Resonant coordinate method for switch-mode power supplies

　　When designing a switch-mode power supply , the most troublesome component is the RCD snubber. The traditional method of designing the RCD snubber does not have a detailed description of the turn-off transient period of the main switch. Therefore, the design equations in the traditional way of design are not completely correct. This article will introduce a new method for designing and analyzing the RCD snubber of a flyback converter. The resonance coordinates provide a simple way to understand the turn-off transient period of the main switch and help to easily design and analyze the RCD snubber.

　　1 Introduction

　　Commercially, the flyback converter is widely used due to its simple structure, compact size, light weight and low cost. However, its main switch performs hard switching operation, resulting in high voltage spikes and ringing on the main switch. The voltage stress of the main switch increases depending on the size of the voltage spike. To reduce the voltage spike so that a lower-cost, low-rated-voltage MOSFET can be used, the most widespread method is the RCD snubber network. Even though the snubber voltage decreases as the snubber resistance decreases, the power dissipation on the snubber network increases, resulting in a decrease in the overall system efficiency. Therefore, the RCD snubber network should be optimized to meet both the main switch voltage stress and the overall system efficiency requirements.

　　This article will first present the conventional analysis of the voltage spike generated by the leakage inductance of the main transformer. A simple way of describing the turn-off transient period will be introduced for further analysis. The snubber current will be analyzed in snubber coordinates in order to provide more detailed design equations.

　　2. RCD snubber design and analysis

　　2.1 General method of RCD snubber design

　　Figure 1 shows a conventional flyback converter with an RCD snubber.

　　Figure 1: Traditional flyback converter

　　RCD缓冲器电路用于箝位由漏电感Llk和主开关漏极至源极的电容CDS之间的谐振导致的电压尖峰。有多种假定来描述工作原理以设计RCD缓冲器，如下所示：

　　(1) Vsn》nVout and Vsn are almost constant due to the large Csn:

　　(2) CDS = COSS + CTRANS, which is constant regardless of vDS(t):

　　(3) When the main switch Q1 is turned off, there is no secondary-side leakage inductance, so iDS(t) can be instantaneously transmitted to the secondary-side diode current iD1(t), where Csn is the snubber capacitor, CDS is the effective capacitance between the drain and source of the main switch, COSS is the output capacitance of the MOSFET, CTRANS is the effective capacitance between the primary circuit terminals of the transformer, vDS(t) is the voltage between the main switches, iDS(t) is the current flowing through the main switch, and Q1 is the main switch.

　　Figure 2 shows the equivalent circuit when the snubber diode is conducting.

　　Figure 2: Equivalent circuit of the snubber diode during on-time

　　当开关Q1关闭时，主电流对Q1的COSS充电(同时对变压器的CTRANS放电)。当COSS被充电至Vin+nVout时，次级端二极管接通，能量传输至次级端，并且对COSS持续充电，因为漏电感Llk仍有一些剩余能量。当Q1的vDS(t)增加至Vin+Vsn，缓冲器二极管Dsn接通，vDS(t)箝位在Vin+Vsn。当Dsn传导时，Llk上的电压为Vsn-nVout，这样Dsn(ts)的导通时间可获取如下：

　　(1)

　　Where Ipeak is the peak drain current before turning off switch Q1. There are two ways to calculate the power dissipation in the snubber network (Psn); the power supplied by Dsn and the power dissipation in Rsn as shown below:

　　(2)

　　Where fsw is the switching frequency of the flyback converter. Therefore, the snubber resistor Rsn can be obtained from the following equation:

　　(3)

　　This is the traditional way to find the snubber resistor Rsn. However, after a few steps of LC resonance, the peak drain current Ipeak is somewhat reduced. Therefore, equation (3) may be misleading for an over-designed system.

　　Let us use the resonance coordinates to derive the actual peak drain current to avoid over-designing the RCD snubber in the next section.

　　2.2 Design and analysis of RCD snubber in resonant coordinates

　　In this section, we will design the RCD snubber using the resonant coordinates. When designing only the snubber, it is not necessary to analyze the entire flyback operation mode. Figure 3 shows the equivalent circuit of each mode, and Figure 4 shows the vDS(t) of the switching MOSFET in the flyback converter.

　　Figure 3: Equivalent circuit for each mode displayed after turning off the main switch (modes 1 to 4 in order)

　　Figure 4: vDS(t) after turning off the switch

　　在模式1中，电感(Llk和Lm)中的电流对CDS充电，直至其电压达到Vin+nVout，其中Lm是变压器的磁化电导。在t1，次级二极管接通，并且磁化电导的两端箝位在反映的输出电压nVout上。在模式2中，通过CDS和Llk之间的谐振，CDS上的电压增加到Vin+Vsn，从而接通缓冲器二极管。因此，漏极电压箝位在Vin+Vsn(在模式3期间)。CDS和Llk之间的谐振由于减幅如模式2一样在模式4中恢复。

　　When the inductor and capacitor resonate in series with a DC voltage source (Vdc), the voltage on the capacitor and the current through the inductor can be plotted in a plane. On the plane, the X-axis is voltage and the Y-axis is current. If the characteristic impedance of the L-C loop is multiplied by the Y-axis so that the units of both axes are the same, the voltage and current trajectories will show a circle with the origin at (Vdc, 0) and the radius being the length between the starting point and the origin. Using this graphical approach to understanding resonance, it is easy to find the actual peak drain current at t2 in Figure 4. During Modes 1 to 4, iDS(t) and vDS(t) are plotted in the resonant coordinates as shown in Figure 5.

　　Figure 5: Mode analysis in resonant coordinates

　　Mode 1 is a circle with its origin at (Vin, 0) and its start at (0, ZmIpeak). It continues until vDS(t) reaches Vin+nVout, as shown in Figure 4. According to Mode 1 of Figure 5, the equation of the circle is as follows:

　　(4)

　　Where Zm is the characteristic impedance of Lm+Llk and CDS, √((Lm+Llk)/CDS).

　　In mode 2, it is an ellipse, with the origin of the ellipse at (Vin+nVout, 0) and the starting point at (A, B). Through coordinate mapping, the circle becomes an ellipse because the characteristic impedance changes from √((Lm+Llk)/CDS) to √(Llk/CDS). According to mode 2 of Figure 5, the equation of the ellipse is as follows:

　　(5)

　　The snubber diode is turned on at the end of mode 2, i.e. point (C, D). Therefore, the actual peak current when the snubber diode is turned on is D/Zm, i.e. D/√((Lm+Llk)/CDS). According to equations (4) and (5), the actual peak current Ipk,sn is as follows:

　　(6)

　　Ipk,sn should be used instead of Ipeak in equation (3) to obtain a more accurate Rsn.

　　Normally, Rsn is chosen based on the Ipeak approximation, and accordingly Rsn is an over-engineered value because Psn is overestimated. Using Ipk,sn, we can get a more accurate and smaller estimate of Psn, and therefore Rsn is also larger.

　　3. Conclusion

　　We can find the exact snubber peak current using the resonant coordinates. According to equations (3) and (6), Llk, Ipk, sn and fsw should be reduced, while CDS should be increased to reduce the snubber loss. But this may bring some side effects such as higher switching losses, larger transformer size, etc. Therefore, all factors must be taken into consideration during design. The exact equations provided in this article will help system designers to easily design the RCD snubber .