Acta Polytechnica https://doi.org/10.14311/AP.2023.63.0050 Acta Polytechnica 63(1):50–64, 2023 © 2023 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague COMPARATIVE STUDY OF STATE SPACE AVERAGING AND PWM WITH EXTRA ELEMENT THEOREM TECHNIQUES FOR COMPLEX CASCADED DC-DC BUCK CONVERTER Sangeeta Shete∗, Prasad Joshi Shivaji University, Government College of Engineering, Department of Electrical Engineering, Karad, Maharashtra, India ∗ corresponding author: sangeetashete2019@gmail.com Abstract. Until now, most commonly used state space averaging and PWM techniques have been applied to different converter topologies and their advantages and disadvantages were stated. However, the superiority of an analytical technique was not justified based on a parametric comparison of these techniques for the same converter topology. Hence, in this paper, first time a comparative evaluation of two commonly used modelling techniques for fourth order converter is presented. The first approach makes use of a state-space averaging of the converter and is based on analytical manipulations using different state representations of the converter. The second approach is based on PWM switch modelling with an extra element theorem and consists of topological manipulations. The two modelling techniques are applied to the same complex cascaded DC-DC Buck converter and a transfer function is obtained. These techniques are compared for different features and the study concludes that state space modelling technique is systematic and less complicated than PWM switch modelling with an extra element theorem for a higher order converter. Keywords: Analytical technique, cascaded DC-DC Buck converter, state space averaging, PWM switch modelling, extra element theorem, transfer function. 1. Introduction Analytical techniques play a vital role in analysing the behaviour of converters under different conditions, performance improvements and design aspects. DC-DC converters have a high number of components and nonlinear behaviour due to the existence of a switch and diode in circuit, which makes them complex in nature. A study of dynamic behaviour and assessment of stability of such nonlinear and complex circuits is a very challenging job. Article [1] presented the review of existing analytical techniques in a structured and meaningful manner and stated that the circuit averaging (CA), state space averaging (SSA), and PWM switch modelling are most commonly used analytical techniques, whereas signal flow graphs, energy factor, switching function, S-Z method are special techniques. SSA is an organised method which allows the inclusion of parasitic elements of a circuit even in the initial stage, permits to explore a linear system with a zero initial condition and nonlinear systems with all initial conditions [2–4]. It is preferred to design a robust controller. However, as it doesn’t consider switching frequency in the analysis, this technique is applied by neglecting the ripple effect on inductor current and output voltage [5, 6], which affects its accuracy [7–9]. State space representation is used for mathematical modelling of SEPIC converter, which is further used to contrivance the controller. The results of PI controls and hysteresis controls are compared using PSIM software [10]. The PWM switch modelling technique employs determination of invariant properties of the PWM switch to obtain the average model of circuit and small signal characteristics can be obtained from this average model [11]. PWM technique with conventional approach is simple and pedagogical approach of analysis and provides complete information about steady state and dynamic properties of the converter. This approach is also useful for frequency domain analysis and quasi resonant converters. PWM technique using Extra Element Theorem (EET) is more efficient and practical for general circuits, as it leads to a faster analysis due to a reduced mathematical manipulation, meaningful form of expressions and it is easy to track the errors [12]. An enormous study has been conducted regarding SSA and PWM switch modelling techniques for converters. Article [2] investigated dynamic modelling of Zeta converter, wherein transfer functions, bode plot, transient response and steady state response were obtained using MATLAB and PSPICE. Article [13] obtained state equations and output equations using SSA technique for Buck, Boost and Buck-Boost converter and observed that the state space model simulation results are comparable with a hardware model with a deviation of only 0.0015 V. However, this technique doesn’t simulate ripple effect of inductor current and output voltage due to the absence of switching frequency. Article [14] analysed SEPIC converter using an average state-space modelling approach taking into account power losses in converter elements and concluded that this methodology 50 https://doi.org/10.14311/AP.2023.63.0050 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 63 no. 1/2023 Comparative study of state space averaging and PWM . . . contributes for converter design as per the requirements and reduces the need for accurate time domain simulations. Mathematical modelling of DC-DC Buck converter using state space averaging and validation of the models was performed using PSIM and MATLAB Simulink [15]. The studies were performed for state space small signal modelling of a double Boost converter integrated with SEPIC converter [3, 16, 17], and SEPIC converter with coupled and uncoupled inductor [18]. Book [19] presented the well-defined and systematic process of small signal analysis of nonlinear circuits using state space analysis to determine the control to output transfer function. The concepts and mathematical modelling of circuits using EET [20] and PWM with EET [21] were studied. The output impedances of three PWM DC-DC converters were analysed in a general and unified manner using PWM switch modelling with EET technique [22]. Article [23] presented the effect of an input filter interaction in the small signal analysis of voltage and current control mode of DC-DC Boost converter and concluded that the methodology is universally adopted. MATLAB Simulink was employed to study the output voltage for multiple DC-DC converters [24] and Boost converter [25]. Article [12] presented PWM with EET to a general circuit and concluded that it is a fast analytical technique which produces a well-ordered polynomial and low entropy form of transfer functions. However, by applying this technique to a complex SEPIC converter, the study concluded that as the circuit becomes complicated, a conventional node and mesh analysis is better instead of the PWM with EET. Nevertheless, the basis of comparison for this statement is not presented in the study, and therefore it is necessary to study the features of PWM with EET technique for higher order circuits. Some of the existing studies have developed the analytical models, obtained simulation results and commented on advantages, disadvantages and applications in a relative manner without comparing the results with a counterpart technique, which doesn’t fit into the acceptable scientific and technical approach. The advantages, disadvantages and superiority of a technique should be based on a comparison of two or more techniques for the same converter topology. Even though numerous studies regarding SSA and PWM with EET techniques for different converters have been performed, surprisingly no efforts were done to compare these techniques for the same converter and to comment on the superiority of the analytical technique. The past few years have perceived notable development in the research of DC-DC converter topologies. Although the conversion efficiency of a single-stage converter is better than that of the two-stage converter, the quadratic converters are proposed for extremely large range of conversion ratios. As a part of this, complex cascaded Buck converter (CBC) is always preferred for many applications, as it has a better conversion ratio, high voltage step down ratio, and large voltage regulation [26]. Therefore, in this paper, a comparative evaluation of two widely preferred modelling tools for a higher order DC-DC converter is presented. The first approach makes use of a state-space averaged model of the converter based on the analytical manipulations, whereas the second approach is based on the PWM switch modelling with EET technique based primarily on the manipulations of circuits. The two modelling techniques are applied to a same complex cascaded DC-DC Buck converter and are compared on the basis of their various features to decide the superiority for a higher order circuit. 2. Materials and methods 2.1. Complex cascaded DC-DC Buck converter The existence of two transistor switches of the conventional cascaded Buck converter can be reduced to a single transistor switch in the complex cascade DC-DC Buck converter as shown in Figure 1, which has two inductors and two capacitors making it a fourth-order circuit and has a quadratic conversion ratio of D2. This single- transistor realisation is the most additional advantage over a straight forward cascade of two basic converters. The switching section consists of one active switch and three diodes. Though the circuit is complex, the conversion ratio of CBC cannot be realised with less than two capacitors, two inductors, and four switches. However, additional complexity of the converter network may compromise the wide conversion ratio. The control strategy of the complex CBC is depicted in Figures 2a and 2b. The ON state and OFF state correspond to the operation of two Buck converters connected in a cascade as depicted in Figure 3. Figure 1. Topology of cascade DC-DC Buck converter. 51 Sangeeta Shete, Prasad Joshi Acta Polytechnica (a) (b) Figure 2. Switching mechanism of complex CBC. Figure 3. Operation of complex CBC. The total output voltage is given by Equation 1. V2 = 1 T ∫ T 0 (−DV1 + DV1(1 + D)) dt V2 = D2V1 (1) Where, D is the duty ratio and D2 = 1 T ∫ T 0 Ď2(t)dt The chopper circuit of CBC has one active switch and three passive switches, which are driven by switching function Ď2(t) and Ď ′2(t), respectively. The switching function is given as, Ď2(t) = { 1 0 < t < TON 0 TON < t < T 2.2. State space averaging technique The state-space averaging method is based on analytical operations for the converter states comprising the determination of linear state model for each possible configuration of circuit and subsequently combining all these elementary models into a single and unified one through a duty factor. The method involves a formulation of the state equation for each state, averaging, perturbation and then rearranging equations to obtain the required transfer function by applying Laplace transform. The complex CBC is considered in continuous conduction mode and switch and diodes are considered in ideal mode. Considering, inductor currents IL1, IL2, and capacitors voltages VC1 and VC2 as the state variables, the state equation and output equation in ON state of the CBC (Figure 2a) are expressed by the Equations 2 and 3. d(IL1) dt = V1 L1 − VC1 L1 d(IL2) dt = V1 L2 − VC2 L2 d(VC1) dt = IL1 C1 d(VC2) dt = IL2 C2 − V2 RC2   (2) V2 = VC2 (3) 52 vol. 63 no. 1/2023 Comparative study of state space averaging and PWM . . . Similarly, the state equation and output equation in OFF state of the converter (Figure 2b) are expressed by the Equations (4) and (5). d(IL1) dt = − VC1 L1 d(IL2) dt = VC1 L2 − VC2 L2 d(VC1) dt = IL1 C1 − IL2 C1 d(VC2) dt = IL2 C2 − V2 RC2   (4) V2 = VC2 (5) The general form of state equations for ON state and OFF state are given as, Ẋ(t) = A1x(t) + B1u(t) During dT Ẋ(t) = A2x(t) + B2u(t) During d’T Y (t) = C1x(t) During dT Y (t) = C2x(t) During d’T Where X(t) is the state vector, x(t) is the state variable matrix, u(t) is the input variable matrix, A1 and A2 are the state matrices and B1 and B2 are the input matrices, C1 and C2 are the output matrices and d is the switching function. Thus, Equations (2) and (3) are represented in matrix form as,   ˙IL1 ˙IL2 ˙VC1 ˙VC2   =   0 0 −1/L1 0 0 0 0 −1/L2 −1/C1 0 0 0 0 1/C2 0 −1/RC2     IL1 IL2 VC1 VC2   +   1/L1 1/L2 0 0  V1 (6) and V2 = [0 0 0 1]   IL1 IL2 VC1 VC2   (7) Equations (4) and (5) are represented in matrix form as,   ˙IL1 ˙IL2 ˙VC1 ˙VC2   =   0 0 −1/L1 0 0 0 1/L2 −1/L2 1/C1 −1/C1 0 0 0 1/C2 0 −1/RC2     IL1 IL2 VC1 VC2   +   0 0 0 0  V1 (8) and V2 = [0 0 0 1]   IL1 IL2 VC1 VC2   . (9) The matrices of Equations (6) to (9) are averaged with respect to switching function D and D ′ and are given in Equations (10) and (11). A = A1D + A2D ′ =   0 0 −1/L1 0 0 0 D ′ /L2 −1/L2 1/C1 −D ′ /C1 0 0 0 1/C2 0 −1/RC2   &B = B1D + B2D′ =   D/L1 D/L2 0 0   (10) C = C1D + C2D ′ = [ 0 0 0 1 ] (11) 53 Sangeeta Shete, Prasad Joshi Acta Polytechnica Thus, a complete averaged state space model of the CBC is given by the Equations (12) and (13).   ˙IL1 ˙IL2 ˙VC1 ˙VC2   =   0 0 −1/L1 0 0 0 D ′ /L2 −1/L2 1/C1 −D ′ /C1 0 0 0 1/C2 0 −1/RC2     IL1 IL2 VC1 VC2   +   D/L1 D/L2 0 0  V1 (12) V2 = [ 0 0 0 1 ]   IL1 IL2 VC1 VC2   (13) To obtain the linear small-signal state-space model, perturbation is added in each state variables and linear steady state model is obtained as in Equation (14).   ˙̂ IL1 ˙̂ IL2 ˙̂ V C1 ˙̂ V C2   =   0 0 −1/L1 0 0 0 D ′ /L2 −1/L2 1/C1 −D ′ /C1 0 0 0 1/C2 0 −1/RC2     ÎL1 ÎL2 V̂C1 V̂C2   +   −V1/D ′ L1 V1/D ′ L2 −DV1/2RC1 0   (14) The general equation of Laplace transform of the linear steady state model is given by the following expression, x̂(s) = [sI −A]−1[(A1 −A2)X + (B1 −B2)U]d̂(s) = [[sI −A]−1Bd]d̂(s). (15) For CBC, (sI −A) = s   1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1  −   0 0 −1/L1 0 0 0 D ′ /L2 −1/L2 1/C1 −D ′ /C1 0 0 0 1/C2 0 −1/RC2   (sI −A) =   s 0 1/L1 0 0 s −D ′ /L2 1/L2 −1/C1 D ′ /C1 s 0 0 −1/C2 0 s + 1/RC2   . (16) Now, [sI −A]−1 = Co-factor of ((sI −A))T Determinant of (sI −A) . (17) Let, Co-factors of (sI −A) =   C11 C12 C13 C14 C21 C22 C23 C24 C31 C32 C33 C34 C41 C42 C43 C44   then, (Co-factors of (sI −A))T =   C11 C21 C31 C41 C12 C22 C32 C42 C13 C23 C33 C43 C14 C24 C34 C44   . (18) Where, co-factors are given in Table 1. 54 vol. 63 no. 1/2023 Comparative study of state space averaging and PWM . . . Co-factor Expression Co- factor Expression C11 S 3+S2 ( 1 RC2 ) +S ( 1 L2C2 + D ′2 L1C1 ) + D ′2 RC1C2L2 C31 −S2 1 L1 −S 1 RC2L1 + 1 C2L1L2 C12 D ′ L2 − 1 RC1C2 −S 1 C1 C32 −S2 D ′ L2 −S D ′ RC2C2 C13 S 2 1 C1 −S 1 RC1C2 + 1 C1C2L2 C33 S3 + S2 ( 1 RC2 ) + S 1 L2C2 C14 − D ′ C1C2L2 C34 −S D ′ C2L2 C21 −S D ′ C1L1 − D ′ RC1C2L1 C41 D ′ C1L1L2 C22 S 3 + S2 ( 1 RC2 ) + S 1 L1C1 + 1 RC1C2L1 C42 −S2 1 L2 − 1 C1L1L2 C23 S2 D ′ C1 + S D ′ RC1C2 C43 −S D ′ C1L2 C24 S 2 1 C2 + 1 C1C2L1 C44 S3 + S D ′2 L2C1 + S 1 C1L1 Table 1. (Co-factors of (sI − A))T . Determinant of (sI −A) is s   s −D′/L2 1/L2D′/C1 s 0 −1/C2 0 s + 1/RC2   + 0   0 D′/L2 1/L2−1/C1 s 0 0 0 s + 1/RC2   +1/L1   0 s 1/L2−1/C1 D′/C1 0 0 −1/C2 s + 1/RC2   + 0   0 s D′/L2−1/C1 D′/C1 s 0 −1/C2 0   . This yields the determinant of (sI −A) as, 1 + s ( L2 + D ′2L1 R ) + s2[L2C2 + L1C1 + L1C2D ′2] + s3 ( L1L2C1 R ) + s4(L1L2C1C2) (19) Substituting Equations (18) and (19) in (15), [sI −A]−1 =   C11 C21 C31 C41 C12 C22 C32 C42 C13 C23 C33 C43 C14 C24 C34 C44   1 + s ( L2+D ′ 2L1 R ) + s2[L2C2 + L1C1 + L1C2D ′2] + s3 ( L1L2C1 R ) + s4(L1L2C1C2) (20) and Bd =   −V1/D ′ L1 V1/D ′ L2 −DV1/2RC1 0   . 55 Sangeeta Shete, Prasad Joshi Acta Polytechnica Thus, from Equation (14), the equation of Laplace transform of complex CBC becomes, x̂(s) = (21)      C11 C21 C31 C41 C12 C22 C32 C42 C13 C23 C33 C43 C14 C24 C34 C44   1 + s ( L2+D ′ 2L1 R ) + s2[L2C2 + L1C1 + L1C2D ′2] + s3 ( L1L2C1 R ) + s4(L1L2C1C2)     −V1/D ′ L1 V1/D ′ L2 −DV1/2RC1 0     d̂(s). The general equation of Laplace transform of the output at a steady state is given by, ŷ = Cx̂(s) + [(C1 −C2)X(s)]d̂(s). From Equation (7) and (9), C1 = C2. Thus, ŷ = Cx̂(s)d̂(s). Hence for complex CBC, V̂2(S) = (22) [ 0 0 0 1 ]      C11 C21 C31 C41C12 C22 C32 C42 C13 C23 C33 C43 C14 C24 C34 C44   1 + s ( L2 +D ′ 2 L1 R ) + s2[L2C2 + L1C1 + L1C2D ′ 2] + s3 ( L1 L2 C1 R ) + s4(L1L2C1C2)     −V1/D ′ L1 V1/D ′ L2 −DV1/2RC1 0     d̂(s). Solving Equation (23), control to output transfer function of complex CBC is, V̂2(s) d̂(s) = ( 1 + sL1 ( DD ′ 2R ) + s2L1C1 (1+D 2D )) 1 + s ( L2+D ′ 2L1 R ) + s2[L2C2 + L1C1 + L1C2D ′2] + s3 ( L1L2C1 R ) + s4(L1L2C1C2) . (23) It is realised that the calculation of matrix (sI − A)−1 becomes complicated for any high order circuits, which is in agreement with what is stated in [27]. 2.3. PWM switch with EET technique The main principle of the PWM switch modelling technique is the elimination of active and passive switches by their time-averaged models and obtain the averaged circuit model for a switched network, which is further inserted into the converter circuit. The final model is a time-averaged equivalent circuit model, where all branch currents and node voltages correspond to averaged values of corresponding original currents and voltages. Replacement of active and passive switches from the basic switch model (Figure 4) by switching functions d and d ′ , respectively, is depicted in Figure 5. Figure 4. Basic circuit model of switch. Figure 5. Replacement of switches. The invariant average equations for Figure 5 are, I1 = dI2 V23 = dV13 } (24) 56 vol. 63 no. 1/2023 Comparative study of state space averaging and PWM . . . Adding a small deviation in duty ratio function d and then differentiating Equation (24), ı̂1 = Dı̂2 + I2d̂ V̂23 = DV̂13 + V13d̂ } , (25) where, D, I2 and V13 are the steady state operating points of PWM switch. The PWM switch model from invariant average Equations (24) is obtained as depicted in Figure 6. Also, using Equation (25), dependent sources Dı̂2 and DV̂23 are replaced with 1:D transformer and moving control source d̂V13 from common terminal side to active terminal side, and the PWM switch model is obtained as in Figure 7. Thus, in PWM switch modelling with EET, the switch and diode are replaced point by point with its equivalent circuit model and further EET is employed to find out TF of the converter circuit. Figure 6. PWM switch model. Figure 7. PWM switch model as 1:D trans- former ratio. Figure 8. Two PWM switches as 1:D transformer ratio for CB. Therefore, for complex CBC, the conditions of ON state and OFF state of switches are employed and two PWM switches are identified, which are represented as S1 and S2. The identified PWM switch models are replaced with 1:D transformer along with their small signals as depicted in Figure 8. In order to determine dc operating point of complex CBC, all inductors are short circuited and capacitors are open circuited. Referring to Figure 8, dc operating point with respect to switch S1 is, Vap1 = V1. Current flowing through terminal ‘c’ of switch S1 is, Ic1 = Ic2 −DIc2 = D ′ Ic2. Similarly, dc operating point with respect to switch S2 is, Vap = V1(1 + D), Ic2 = V2 R = D2V1 R . Once the dc operating point is determined, 4-EET is applied that provides separate, independent steps of the analytical technique. The general form of control to output transfer function is [ v̂2(s) d̂(s) ] . In the very first step of 4-EET, denominator d̂(s) is determined, for which each impedance element of the converter is treated as extra element and input sources are set to zero. With respect to Figure 3 C1, C2, L2 and L1 are considered as 4 ports and numbered as port (1), (2), (3) and (4), respectively. Treating these ports as an extra element, 57 Sangeeta Shete, Prasad Joshi Acta Polytechnica various time constants are determined by removing port elements and observing the remaining circuit through the removed port. The denominator of the transfer function is given by the following equation: d̂(s) = 1 + a1s + a2s2 + a3s3 + a4s4, (26) where a1 = τ1 + τ2 + τ3 + τ4 a2 = τ1τ21 + τ1τ31 + τ1τ41 + τ2τ32 + τ2τ42 + τ3τ43 a3 = τ1τ31 τ231 + τ4τ14 τ241 + τ4τ14 τ341 + τ3τ32 τ423 a4 = τ4τ14 τ341τ2413   (27) Figure 9 is referred for determination of (τ) for a1, which is explained in Table 2. Figure 9. Determination of (a) τ1 (b) τ2 (c) τ3 (d) τ4. τ Expression τ1 Time constant of first port is Req x C1, determined by observing the circuit through port (1), capacitor of this port 1 is assumed as extra element and is removed temporarily, L1, L2 and C2 are assumed in dc condition. By observing through port 1, the Req is 0, therefore, τ1 = Req x C1 = 0. τ2 Time constant of second port is Req x C2, determined by observing the circuit through port (2), capacitor of this port is now extra element and is removed temporarily, L1, L2 and C1 are assumed in dc condition. After observation, Req is 0, therefore, τ2 = Req x C2 = 0. τ3 Time constant of third port is ( L2 Req ) , which is determined by observing the circuit through port (3). Inductor of this port is extra element and is removed temporarily and C1, C2 and L1 are kept in dc condition. By observation, Req is R. Hence, τ3 = L2R . τ4 Time constant of fourth port is ( L1 Req ) , which is determined by observing the circuit through port (4). Inductor of this port is extra element and is removed temporarily and C1, C2 and L2 are kept in dc condition. By observing the circuit it is seen that Req is RD′ 2 . Hence, τ4 = L1D ′ 2 R . Table 2. Determination of (τ) for a1. By summarizing the results of Table 2, we get: a1 = L2 R + L1D ′2 R (28) First two terms and the last term of the a2 (Equation (27)) are zero. The calculation of a2 is only addition of middle terms, i.e. (τ1τ41 + τ2τ32 + τ2τ42 ), which are indeterminate and is removed by changing the port sequence as [τ4τ14 + τ3τ23 + τ4τ24 ]. Referring the Figure 10, terms τ14 , τ23 and τ24 are determined as described in Table 3. By summing up the results from the Table 3, we get: a2 = L1C1 + L2C2 + L1C2D ′2. (29) The first term of the a3 (Equation (27)) cannot be determined as it gives a ratio of (0/0), this indeterminacy in time constants is removed with new sequence of ports for a3 which is, a3 = τ1τ31 τ 2 31 + τ4τ 1 4 τ 2 41 + τ4τ 1 4 τ 3 41 + τ3τ 3 2 τ 4 23. (30) 58 vol. 63 no. 1/2023 Comparative study of state space averaging and PWM . . . Figure 10. Determination of (a) τ14 (b) τ23 (c) τ24 . τ Expression τ14 Observing the circuit through port (1), port (4) is considered in high frequency state and ports (2) and (3) are kept in dc condition, capacitor of port (1) is treated as extra element and removed temporarily, equivalent resistance Req is RD′ 2 . Hence τ4τ 1 4 = L1C1. τ23 Observing the circuit through port (2), port (3) is considered in high frequency state and ports (1) and (4) are kept in dc condition, capacitor of port (2) is treated as extra element and removed temporarily, equivalent resistance Req is R. Therefore τ3τ23 is L2C2. τ24 Observing the circuit through port (2), port (4) is considered in high frequency state and ports (1) and (3) are kept in dc condition, capacitor of port (2) is treated as extra element and removed temporarily, equivalent resistance Req is R. Therefore τ4τ24 is L1C2D ′2. Table 3. Determination of (τ) for a2. The first two terms and last term of the Equation (30) are zero. To calculate τ341 , the circuit in Figure 11 is observed through port (3), while ports (1) and (4) are kept in a high frequency condition. Here Req is R which results in τ341 = L3 Req = L3 R . Hence, a3 = τ4τ14 τ 3 41 = L1L2C1 R . (31) Figure 11. Determination of τ341. Figure 12. Determination of τ2413. For the determination of a4, referring to Figure 12, τ2413 is determined by observing the circuit through port (2), while ports (4), (1) and (3) are kept in high frequency condition. In such situation, Req is R and τ2413 is RC2. The calculation of the term (τ4τ14 τ341τ2413) results in (L1L2C1C2). Hence, a4 = L1L2C1C2. (32) Using Equations (28), (29), (31) and (32) in Equation (26), the denominator of control to output transfer function of complex CBC becomes, d̂(s) = 1 + ( L2 R + L1D ′2 R ) s + (L1C1 + L2C2 + L1C2D ′2)s2 + ( L1L2C1 R ) s3 + L1L2C1C2s4. (33) Now, the excitation applied to the port is retained as it is and additional excitation is applied to the port of the extra element with the extra element removed. With these two excitations, the response is considered as null. Simultaneously, the conversion ratio D2 is differentiated with respect to D. Considering this differentiation and null response condition altogether, the numerator of transfer function is determined. 59 Sangeeta Shete, Prasad Joshi Acta Polytechnica Figure 13. Null condition at the output of complex CBC for determination of numerator. By considering a null response at the output of complex CBC (Figure 13), it is observed that the presence of ı̂L2(s) creates the condition as represented by the equation, ı̂p2(s) = ı̂c2d̂(s) and the resultant voltage across C1 is, V̂c1(s) = Vapd̂ (1/(sC1) (SL1 + 1/sC1) + Ic2d̂ [sL1||(1/sC1)] . (34) This voltage must be the same as that of D side of the second PWM switch. Hence, V̂c1(s) = − [ Vap2 D d̂− V̂c1(s) ] D. (35) From Equations (34) and (35), D ′ Vap1 1 1 + s2L1C1 + D ′ IC2 sL1 1 + s2L1C1 + Vap2 = 0. (36) Which yields the numerator of the control to output transfer function of complex CBC as given by, V̂2(s) = 1 + sL1 DD ′ 2R + s2L1C1 1 + D 2D . (37) Using Equations (33) and (37), the overall transfer function of complex CBC is given by the Equation (38): V̂2(s) d̂(s) = ( 1 + sL1 ( DD ′ 2R ) + s2L1C1 (1+D 2D )) 1 + s ( L2+D ′ 2L1 R ) + s2[L2C2 + L1C1 + L1C2D ′2] + s3 ( L1L2C1 R ) + s4(L1L2C1C2) . (38) 3. Simulation of complex CBC The simulation circuit is developed; voltage and current graphs (Figure 14), time response (Figure 15) and frequency response plot (Figure 16) are obtained for complex CBC using MATLAB simulation to study the behaviour of CBC. The circuit parameters for the simulation are selected as: Inductors L1 = 524 µH, L2 = 1200 µH, capacitors C1 = C2= 5 µF, output resistance R0 = 8.05 ohm, switch (Q1) parameters: FET ON resistance = 0.1 Ω, internal diode resistance = 0.01 Ω, internal diode forward voltage = 0.7 V, diode (D1, D2, D3) parameters: resistance = 0.01 Ω, forward voltage = 0.8 V, switching frequency = 50 KHz. For a duty ratio of 0.35, input voltage V1 = 25 V, and source current = 0.05746 A, output voltage V2 = 3 V and load current = 0.367 A was observed (Figure 14). Also output voltage and load currents for different duty ratios of complex CBC are depicted in Table 4. From the transient and steady state response, it is observed that the output voltage has settled to about 3 volts with the settling time of approximately 0.9 ms after the application of input voltage. The system is stable (Figure 16), as gain margin is positive (∞db) and phase margin is also positive (450°). 60 vol. 63 no. 1/2023 Comparative study of state space averaging and PWM . . . Figure 14. Voltage and current graphs of complex CBC. Figure 15. Transient and steady state response of complex CBC. Figure 16. Bode plot of control to output transfer function of complex CBC. 61 Sangeeta Shete, Prasad Joshi Acta Polytechnica Duty ratio Output voltage [V] Load current [A] 0.3 2.47 0.3076 0.4 3.434 0.4266 0.5 5.44 0.6762 0.6 8.601 1.068 0.7 11.78 1.464 Table 4. Load current and output voltage for different duty rates. 4. Results and discussion SSA and PWM switch modelling with EET technique for CBC (Section 2.2 and 2.3) are brainstormed, compared for different features and the results are summarized in Table 5. In SSA, a linearisation of all the components of the converter is performed, whereas in the PWM switch model method, only non-linear switching devices are linearised and linear components remain unchanged. In SSA technique, a set of differential equations are written which are further averaged with respect to the duty ratio function. The averaged equations are then linearised by using principle of perturbation, Laplace transform is applied to the linear state space model of CBC and the control to output transfer function is obtained. Unlike SSA technique, differential equations are not required in PWM switch modelling technique. Averaging and linearisation with the help of perturbation is accomplished using the PWM switch itself. Instead of Laplace transform, EET is applied to the linearised model of CBC to derive the control to output transfer function. The SSA model of CBC is a systematic procedure involving mathematics without any circuit transformation. It involves complex mathematical transformations and a higher number of mathematical equations. In contrast, PWM switch modelling technique with EET consists of simple mathematical transformations. However, expertise is required for proper circuit transformations. Based on the parametric comparison (Table 5) of two analytical techniques, it is observed that SSA is better than the PWM with EET for a high-order circuit, due to systematic, straightforward procedural steps. PWM with EET for a high-order circuit requires a higher number of circuit transformations and a high expertise, which makes it time consuming and difficult. Features SSA technique PWM switch modelling with EET Analysis approach Mathematical Mainly circuit oriented Structure of analysis Systematic Less systematic Equivalent diagram Not required Required Circuit transformation Not required Required Number of mathematical transforma- tions More Less Number of mathematical equations More Less Complexity of mathematical transfor- mations High Low Expertise required Mathematical Identification of PWM switch, changing the port sequences in case of indetermi- nacy and application of EET Identification of PWM switch Not required Required Extra Element Theorem Not applicable Applicable Form of transfer function Low entropy Low entropy The complexity of the modelling process Low High Steady state analysis Yes Yes Transient state analysis Yes Yes Table 5. Results of comparative evaluation of the modelling techniques. 5. Conclusion This paper presents a comparative study of two modelling techniques commonly used for modelling of DC-DC converters, which is first in its kind. The transfer functions of a complex CBC converter were developed using 62 vol. 63 no. 1/2023 Comparative study of state space averaging and PWM . . . state space averaging and PWM switch modelling with an extra element theorem and the simulation results were obtained. The state space modelling approach is purely mathematical and systematic. However, it involves complex mathematics for high-order circuits. The PWM switch modelling approach is mainly circuit oriented approach, which involves relatively simple mathematical transformations. However, circuit transformations become complex for a high-order circuit. 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In 2015 10th Asian Control Conference (ASCC), pp. 1–6. 2015. https://doi.org/10.1109/ASCC.2015.7244716. 64 https://doi.org/10.1109/INDEL.2016.7797807 https://doi.org/10.23919/ICPE2019-ECCEAsia42246.2019.8796492 https://doi.org/10.1109/TPEL.2006.889925 https://doi.org/10.31803/tg-20190427093441 https://hrcak.srce.hr/file/238913 https://doi.org/10.1109/63.65013 https://doi.org/10.1109/ASCC.2015.7244716 Acta Polytechnica 63(1):50–64, 2023 1 Introduction 2 Materials and methods 2.1 Complex cascaded DC-DC Buck converter 2.2 State space averaging technique 2.3 PWM switch with EET technique 3 Simulation of complex CBC 4 Results and discussion 5 Conclusion References