Application Report SLVA589 β July 2013 AC Modeling of Power Stage in Flyback Converter Rushil K K PWR-DCDC Controllers ABSTRACT This application report provides a step-by-step procedure for constructing the AC model for the power stage of a Flyback DC/DC Converter in Continuous Conduction Mode. The model is used in the TIna simulator to plot and verify the control-to-output transfer function. 1 2 3 4 5 Contents Introduction .................................................................................................................................. 1 1.1 How it Works .......................................................................................................................... 2 Circuit Manipulations ................................................................................................................... 3 Construction of the Mathematical Model .................................................................................... 5 Simulation and Results ................................................................................................................ 8 Conclusion: ................................................................................................................................ 11 Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Figure 8. 1 Figures Basic Topology of Flyback Converter ............................................................................ 2 Separating Out the Non-Linear Switch Elements .......................................................... 3 Voltage Waveforms for the 2-Port Network.................................................................... 4 Current Waveforms for the 2-Port Network.................................................................... 4 Replacing the Non-Linear Switch Network with Equivalent Linear Network ............... 7 Complete Linear Circuit for Flyback Converter ............................................................. 8 Implementation of Example Case in Tina Simulator ................................................... 10 Control-to-Output Transfer Function Plotted using TIna ............................................ 11 Introduction The flyback DC-DC converter is derived from the basic buck-boost converter. The conversion ratio for this converter is given by, π· πππ’π‘ = π. 1βπ· πππ As seen from this equation, the conversion ratio is similar to that of the buck-boost converter with an included factor of n, which is the turns ratio of the flyback transformer. 1 SLVA589 1.1 How it Works In a flyback converter, when the current ramps up in the primary, energy is built up in the transformer core. When the FET switches off, this energy is dumped into the secondary, allowing current to flow in the secondary, so the primary current collapses immediately to zero. This causes an immediate rise in current in the secondary, after which, the current in the secondary ramps down linearly as it discharges, supplying energy to the load. The current in the secondary when the FET switches off, is proportional to the peak in the primary, and is determined by the turns ratio. The circuit topology of the flyback converter is as shown in Figure 1. Figure 1. 2 Basic Topology of a Flyback Converter AC Modeling of Power Stage in Flyback Converter SLVA589 2 Circuit Manipulations The circuit can be re-drawn, to separate out the reactive components and the switching components, as shown in Figure 2. Figure 2. Separating Out the Non-Linear Switch Elements The switching elements (MOSFET and Diode) are shown in the dashed rectangular box. These are non-linear switching elements. The box is treated as a 2-port network with input voltage π£1 (π‘), input current π1 (π‘), output voltage π£2 (π‘), and output current π2 (π‘). The waveforms for π£1 (π‘) and π£2 (π‘) over one switching period, ππ , are shown in Figure 3. A simplifying assumption of zero MOSFET and diode conduction drop has been made for obtaining the waveform shown below. AC Modeling of Power Stage in Flyback Converter 3 SLVA589 Note: The slope of the voltage waveform in the figure is exaggerated for the sake of clarity. Figure 3. Voltage Waveforms for the 2-Port Network Similarly, the waveforms for π1 (π‘) and π2 (π‘) are shown in Figure 4. Figure 4. 4 Current Waveforms for the 2-Port Network AC Modeling of Power Stage in Flyback Converter SLVA589 3 Construction of the Mathematical Model The waveforms in Figure 3 and Figure 4 are averaged over one switching period, which effectively removes the switching harmonics. The switching harmonics have no role of consequence in the development of the AC model. Observing the waveform, it can be seen that: < π½π (π) >π»π = π β² (π) β {π½ππ + π½ποΏ½π } < π½π (π) >π»π = π (π) β {ππ½ππ + π½π } Dividing the above equations, we get the following relation: Hence, we have: π π β² (π) < π½π (π) >π»π οΏ½< π½ (π) > = β π π (π) π π»π < π½π (π) >π»π = π π β π β² (π) π (π) β< π½π (π) >π»π (1) Equation 1 is a non-linear one, since it contains products of time-dependent variables. To obtain a linear relation, the variables are perturbed about the quiescent operating point. π (π) = π· + πΜ < π½π (π) >π»π = π1 + π£ οΏ½1 Using the preceding equations in (1): οΏ½2 < π½π (π) >π»π = π2 + π£ π π·β² β πΜ β β (π½π + ποΏ½π ) π π· + πΜ οΏ½ +π οΏ½ = π β {π½ β π«β² β π½ β π οΏ½ +π οΏ½} οΏ½π β π« + π οΏ½π β π οΏ½π β π«β² β π οΏ½π β π π½π β π« + π½π β π π π π οΏ½π = π½π + π οΏ½ and π οΏ½ are small enough to οΏ½π β π οΏ½π β π Since the perturbations are small, the non-linear terms π be ignored. Upon rearranging and grouping terms appropriately: (2) The equation is re-arranged into the below form, (3) π Note: Equating the quiescent terms in Equation 2, we have π½π β π· = π π½π β π«β². Substituting this relation for π½π in Equation 3, AC Modeling of Power Stage in Flyback Converter 5 SLVA589 Now, D+Dβ=1. So the above equation finally reduces to: (A) Using the same procedure for the 2-port network currents π1 (π‘) and π2 (π‘), Equation B is obtained: (B) Equations (A) and (B) are the relations for the input voltage and output current of the 2-port network and are used to obtain the linear model for the averaged switch network. These relations can be implemented by a network consisting of an ideal transformer and an independent voltage and current source, both driven by duty cycle variations, d(t), as shown in Figure 5. 6 AC Modeling of Power Stage in Flyback Converter SLVA589 Figure 5. Replacing the Non-Linear Switch Network with an Equivalent Linear Network The linear model for the switching element thus obtained is used in place of the MOSFET and diode in the rectangular dashed box of Figure 2 to obtain the circuit shown in Figure 6. AC Modeling of Power Stage in Flyback Converter 7 SLVA589 Figure 6. Complete Linear Circuit for Flyback Converter The inductance πΏπ indicates the magnetizing inductance, and is used with an ideal transformer of turns ratio 1:n. 4 Simulation and Results Example: Consider a flyback converter designed for the following specifications: πππ = 5π, πππ’π‘ = 10π. A load resistance of 10 Ξ© is connected at the output, so the load current drawn is 1 A. The magnetizing inductance of the flyback transformer is 6 µH, and load capacitance used is 500 µF. The turns ratio n=4, hence, the duty cycle, D, required to produce a 10-V output is D = 0.33. The quiescent value for π£1 (t), is π1 = π· β² . (πππ + ππ /π), as seen from waveform. Similarly, πΌ2 = πΌππππ Hence, π1 = 4.95 V. Also, πΌ2 = 1 A. Now, for modeling in TIna, the voltage and current sources controlled by d(t) are implemented using the VCVS and VCCS sources respectively, as shown in Figure 7. The gain of the VCVS source is set to be ac model is π1 π·.π·β² β πΜ ) π1 π·.π·β² (since the voltage source obtained in the derived so, the gain of VCVS source is 22.31. 8 AC Modeling of Power Stage in Flyback Converter SLVA589 The gain of VCCS source must be πΌ2 π·.π·β² (again, according to derived model). The gain of VCCS source is calculated to be 4.5. The turns ratio of the transformer (of the switch model) is nD/Dβ = 2. Verification: For a flyback converter, the location of RHPZ and LC resonant pole, is at(1): ππ π»ππ = Also, the gain at DC is, (π π β π·β²2 ) οΏ½ 2 (π β 2π β πΏπ β π·) ππΏπΆ = π·β²οΏ½ (π β 2π β οΏ½πΏπ πΆ) πΊπ0 = π0 /π·π·β² Using the example values in the above equations, we should have the following: ππ π»ππ = 21.88 kHz πΊπ0 = 45.09 ππΏπΆ = 883.3π»π§ πΊπ0 |ππ΅ = 33.2ππ΅ AC Modeling of Power Stage in Flyback Converter 9 SLVA589 The circuit is constructed in TIna Simulator as shown in Figure 7. Figure 7. Implementation of the Example Case in TIna Simulator Running the AC analysis, the bode plot for the control-to-output transfer function is obtained as shown in Figure 8. 10 AC Modeling of Power Stage in Flyback Converter SLVA589 Figure 8. Control-to-Output Transfer Function Plotted using TIna As seen from Figure 8, the LC resonant pole appears at around 862 Hz. The RHPZ is located at around 22.13 kHz. Gain at DC is 33 dB. Hence, it is seen that the response obtained from the model matches closely with the theoretical calculations. 5 Conclusion The AC model of the power stage for a flyback converter can be constructed following the steps shown in this application report. The response of the model, implemented in a TIna Simulator, is found to be in alignment with what is theoretically expected. 6 References 1. Basso, C., βHow to Keep a FLYBACK Switch Mode Supply Stable with a Critical-Mode Controller,β ON Semiconductorβ’, AN1681/D Application Note, September 2000 β Rev. 0 AC Modeling of Power Stage in Flyback Converter 11 IMPORTANT NOTICE Texas Instruments Incorporated and its subsidiaries (TI) reserve the right to make corrections, enhancements, improvements and other changes to its semiconductor products and services per JESD46, latest issue, and to discontinue any product or service per JESD48, latest issue. Buyers should obtain the latest relevant information before placing orders and should verify that such information is current and complete. All semiconductor products (also referred to herein as βcomponentsβ) are sold subject to TIβs terms and conditions of sale supplied at the time of order acknowledgment. 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