《Switching Power Supply Simulation and Design》-Small Signal Modeling

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《Switching Power Supply Simulation and Design》-Small Signal Modeling [Copy link]

In the time domain, SPICE simulation determines the simulation conditions through the .TRAN statement, that is, transient analysis. SPICE adjusts the time step according to the signal changes during the simulation process to capture more data points. It takes a long time to evaluate the transient response of low-bandwidth systems. Simulation engines such as PSIM that use fixed steps can solve such problems faster. Another solution is to derive an average model, cancel the switching behavior, and generate a smooth continuous signal. Applying an average model to a time-varying signal can obtain a continuous description of discrete values.

There are two main control methods for switching power supplies: voltage mode and current mode. Voltage mode control is simple and does not require detection of the inductor current, but it has disadvantages such as poor input voltage suppression and difficulty in stable operation under CCM. Current mode control always detects the inductor current and adjusts the current value according to the output power requirements. It has better input voltage suppression capability and load response speed. However, under CCM conditions, current mode control may cause subharmonic oscillations, and a slope compensation signal is required to increase stability. Whether it is voltage mode or current mode control, special PWM switch models are required to simulate the behavior of switching elements. These models can accurately reflect the nonlinear characteristics of switching elements during the switching cycle, thereby ensuring the accuracy and effectiveness of small signal models. Let's take a closer look at these two modes.

PWM Switching Mode - Voltage Mode

SSA technology has opened up a fascinating and challenging path for exploring small-signal models of power converters. Its complexity comes from the widespread application of SSA in the converter, which requires many state variables to participate in the operation compared to negligible variables, which undoubtedly increases the difficulty of analysis. Fortunately, however, a series of innovative technologies have emerged to greatly simplify the process of studying converter small-signal models. Among them, the PWM switching mode stands out with its unique value. The beauty of this model is that it does not require tedious averaging and linearization. It only needs to convert the circuit into a small-signal model and directly solve for the selected parameters. Even more commendable is that once the model is established, its universality is revealed and it can easily adapt to the conversion needs of different structures by simply adjusting the model configuration. The concept of invariance lies in the uniformity of the PWM switch structure at the electrical level, even if its design covers dual-switch converters. This switch design cleverly integrates the operation of the main power switch (i.e., Sw mentioned earlier) and the diode D into a compact switch model. Clearly identify three key nodes in the switch arrangement: active nodes - switch endpoints that have no direct connection to the diode; passive nodes - diode endpoints that have no direct connection to the switch; and common nodes - where the diode meets the power switch. The characteristics of this circuit structure are not affected by its operating mode (CCM or DCM).

Waveform averaging is the core of the derivation of PWM (pulse width modulation) switching model. Its essence lies in capturing the shape of voltage and current waveforms between the endpoints and calculating their average values in the entire switching cycle to achieve the conversion from instantaneous value to average value. At the mathematical level, this sampling process can be formalized into a specific mathematical expression, which accurately describes the internal mechanism of waveform averaging.

Similar to the inductor-based component, the performance of the boost converter can be optimized by correcting the inductor ohmic losses. This adjustment is simple and easy to make. It only requires a resistor to be connected in series with the inductor L. To simplify the calculation process, the value of the resistor R can be converted to the opposite side of the transformer. The circuit structure consists of a resistor divider in series with the transformer. By reconfiguring the DCM PWM switch, based on the original common passive node structure, an extremely simple model can be constructed. The model can achieve seamless transition between DCM and CCM, greatly improving the design flexibility and efficiency.

Given the above discussion, clamping of various generators is critical. The first task is to ensure that the model duty cycle input node d (which is consistent with DCM) varies strictly between 0 and 100%. If 1V represents 100% duty cycle, the control circuit must avoid outputting voltage values exceeding 1V to prevent non-convergence issues or deviations in the calculated operating point. In SPICE, signal source clamping can be implemented in a variety of ways, one of which is a single built-in method. Suppose we need to limit the duty cycle deviation to less than 10mV (or 1%). How a PWM modulator generates a variable pulse width signal from a gain voltage mode power supply to regulate the output. This process is achieved by comparing the DC error signal (output by the operational amplifier) with a fixed amplitude sawtooth wave to determine the appropriate pulse width.

模式转换的显著特征之一在于其能在CCM（连续导通模式）与DCM（不连续导通模式）之间实现自动切换。为实现这一点，可借助降压变换器对输出电流进行监测，以验证触发点是否对应于特定的负载条件。通过连接一个探头至发生器，能够捕捉到CCM与DCM之间的转换情况。

PWM Switching Mode - Current Mode

Current mode control, as one of the most popular control strategies at present, is based on the direct regulation of current. Through simple deduction, a new model that combines the characteristics of CCM and DCM is constructed. The physical architecture of the current mode control (CMC) converter is different from the duty cycle regulation logic. Under voltage mode control, the generation of the switching signal comes from the comparison of the error voltage and the sawtooth wave, which has no direct connection with the inductor current. However, for safety reasons, modern controllers generally add an inductor current monitoring function to ensure that it operates above the set threshold. In current mode control, the switching clock first triggers the power switch action, and then the error voltage Ver sets the peak value of the inductor current before the switch is turned off. The sensing resistor R is responsible for real-time monitoring of the current. Once the threshold set by the error voltage is reached, the comparator triggers the reset operation to achieve precise control.

Current mode instability, also known as subharmonic oscillation, is a hot topic in the electrical field. In continuous conduction mode (CCM), the inherent instability of current mode converters becomes apparent when the duty cycle exceeds the 50% threshold (in contrast, in discontinuous conduction mode (DCM), such subharmonic instability does not occur).

The figure above intuitively presents the waveform of the inductor current (without considering the specific topology of the converter). Since the system is in a steady state, the current waveform starts at 0 and returns to the same end value after a cycle Tsw. Based on this steady-state characteristic, the expression of the relationship between the current peak and the cycle can be derived. Imagine that the system is subjected to a small disturbance. At this time, the cycle is no longer strict. In this situation, the controller will adjust the strategy to keep the power switch on until the inductor current climbs to a new peak peak, and then trigger the internal mechanism to disconnect the switch. The current decreases along the slope of S2 until the next cycle starts. It is worth noting that the starting value L(Tsw) of the inductor current in the new cycle will be different from the previous moment. This unbalanced dynamic process is a disturbance (instability).

变换器虽能耐受非稳定性次谐波振荡，但其引发的输出纹波及变压器内的音频噪声却不容忽视，需要采取有效措施加以遏制。斜坡补偿技术，亦称斜率补偿，作为一种成熟策略，能够显著增强变换器在宽占空比范围内的稳定性，有效防止次谐波振荡。此技术可融合电流检测信息（常借助传感电阻或电流变压器实现），并可通过从反馈信号Ver中减去相应值来实施。

In a CCM (continuous conduction mode) environment, if the duty cycle of a current mode converter exceeds the 50% threshold, it must rely on slope compensation technology to maintain stable operation, which is a hallmark of fixed frequency PWM converters. However, it is difficult to understand the control strategy behind it by simply displaying the static waveform on an oscilloscope. In view of this, the constant law in the voltage mode PWM switch also applies to the current mode, that is, the changing relationship between current and duty cycle and voltage and duty cycle still holds.

在DCM（不连续导电模式）下，升压及升降压变换器的电流模式展现出固有的稳定性，无需额外的斜率补偿措施。然而，鲜为人知的是，当降压变换器在DCM条件下运作，且变换系数M攀升至超过2/3的阈值时，其直流工作将遭遇不稳定性挑战。这一现象的根源可追溯到降压变换器小信号模型中的极点表达式，其分母在M超越2/3时转变为负值。DCM模型精准地预测了此异常行为，下图直观展示了不同输入电压条件下，采用电流模式控制的DCM降压变换器开环增益的变化情况，其中涵盖了不同传输系数M的影响。尤为显著的是，当M逼近或超越2/3时，相位发生急剧反转，需微量的斜坡补偿化解这一难题，确保变换器稳定运行。

PWM Switching Mode - Parasitic Component Effects

在探讨电流模式或电压模式驱动的PWM开关模型时，常将V电压简化为理想的直流电压，实则不然。Vap电压的特性紧密关联于模型的具体架构。以直流输入电压为例，它需穿越LC滤波器，其中输出电容有等效串联电阻(ESR)，这在降压变换器中尤为显著。转向升压变换器，Vap则直接映射为输出电压，其负载端由负载与输出电容共同构成，同样受到ESR的微妙影响。这一效应催生了脉动电压，即在Vap上叠加了一层纹波电压。无论是降压还是升压场景，Vap电压均遭受电压降落的干扰。纹波幅度与阻值Re紧密相关，而Re的值则依据所分析的拓扑结构而异。特别在升压变换器中，所有变量均携带负号，此乃特定开关布局所致。

在理想的电压模式模型中，简化了导通开关（包括Sw与二极管D）的压降，视其为零。然而，在实际应用中，MOSFET的Rds(on)或双极型晶体管的Vcesat，以及二极管的压降Vd，均对电路的最终效率产生显著影响。当电流流经开关时，其在导通期间会产生压降；而在截止期间，电流则通过二极管并产生相应的压降。为了更直观地理解这一过程，我们可以设想一个电流源，它通过为电阻（或直接选择MOSFET）提供偏置电流，来模拟Sw上的压降。类似地，二极管上的压降Vd（标记为VDo）也可通过类似方式分析。在电流模式下探讨欧姆损耗与压降时，我们发现其与电压模式下的情况并无本质区别。MOSFET与续流二极管同样存在压降。寄生源保持不变，且仍与端口串联。值得注意的是，在计算占空比时，我们应参考原始端口，而非与电感直接相连的端点。

PWM Model in Boundary Conduction Mode

Boundary conduction mode (BCM), also known as critical conduction mode, has the outstanding feature of accurately capturing the inductor reset moment (that is, the inductor current returns to zero), thereby constructing a converter architecture that is free from the troubles of CCM mode and is not afraid of reverse recovery effect. This mode can remain stable even in the short circuit or startup phase. In the DCM state, the system maintains first-order characteristics, which is convenient for building a closed-loop feedback mechanism. If there is a slight delay before conduction, the transistor can use the sinusoidal oscillation between the drain and the source to approach the voltage to zero, realizing zero voltage switching (ZVS). This technology is particularly common in flyback converters and power factor correction (PFC) circuits.

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