Average and Small Signal Modeling of Negative-Output KY Boost Converter in CCM Operation

Negative-output KY Boost converter, which can obtain the negative output voltage and could be driven easily, is a good topology to overcome traditional Boost and Buck-Boost converters and it is believed that this converter will be widely used in engineering applications in the future. In this study, by using the averaging method and geometrical technique, the average and small signal model of the negative-output KYBoost converter are established.TheDC equilibriumpoint and transfer functions of the system are derived and analyzed. Finally, the effectiveness of the establishedmodel and the correctness of the theoretical analysis are confirmed by the circuit experiment.


Introduction
In recent years, exploring new topologies of DC-DC converters has attracted much attention of many researchers since it plays an important role in DC-DC switching power supplies to improve their performance, and different kinds of DC-DC converters have been proposed [1][2][3][4][5][6][7][8][9], such as Buck converter [1,2], Boost converter [3], Buck-Boost converter [3], Cuk converter [3], Sepic converter [4], Luo converter [5], Superbuck converter [6], and single-inductor-multipleoutput converter [7,8].Among them, a good topology of DC-DC converter, named negative-output KY Boost converter, has been proposed by Hwu et al. in 2009 [9].For this new converter, it can obtain negative-output voltage; that is, the polarity of output voltage is in contrast to its input voltage so that it makes up the drawbacks of the traditional Boost converter who obtain the positive output voltage only.Additionally, this new converter could be driven easily since its power switch is connected to ground directly.Therefore, to obtain the negative-output voltage, this new converter is superior to the traditional Buck-Boost converter since the latter one is floating.Accordingly, it is believed that the negative-output KY Boost converter will gain the popularity in the field of needing the negative electricity source, which is indispensable for the audio amplifier, signal generator, or data transmission interface and so forth.Therefore, modeling and analysis of this new converter are an important and valuable work for its real designing in practical engineering.But, until now, there are only a few studies on this new converter [9,10].For example, in [9], the negative-output KY Boost converter and its operational principle have been presented.Under the assumption that the energy-transferring capacitor is large enough to keep the voltage on itself constant at the output voltage, the DC equilibrium point of the negative-output KY Boost converter has been derived; that is, the DC output voltage is equal to the negative DC input voltage over the results of one minus the DC duty cycle.In [10], the discrete model of the voltage-controlled negative-output KY Boost converter has been derived for investigating its stability and bifurcation, and the result indicates that the energy-transferring capacitor has an important influence on its dynamical behaviors; that is, the energy-transferring capacitor is bigger; the output voltage ripple is smaller whereas the instability region is larger.Thus, a compromise must be done when choosing the energy-transferring capacitor.So, the assumption in [9] is only an extreme case.If the condition of this extreme case is not satisfied, the result about the DC equilibrium point in [9] will not be accurate, and this point will be investigated in this paper by establishing the average model of the negative-output KY Boost converter.Moreover, the small signal model of the negative-output KY Boost converter is derived and corresponding transfer functions are also presented, analyzed, and confirmed.
The rest of this paper is organized as follows.In Section 2, the circuit operation, mathematical model, and some PSIM simulations of the negative-output KY Boost converter are briefly given.In Section 3, by using the averaging method and geometrical technique, the average and small signal model of the system are established.The corresponding transfer functions are also derived and analyzed.In Section 4, the circuit experimental results are given for confirmation.Finally, some concluding remarks and comments are given in Section 5.

Circuit Operation, Mathematical Model, and PSIM Simulations
The circuit schematic of the negative-output KY Boost converter is shown in Figure 1.It consists of one input voltage V in , one power switch , two diodes:  1 and  2 , one energytransferring capacitor   , one input inductor , one output capacitor  0 , and one load .Note that the power switch  is directly controlled by the PWM signal V  and all the components are assumed as ideal.The current through  and   are defined as   and   , respectively.The voltage across   ,  0 , and  is defined as V  , V 0 , and V  , respectively.Also, this paper only takes the continuous conduction mode (CCM) operations into consideration, in which there are two operation modes (Figure 2) in this converter.The circuit parameters here are chosen as V in = 8 V,  = 1 mH,   = 2 F,  0 = 40 F,  = 100 Ω,  = 1/,  = 25 kHz,  = 1/, and  = 0.5.Mode 1 (Figure 2(a)) shows that the power switch  is turned on and the diode  1 is conducted whereas the diode  2 is opened.Accordingly, the voltage across the inductor  is equal to the input voltage V in , thereby causing the inductor  to be magnetized.The voltage across the energy-transferring capacitor   is the same as the voltage across the output capacitor  0 .The sum of the current through the energytransferring capacitor   , the current through the output capacitor  0 , and the current through the load  equals zero.
Therefore, the mathematical model for this mode can be derived as follows: Mode 2 (Figure 2(b)) shows that the power switch  is turned off and the diode  1 is opened whereas the diode  2 is conducted.In this mode, the input voltage V in plus the voltage across the energy-transferring capacitor   equals the voltage across the inductor .The current through the output capacitor  0 plus the current through the load  equals zero.The current through the energy-transferring capacitor   plus the current through the inductor  equals zero.Thus, the mathematical model for this mode can be derived as follows: According to Figure 1 and using PSIM which is widely used in power electronics and motor drives simulating [11,12], the time-domain waveforms for the voltage V  , V 0 , and V  can be obtained and shown in Figures 3(a) and 3(b), respectively.Note that the voltage V  reflects the PWM signal V  .That is, if the voltage V  equals zero, it describes that the power switch  is turned on; that is, the PWM signal V  is high level.But, when the power switch  is turned off, that is, the PWM signal V  is low level, V  is equal to the voltage V  .Hereafter, the average and small signal model for the negative-output KY Boost converter are going to be derived and analyzed.

Derivations for Average and Small Signal Model
By using the averaging method [13], the average model of the negative-output KY Boost converter can be derived by averaging circuit variables of the system within each switching period: where ⟨  ⟩, ⟨V 0 ⟩, ⟨V  ⟩, ⟨  ⟩, and ⟨V in ⟩ are the average values of   , V 0 , V  ,   , and V in , respectively.Obviously, it is necessary to derive the expressions for ⟨  ⟩ and ⟨V  ⟩ in (3) to obtain the average model for the negative-output KY Boost converter completely.
By applying the ampere-second balance on the capacitor   , ⟨  ⟩ can be expressed as a function of ⟨  ⟩ to be Here, in order to see the relationship between the voltage V  and V 0 clearly and derive the expression for ⟨V  ⟩, the closedup view of the voltage V 0 , V  , and V  is replotted in Figure 4.One can see that the voltage V  is the same as V 0 within (, ( + )) and they are denoted as   at the instant  and   at ( + ).But, within (( + ), ( + 1)), these two voltages (V  and V 0 ) are not equal to each other and they are denoted as   and  0 at the ( + 1) before abruptly changing point, respectively.After abruptly changing point, they are denoted as  +1 and   =  +1 for the characteristic of the negative-output KY Boost converter.
Therefore, the voltage V  cannot be considered as equaling V 0 within the whole switching period.That is, the assumption that the energy-transferring capacitor is large enough to keep the voltage on itself at the output voltage in [9] is only the extreme case and not suitable here.
Assuming that both the voltage V  and voltage V 0 increase linearly within (, ( + )) and V  decreases and V 0 increases linearly within (( + ), ( + 1)), the following equation for ⟨V 0 ⟩ can be obtained by applying the geometric technique: where In the same way, the equation for ⟨V  ⟩ can also be derived and shown as follows: where  Thus, the voltage ⟨V  ⟩ can be derived by taking ( 7) minus ( 5) and using ( 6) and ( 8), and its expression is Accordingly, the overall average equations can be obtained by substituting ( 4) and ( 9) into (3): In order to obtain the small signal model of the system, it is necessary to use the perturbation and linearization of (10).Assume that   ,  0 ,  in , and  are the DC value of ⟨  ⟩, ⟨V 0 ⟩, ⟨V in ⟩, and , respectively, and î , V0 , Vin , and d are their small AC values.The following equations are also assumed to be: Therefore, the DC equilibrium point of the negativeoutput KY Boost converter can be calculated by substituting (11) into (10) and then extracting the DC values: From (12), one can see that the DC output voltage and DC inductor current include not only the DC duty cycle and DC input voltage, but also the switching frequency , load , energy-transferring capacitor   , and output capacitor  0 .Thus, (12) are very different from the results in [9].
By substituting (11) into (10) and detaching the AC values and neglecting the second and higher order AC terms since their values are very small, the small signal model of the negative-output KY Boost converter can be obtained: where  = (1 − ) 2 /(2 0 ) and  = (1 − ) 2 /(2  ).Therefore, the transfer functions of the negative-output KY Boost converter can be derived by using Laplace transform on (13).Here, the transfer function from the duty cycle to the output voltage of the negative-output KY Boost converter is concerned.Its expression can be derived by making the perturbation of the input voltage be zero and then calculating  V () = V0 ()/ d(): Under the given circuit parameters in Section 2, Bode diagram of  V () is calculated and shown in Figure 5. Also, the PSIM simulations for Bode diagram of  V (), which is obtained from the original switch mode form, where no average model is required [14,15], are shown in Figure 5. Thus, the theoretical calculations are in basic agreement with PSIM simulations.The subtle discrepancy is mainly caused by the assumption in theoretical derivations.
Additionally, the transfer functions from the input to the output voltage ( VV ()), the input voltage to the inductor current ( V ()), and the duty cycle to the inductor current (  ()) are listed in appendix.

Circuit Experiments
According to Figure 1, the hardware circuit of the negativeoutput KY Boost converter is constructed by using IRFP460 and MUR1560 for realizing the power switch  and two diodes ( 1 and  2 ), respectively.The digital oscilloscope GDS 3254 is applied to capture the measured time-domain waveforms in the probes and the Agilent E5061B LF-RF network analyzer is employed to capture the measured gain and phase in the probes.Under the given circuit parameters in Section 2, the time-domain waveforms from the circuit experiment for the voltage V  and V  are shown in Figure 6(a), and the voltage V 0 and voltage V  are shown in Figure 6(b).Comparing Figures 6(a) and 6(b) with Figures 3(a) and 3(b), respectively, it is found that they are in basic agreement with each other and it again demonstrates that the voltage V  cannot be really considered as equaling V 0 within the whole switching period.
Moreover, Bode diagram of  V () from the circuit experiment is tested by using Agilent E5061B LF-RF network analyzer and shown in Figure 7. Also, the corresponding theoretical calculations and PSIM simulations for Bode diagram of  V () are shown in Figure 7. Thus, they are in basic agreement with each other.The subtle discrepancy is mainly caused by the assumption in theoretical derivations and PSIM simulations.

Conclusions
By using the averaging method and the geometric technique to calculate the average value, the average model and small signal model of the negative-output KY converter are established.The obtained DC equilibrium point shows that the DC output voltage and DC inductor current are affected by not only the DC duty cycle and DC input voltage, but also the switching frequency, load, energy-transferring capacitor, and output capacitor.Moreover, the transfer function from the duty cycle to the output voltage is also derived, and the theoretical calculations and PSIM simulations are in basic agreement with the circuit experiments.The obtained results here will be helpful for designing the negative-output KY Boost converter in practical engineering.

Figure 4 :
Figure 4: Closed up view of the voltage V 0 (dashed line), V  (solid line), and V  .

Figure 5 :
Figure 5: Bode diagram of  V () from the theoretical calculations and PSIM simulations.