A Comparison of the Ladder and Full-Order Magnetic Models

Transcription

1 A Comparison of the Ladder and Full-Order Magnetic Models Kusumal Changtong Robert W. Erickson Dragan Maksimovic Colorado Power Electronics Center University of Colorado Boulder, Colorado ABSTRACT Predictions of the reduced-order ladder model and the full-order extended cantilever model are compared the measured behavior of an experimental multiple-output flyback converter. The flyback transformer geometry was chosen to closely follow the idealizing assumptions that lead to the ladder approximation, including use of a pot core with full-width copper foil windings. In spite of this, the ladder model fails to predict the observed waveforms and the mode of operation. The full-order extended cantilever model does indeed correctly predict the operating mode and the observed waveforms. Model parameters are measured using a network analyzer. Conditions on the validity of these measurements are derived using Middlebrook s Extra Element Theorem. I. INTRODUCTION The electrical models of multiwinding transformers, such as the four-winding pot-core example shown in Fig. (a), can be expressed in several different ways. The ladder model [] illustrated in Fig. ( is one of the most popular and widely used magnetics models in power electronics. It consists of a magnetizing inductance and a series of leakage inductances connected between the adacent windings. In relatively simple magnetics structures, such as the pot-core transformer of Fig.(a), the parameters of the ladder model can be easily related to the geometry of the core and the windings. The inductances can be computed by relating the flux pattern to the physical arrangements of the core and the windings in the transformer. For an n-winding transformer, the ladder model has n- independent parameters. However, the inductance matrix in the general magnetics model has n(n+)/ independent parameters. Therefore, it should be noted that the ladder model is an approximate equivalent circuit model. Figure ( shows the extended cantilever model [], which is a full-order model with the correct number of independent parameters. For the example of Fig. (a), compared to the ladder model of Fig. (, the full-order extended cantilever model has three additional leakage inductances l 3, l 4, and l 4. These inductances represent couplings between nonadacent windings. In the pot-core example, we expect that these couplings are relatively small, i.e., that these additional inductances in the full-order model should be much larger in value than the inductances that represent couplings between adacent windings. Based on this, one can argue that the additional inductances in the full-order model can be neglected, and that the ladder model represents an adequate a) Pot core 49 l 4 l 3 l l 3 l l l 3 l Fig.. Circuit model of a multiwinding transformer: (a) A four-winding pot-core transformer example, ( The ladder model corresponding to the four-winding transformer, ( The full-order model (extended cantilever form) of the four-winding transformer. approximation in analyses of converters with multiplewinding magnetics. In this paper, we question the validity of this argument, and show that the reduced-order ladder model can lead to incorrect conclusions in cases of practical interest. The pot-core example of Fig.(a) is chosen because this configuration is closest to the ideal assumptions that lead to the ladder model: the shape of the core is closed, and the magnetic flux, including leakage flux, is enclosed within the l 4

2 structure. Furthermore, the winding arrangement in the core is top-to-bottom and copper foil is used to increase the coupling between adacent windings. We show that even in this case, the ladder model can fail to predict experimentally observed behavior in a multiple-output flyback converters. Therefore, our conclusions from the comparison of the ladder and full-order models are even stronger for other core/windings configurations. The conclusions apply equally well to other similar full-order equivalent circuit models such as [8] and [9]. Measurement of parameters in the full order extended cantilever and the ladder models is discussed in Section II. In Section III, limitations of the measurement setup are examined using Middlebrook s Extra Element Theorem [3]. In Section IV, an experimental multiple-output flyback converter is used to validate results obtained using the ladder model and the extended cantilever model. In this example, we show how the ladder model leads to qualitatively incorrect results for the shapes of current waveforms in the secondary windings. II. MEASUREMENT OF INDUCTANCE PARAMETER IN THE MAGNETICS MODELS Each inductance parameter and effective turn ratio in the extended cantilever model can be directly measured using a network analyzer. The leakage inductance parameters are determined by driving a given winding with a voltage source, short circuiting all other windings, and measuring the short circuit current. For example, Fig. shows how to measure the inductance parameter l 3 by using the network analyzer and Fig. 3 illustrates the magnitude and phase of the Bode plot of the measured inductance parameter l 3. The measurement frequency is selected on a segment where the slope of the magnitude response is approximately +db/dec, and the phase is approximately +9 (or 9) degrees. The inductance is computed from Eq. (). Inect ac voltage + V - Network Analyzer Short all other windings Current probe Fig.. The measurement method of leakage inductance parameter Fig. 3. Measurement of the inductance parameter l 3 : magnitude and phase responses of v / i 3 vi l = i π fnin i () The results of the measured inductance parameters and the effective turn ratios of the extended cantilever model are summarized in Table. Each leakage inductance parameter was determined as both l i and l i to check for consistency. Moreover, additional measurements with open and shorted windings were performed and the results were compared to verify model predictions. Table. Measured inductance parameters. Inductance parameters Measured values n.396 n n 4.3 L 35 µh l.3 µh l 3 7 µh l 4 µh l 3.5 µh l 4 8 µh l 3 µh Following the expectations, based on the physical winding geometry, we can see that l << l 3 << l 4 because primary winding is much better coupled to winding than to windings 3 or 4. As a result, the effective leakage inductance l 4 is much larger than l and l 3. The relative values of the inductances l 3, l 4 and l can be explained in the same manner. The complete extended cantilever model, where all inductance parameters are obtained by direct measurements, is shown in Fig 4. The ladder model parameters are the same as the extended cantilever model parameters, except that the additional leakage inductances are removed as shown in Fig. 5.

3 l 4 = µh parameters ( D and N ) can be found in general form in terms of the inductance parameters as follows: l 3 = 7 µh l 4 =8 µh l =.3 µh l 3 =.5 µh l = 3 µh W W W 3 Fig. 4. The extended cantilever model. l =.3 µh l 3 =.5 µh l = 3 µh N = nk s lik lk lnk... lik D = nks lk lnk... li + lk where n is number of windings and n i k a) (3) v W W W 3 Fig.5. The reduced-order model (ladder model) Based on the measured values of the leakage inductances, one might argue that the approximation made by removing the leakage inductances l 3, l 4, l 4 is well ustified. 5 3 µh an extra element D N III. LIMITATIONS ON THE MEASUREMENT OF LEAKAGE INDUCTANCES In the measurement of inductance parameters of the extended cantilever model (or the ladder model), imperfect shortcircuit impedance may degrade the accuracy of the measured inductance parameters. To examine the effects of non-zero short-circuit impedance, Middlebrook s Extra Element Theorem [3] is applied. This method is used to investigate the effect of a parasitic element (s), not included in the original analysis, and to establish conditions on (s) to ensure that the quantity of interest is unchanged. In the presence of (s), the transfer function measured to determine the leakage inductance l i becomes: () s i + N() s = v ( ) () s i nn i sli + D() s The correction factor in square brackets shows how the ideal response is modified by the parasitic element. In Eq. (), D is the output impedance of the system at the port where is connected, whereas N is the impedance looking from the extra element s port with the current i nulled to zero. In the cantilever model, the extra element theorem is applied for each measured inductance parameter l i with the extra impedance connected in port k. The correction factor () Frequency ( rad/se ( ) Frequency (rad/se Fig.6. (a) The equivalent circuit to find D and N with an extra element inserted to winding 3, ( The Bode plot of N, D and, ( The Bode plot of the correction factor. An example of using the extra element theorem in the measurement of the leakage impedance parameter l 4 is shown in Fig. 6(a). An extra short circuit impedance 3 = R 3 is approximately.7 Ω (which is equal to the winding dc

4 resistance) instead of the ideal short. Impedances N and D can be calculated as follows: D N = n 3 = n s 3 [ l l l ] 3 l3 s l + l 4 l 3 l These magnitudes are plotted in Fig. 6(. To obtain an accurate measurement, it is necessary that the effective short circuit impedance (s) be much smaller in magnitude that the impedances D and N of the above equation. From Fig. 6(, the effective short circuit impedance (s) is much smaller in magnitude than the impedances D and N for frequencies above 4 6 rad/sec or 64 khz. The correction factor is plotted in Fig. 6(; it can be seen that this factor is close to unity for frequency > 64 khz. Therefore, the short circuit resistance does not significantly affect the measurement of the impedance parameter l 4 when the measurement is made at frequencies above 64 khz. For this example similar calculations for other impedance parameters show that winding resistance have negligible effect on the measurement provides that the measurement is made at a frequency of at least 7 khz. From Eq. () and (3), if the extra element is taken to be an inductance L instead of a resistance R, the correction factor is a constant value. We can find limits on the value of the extra inductance has negligible. For this example, the values of D and N can be calculated from Eq. 4 as.5 µh and.3 µh respectively, hence L must be smaller than both of these values. The results of this Section show that measurements should be performed so that the results are not significantly affected by imperfect shorts or the winding resistances. The measurement and model are also both limited by effects such as proximity losses, interwinding capacitances, etc. If desired, a more complex approach such as [9] can be employed to model these effects. IV. COMPARISON OF THE MODEL PREDICTIONS IN A MULTIPLE-OUTPUT FLYBACK CONVERTER The four-winding transformer of Fig. (a) was used in the khz flyback converter shown in Fig. 7 with the following specifications: Input:3V (winding W ) Output : + V (winding W ) Output : - V (winding W 3 ) Output 3: +3.3 V (winding W 4 ) Switching Frequency: khz Duty ratio:.5 3 (4) Fig. 7. The prototype multiple output flyback converter The multiple-output flyback converter and the extended cantilever model or the ladder model of the transformer were used in computer simulations. The simulated and measured secondary winding current waveforms are illustrated in Fig. 8. The same simulations were repeated with added winding resistances, and the obtained results and conclusions were not significantly different. From Fig. 8(a), the output waveforms predicted by the extended cantilever model show that all outputs operate in continuous conduction mode (CCM). During the diode conduction interval i and i 3 are negative (as usual) while the slope of i 4 is positive. Based on the argument that significantly larger leakage inductances neglected in the ladder model should not affect the model accuracy, one would expect that the waveforms obtained by simulation of the converter with the ladder model should not be significantly different. However, Fig. 8( shows the substantial disagreement between the prediction of the extended cantilever model and that of the reduced-order ladder model. The reduced order model predicts that winding 3 operates in discontinuous conduction mode, even though the operating conditions are the same. Furthermore, the polarities of the slopes of the output currents are different. In comparison to the experimental waveforms in Fig. 8(, we observe that the extended cantilever model correctly predicts the behavior of the multiple-output flyback converter, while the ladder model fails. It can be concluded that we cannot simply neglect parameters in the full order model based on ust only their relative values. Any approximations to the model should be ustified by waveforms of the specific application. The ladder model may lead to incorrect conclusions in practical applications of interest, even in this near-ideal case of the potcore transformer with full width foil windings.

5 a) -. i i i In this paper, we first examine measurements of model inductance parameters by using network analyzer and determine the frequency limitation of the measurement procedure, which is taken into account to perform an accurate measurement of model parameters by using Middlebrook s Extra Element Theorem. The limiting frequency is performed above 6 khz. The measured inductance parameters are reasonable values based on the physical winding geometry. The inductance parameters in adacent windings: l, l 3, and l are relatively much smaller that the inductance parameters in non-adacent windings: l 3, l 4, and l 4. The ladder model can be reasonably obtained by removing additional leakage inductances in non-adacent windings. Finally, we compare predictions in the multiple-output flyback converter application of the ladder model to predictions of the full-order, extended cantilever model. It is shown that the ladder model leads to incorrect predictions in terms of operating mode and signs of the slopes of secondary winding currents. These results were obtained even for a nearideal case of a pot-core transformer with top-to-bottom foil windings, where the ladder model should work the best..8 i i i Fig. 8. (a) The simulated waveforms using the extended cantilever model (Fig 3), ( Simulated waveforms predicted by the reduced-order ladder model (Fig. 4), and ( Measured output waveforms. V. CONCLUSIONS The widely-used ladder model is an approximation based on winding geometry: the couplings between non-adacent windings are neglected. We have shown that geometry alone is insufficient to ustify this approximation, and that circuit conditions must also be considered. In this proect, a pot-core transformer having foil copper windings, with geometry following the idealizing assumptions behind the ladder model approximation was constructed and tested. It was found that the ladder model does not predict observed phenomena such as operation in discontinuous conduction mode. Full-order model such as the extended cantilever model does indeed predict these phenomena. REFERENCES [] Dauhare, "Modelling and Estimation of Leakage Phenomena in Magnetic Circuits," Ph.D. Thesis, California Institute of Technology, Pasadena, California, April 986 [] Erickson and D. Maksimovic, "A Multiple-Winding Magnetics Model Having Directly Measurable Parameters," IEEE Power Electronics Specialists Conference, May 998 Record, pp [3] D.Middlebrook, "Null Double Inection and the Extra Element Theorem," IEEE Transactions on Education, Vol. 3, No. 3, pp. 67-8, August 989. [4] D. Middlebrook and Slobodan Cuk, "Isolation and Multiple Output Extensions of a New Optimum Topology Switching DC-to-DC Converter," IEEE Power Electronics Specialists Conference, 978 Record, pp [5] William M. Polivka, "Applications of Magnetics to Problem in Switched-Mode Power Converters," Ph.D. Thesis, California Institute of Technology, Pasadena, California, February 984. [6] D. Maksimovic and R. Erickson, "Modeling of Cross Regulation in Multiple-Output Flyback Converters," IEEE Applied Power Electronics Conference, 999 Record, pp [7] D. Maksimovic, R. Erickson, C. Griesbach, "Modeling of crossregulation in converters containing coupled inductors," IEEE Applied Power Electronics Conference, 998 Record, pp [8] F.M. Stars. Equivalent Circuits I, AIEE Transactions, June 93, pp [9] V. A. Neimela, Analysis and Modeling of Leakage Inductance and AC winding Resistance in High-Frequency Multiple-Winding Transformers, Ph.D. Thesis, Duke Univ., 99.