they incorporate different averaging processes, thus yielding different small signal transfer functions even under identical operating conditions. Some research studies claimed that the improved SSA can obtain the highest accuracy among all the modeling methods, but this paper discovers and clearly verifies that this is not the case. In this paper, we first review and study these two modeling methods for various DCDC converters operating in the discontinuous conduction mode (DCM). We also streamline the general modelderiving processes for DCDC converters, and test and compare the accuracy of these two methods under various conditions. Finally, we provide a selection strategy for a highaccuracy modeling method for different DCDC converters operating in DCM and verified by simulations, which revealed necessary and beneficial for designing a more accurate DCM closedloop controller for DCDC converters, thus achieving better stability and transient response.
1. Introduction
A PWM DCDC converter operates either in the continuous conduction mode (CCM) or in the discontinuous conduction mode (DCM). For small inductances or light loads, DCM operation is occasionally unavoidable in DCDC converters, their design being intentionally in DCM to reduce both the inductor’s size and the switching frequency. Then, it is necessary to develop a modeling analysis for the DCDC converter operation in DCM to design a closedloop controller. In previous works [1–22], various small signal modeling methods of PWM DCDC converters in CCM and DCM are proposed, where different modeling methods provide either an analytical equation or an equivalent circuit and can be categorized as reducedorder or fullorder models, as summarized in Table 1.

Among these small signal modeling methods, the improved statespace averaging (SSA) and the circuit averaging (CA) are the latest to present, with high accuracy, an analytical equation, and an equivalent circuit, respectively. The improved SSA method [6] claims that it has the highest accuracy among all the other modeling methods and can be applied to any circuit composed by inductors and capacitors. But, this conclusion is only proven and verified in a boost converter under specific operating conditions [6]. Besides, this may not hold in other DCDC converters and under different operating conditions. Moreover, Zeng et al. [13] used the CA method from [5] to deduce the small signal model of a KY converter, but when applying the improved SSA method to the same case, the CA method yields a better modeling accuracy under some conditions that are discussed in this paper. Contradictions met in CA and SSA methods motivate further analysis and retesting their accuracy in various DCDC converters under different operating conditions in DCM, in order to derive a selection criterion that allows higher accuracy in the small signal modeling method.
Also, the rather complicated (and not general) original derivation processes of the CA and the improved SSA methods, in which the whole small signal model derivation process should be repeated whenever the DCDC circuit topology changes, whenever the DCDC circuit topology changes, leads to unnecessary and timeconsuming efforts. Then, another motivation for this paper is to generalize the whole derivation process to attain a general and intuitive modelderiving solution.
The main contributions of this paper are as follows:(1)To propose and deduct a general and intuitive derivation process of the improved SSA and CA modeling methods for different DCDC converters, such that their corresponding DCM small signal models can be easily determined.(2)To study, retest, and compare, through simulations, the accuracy of the improved SSA and CA modeling methods for various DCDC converters under different operating conditions. Since most of the previous works are either based on the improved SSA or CA, they lack a detailed comparison among them.(3)To propose a selection strategy of the DCM small signal modeling method in order to obtain high accuracy for different DCDC converters under different operating conditions.
This paper contributes significantly to the design of a stable and fast transient response DCM closedloop controller for different DCDC converters. Section 2 presents the DC analysis of different DCDC converters in DCM. Section 3 introduces two small signal relationship calculation methods based on the largesignal equation. Section 4 discusses the general DCM largesignal and small signal modeling deduction based on the improved SSA method applied to different DCDC converters. Then, Section 5 determines the general DCM largesignal and small signal modeling deduction based on the CA method. Section 6 compares the simulation results of the small signal modeling obtained through the improved SSA and the CA methods. Section 7 presents a selection strategy of the DCM small signal modeling for different DCDC converters. Finally, the conclusions are drawn in Section 8.
2. DC Analysis of DCDC Converters in DCM
In the discussion hereafter, if we use the capital letter to denote the averaged value of a specific variable in one switching frequency (DC value), then the lowercase letter denotes its largesignal value and denotes its small signal value. The relationship between these three variables is [10]
2.1. DCM Operation
The DCM operation of the DCDC converters consists of three intervals. Here, we use D_{1}, D_{2}, and D_{3} to denote the duty ratio of each interval, respectively, and T_{s} denotes the switching period. Figure 1 shows the inductor current, i_{L}, waveform of the DCDC converters operating in DCM [10].
For a typical DCDC converter, i_{L} starts to rise during the charging of the inductor L and begins to drop while the L is discharging. In this paper, we use (charging) and (discharging) to denote the voltage across the L in the first and the second intervals, respectively. From Figure 1, we can obtainwhere denotes the peak value of inductor current i_{L}, D_{1} denotes the duty ratio of charging interval, and T_{s} denotes the switching period. From Figure 1, the relationship between the peak value () and average value () of the inductor current can be expressed aswhere denotes the average value of the inductor current i_{L} while T_{s} and D_{2} denotes the duty ratio of the discharging interval. From Figure 1,
Equation (4) yields the followingwhere (charging) and (discharging) denote the voltage across the L in the first and the second intervals, respectively. Considering (2), (3), and (5), can be expressed as
2.2. DC Parameters Calculation for Different DCDC Converters in DCM
Before we derive the small signal model, some DC parameters of the circuit operating in DCM need to be calculated beforehand. Figure 2 shows the four DCDC converters that will be studied throughout this paper; they are the buck converter, the boost converter, the buckboost converter, and the KY converter [23], where and represent the input and output voltage/current of the converter, respectively, and L represents the inductor, D the diode, S the switch, C the output capacitor, R the loading, C_{f} the flying capacitor, i_{L} the inductor current, i_{D} the diode current, and i_{S} the switch current. As the equivalent series resistance (ESR) for the passive components are usually small due to high Q design, which are neglected in this paper for simplification. Thus, the lefthand plane zero caused by the ESR of the output capacitor C will not present in the small signal transfer functions for different DCDC converters.
(a)
(b)
(c)
(d)
For calculating the necessary DC parameters of different DCDC converters, we can just input different V_{on} and V_{off} into (5) and (6). For the buck converter, V_{on} = V_{i}− V_{o} and V_{off} = −V_{o}; for the boost converter, V_{on} = V_{i} and V_{off} = V_{i}− V_{o}; for the buckboost converter, V_{on} = V_{i} and V_{off} = −V_{o}, and for the KY converter, V_{on} = 2V_{i}−V_{o} and V_{off} = V_{i}− V_{o}. On the other hand, using to denote the voltage gain, then D_{2} and can be calculated via (5) and (6), as summarized in Table 2.

For the DC relationship between D_{1} and M, according to [4], the DC relationships among the inductor current i_{L}, the switch current i_{S}, and the diode current i_{D} are
For different DCDC converters’ configurations as in Figure 2, the output current I_{o} = I_{L} for the buck and the KY converters and I_{o} = I_{D} for the boost and the buckboost converters. With the D_{2} and I_{L} in Table 2 and the help of (7) and (8), the relationship between D_{1} (expressed as ) and M can be calculated as presented in Table 2. These parameters are essential for the smallsignal transfer function calculation as shown in Sections 4 and 5.
3. Calculation of the SmallSignal Relationship with the Proposed Differentiation Method
If f is a largesignal function of some variables x, y, and z, to attain the smallsignal model of these variables, we can express them as the sums of DC and small signal components [10],
By neglecting the DC terms and the highorder small signal terms, we can realize the linear approximation. During the small signal calculation, the following approximation can be used:
For example, the largesignal inductor current of the buck converter given by Kazimierczuk [10] is shown in (14), which can also be given by referring to the expression of I_{L} in Table 2 and replacing all DC quantities with largesignal quantities. The basis for this replacement is that largesignal analysis is based on DC relationship of circuit variables [10, 15, 19, 21]:where m denotes the largesignal voltage gain and denotes the small signal input voltage. Expressing the variables as the sums of DC and AC components as (9)–(12) do yield (15), after cancelling the DC components, the small signal expression of the inductor current will become (16):
However, the above deduction process is quite complicated and timeconsuming. To simplify the analysis, we can utilize the calculation of the small signal perturbation as shown in Figure 3.
With a small variation of , the corresponding variation of is also very small such that the value is equal to the derivative of f to x, thus yielding
Following (17) and taking the derivative of (14) with respect to each variable, we can easily getwhere the results in (16) and (18) are equivalent. In the following section, we will use the proposed differentiation method (17) to calculate the small signal relationship.
4. Improved StateSpace Averaging Method for LargeSignal and Small Signal Modeling Deduction
In this section, the approach for deducing the DCM small signal models for different DCDC converters by using the improved SSA method will be discussed.
4.1. General LargeSignal and Corresponding Small Signal Modeling Using the SSA Method
According to the conclusion from [10], the largesignal value relationship and the DC value relationship are identical. Then, (2) and (3) lead to the following largesignal duty ratio d_{2}:
By applying the SSA to the inductor voltage and using (19), we get
Taking the derivative of (20) with respect to , , , and , which are variables in (20), and using (6), we will have the corresponding small signal model given by
Then, using the SSA to the capacitor current yields
If i_{o} = i_{L}, and taking the derivative of (22), the corresponding small signal model becomes
If i_{o} = i_{D}, then (8), (19), and (22) lead to
Finally, taking the derivative of (24), we obtain the corresponding small signal model as follows:where (21), (23), and (25) are the three general equations for deducing the DCM small signal models of different DCDC converters. Then, after input, the values of the inductor voltage drops during the charging (V_{on}) and discharging (V_{off}) cycles, as well as the DC parameters of each DCDC converter from Table 2, into (21), (23), and (25), and the corresponding DCM small signal models can be deduced easily, as detailed next.
4.2. Buck Converter Small Signal Transfer Function
For the buck converter, V_{on} = V_{i}− V_{o}, V_{off} = −V_{o}, and i_{o} = i_{L}. Besides, Table 2 gives and , and substituting them into (21) and (23), gives
Let , then (26) becomes
Further, if u_{c} = , then (27) will be
Considering (28) and (29) simultaneously, the transfer function ( over d_{1}) of the buck converter can be obtained as
4.3. Boost Converter Small Signal Transfer Function
For the boost converter, V_{on} = V_{i}, V_{off} = V_{i}− V_{o}, and i_{o} = i_{D}, and Table 2 gives and ; by substituting them into (21) and (25), it leads to
Let , then (31) will beand if u_{c} = and , then we can obtain the following equation from (32):
Also considering (33) and (34) simultaneously, in this case, the transfer function of the boost converter will become
4.4. BuckBoost Converter Small Signal Transfer Function
For the boost converter, V_{on} = V_{i}, V_{off} = −V_{o}, and i_{o} = i_{D}, and from Table 2, and . Then, by substituting them into (21) and (25), we have
With , then (36) will lead toand again if u_{c} = and , (37) yields
Finally, considering (38) and (39) simultaneously, again the transfer function of the buckboost converter will be
4.5. KY Converter Small Signal Transfer Function
For the KY converter, V_{on} = 2V_{i}− V_{o}, V_{off} = V_{i}− V_{o}, and i_{o} = i_{L}, and from Table 2, and . Then, substituting them into (21) and (23) will lead to
Let , then (41) becomes
Then with uc = vo, (42) gives
Again with (43) and (44) considered simultaneously, the transfer function of the KY converter can be obtained as
5. Circuit Averaging Method for LargeSignal and Small Signal Modeling Deduction
In this section, the approach for deducing the DCM small signal models for different DCDC converters by using the CA method will be discussed.
5.1. General LargeSignal and Corresponding Small Signal Modeling Using the CA Method
With the help of Reference [10], according to (7) and (8), the switching network of the DCDC converter can be transformed into the circuit as shown in Figure 4.
When the circuit reaches the steadystate, the average voltage across the inductor is zero, then Figure 4 imposes,
From Figure 4, we can obtain
By using (5)–(8), they yield the following largesignal equations (considering identical the largesignal value and the DC value relationships):
Taking the derivative of (49) and (50), the corresponding small signal equations will emerge as follows:and from (47) and (48), V_{on} = V_{LS} = V_{MS} and V_{off} = V_{LD} = V_{MD}. Then, the small signal circuit of the switching network in Figure 4 can be transformed into Figure 5, where
From Figure 5, we can write
Then, with k_{d1} = k_{S} + k_{D} and , we can obtain the factors k_{S}, k_{D}, k_{d1}, of the four DCDC converters (buck, boost, buckboost, and KY) as shown in Table 3.

With these factors, we can substantially simplify the calculation process, as shown next. Further, we can deduce that
Then (53) can be rewritten asand considering the capacitor, its current becomes
If i_{o} = i_{L}, (57) will become
Then, with u_{c} = , we haveand with i_{o} = i_{D}, (57) yields
Finally, with (54)–(56), u_{c} = , (60) can be rewritten as
Here, (56), (59), and (61) are the 3 general equations to deduce the DCM small signal models for the different DCDC converters. By just input, the voltage drops V_{LS} and V_{SD} from Figure 5, plus the factors (Table 3) and the DC parameters (Table 2) into (56), (59) and (61), and the corresponding DCM small signal transfer functions can be calculated easily in the following.
5.2. Buck Converter Small Signal Transfer Function
For the buck converter, from Figure 5, V_{LS} = V_{i} − V_{o}, V_{SD} = −V_{i}, and i_{o} = i_{L}, and from Table 2, and . Then, with , (56) and (59) lead to
And from (62) and (63) simultaneously, the transfer function of the buck converter can be obtained as
5.3. Boost Converter Small Signal Transfer Function
For the boost converter, from Figure 5, V_{LS} = V_{i}, V_{SD} = −V_{o}, i_{o} = i_{D}, and from Table 2, and . Then, with , (56) and (61) imply,
Again, from (65) and (66) simultaneously, the transfer function of the boost converter will become,
5.4. BuckBoost Converter Small Signal Transfer Function
For the buckboost converter, from Figure 5, V_{LS} = V_{i}, V_{SD} = −(V_{i}+V_{o}), and i_{o} = i_{D}, and from Table 2, and . Then, with , (56) and (61) impose
Similarly, with (68) and (69) considered simultaneously, the transfer function of the buckboost converter will be
6.5. KY Converter Small Signal Transfer Function
For the KY converter, from Figure 5, V_{LS} = 2V_{i} − V_{o}, V_{SD} = −V_{i}, and i_{o} = i_{L}, and from Table 2, and . Then, with , (56) and (59) entail
Finally, from (71) and (72) simultaneously, the transfer function of the KY converter will become
6. Simulation Results
We simulated the four DCDC converters within a Cadence environment of a 65 nm CMOS process and also MATLAB to verify and compare the accuracy of the small signal transfer functions (30), (35), (40), (45), (64), (67), (70), and (73) deduced by the improved SSA and CA methods.
6.1. Buck Converter
The relevant parameters of the buck converter (Figure 2) are V_{i} = 1.2 V, f_{s} = 100 MHz, C = 10 nF, L = 36 nH, and R = 40 Ω. The Bode plot comparison between the transfer functions with the SSA and the CA methods ((30) and (64)), and the simulated circuit power stage (PS), is shown in Figure 6, for d_{1} = 0.3, 0.5, and 0.7.
(a)
(b)
(c)
From Figure 6, we can conclude that in the buck converter, the CA method provides higher accuracy than the SSA method in any duty ratio range d_{1}, with clearer emphasis at larger d_{1} values.
6.2. Boost Converter
The relevant parameters of the boost converter (Figure 2) are V_{i} = 1.2 V, f_{s} = 100 MHz, C = 10 nF, L = 13.5 nH, and R = 60 Ω. The Bode plot comparison between the transfer functions with the SSA and the CA methods ((35) and (67)), and the simulated circuit PS, is shown in Figure 7, for d_{1} = 0.3, 0.5, and 0.7.
(a)
(b)
(c)
From Figure 7, we can conclude that in the boost converter with a small duty ratio value (d_{1} = 0.3), the SSA method contains slightly better accuracy than the CA method. But, as the d_{1} increases, the accuracy of the CA method will improve over the SSA method, again being more evident for larger d_{1} values.
6.3. BuckBoost Converter
The relevant parameters of the buckboost converter (Figure 2) are V_{i} = 1.2 V, f_{s} = 100 MHz, C = 40 nF, L = 15 nH, and R = 150 Ω. The Bode plot comparison between the transfer functions with the SSA and the CA methods ((40) and (70)), and the simulated circuit PS, is shown in Figure 8, for d_{1} = 0.3, 0.5, and 0.7.
(a)
(b)
(c)
From Figure 8, the analysis of the simulation results of the boost converter is similar to those of the buckboost converter, leading exactly to the same conclusions.
6.4. KY Converter
Finally, for the KY converter (Figure 2), V_{i} = 1.2 V, f_{s} = 100 MHz, C = 10 nF, C_{f} = 10 nF, L = 3.6 nH, and R = 60 Ω, and the Bode plot comparison between the transfer functions with the SSA and the CA methods ((45) and (73)), and the simulated circuit PS, is shown in Figure 9, for d_{1} = 0.3, 0.5, and 0.7.
(a)
(b)
(c)
From Figure 9, the analysis of the simulation results of the KY converter are similar to those of the buck converter, leading exactly to the same conclusions.
7. HighAccuracy Modeling Method for Different DCDC Converters in DCM—Selection Strategy
7.1. Derivation
By using the methods presented in [3] and [6], the approximate poles and zeros for different DCDC converters in DCM with the SSA and CA methods can be calculated and summarized in Table 4. The previous Bode plot simulation results clearly demonstrate that the phasefrequency responses of the DCDC converters power stages generally show a larger phase lag than the small signal models given by both the SSA and CA methods. Based on this, if the modeling method presents a smaller value of the second pole or zero (leading to a larger phase lag), it will exhibit a better accuracy of the system phasefrequency response. From Table 4, we propose a selection strategy of highaccuracy small signal modeling method for the DCDC converters as in Table 5.


7.2. Verification
For verification of the proposed selection strategy, we simulated a buck converter and a boost converter with the Cadence Spectre simulator to demonstrate that the converter operates with higher stability with a more accurate small signal modeling method used in the compensator design. The simulated results of the system phasefrequency response and the closedloop controlled converters’ load transient response are presented in Figures 10–13. These figures confirm the correctness of the proposed selection strategy in Table 5.
(a)
(b)
(a)
(b)
Figure 12 presents the simulated output voltage V_{o} and load current i_{o} of the designed closedloop controlled buck converter during a load transient, applying a Type II compensator. As indicated in Table 5, the CA method is more accurate with the condition and the simulated result (Figure 12) also confirms that the closedloop controller designed with the CA method exhibits better stability and transient response than the SSA method, even though both cases have the same phase margin (PM) of 45°. On the other hand, Figure 13 shows the simulated V_{o} and the i_{o} of the closedloop controlled boost converter during a load transient, applying also a Type II compensator. In this case, as indicated in Table 5, the SSA modeling method is more accurate with the condition and the simulated result (Figure 13) also confirms that the closedloop controller designed with SSA method obtain a better stability and transient response. When there is a sudden change of the i_{o}, the boost converter with the SSA method responds faster than that with the CA method.
Unlike the conclusion made in [6], this paper shows that, in some cases, the CA method exhibits better accuracy than the SSA method. Figures 12 and 13 confirm that an accurate modeling method is critical to design the appropriate closedloop controller of the DCDC converter, demonstrating that the selection strategy given in Table 5 is essential and necessary in the design. The general and streamlined small signal deduction process for both modeling methods can be further applied conveniently to similar DCDC converter topologies.
8. Conclusions
This paper presented the review, study, DCM small signal modeling deduction and simulation verification by using the improved SSA and CA methods for four DCDC converters. This paper first proposed a general and intuitive deriving process for the improved SSA and CA modeling methods, such that the corresponding DCM small signal models for DCDC converters can be easily determined. Then, this paper discovers that the CA can obtain higher accuracy than the improved SSA at some operating conditions, as some research studies claimed that the improved SSA can obtain the highest accuracy among all the modeling methods. Finally, this paper provided a selection strategy for a highaccuracy modeling method for various DCDC converters operating in DCM, verified by simulations, which is necessary and beneficial in the design of a more accurate DCM closedloop controller for DCDC converters, achieving better stability and transient response.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported in part by the Science and Technology Development Fund, Macao SAR (FDCT) (120/2016/A3) and in part by the Research Committee of the University of Macau (MYRG201500030AMSV and MYRG201700090AMSV).
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Copyright © 2018 YuJun Mao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.