Comparison of Different Small Signal Modeling Methods ... - IEEE Xplore

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Comparison of Different Small Signal Modeling Methods for Bidirectional DC-DC Converter Enes Ugur∗ and Bulent Vural Department of Electrical Engineering, Yildiz Technical University, Istanbul, Turkey [email protected], [email protected]

Abstract—Bidirectional dc-dc converters are widely used in applications requiring bidirectional power flow. Control of these converters in a more suitable way and modeling and analyzing of them in a simpler method is still a research topic. In this paper different small signal modeling methods for bidirectional dc-dc converters are compared. Mathematical averaged small signal modeling and ac sweep tool on PSIM simulation software are used to obtain ac response of the converter. These results are compared in terms of bode plots to evaluate accuracy and convenience of the modeling techniques. Step-by-step calculations and simulations are presented together with detailed comparison of simulation results.

I. INTRODUCTION Bidirectional buck-boost DC/DC converters are extensively required to transfer power in either direction in many applications, such as renewable energy systems, hybrid electric vehicle energy systems, uninterrupted power supplies and battery chargers. Their low cost and high efficiency with decreased number of devices also makes them popular for industrial applications [1] and [2]. Numerous studies have been investigated bidirectional dcdc converter topologies and their topologies, control methods, related technologies like soft start method, parameters optimization have been discussed [3]. Although their topology is well understood, there is numerous ways for controlling these converters. This can be achieved by building a feedback circuit that adjusts the converter control input in such a way that the output voltage is precisely regulated, and is unaffected from disturbances in the input voltage or in the load current [4]. Accurate models which cover the dynamics of the power conversion system are needed to understand the switching period modulation fully in terms of dynamic transient performance and to ensure stability in all conditions. Small signal ac models are aimed to calculate low-frequency components of converter waveforms with removing the switching ripples by averaging over one switching period. The magnitude and phase of these components mainly depends on frequency response of the converter. There are several different ac modeling methods and most of them are mathematical models while some others are simulation based. Mathematical methods are attributed on the use of an averaged mathematical model devoted to a specific power topology. Occasionally the topology under inspection does not have an averaged model, and deriving its ac response becomes a tedious job. PSIM offers a possibility to ac sweep

a cycle by cycle model. Thanks to its execution speed, one can directly build transient models and sweep them using a variable-frequency source [5]. The rest of the paper is organized as follows. In section II, the mathematical small-signal averaged model of the buckboost bidirectional converter is derived. In section III, the small-signal model is established with PSIMs ac sweep tool. In section IV, results of each modeling methods are compared in terms of bode diagrams to confirm the effectiveness of the each small-signal modeling method. Finally, some concluding remarks and comments are given in section V. II. MATHEMATICAL SMALL SIGNAL AVARAGED MODEL In this section, mathematical small-signal averaged and linearized model of the buck-boost bidirectional converter will be derived. As in electric vehicle applications the load can be positive (i.e. during acceleration) or negative (i.e. during regenerative braking) so the load is depicted as a black box in the schematic of the converter which is illustrated in Fig.1. In order to simplify the modeling process, the resistance of inductor and capacitor (RL and RC ) will be neglected at calculations. The converter has two working modes which are defined by load condition. The converter can operate in boost mode to forward direction while the load is supplied by the input source or it may work in buck mode to backward direction while the input source is supplied by the load. Only boost mode of the converter will be analyzed. Note that only the continuous conduction mode (CCM) operation of the converter is examined in this paper.

Fig. 1.

Buck-boost bidirectional DC-DC converter

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In boost mode, only SF switch is controlled and SB switch is always open. During SF switch is conducting the inductor voltage and capacitor current is given by diL = vg (t)Ts vL (t) = L dt ic (t) = C

vo (t)Ts dv(t) =− dt RLoad

diL = vg (t)Ts − vo (t)Ts dt

vo (t)Ts dvo (t) ic (t) = C =− dt RLoad

C

d iL (t)Ts = vg (t)Ts + d v0 (t)Ts d(t)

d vg (t)Ts d(t)

=−

vg (t)Ts RLoad

ˆ sLˆıL (s) = vˆg (s) − D vˆo (s) + Vo d(s) vˆo (s) ˆ sC vˆo (t) = DˆıL (s) − d(s)I L− RLoad ˆıg (s) = ˆıL (s)

(2)

(4)

(5)

ˆ + Gvg (s)ˆ vˆo (s) = Gvd (s)d(s) vg (s)

(10)

where Gvd (s) is control-to-output transfer function and Gvg (s) is line-to-output transfer function. Hence, we can define the line-to-output transfer function by setting control signal variation to zero and the control-to-output transfer function by setting input voltage variation to zero. Gvg1 (s) =

1 vˆo (s) =  vˆg (s) D 1+s

Gvd1 (s) =

1 − s VIoLDL vˆo (s) Vo =  ˆ D 1 + s L2 + s2 LC d(s) 2

1 L

D 2 R

LC + s2 D 2

D R

+ d (t) iL (t)Ts

(9)

From these equations we can build the small signal equivalent circuit model as depicted in Fig.2. The input voltage and control signal are two independent inputs of the converter; therefore the output voltage variation of the converter can be represented as the superposition of terms originated from two inputs.

(3)

where iL (t)Ts is low-frequency averaged value of inductor current using small-ripple approximation. By averaging waveforms of inductor voltage and capacitor current, the following formulas can be obtained where d’(t)=1-d(t) L

related parts and simplifying, the following formulas can be obtained in s-domain

(1)

where vg (t)Ts and vo (t)Ts are low-frequency averaged values of input and output voltages respectively. For the remaining time SF switch is off and body diode of SB (DB ) is conducting. The inductor voltage and capacitor current for this subinterval can be described as vL (t) = L

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(11)

(12)

D

(6)

The input current of the converter is equal to the inductor current in both subintervals so DC and low-frequency AC variations in the converter input current can be described as ig (t)Ts = iL (t)Ts

(7)

If we define each signal as sum of DC values (capital letter quantities) and small AC variations (denoted with a small ”hat”) and insert these definitions into (5-7), we can obtain the followings 

 dIL dˆıL (t) L + = (Vg − D Vo ) dt dt ˆ ˆ vo (t)) +(ˆ vg (t) − D vˆo (t) + Vo d(t)) + (d(t)ˆ   dVo dˆ vo (t) Vo C + = (D IL − ) dt dt RLoad ˆ L − vˆo (t) ) − (d(t)ˆ ˆ ıL (t)) +(DˆıL (t) − d(t)I RLoad Ig + ˆıg (t) = IL + ˆıL (t)

Small signal equivalent circuit model for boost mode

The obtained transfer functions can be reorganized in the following standard normalized form where Gg0 and Gd0 are DC gains for each transfer function, w0 is the angular frequency, Q is quality factor and wz is angular frequency of zero. Gvgi (s) = Gg0 Gvdi (s) = Gd0

(8)

First parts in (8) contain DC terms; second parts include first order ac terms and the last ones have second order AC terms which are nonlinear. DC terms are equal on both sides of the equations and we can neglect second order ac terms due to their being very small in magnitude. By taking apart from

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Fig. 2.

1+

1 + ( ws0 )2

1− 1+

(13)

s Qw0

s Qw0

s wz

(14)

+ ( ws0 )2

Then, salient feature parameters of each mode are given in Table.I. TABLE I C ALCULATED PARAMETERS OF TRANSFER FUNCTION Mode

Gg0

Gd0

Boost

1/D 

V0 /D 

w0  √D LC

Q 

D R

C L

wz D 2 R L



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III. M ODELING WITH AC SWEEP TOOL OF PSIM In this section, the open loop transfer functions of each mode are provided in terms of bode plots with the help of PSIM. The principle of the AC analysis based on injecting a small AC excitation signal as perturbation into the system input signal and extracting the signal from output. By this excitation, the frequency response of a circuit can be evaluated. PSIM enables any circuit to be analyzed in its original switch mode form by ac sweep tool. Furthermore neglected terms in average modeling, i.e. RL and RC for simplifying modeling process, can be taken into account. A proper sinusoidal voltage source should be identified for obtaining accurate ac analysis results. The amplitude of the source must be small enough to keep the perturbation in the linear region and it must be large enough for output signal not affecting by numerical errors [6]. The excitation source amplitude between 5-10% of reference voltage is appropriate for accurate ac analysis results. The converter parameters used for controller design is given in Table.II. The schematic of the buck-boost bidirectional converter for obtaining the open loop transfer function of boost mode is given in Fig.3 together with parameters of ac sweep block. After running the simulation, PSIM provides the AC solution in a very short time. TABLE II S PECIFICATIONS OF BIDIRECTIONAL DC/DC CONVERTER Parameters Nominal Value Input Voltage 24V Output Voltage 48V Inductor 600uH Output Capacitor 120uF Output Power 100W Switching Frequency 20kHz

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It can be seen that both modeling techniques has quite similar results. It should be noted here that, RL and RC elements are taken as very small amount (1n) for decreasing their effects on the results and to make the results comparable. On the other hand, even if they are very small, these parasitic loss elements cause a lower Q-factor to be observed. Because of this we observe some differentiation between graphs around the resonance frequency.

Fig. 4.

Bode diagrams of the converter at boost mode

V. C ONCLUSION Different small signal modeling methods for bidirectional dc-dc converters are compared. In mathematical modeling, heavy mathematical calculation is needed for each circuit. In addition to this, it is necessary to recalculate even with small changes in parameters. Moreover, all parasitic elements should be neglected to make the calculation possible for complex circuits. In PSIM, any circuit to be analyzed in its original switch mode form by ac sweep tool and there is no need to neglect some terms for simplify the modeling process and no need for recalculation with small variations. Both of modeling methods are compared to evaluate precision and suitability of them. Results show that both methods gives the similar results while the effort to obtain them differs extensively.The paper shows that simulation based modeling is a promising option and new topologies will be able designed easily and quickly with simulation based controller design. ACKNOWLEDGMENT This study is supported in part by TUBITAK under Grant 113M088.

Fig. 3.

Small signal equivalent circuit model for boost mode

IV. S IMULATION R ESULTS In this section, bode plots of each modeling techniques are compared to evaluate accuracy and convenience of the modeling techniques. The bode plot of mathematical transfer function are constructed by using Matlab. Then they are compared with bode plots obtained with AC sweep tool. Fig.4 depicts the comparison of bode plots for boost mode. The bode plot of mathematical transfer function has a crossover frequency of 5.73 kHz, with a phase margin of 74.5 degrees, while bode plot of AC sweep tool has a crossover frequency of 5.66 kHz, with a phase margin of -75 degrees.

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R EFERENCES [1] X. Hu and C. Gong, “A High Gain Input-Parallel Output-Series DC/DC Converter with Dual Coupled-Inductors,” IEEE Transactions on Power Electronics, vol. PP, no. 99, pp. 1–1, 2014. [2] Lung-Sheng Yang and Tsorng-Juu Liang, “Analysis and Implementation of a Novel Bidirectional DCDC Converter,” IEEE Transactions on Industrial Electronics, vol. 59, no. 1, pp. 422–434, Jan. 2012. [3] W. Jianhua, Z. Fanghua, G. Chunying, and C. Ran, “Modeling and analysis of a buck/boost bidirectional converter with developed PWM switch model,” in 8th International Conference on Power Electronics ECCE Asia. IEEE, May 2011, pp. 705–711. [4] R. W. Erickson and D. Maksimovi´c, Fundamentals of Power Electronics, 2nd ed. Boston, MA: Springer US, 2001. [5] C. Basso, Switch-Mode Power Supplies Spice Simulations and Practical Designs, 2008. [6] “PSIM and SmartControl Tutorials,” 2004. [Online]. Available: http://www.psim-europe.com/learningcenter tutorials.php