# Review and Selection Strategy for High-Accuracy Modeling of PWM Converters in DCM

Authors: Yu-Jun Mao; Chi-Seng Lam; Sai-Weng Sin; Man-Chung Wong; Rui Paulo Martins
Journal: Journal of Electrical and Computer Engineering
 (2018)

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## Abstract

Among various modeling methods for DC-DC converters introduced in the past two decades, the state-space 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 DC-DC converters operating in the discontinuous conduction mode (DCM). We also streamline the general model-deriving processes for DC-DC converters, and test and compare the accuracy of these two methods under various conditions. Finally, we provide a selection strategy for a high-accuracy modeling method for different DC-DC converters operating in DCM and verified by simulations, which revealed necessary and beneficial for designing a more accurate DCM closed-loop controller for DC-DC converters, thus achieving better stability and transient response.


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## Body

## 1. Introduction

A PWM DC-DC 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 DC-DC 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 DC-DC converter operation in DCM to design a closed-loop controller. In previous works [1–22], various small signal modeling methods of PWM DC-DC 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 reduced-order or full-order models, as summarized in Table 1.Table 1
Summary of small signal modeling methods.
Method name
Order
Model type
Conventional state-space averaging (SSA) [1]
Reduced order (low accuracy)
Analytical equation
Converter cell [2]
Reduced order (low accuracy)
Equivalent circuit
Full-order SSA [3, 8, 11, 12, 20]
Full order (high accuracy)
Analytical equation
Circuit averaging (CA) [4, 5, 8–10, 13, 14, 17–19, 21, 22]
Full order (high accuracy)
Equivalent circuit
Improved SSA [6, 7, 15, 16]
Full order (high accuracy)
Analytical equationAmong these small signal modeling methods, the improved state-space 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 DC-DC 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 DC-DC 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 DC-DC circuit topology changes, whenever the DC-DC circuit topology changes, leads to unnecessary and time-consuming efforts. Then, another motivation for this paper is to generalize the whole derivation process to attain a general and intuitive model-deriving 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 DC-DC 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 DC-DC 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 DC-DC converters under different operating conditions.This paper contributes significantly to the design of a stable and fast transient response DCM closed-loop controller for different DC-DC converters. Section2 presents the DC analysis of different DC-DC converters in DCM. Section 3 introduces two small signal relationship calculation methods based on the large-signal equation. Section 4 discusses the general DCM large-signal and small signal modeling deduction based on the improved SSA method applied to different DC-DC converters. Then, Section 5 determines the general DCM large-signal 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 DC-DC converters. Finally, the conclusions are drawn in Section 8.


## 2. DC Analysis of DC-DC Converters in DCM

In the discussion hereafter, if we use the capital letterX to denote the averaged value of a specific variable in one switching frequency (DC value), then the lowercase letter x denotes its large-signal value and x^ denotes its small signal value. The relationship between these three variables is [10](1)x=X+x^.


### 2.1. DCM Operation

The DCM operation of the DC-DC converters consists of three intervals. Here, we useD1, D2, and D3 to denote the duty ratio of each interval, respectively, and Ts denotes the switching period. Figure 1 shows the inductor current, iL, waveform of the DC-DC converters operating in DCM [10].Figure 1
Inductor current of DC-DC converters operating in DCM.For a typical DC-DC converter,iL starts to rise during the charging of the inductor L and begins to drop while the L is discharging. In this paper, we use Von (charging) and Voff (discharging) to denote the voltage across the L in the first and the second intervals, respectively. From Figure 1, we can obtain(2)Ipk=VonLD1Ts,where Ipk denotes the peak value of inductor current iL, D1 denotes the duty ratio of charging interval, and Ts denotes the switching period. From Figure 1, the relationship between the peak value (Ipk) and average value (IL) of the inductor current can be expressed as(3)12IpkD1+D2Ts=ILTs,where IL denotes the average value of the inductor current iL while Ts and D2 denotes the duty ratio of the discharging interval. From Figure 1,(4)VonLD1Ts=−VoffLD2Ts=Ipk.Equation (4) yields the following(5)D2=−VonVoffD1,where Von (charging) and Voff (discharging) denote the voltage across the L in the first and the second intervals, respectively. Considering (2), (3), and (5), IL can be expressed as(6)IL=D12TsVonVoff−Von2LVoff.


### 2.2. DC Parameters Calculation for Different DC-DC Converters in DCM

Before we derive the small signal model, some DC parameters of the circuit operating in DCM need to be calculated beforehand. Figure2 shows the four DC-DC converters that will be studied throughout this paper; they are the buck converter, the boost converter, the buck-boost converter, and the KY converter [23], where vi/ii and vo/io 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, Cf the flying capacitor, iL the inductor current, iD the diode current, and iS 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 left-hand plane zero caused by the ESR of the output capacitor C will not present in the small signal transfer functions for different DC-DC converters.Figure 2
Circuit configuration of (a) buck, (b) boost, (c) buck-boost, and (d) KY converters.
(a)
(b)
(c)
(d)For calculating the necessary DC parameters of different DC-DC converters, we can just input differentVon and Voff into (5) and (6). For the buck converter, Von = Vi − Vo and Voff = −Vo; for the boost converter, Von = Vi and Voff = Vi − Vo; for the buck-boost converter, Von = Vi and Voff = −Vo, and for the KY converter, Von = 2Vi −Vo and Voff = Vi − Vo. On the other hand, using M=Vo/Vi to denote the voltage gain, then D2 and IL can be calculated via (5) and (6), as summarized in Table 2.Table 2
DC parameters for different DC-DC converters.
DC-DC converter
D
2
I
L
D
1
2
T
s
R
/
2
L
Buck
1
−
M
/
M
D
1
D
1
2
T
s
V
i
1
−
M
/
2
L
M
M
2
/
1
−
M
Boost
1
/
M
−
1
D
1
D
1
2
T
s
V
i
M
/
2
L
M
−
1
M
2
−
M
Buck-boost
1
/
M
D
1
D
1
2
T
s
V
i
1
+
M
/
2
L
M
M
2
KY
2
−
M
/
M
−
1
D
1
D
1
2
T
s
V
i
2
−
M
/
2
L
M
−
1
M
M
−
1
/
2
−
MFor the DC relationship betweenD1 and M, according to [4], the DC relationships among the inductor current iL, the switch current iS, and the diode current iD are(7)IS=D1D1+D2IL,(8)ID=D2D1+D2IL.For different DC-DC converters’ configurations as in Figure2, the output current Io = IL for the buck and the KY converters and Io = ID for the boost and the buck-boost converters. With the D2 and IL in Table 2 and the help of (7) and (8), the relationship between D1 (expressed as D12TsR/2L) and M can be calculated as presented in Table 2. These parameters are essential for the small-signal transfer function calculation as shown in Sections 4 and 5.


## 2.1. DCM Operation

The DCM operation of the DC-DC converters consists of three intervals. Here, we useD1, D2, and D3 to denote the duty ratio of each interval, respectively, and Ts denotes the switching period. Figure 1 shows the inductor current, iL, waveform of the DC-DC converters operating in DCM [10].Figure 1
Inductor current of DC-DC converters operating in DCM.For a typical DC-DC converter,iL starts to rise during the charging of the inductor L and begins to drop while the L is discharging. In this paper, we use Von (charging) and Voff (discharging) to denote the voltage across the L in the first and the second intervals, respectively. From Figure 1, we can obtain(2)Ipk=VonLD1Ts,where Ipk denotes the peak value of inductor current iL, D1 denotes the duty ratio of charging interval, and Ts denotes the switching period. From Figure 1, the relationship between the peak value (Ipk) and average value (IL) of the inductor current can be expressed as(3)12IpkD1+D2Ts=ILTs,where IL denotes the average value of the inductor current iL while Ts and D2 denotes the duty ratio of the discharging interval. From Figure 1,(4)VonLD1Ts=−VoffLD2Ts=Ipk.Equation (4) yields the following(5)D2=−VonVoffD1,where Von (charging) and Voff (discharging) denote the voltage across the L in the first and the second intervals, respectively. Considering (2), (3), and (5), IL can be expressed as(6)IL=D12TsVonVoff−Von2LVoff.


## 2.2. DC Parameters Calculation for Different DC-DC Converters in DCM

Before we derive the small signal model, some DC parameters of the circuit operating in DCM need to be calculated beforehand. Figure2 shows the four DC-DC converters that will be studied throughout this paper; they are the buck converter, the boost converter, the buck-boost converter, and the KY converter [23], where vi/ii and vo/io 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, Cf the flying capacitor, iL the inductor current, iD the diode current, and iS 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 left-hand plane zero caused by the ESR of the output capacitor C will not present in the small signal transfer functions for different DC-DC converters.Figure 2
Circuit configuration of (a) buck, (b) boost, (c) buck-boost, and (d) KY converters.
(a)
(b)
(c)
(d)For calculating the necessary DC parameters of different DC-DC converters, we can just input differentVon and Voff into (5) and (6). For the buck converter, Von = Vi − Vo and Voff = −Vo; for the boost converter, Von = Vi and Voff = Vi − Vo; for the buck-boost converter, Von = Vi and Voff = −Vo, and for the KY converter, Von = 2Vi −Vo and Voff = Vi − Vo. On the other hand, using M=Vo/Vi to denote the voltage gain, then D2 and IL can be calculated via (5) and (6), as summarized in Table 2.Table 2
DC parameters for different DC-DC converters.
DC-DC converter
D
2
I
L
D
1
2
T
s
R
/
2
L
Buck
1
−
M
/
M
D
1
D
1
2
T
s
V
i
1
−
M
/
2
L
M
M
2
/
1
−
M
Boost
1
/
M
−
1
D
1
D
1
2
T
s
V
i
M
/
2
L
M
−
1
M
2
−
M
Buck-boost
1
/
M
D
1
D
1
2
T
s
V
i
1
+
M
/
2
L
M
M
2
KY
2
−
M
/
M
−
1
D
1
D
1
2
T
s
V
i
2
−
M
/
2
L
M
−
1
M
M
−
1
/
2
−
MFor the DC relationship betweenD1 and M, according to [4], the DC relationships among the inductor current iL, the switch current iS, and the diode current iD are(7)IS=D1D1+D2IL,(8)ID=D2D1+D2IL.For different DC-DC converters’ configurations as in Figure2, the output current Io = IL for the buck and the KY converters and Io = ID for the boost and the buck-boost converters. With the D2 and IL in Table 2 and the help of (7) and (8), the relationship between D1 (expressed as D12TsR/2L) and M can be calculated as presented in Table 2. These parameters are essential for the small-signal transfer function calculation as shown in Sections 4 and 5.


## 3. Calculation of the Small-Signal Relationship with the Proposed Differentiation Method

Iff is a large-signal function of some variables x, y, and z, to attain the small-signal model of these variables, we can express them as the sums of DC and small signal components [10],(9)f=F+f^,(10)x=X+x^,(11)y=Y+y^,(12)z=Z+z^.By neglecting the DC terms and the high-order small signal terms, we can realize the linear approximation. During the small signal calculation, the following approximation can be used:(13)11+x^≈1−x^.For example, the large-signal 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 IL in Table 2 and replacing all DC quantities with large-signal quantities. The basis for this replacement is that large-signal analysis is based on DC relationship of circuit variables [10, 15, 19, 21]:(14)iL=d12Tsvi1−m2Lm,where m denotes the large-signal voltage gain and vi 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 i^L will become (16):(15)IL+i^L=D1+d^12TsVi+v^i1−M−m^2LM+m^≈TsD12+2D1d^1Vi+v^i1−M−m^1−m^/M2LM,(16)i^L=Ts2D1Vi1−Md^1+D121−Mv^i−D12Vim^−D12Vi1−Mm^/M2LM=D1TsVi1−MLMd^1+D12Ts1−M2LMv^i−D12TsVi2LM2m^.However, the above deduction process is quite complicated and time-consuming. To simplify the analysis, we can utilize the calculation of the small signal perturbation as shown in Figure3.Figure 3
Small signal perturbation.With a small variation ofx^, the corresponding variation of f^ is also very small such that the value f^/x^ is equal to the derivative of f to x, thus yielding(17)f^=fx′x^+fy′y^+fz′z^.Following (17) and taking the derivative of (14) with respect to each variable, we can easily get(18)i^L=D1TsVi1−MLMd^1+D12Ts1−M2LMv^i−D12TsVi2LM2m^,where 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 State-Space Averaging Method for Large-Signal and Small Signal Modeling Deduction

In this section, the approach for deducing the DCM small signal models for different DC-DC converters by using the improved SSA method will be discussed.


### 4.1. General Large-Signal and Corresponding Small Signal Modeling Using the SSA Method

According to the conclusion from [10], the large-signal value relationship and the DC value relationship are identical. Then, (2) and (3) lead to the following large-signal duty ratio d2:(19)d2=2LiLvond1Ts−d1.By applying the SSA to the inductor voltage and using (19), we get(20)LdiLdt=vond1+voffd2=von−voffd1+2voffLiLvond1Ts.Taking the derivative of (20) with respect to von, voff, d1, and iL, which are variables in (20), and using (6), we will have the corresponding small signal model given by(21)Ldi^Ldt=2Von−VoffVonD1v^on+−VonVoffD1v^off+2Von−Voffd^1+Voff−VonILD1i^L.Then, using the SSA to the capacitor current yields(22)CduCdt=io−uCR.Ifio = iL, and taking the derivative of (22), the corresponding small signal model becomes(23)Cdu^Cdt=i^L−u^CR.Ifio = iD, then (8), (19), and (22) lead to(24)CduCdt=iL−d12Tsvon2L−uCR.Finally, taking the derivative of (24), we obtain the corresponding small signal model as follows:(25)Cdu^Cdt=i^L−D1TsVonLd^1−D12Ts2Lv^on−u^CR,where (21), (23), and (25) are the three general equations for deducing the DCM small signal models of different DC-DC converters. Then, after input, the values of the inductor voltage drops during the charging (Von) and discharging (Voff) cycles, as well as the DC parameters of each DC-DC 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,Von = Vi − Vo, Voff = −Vo, and io = iL. Besides, Table 2 gives IL=D12TsVi1−M/2LM and D12TsR/2L=M2/1−M, and substituting them into (21) and (23), gives(26)Ldi^Ldt=2Vi−VoVi−VoD1v^i−v^o−Vi−VoVoD1v^o+2Vid^1−ViILD1i^L,(27)Cdu^Cdt=i^L−u^CR.Letv^i=0, then (26) becomes(28)sL+2LMD1Ts1−Mi^L=1M2−MD1v^o+2Vid^1.Further, ifuc = vo, then (27) will be(29)sC+1Rv^o=i^L.Considering (28) and (29) simultaneously, the transfer function (Vo over d1) of the buck converter can be obtained as(30)v^osd^1s=2Vis2LC+sL/R+2LCM/D1Ts1−M+2−MD1/1−MM.


### 4.3. Boost Converter Small Signal Transfer Function

For the boost converter,Von = Vi, Voff = Vi − Vo, and io = iD, and Table 2 gives IL=D12TsViM/2LM−1 and D12TsR/2L=M2−M; by substituting them into (21) and (25), it leads to(31)Ldi^Ldt=Vi+VoViD1v^i−ViVi−VoD1v^i−v^o+2Vod^1−VoILD1i^L,(32)Cdu^Cdt=i^L−D1TsViLd^1−D12Ts2Lv^i−u^CR.Letv^i=0, then (31) will be(33)sL+2LM−1D1Tsi^L=11−MD1v^o+2Vod^1,and if uc = vo and v^i=0, then we can obtain the following equation from (32):(34)sC+1Rv^o+D1TsViLd^1=i^L.Also considering (33) and (34) simultaneously, in this case, the transfer function of the boost converter will become(35)v^osd^1s=D1TsVi2/D1Ts−ss2LC+sL/R+2LCM−1/D1Ts+2M−1D1/MM−1.


### 4.4. Buck-Boost Converter Small Signal Transfer Function

For the boost converter,Von = Vi, Voff = −Vo, and io = iD, and from Table 2, IL=D12TsVi1+M/2LM and D12TsR/2L=M2. Then, by substituting them into (21) and (25), we have(36)Ldi^Ldt=2Vi+VoViD1v^i−ViVoD1v^o+2Vi+Vod^1−Vi+VoILD1i^L,(37)Cdu^Cdt=i^L−D1TsViLd^1−D12Ts2Lv^i−u^CR.Withv^i=0, then (36) will lead to(38)sL+2LMD1Tsi^L=−1MD1v^o+2Vi+Vod^1,and again if uc = vo and v^i=0, (37) yields(39)sC+1Rv^o+D1TsViLd^1=i^L.Finally, considering (38) and (39) simultaneously, again the transfer function of the buck-boost converter will be(40)v^osd^1s=D1TsVi2/D1Ts−ss2LC+sL/R+2LCM/D1Ts+2D1/M.


### 4.5. KY Converter Small Signal Transfer Function

For the KY converter,Von = 2Vi − Vo, Voff = Vi − Vo, and io = iL, and from Table 2, IL=D12TsVi2−M/2LM−1 and D12TsR/2L=MM−1/2−M. Then, substituting them into (21) and (23) will lead to(41)Ldi^Ldt=3Vi−Vo2Vi−VoD12v^i−v^o−2Vi−VoVi−VoD1v^i−v^o+2Vid^1−ViILD1i^L,(42)Cdu^Cdt=i^L−u^CR.Letv^i=0, then (41) becomes(43)sL+2LM−1D1Ts2−Mi^L=12−M1−MD1v^o+2Vid^1.Then with uc = vo, (42) gives(44)sC+1Rv^o=i^L.Again with (43) and (44) considered simultaneously, the transfer function of the KY converter can be obtained as(45)v^osd^1s=2Vis2LC+sL/R+2LCM−1/D1Ts2−M+M2−4M+2D1/2−M1−MM.


## 4.1. General Large-Signal and Corresponding Small Signal Modeling Using the SSA Method

According to the conclusion from [10], the large-signal value relationship and the DC value relationship are identical. Then, (2) and (3) lead to the following large-signal duty ratio d2:(19)d2=2LiLvond1Ts−d1.By applying the SSA to the inductor voltage and using (19), we get(20)LdiLdt=vond1+voffd2=von−voffd1+2voffLiLvond1Ts.Taking the derivative of (20) with respect to von, voff, d1, and iL, which are variables in (20), and using (6), we will have the corresponding small signal model given by(21)Ldi^Ldt=2Von−VoffVonD1v^on+−VonVoffD1v^off+2Von−Voffd^1+Voff−VonILD1i^L.Then, using the SSA to the capacitor current yields(22)CduCdt=io−uCR.Ifio = iL, and taking the derivative of (22), the corresponding small signal model becomes(23)Cdu^Cdt=i^L−u^CR.Ifio = iD, then (8), (19), and (22) lead to(24)CduCdt=iL−d12Tsvon2L−uCR.Finally, taking the derivative of (24), we obtain the corresponding small signal model as follows:(25)Cdu^Cdt=i^L−D1TsVonLd^1−D12Ts2Lv^on−u^CR,where (21), (23), and (25) are the three general equations for deducing the DCM small signal models of different DC-DC converters. Then, after input, the values of the inductor voltage drops during the charging (Von) and discharging (Voff) cycles, as well as the DC parameters of each DC-DC 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,Von = Vi − Vo, Voff = −Vo, and io = iL. Besides, Table 2 gives IL=D12TsVi1−M/2LM and D12TsR/2L=M2/1−M, and substituting them into (21) and (23), gives(26)Ldi^Ldt=2Vi−VoVi−VoD1v^i−v^o−Vi−VoVoD1v^o+2Vid^1−ViILD1i^L,(27)Cdu^Cdt=i^L−u^CR.Letv^i=0, then (26) becomes(28)sL+2LMD1Ts1−Mi^L=1M2−MD1v^o+2Vid^1.Further, ifuc = vo, then (27) will be(29)sC+1Rv^o=i^L.Considering (28) and (29) simultaneously, the transfer function (Vo over d1) of the buck converter can be obtained as(30)v^osd^1s=2Vis2LC+sL/R+2LCM/D1Ts1−M+2−MD1/1−MM.


## 4.3. Boost Converter Small Signal Transfer Function

For the boost converter,Von = Vi, Voff = Vi − Vo, and io = iD, and Table 2 gives IL=D12TsViM/2LM−1 and D12TsR/2L=M2−M; by substituting them into (21) and (25), it leads to(31)Ldi^Ldt=Vi+VoViD1v^i−ViVi−VoD1v^i−v^o+2Vod^1−VoILD1i^L,(32)Cdu^Cdt=i^L−D1TsViLd^1−D12Ts2Lv^i−u^CR.Letv^i=0, then (31) will be(33)sL+2LM−1D1Tsi^L=11−MD1v^o+2Vod^1,and if uc = vo and v^i=0, then we can obtain the following equation from (32):(34)sC+1Rv^o+D1TsViLd^1=i^L.Also considering (33) and (34) simultaneously, in this case, the transfer function of the boost converter will become(35)v^osd^1s=D1TsVi2/D1Ts−ss2LC+sL/R+2LCM−1/D1Ts+2M−1D1/MM−1.


## 4.4. Buck-Boost Converter Small Signal Transfer Function

For the boost converter,Von = Vi, Voff = −Vo, and io = iD, and from Table 2, IL=D12TsVi1+M/2LM and D12TsR/2L=M2. Then, by substituting them into (21) and (25), we have(36)Ldi^Ldt=2Vi+VoViD1v^i−ViVoD1v^o+2Vi+Vod^1−Vi+VoILD1i^L,(37)Cdu^Cdt=i^L−D1TsViLd^1−D12Ts2Lv^i−u^CR.Withv^i=0, then (36) will lead to(38)sL+2LMD1Tsi^L=−1MD1v^o+2Vi+Vod^1,and again if uc = vo and v^i=0, (37) yields(39)sC+1Rv^o+D1TsViLd^1=i^L.Finally, considering (38) and (39) simultaneously, again the transfer function of the buck-boost converter will be(40)v^osd^1s=D1TsVi2/D1Ts−ss2LC+sL/R+2LCM/D1Ts+2D1/M.


## 4.5. KY Converter Small Signal Transfer Function

For the KY converter,Von = 2Vi − Vo, Voff = Vi − Vo, and io = iL, and from Table 2, IL=D12TsVi2−M/2LM−1 and D12TsR/2L=MM−1/2−M. Then, substituting them into (21) and (23) will lead to(41)Ldi^Ldt=3Vi−Vo2Vi−VoD12v^i−v^o−2Vi−VoVi−VoD1v^i−v^o+2Vid^1−ViILD1i^L,(42)Cdu^Cdt=i^L−u^CR.Letv^i=0, then (41) becomes(43)sL+2LM−1D1Ts2−Mi^L=12−M1−MD1v^o+2Vid^1.Then with uc = vo, (42) gives(44)sC+1Rv^o=i^L.Again with (43) and (44) considered simultaneously, the transfer function of the KY converter can be obtained as(45)v^osd^1s=2Vis2LC+sL/R+2LCM−1/D1Ts2−M+M2−4M+2D1/2−M1−MM.


## 5. Circuit Averaging Method for Large-Signal and Small Signal Modeling Deduction

In this section, the approach for deducing the DCM small signal models for different DC-DC converters by using the CA method will be discussed.


### 5.1. General Large-Signal 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 DC-DC converter can be transformed into the circuit as shown in Figure 4.Figure 4
Equivalent circuit of the switching network (current source).When the circuit reaches the steady-state, the average voltage across the inductor is zero, then Figure4 imposes,(46)VLM=0.From Figure4, we can obtain(47)VLS=VLM+VMS=VMS,(48)VLD=VLM+VMD=VMD.By using (5)–(8), they yield the following large-signal equations (considering identical the large-signal value and the DC value relationships):(49)iS=d12Tsvon2L,(50)iD=−d12Tsvon22Lvoff.Taking the derivative of (49) and (50), the corresponding small signal equations will emerge as follows:(51)i^S=D1TsVonLd^1+D12Ts2Lv^on,i^D=−D1TsVon2LVoffd^1−D12TsVonLVoffv^on+D12TsVon22LVoff2v^off,and from (47) and (48), Von = VLS = VMS and Voff = VLD = VMD. Then, the small signal circuit of the switching network in Figure 4 can be transformed into Figure 5, where(52)kS=D1TsVonL,gS=D12Ts2L,kD=−D1TsVon2LVoff,gD=−D12TsVonLVoff,gM=D12TsVon22LVoff2.Figure 5
Small signal equivalent circuit of the switching network.From Figure5, we can write(53)i^L=kS+kDd^1+gS+gDv^MS+gMv^MD,=kS+kDd^1+gS+gD+gMv^MS+gMv^SD.Then, withkd1 = kS + kD and gMS=gS+gD+gM, we can obtain the factors kS, kD, kd1, gS,gD,gM,andgMS of the four DC-DC converters (buck, boost, buck-boost, and KY) as shown in Table 3.Table 3
Relevant factors of the circuit averaging (CA) method.
DC-DC converter
Buck
Boost
Buck-boost
KY
k
S
2
M
2
V
i
/
D
1
R
2
M
M
−
1
V
i
/
D
1
R
2
M
2
V
i
/
D
1
R
2
M
M
−
1
V
i
/
D
1
R
k
D
2
M
1
−
M
V
i
/
D
1
R
2
M
V
i
/
D
1
R
2
M
V
i
/
D
1
R
2
M
2
−
M
V
i
/
D
1
R
k
d1
2
M
V
i
/
D
1
R
2
M
2
V
i
/
D
1
R
2
M
M
+
1
V
i
/
D
1
R
2
M
V
i
/
D
1
R
g
S
M
2
/
1
−
M
R
M
2
−
M
/
R
M
2
/
R
M
M
−
1
/
2
−
M
R
g
D
2
M
/
R
2
M
/
R
2
M
/
R
2
M
/
R
g
M
1
−
M
/
R
M
/
M
−
1
R
1
/
R
M
2
−
M
/
M
−
1
R
g
MS
1
/
1
−
M
R
M
3
/
M
−
1
R
M
+
1
2
/
R
M
/
2
−
M
M
−
1With these factors, we can substantially simplify the calculation process, as shown next. Further, we can deduce that(54)v^LM=sLi^L,(55)v^MS=v^LS−v^LM.Then (53) can be rewritten as(56)sL+1gMSi^L=kd1gMSd^1+v^LS+gMgMSv^SD,and considering the capacitor, its current becomes(57)Cdu^Cdt=i^o−u^CR.Ifio = iL, (57) will become(58)Cdu^Cdt=i^L−u^CR.Then, withuc = vo, we have(59)i^L=SC+1Rv^o,and with io = iD, (57) yields(60)Cdu^Cdt=kDd^1+gDv^MS+gMv^MD−u^CR.Finally, with (54)–(56), uc = vo, (60) can be rewritten as(61)i^L=gMSgD+gMSC+1Rv^o+kd1−kD⋅gMSgD+gMd^1+gM−gM⋅gMSgD+gMv^SD.Here, (56), (59), and (61) are the 3 general equations to deduce the DCM small signal models for the different DC-DC converters. By just input, the voltage drops VLS and VSD 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 Figure5, VLS = Vi − Vo, VSD = −Vi, and io  =  iL, and from Table 2, IL=D12TsVi1−M/2LM and D2TsR/2L=M2/1−M. Then, with v^i=0, (56) and (59) lead to(62)sL+1−MRi^L=2M1−MViD1d^1−v^o,(63)i^L=sC+1Rv^o.And from (62) and (63) simultaneously, the transfer function of the buck converter can be obtained as(64)v^osd^1s=M1−M2Vi/D1s2LC+sL/R+RC1−M+2−M.


### 5.3. Boost Converter Small Signal Transfer Function

For the boost converter, from Figure5, VLS = Vi, VSD = −Vo, io = iD, and from Table 2, IL=D12TsViM/2LM−1 and D12TsR/2L=M2−M. Then, with v^i=0, (56) and (61) imply,(65)sL+M−1RM3i^L=2M−1ViD1Md^1−1M2v^o,(66)i^L=M22M−1sC+2M−1MRv^o+2M−1M2Vi2M−1D1Rd^1.Again, from (65) and (66) simultaneously, the transfer function of the boost converter will become,(67)v^osd^1s=D1TsVi/M2M−1/D12TsM−ss2LC+sL2M−1/RM+RCM−1/M3+2M−1/M3.


### 5.4. Buck-Boost Converter Small Signal Transfer Function

For the buck-boost converter, from Figure5, VLS = Vi, VSD = −(Vi +Vo), and io = iD, and from Table 2, IL=D12TsVi1+M/2LM and D12TsR/2L=M2. Then, with v^i=0, (56) and (61) impose(68)sL+RM+12i^L=2MViD1M+1d^1−1M+12v^o,(69)i^L=M+122M+1SC+2M2+2M+1M+12Rv^o+2M3+M2Vi2M+1D1Rd^1.Similarly, with (68) and (69) considered simultaneously, the transfer function of the buck-boost converter will be(70)v^osd^1s=D1TsVi/M+12M/D12TsM+1−ss2LC+sL2M2+2M+1/RM+12+RC/M+12+2/M+12.


### 6.5. KY Converter Small Signal Transfer Function

For the KY converter, from Figure5, VLS = 2Vi − Vo, VSD = −Vi, and io = iL, and from Table 2, IL=D12TsVi2−M/2LM−1 and D12TsR/2L=MM−1/2−M. Then, with v^i=0, (56) and (59) entail(71)sL+2−MM−1RMi^L=22−MM−1ViD1d^1−v^o,(72)i^L=sC+1Rv^o.Finally, from (71) and (72) simultaneously, the transfer function of the KY converter will become(73)v^osd^1s=2−MM−12Vi/D1s2LC+sL/R+RC2−MM−1/M+2−MM−1+M/M.


## 5.1. General Large-Signal 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 DC-DC converter can be transformed into the circuit as shown in Figure 4.Figure 4
Equivalent circuit of the switching network (current source).When the circuit reaches the steady-state, the average voltage across the inductor is zero, then Figure4 imposes,(46)VLM=0.From Figure4, we can obtain(47)VLS=VLM+VMS=VMS,(48)VLD=VLM+VMD=VMD.By using (5)–(8), they yield the following large-signal equations (considering identical the large-signal value and the DC value relationships):(49)iS=d12Tsvon2L,(50)iD=−d12Tsvon22Lvoff.Taking the derivative of (49) and (50), the corresponding small signal equations will emerge as follows:(51)i^S=D1TsVonLd^1+D12Ts2Lv^on,i^D=−D1TsVon2LVoffd^1−D12TsVonLVoffv^on+D12TsVon22LVoff2v^off,and from (47) and (48), Von = VLS = VMS and Voff = VLD = VMD. Then, the small signal circuit of the switching network in Figure 4 can be transformed into Figure 5, where(52)kS=D1TsVonL,gS=D12Ts2L,kD=−D1TsVon2LVoff,gD=−D12TsVonLVoff,gM=D12TsVon22LVoff2.Figure 5
Small signal equivalent circuit of the switching network.From Figure5, we can write(53)i^L=kS+kDd^1+gS+gDv^MS+gMv^MD,=kS+kDd^1+gS+gD+gMv^MS+gMv^SD.Then, withkd1 = kS + kD and gMS=gS+gD+gM, we can obtain the factors kS, kD, kd1, gS,gD,gM,andgMS of the four DC-DC converters (buck, boost, buck-boost, and KY) as shown in Table 3.Table 3
Relevant factors of the circuit averaging (CA) method.
DC-DC converter
Buck
Boost
Buck-boost
KY
k
S
2
M
2
V
i
/
D
1
R
2
M
M
−
1
V
i
/
D
1
R
2
M
2
V
i
/
D
1
R
2
M
M
−
1
V
i
/
D
1
R
k
D
2
M
1
−
M
V
i
/
D
1
R
2
M
V
i
/
D
1
R
2
M
V
i
/
D
1
R
2
M
2
−
M
V
i
/
D
1
R
k
d1
2
M
V
i
/
D
1
R
2
M
2
V
i
/
D
1
R
2
M
M
+
1
V
i
/
D
1
R
2
M
V
i
/
D
1
R
g
S
M
2
/
1
−
M
R
M
2
−
M
/
R
M
2
/
R
M
M
−
1
/
2
−
M
R
g
D
2
M
/
R
2
M
/
R
2
M
/
R
2
M
/
R
g
M
1
−
M
/
R
M
/
M
−
1
R
1
/
R
M
2
−
M
/
M
−
1
R
g
MS
1
/
1
−
M
R
M
3
/
M
−
1
R
M
+
1
2
/
R
M
/
2
−
M
M
−
1With these factors, we can substantially simplify the calculation process, as shown next. Further, we can deduce that(54)v^LM=sLi^L,(55)v^MS=v^LS−v^LM.Then (53) can be rewritten as(56)sL+1gMSi^L=kd1gMSd^1+v^LS+gMgMSv^SD,and considering the capacitor, its current becomes(57)Cdu^Cdt=i^o−u^CR.Ifio = iL, (57) will become(58)Cdu^Cdt=i^L−u^CR.Then, withuc = vo, we have(59)i^L=SC+1Rv^o,and with io = iD, (57) yields(60)Cdu^Cdt=kDd^1+gDv^MS+gMv^MD−u^CR.Finally, with (54)–(56), uc = vo, (60) can be rewritten as(61)i^L=gMSgD+gMSC+1Rv^o+kd1−kD⋅gMSgD+gMd^1+gM−gM⋅gMSgD+gMv^SD.Here, (56), (59), and (61) are the 3 general equations to deduce the DCM small signal models for the different DC-DC converters. By just input, the voltage drops VLS and VSD 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 Figure5, VLS = Vi − Vo, VSD = −Vi, and io  =  iL, and from Table 2, IL=D12TsVi1−M/2LM and D2TsR/2L=M2/1−M. Then, with v^i=0, (56) and (59) lead to(62)sL+1−MRi^L=2M1−MViD1d^1−v^o,(63)i^L=sC+1Rv^o.And from (62) and (63) simultaneously, the transfer function of the buck converter can be obtained as(64)v^osd^1s=M1−M2Vi/D1s2LC+sL/R+RC1−M+2−M.


## 5.3. Boost Converter Small Signal Transfer Function

For the boost converter, from Figure5, VLS = Vi, VSD = −Vo, io = iD, and from Table 2, IL=D12TsViM/2LM−1 and D12TsR/2L=M2−M. Then, with v^i=0, (56) and (61) imply,(65)sL+M−1RM3i^L=2M−1ViD1Md^1−1M2v^o,(66)i^L=M22M−1sC+2M−1MRv^o+2M−1M2Vi2M−1D1Rd^1.Again, from (65) and (66) simultaneously, the transfer function of the boost converter will become,(67)v^osd^1s=D1TsVi/M2M−1/D12TsM−ss2LC+sL2M−1/RM+RCM−1/M3+2M−1/M3.


## 5.4. Buck-Boost Converter Small Signal Transfer Function

For the buck-boost converter, from Figure5, VLS = Vi, VSD = −(Vi +Vo), and io = iD, and from Table 2, IL=D12TsVi1+M/2LM and D12TsR/2L=M2. Then, with v^i=0, (56) and (61) impose(68)sL+RM+12i^L=2MViD1M+1d^1−1M+12v^o,(69)i^L=M+122M+1SC+2M2+2M+1M+12Rv^o+2M3+M2Vi2M+1D1Rd^1.Similarly, with (68) and (69) considered simultaneously, the transfer function of the buck-boost converter will be(70)v^osd^1s=D1TsVi/M+12M/D12TsM+1−ss2LC+sL2M2+2M+1/RM+12+RC/M+12+2/M+12.


## 6.5. KY Converter Small Signal Transfer Function

For the KY converter, from Figure5, VLS = 2Vi − Vo, VSD = −Vi, and io = iL, and from Table 2, IL=D12TsVi2−M/2LM−1 and D12TsR/2L=MM−1/2−M. Then, with v^i=0, (56) and (59) entail(71)sL+2−MM−1RMi^L=22−MM−1ViD1d^1−v^o,(72)i^L=sC+1Rv^o.Finally, from (71) and (72) simultaneously, the transfer function of the KY converter will become(73)v^osd^1s=2−MM−12Vi/D1s2LC+sL/R+RC2−MM−1/M+2−MM−1+M/M.


## 6. Simulation Results

We simulated the four DC-DC 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 (Figure2) are Vi = 1.2 V, fs = 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 d1 = 0.3, 0.5, and 0.7.Figure 6
Small signal models comparison of the buck converter with SSA and CA methods: (a)d1 = 0.3; (b) d1 = 0.5; and (c) d1 = 0.7.
(a)
(b)
(c)From Figure6, we can conclude that in the buck converter, the CA method provides higher accuracy than the SSA method in any duty ratio range d1, with clearer emphasis at larger d1 values.


### 6.2. Boost Converter

The relevant parameters of the boost converter (Figure2) are Vi = 1.2 V, fs = 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 d1 = 0.3, 0.5, and 0.7.Figure 7
Small signal models comparison of the boost converter with SSA and CA methods: (a)d1 = 0.3; (b) d1 = 0.5; and (c) d1 = 0.7.
(a)
(b)
(c)From Figure7, we can conclude that in the boost converter with a small duty ratio value (d1 = 0.3), the SSA method contains slightly better accuracy than the CA method. But, as the d1 increases, the accuracy of the CA method will improve over the SSA method, again being more evident for larger d1 values.


### 6.3. Buck-Boost Converter

The relevant parameters of the buck-boost converter (Figure2) are Vi = 1.2 V, fs = 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 d1 = 0.3, 0.5, and 0.7.Figure 8
Small signal models comparison of the buck-boost converter with SSA and CA methods: (a)d1 = 0.3; (b) d1 = 0.5; and (c) d1 = 0.7.
(a)
(b)
(c)From Figure8, the analysis of the simulation results of the boost converter is similar to those of the buck-boost converter, leading exactly to the same conclusions.


### 6.4. KY Converter

Finally, for the KY converter (Figure2), Vi = 1.2 V, fs = 100 MHz, C = 10 nF, Cf = 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 d1 = 0.3, 0.5, and 0.7.Figure 9
Small signal models comparison of the KY converter with SSA and CA methods: (a)d1 = 0.3; (b) d1 = 0.5; and (c) d1 = 0.7.
(a)
(b)
(c)From Figure9, the analysis of the simulation results of the KY converter are similar to those of the buck converter, leading exactly to the same conclusions.


## 6.1. Buck Converter

The relevant parameters of the buck converter (Figure2) are Vi = 1.2 V, fs = 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 d1 = 0.3, 0.5, and 0.7.Figure 6
Small signal models comparison of the buck converter with SSA and CA methods: (a)d1 = 0.3; (b) d1 = 0.5; and (c) d1 = 0.7.
(a)
(b)
(c)From Figure6, we can conclude that in the buck converter, the CA method provides higher accuracy than the SSA method in any duty ratio range d1, with clearer emphasis at larger d1 values.


## 6.2. Boost Converter

The relevant parameters of the boost converter (Figure2) are Vi = 1.2 V, fs = 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 d1 = 0.3, 0.5, and 0.7.Figure 7
Small signal models comparison of the boost converter with SSA and CA methods: (a)d1 = 0.3; (b) d1 = 0.5; and (c) d1 = 0.7.
(a)
(b)
(c)From Figure7, we can conclude that in the boost converter with a small duty ratio value (d1 = 0.3), the SSA method contains slightly better accuracy than the CA method. But, as the d1 increases, the accuracy of the CA method will improve over the SSA method, again being more evident for larger d1 values.


## 6.3. Buck-Boost Converter

The relevant parameters of the buck-boost converter (Figure2) are Vi = 1.2 V, fs = 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 d1 = 0.3, 0.5, and 0.7.Figure 8
Small signal models comparison of the buck-boost converter with SSA and CA methods: (a)d1 = 0.3; (b) d1 = 0.5; and (c) d1 = 0.7.
(a)
(b)
(c)From Figure8, the analysis of the simulation results of the boost converter is similar to those of the buck-boost converter, leading exactly to the same conclusions.


## 6.4. KY Converter

Finally, for the KY converter (Figure2), Vi = 1.2 V, fs = 100 MHz, C = 10 nF, Cf = 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 d1 = 0.3, 0.5, and 0.7.Figure 9
Small signal models comparison of the KY converter with SSA and CA methods: (a)d1 = 0.3; (b) d1 = 0.5; and (c) d1 = 0.7.
(a)
(b)
(c)From Figure9, 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. High-Accuracy Modeling Method for Different DC-DC Converters in DCM—Selection Strategy

### 7.1. Derivation

By using the methods presented in [3] and [6], the approximate poles and zeros for different DC-DC 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 phase-frequency responses of the DC-DC 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 phase-frequency response. From Table 4, we propose a selection strategy of high-accuracy small signal modeling method for the DC-DC converters as in Table 5.Table 4
Approximate poles and zeros for various DC-DC converters in DCM.
Name
Method
First pole
Second pole
Zero
Buck
SSA
2
−
M
/
R
C
1
−
M
2
M
/
d
1
T
s
1
−
M
CA
2
−
M
/
R
C
1
−
M
2
M
2
/
d
1
2
T
s
Boost
SSA
2
M
−
1
/
R
C
M
−
1
2
M
−
1
/
d
1
T
s
2
/
d
1
T
s
CA
2
M
−
1
/
R
C
M
−
1
2
M
−
1
/
M
2
/
d
1
2
T
s
2
M
−
1
/
M
/
d
1
2
T
s
Buck-boost
SSA
2
/
R
C
2
M
/
d
1
T
s
2
/
d
1
T
s
CA
2
/
R
C
2
M
/
M
+
1
2
/
d
1
2
T
s
2
M
/
M
+
1
/
d
1
2
T
s
KY
SSA
M
2
−
4
M
+
2
/
R
C
M
2
−
3
M
+
2
2
M
−
1
/
d
1
T
s
2
−
M
CA
M
2
−
4
M
+
2
/
R
C
M
2
−
3
M
+
2
2
M
−
1
2
/
d
1
2
T
sTable 5
Selection strategy between SSA and CA for high-accuracy small signal modeling.
Name
Criterion
Modeling selection
Buck
2
M
/
d
1
T
s
1
−
M
<
2
M
2
/
d
1
2
T
s
SSA
2
M
/
d
1
T
s
1
−
M
>
2
M
2
/
d
1
2
T
s
CA
Boost
min
2
M
−
1
/
d
1
T
s
,
2
/
d
1
T
s
<
2
M
−
1
/
M
2
/
d
1
2
T
s
SSA
min
2
M
−
1
/
d
1
T
s
,
2
/
d
1
T
s
>
2
M
−
1
/
M
2
/
d
1
2
T
s
CA
Buck-boost
min
2
M
/
d
1
T
s
,
2
/
d
1
T
s
<
2
M
/
M
+
1
2
/
d
1
2
T
s
SSA
min
2
M
/
d
1
T
s
,
2
/
d
1
T
s
>
2
M
/
M
+
1
2
/
d
1
2
T
s
CA
KY
2
M
−
1
/
d
1
T
s
2
−
M
<
2
M
−
1
2
/
d
1
2
T
s
SSA
2
M
−
1
/
d
1
T
s
2
−
M
>
2
M
−
1
2
/
d
1
2
T
s
CA


### 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 phase-frequency response and the closed-loop controlled converters’ load transient response are presented in Figures10–13. These figures confirm the correctness of the proposed selection strategy in Table 5.Figure 10
Simulation verification with the buck converter (Vi = 1.2 V, fs = 100 MHz, C = 10 nF, L = 40 nH, and R = 40 Ω): (a) d1=0.2, 2M/d1Ts1−M<2M2/d12Ts; and (b) d1=0.5, 2M/d1Ts1−M>2M2/d12Ts.Figure 11
Simulation verification with the boost converter (Vi = 1.2 V, fs = 100 MHz, C = 20 nF, L = 2 nH, and R = 15 Ω): (a) d1=0.2, min2M−1/d1Ts,2/d1Ts<2M−1/M2/d12Ts; and (b) d1=0.5, min2M−1/d1Ts,2/d1Ts>2M−1/M2/d12Ts.Figure 12
V
o and io of the designed buck converter with 2M/d1Ts1−M>2M2/d12Ts (Vi = 1.2 V, fs = 100 MHz, C = 10 nF, L = 40 nH, and R = 80/40 Ω) and PM = 45°. (a) Compensator based on CA. (b) Compensator based on SSA.
(a)
(b)Figure 13
V
o and io of the designed boost converter with min2M−1/d1Ts,2/d1Ts<2M−1/M2/d12Ts (Vi = 1.2 V, fs = 100 MHz, C = 20 nF, L = 2 nH, and R = 30/15 Ω) and PM = 45°. (a) Compensator based on SSA. (b) Compensator based on CA.
(a)
(b)Figure12 presents the simulated output voltage Vo and load current io of the designed closed-loop 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 2M/d1Ts1−M>2M2/d12Ts, and the simulated result (Figure 12) also confirms that the closed-loop 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 Vo and the io of the closed-loop 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 min2M−1/d1Ts,2/d1Ts<2M−1/M2/d12Ts and the simulated result (Figure 13) also confirms that the closed-loop controller designed with SSA method obtain a better stability and transient response. When there is a sudden change of the io, 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 closed-loop controller of the DC-DC 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 DC-DC converter topologies.


## 7.1. Derivation

By using the methods presented in [3] and [6], the approximate poles and zeros for different DC-DC 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 phase-frequency responses of the DC-DC 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 phase-frequency response. From Table 4, we propose a selection strategy of high-accuracy small signal modeling method for the DC-DC converters as in Table 5.Table 4
Approximate poles and zeros for various DC-DC converters in DCM.
Name
Method
First pole
Second pole
Zero
Buck
SSA
2
−
M
/
R
C
1
−
M
2
M
/
d
1
T
s
1
−
M
CA
2
−
M
/
R
C
1
−
M
2
M
2
/
d
1
2
T
s
Boost
SSA
2
M
−
1
/
R
C
M
−
1
2
M
−
1
/
d
1
T
s
2
/
d
1
T
s
CA
2
M
−
1
/
R
C
M
−
1
2
M
−
1
/
M
2
/
d
1
2
T
s
2
M
−
1
/
M
/
d
1
2
T
s
Buck-boost
SSA
2
/
R
C
2
M
/
d
1
T
s
2
/
d
1
T
s
CA
2
/
R
C
2
M
/
M
+
1
2
/
d
1
2
T
s
2
M
/
M
+
1
/
d
1
2
T
s
KY
SSA
M
2
−
4
M
+
2
/
R
C
M
2
−
3
M
+
2
2
M
−
1
/
d
1
T
s
2
−
M
CA
M
2
−
4
M
+
2
/
R
C
M
2
−
3
M
+
2
2
M
−
1
2
/
d
1
2
T
sTable 5
Selection strategy between SSA and CA for high-accuracy small signal modeling.
Name
Criterion
Modeling selection
Buck
2
M
/
d
1
T
s
1
−
M
<
2
M
2
/
d
1
2
T
s
SSA
2
M
/
d
1
T
s
1
−
M
>
2
M
2
/
d
1
2
T
s
CA
Boost
min
2
M
−
1
/
d
1
T
s
,
2
/
d
1
T
s
<
2
M
−
1
/
M
2
/
d
1
2
T
s
SSA
min
2
M
−
1
/
d
1
T
s
,
2
/
d
1
T
s
>
2
M
−
1
/
M
2
/
d
1
2
T
s
CA
Buck-boost
min
2
M
/
d
1
T
s
,
2
/
d
1
T
s
<
2
M
/
M
+
1
2
/
d
1
2
T
s
SSA
min
2
M
/
d
1
T
s
,
2
/
d
1
T
s
>
2
M
/
M
+
1
2
/
d
1
2
T
s
CA
KY
2
M
−
1
/
d
1
T
s
2
−
M
<
2
M
−
1
2
/
d
1
2
T
s
SSA
2
M
−
1
/
d
1
T
s
2
−
M
>
2
M
−
1
2
/
d
1
2
T
s
CA


## 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 phase-frequency response and the closed-loop controlled converters’ load transient response are presented in Figures10–13. These figures confirm the correctness of the proposed selection strategy in Table 5.Figure 10
Simulation verification with the buck converter (Vi = 1.2 V, fs = 100 MHz, C = 10 nF, L = 40 nH, and R = 40 Ω): (a) d1=0.2, 2M/d1Ts1−M<2M2/d12Ts; and (b) d1=0.5, 2M/d1Ts1−M>2M2/d12Ts.Figure 11
Simulation verification with the boost converter (Vi = 1.2 V, fs = 100 MHz, C = 20 nF, L = 2 nH, and R = 15 Ω): (a) d1=0.2, min2M−1/d1Ts,2/d1Ts<2M−1/M2/d12Ts; and (b) d1=0.5, min2M−1/d1Ts,2/d1Ts>2M−1/M2/d12Ts.Figure 12
V
o and io of the designed buck converter with 2M/d1Ts1−M>2M2/d12Ts (Vi = 1.2 V, fs = 100 MHz, C = 10 nF, L = 40 nH, and R = 80/40 Ω) and PM = 45°. (a) Compensator based on CA. (b) Compensator based on SSA.
(a)
(b)Figure 13
V
o and io of the designed boost converter with min2M−1/d1Ts,2/d1Ts<2M−1/M2/d12Ts (Vi = 1.2 V, fs = 100 MHz, C = 20 nF, L = 2 nH, and R = 30/15 Ω) and PM = 45°. (a) Compensator based on SSA. (b) Compensator based on CA.
(a)
(b)Figure12 presents the simulated output voltage Vo and load current io of the designed closed-loop 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 2M/d1Ts1−M>2M2/d12Ts, and the simulated result (Figure 12) also confirms that the closed-loop 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 Vo and the io of the closed-loop 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 min2M−1/d1Ts,2/d1Ts<2M−1/M2/d12Ts and the simulated result (Figure 13) also confirms that the closed-loop controller designed with SSA method obtain a better stability and transient response. When there is a sudden change of the io, 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 closed-loop controller of the DC-DC 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 DC-DC 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 DC-DC 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 DC-DC 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 high-accuracy modeling method for various DC-DC converters operating in DCM, verified by simulations, which is necessary and beneficial in the design of a more accurate DCM closed-loop controller for DC-DC converters, achieving better stability and transient response.



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*Source: 3901693-2018-10-28.xml*