功率因数校正（PFC）级的环路补偿通常不被视为主要问题。
实际上，PFC电路必须对作为低频信号的线电流进行整形，因此，从本质上讲，这是一个极其缓慢的系统。
这就是为什么PFC环路的补偿通常不被认为是设计电源时的关键步骤：“大幅缩减带宽，就是这样！”
然而，作为任何控制系统，必须补偿PFC级，并且必须适当地执行这个不可避免的步骤，以实现下游转换器的优化操作和令人满意的功率因数。
因此必须导出传递函数。
PFC阶段是慢速系统
让我们参考图1.由于PFC功能，线电流（黑色迹线）和线电压（蓝色迹线）都是正弦曲线。
因此，提供给负载的功率具有正方形正弦曲线的形状，如红色迹线所示。
现在，如黑色虚线所示，输出功率通常是恒定的，并且所提供的功率仅平均匹配负载需求。
因此，PFC级提供的功率超过了所需的时间段。
请参阅“+”符号区域。
在此阶段，输出电容吸收这些过剩的能量，从而充电。
另一方面，当PFC级提供的功率低于所需的功率时，接着是第二个序列。
请参阅“ - ”符号区域。
在第二种情况下，输出电容器在负载中放电以补偿能量的不足。
因此，输出电压（绿色迹线）在线频率的两倍处呈现低频纹波（欧洲通常为100 Hz，美国为120 Hz）。
这种纹波是PFC功能所固有的。
图1 - PFC级的输出电压表现出低频纹波
现在，传统的电流整形技术要求控制信号（由调节电路提供）是缓慢变化的信号。
否则，线电流将失真。
因此，调节环必须抑制低频输出纹波。
实际上，这是通过设置20 Hz范围内的低带宽来实现的。
这就是为什么PFC系统本质上是慢速系统的原因。
一个简单的平均模型
在研究PFC级的动态特性时，我们可能倾向于考虑两种不同的机制及其各自的频率范围：第一，功率传输的切换过程，第二，输入电流整形的线路频率调制。
实际上，我们可以使用忽略这两个方面的简化方法。
我们的PFC级的大信号模型可以完全基于线周期内提供的平均功率。
PFC级被建模为连续提供该计算的平均功率的设备。
切换事件和输入电流整形调制仅被视为假设稳定的发电机制。
因此，我们忽略了在线路频率之上发生的所有事情，这很好，因为环路带宽设置如下。
如果您的系统在连续导通模式而非临界导通模式下工作，在电流模式而非电压模式下，无关紧要。
我们只考虑PFC阶段提供的平均功率。
该方法如图2所示。
图2 - 一个简单的平均过程
公式1给出了PFC阶段提供的平均功率的一般表达式。
功率表达式取决于控制信号（Vcontrol）与其最小值（Vcontrol，min）之间的电压差：
（1）
在没有前馈的系统中，平均输入功率也与平方线幅度（m = 2）成比例，或者仅在前馈部分实现时（m = 1）与其幅度成比例。
通常，功率表达式与输出电压电平无关（n = 0）。
然而，有一些例外，其中功率与Vout成反比。
这是一些连续导通模式（CCM）系统的情况，其中n = 1。
在NCP1605中，它嵌入的跟随器增强技术使功率表达式与输出电压平方成反比（n = 2）。
然而，无需担心：这种功率表达通常在PFC控制器数据表中给出，或者可以在知道电路操作模式的情况下计算。
实际上，功率可能不是最好考虑的变量。
相反，我们将使用电流源iB（t），其值为[pout（avg）（t）/ vout（t）]。
换句话说，考虑到输出电压电平，该电流提供计算的功率。
这里假设负载是电阻（RL）。
这就对了！
我们的模型由图3提供。
图3 - 一个非常简单的大信号平均模型
导出传递函数
需要额外的努力来获得传递函数：提取小信号模型。
为此，传统技术应如图4所示。第一步包括在系统变量的微小变化激发时考虑电流源iB（t）在其dc值附近的微小变化。
请注意，由于我们通常对线变化不感兴趣，因此线幅度被认为是常数。
因此，忽略输入电压的小变化，并且这里仅关注输出电压（Vout）和Vcontrol的小变化。
它们对iB（t）的影响是通过计算其偏导数来计算的。
（2）
完成后，我们获得图4的上部表示，其中考虑了iB和Vout的dc值及其小信号变化。
下一步是消除这些在小信号尺度上不起作用的直流值。
Vout ac贡献可以如下计算：
（3）
式。
图3示出了对由小Vout变化引起的iB变化建模的电流源可以由电阻RL /（n + 1）代替。
回想一下，n是输出电压干扰PFC功率表达式的系数：n一般为0（例如NCP1608驱动的PFC级），在某些CCM PFC的情况下为1或在控制时为2
由NCP1605。
图4 - 推导小信号模型的步骤
最后，两个并联的电阻器可以用单个电阻器代替，并且获得图5的非常简单的模型，其中电流源馈送与大容量电容器并联放置的电阻器。
注意，这里显示的是大容量电容器ESR，即使通常它在我们感兴趣的频率范围（低于线路频率）中也不起作用。
图5 - 小信号模型
我们现在有了补偿PFC阶段循环所需的传递函数：
（4）
NCP1654是一款8引脚CCM PFC控制器。
举个例子，让我们考虑一下这个电路驱动的PFC级。
NCP1654数据手册[1]的公式16给出了以下幂表达式：
（3）
其中n是效率，而Rcs，RM和Rsense，kBO和VREF是系统常数（Rcs，RM和Rsense表示设置最大电流和功率电平的外部电阻，kBO，外部电压传感网络的缩小因子，VREF
，内部电压参考）。
注意，由于NCP1654部分前馈和（n = 1），（m = 1）。
由此，我们推断：
（6）
导致以下传递功能：
（7）
请注意，我们已经处理了唯一的电压环（或外环），它调节PFC级的输出电压。
在此不考虑在一些CCM PFC中进一步实现以主动迫使输入电流匹配正弦参考的电流（或内部）环路。
更多信息可以在[2]和[3]中找到。
参考
[1] NCP1654数据表，http：//www.onsemi.com/pub_link/Collateral/NCP1654-D.PDF
[2]JoëlTurchi和Dhaval Dalal，专业教育研讨会“功率因数校正：从基础到优化”，APEC2014，FORT WORTH，TX
[3]JoëlTurchi，教程TND382 / D，“补偿PFC阶段”：http：//www.onsemi.com/pub_link/Collateral/TND382-D.PDF
以上来自于谷歌翻译
以下为原文
Loop compensation of a power factor correction (PFC) stage is generally not viewed as a major concern. Indeed, a PFC circuit has to shape the line current which is a low frequency signal and hence, by essence, this is an extremely slow system. That is why, the compensation of the PFC loop is generally not considered as a critical step when designing a power supply: “wildly shrink your bandwidth, that’s it!”
However, as any control system, a PFC stage must be compensated and this unavoidable step must be properly performed for an optimized operation of the downstream converter and a satisfactory power factor. The transfer function must hence be derived.
PFC stages are slow systems
Let’s refer to Figure 1. As a result of the PFC function, both the line current (black trace) and the line voltage (blue trace) are sinusoidal. Thus, the power provided to the load has the shape of a square sinusoid as shown by the red trace. Now, as suggested by the black dashed line, the output power is generally constant and the provided power matches the load demand in average only. There are hence periods of time for which the PFC stage provides more power than needed. See “+” sign area. During this phase, the output capacitor absorbs this excess of energy and thus charges up. On the other hand, a second sequence follows when the PFC stage supplies less power than necessary. See “-” sign area. In this second case, the output capacitor discharges in the load to compensate for the lack of energy. As a consequence, the output voltage (green trace) exhibits a low frequency ripple at twice the line frequency (typically 100 Hz in Europe, 120 Hz in US). This ripple is inherent to the PFC function.
![]()
Figure 1 – The output voltage of PFC stages exhibits a low-frequency ripple
Now, traditional current shaping techniques require the control signal (provided by the regulation circuitry) to be a slowly-varying signal. If not, the line current will be distorted. Hence, the regulation loop must reject the low-frequency output ripple. Practically, this is achieved by setting a low bandwidth, in the range of 20 Hz. That is why PFC systems, by nature, are slow systems.
A simple averaged model
When studying the dynamic characteristics of a PFC stage, we may be inclined to consider the two different mechanisms in play and their respective frequency range: first, the switching process for power transfer, second, the modulation at the line frequency for input current shaping. Actually, we can use a simplified approach which ignores these two aspects.
A large-signal model of our PFC stage can be solely based on the averaged power delivered within a line period. The PFC stage is modelled as a device continuously delivering this computed averaged power. Switching events and the input current shaping modulation are only viewed as the mechanism of the power generation which is assumed to be stable. We hence ignore everything happening above the line frequency, which is good enough since the loop bandwidth is set below. Should your system operate in continuous conduction mode rather than in critical conduction mode, in current rather than voltage mode, it does not matter. We just consider the average power delivered by the PFC stage. This method is illustrated by Figure 2.
![]()
Figure 2 – A simple averaging process
Eq.1 gives the general expression of the average power delivered by the PFC stage. The power expression depends on the voltage difference between the control signal (Vcontrol) and its minimum value (Vcontrol,min):
![]() (1)
In systems devoid of feedforward, the average input power is also proportional to the squared line magnitude (m=2) or simply to its magnitude when feedforward is partially implemented (m=1). In general, the power expression is independent of the output voltage level (n=0). There are exceptions however where the power is inversely dependent on Vout. This is the case of some Continuous Conduction Mode (CCM) systems where n=1. In the NCP1605, the follower boost technique it embeds leads the power expression to be inversely proportional to the output voltage square (n=2).
No need to worry however: such a power expression is generally given in the PFC controller data sheet or can be computed knowing the circuit operation mode.
Actually, the power may not be the best variable to consider. Instead, we will use a current source iB(t) which value is [pout(avg)(t)/vout(t)]. In other words, this current delivers the computed power considering the output voltage level. The load is here assumed to be a resistance (RL). That is it! Our model is provided by Figure 3.
![]()
Figure 3 – A very simple large-signal averaged model
Deriving the transfer function
An additional effort is necessary to get the transfer function: extracting the small-signal model.
To do so, traditional techniques are to be engaged as summarized by Figure 4. The first step consists of considering small variations of the current source iB(t) around its dc value when excited by small variations of the system variables. Note that since we are generally not interested in line variations, the line magnitude is considered constant. Hence, input voltage small variations are ignored and only small variations of the output voltage (Vout) and of Vcontrol are of interest here. Their impact on iB(t) is computed by calculating its partial derivatives.
![]() (2)
When done, we obtain the upper representation of Figure 4 which takes into account the dc values and their small signal variations for iB and Vout. The next step consists of eliminating these dc values which play no role at the small-signal scale.
The Vout ac contribution can be computed as follows:
![]() (3)
Eq. 3 shows that the current source modeling the iB variations caused by small Vout variations can be replaced by a resistance RL/(n+1). Recall that n is the coefficient at which the output voltage interferes in the PFC power expression: n is 0 in general (with a NCP1608-driven PFC stage for instance), is 1 in the case of some CCM PFC or can be 2 when controlled by a NCP1605.
![]()
Figure 4 – Steps to derive the small-signal model
Finally, the two paralleled resistors can be replaced by a single resistor and the very simple model of Figure 5 is obtained where a current source feeds a resistor placed in parallel with the bulk capacitor. Note that rC, the bulk capacitor ESR, is shown here even if in general, it plays no role in our frequency range of interest (below the line frequency).
![]()
Figure 5 – The small-signal model
We have now the transfer function we need to compensate the loop of our PFC stage:
![]() (4)
The NCP1654 is a 8-pin CCM PFC controller. As an example, let’s consider a PFC stage driven by this circuit. Equation 16 of the NCP1654 data sheet [1] gives the following power expression:
![]() (3)
Where n is the efficiency while Rcs, RM and Rsense, kBO and VREF are system constants (Rcs, RM and Rsense denote external resistances setting the maximum current and power levels, kBO, the scale-down factor of an external voltage sensing network, VREF, an internal voltage reference). Note that (m=1) due to the NCP1654 partial feedforward and (n=1).
From this, we deduce:
![]() (6)
Leading to the following transfer function:
![]() (7)
Note that we have dealt with the only voltage loop (or outer loop) which regulates the output voltage of the PFC stage. The current (or inner) loop which is further implemented in some CCM PFC to actively force the input current to match a sinusoidal reference, is not considered here.
More information can be found in [2] and [3].
References
[1] NCP1654 Data sheet, http://www.onsemi.com/pub_link/Collateral/NCP1654-D.PDF
[2] Joël Turchi and Dhaval Dalal, Professional Educational Seminar “Power Factor Correction:
From Basics to Optimization”, APEC2014, FORT WORTH, TX
[3] Joël Turchi, Tutorial TND382/D, “Compensating a PFC stage”: http://www.onsemi.com/pub_link/Collateral/TND382-D.PDF
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