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多电源系统的源-负载相互作用
2017-04-19
Abstract:摘要
Switching regulators have negative input impedance, at least at low frequency. This is evidenced by the fact that as the input voltage increases, the input current decreases, and vice-versa. Before switching regulators came into such widespread use, they were mainly stand-alone units operating from a low source impedance and driving a passive load. Oscillation was rarely a problem. Now, as switching regulators become lower in cost and more widespread in application, there are many instances where switching regulators serve as both source and load. This is happening both at low power, as in distributed power systems, and at high power, in utility systems with DC links. The space station power system is a good example of switching regulators driving other switching regulators. If a source has a resonant output characteristic, connecting a negative resistance to that source can allow the resonant output to oscillate. This characteristic is shared by both L-C input filters and switching power supply outputs. This paper discusses various criteria that have been developed recently to assure stability under these source-load conditions and gives practical suggestions of design and testing criteria to assure stability under all operating conditions.
开关稳压器具有负的输入阻抗，至少在较低的频率。这是因为随着输入电压的增加，输入电流减小，反之亦然。开关稳压器进入广泛使用之前，他们主要是通过一个较低的源阻抗和驱动无源负载来进行独立的单元操作。振荡是很少的问题。现在，随着开关稳压器的成本降低和更广泛应用，有许多情况下，开关稳压器作为电源和负载。这是发生在低功率，如分布式电源系统，并且还存在于高功率，例如在直流环节的实用系统。空间站电源系统的开关稳压器驱动其他开关稳压器是很好的例子。如果源具有谐振输出特性，连接到源的负电阻可以允许谐振输出出现振荡。这一特点是由L-C输入滤波器和开关电源输出共享。本文讨论了各种标准最近已被开发，以确保这些源负载条件下的稳定性并给出了实际的建议和测试标准的设计保证在所有操作条件下的稳定性
Introduction：简介
There is increasing interest in calculating or measuring the stability of complex power systems. This interest originally took the form of input filter considerations in power supplies. Papers on this subject extend back at least to 1971 with a paper by Yuan Yu and John Biess1. In 1976 and 1978 Dr. David Middlebrook wrote papers on the subject 2, 3, the former of which has become a classic in the field. Middlebrook pointed out that oscillation could be positively prevented by assuring that the magnitude of the output impedance of the source (filter) remained lower than the magnitude of the input impedance of the power supply (measured beyond the filter).
对于复杂的电力系统稳定性的计算或测量已经引起人们越来越大的兴趣。这种兴趣最初以输入滤波器的表现形式被考虑在电源中。有关这个问题的论文至少可以追溯到1971年来自YUAN YU和John biess1 的一篇论文中。在1976年和1978年，Dr. David Middlebrook 写的论文主题2和3，其中前者已经成为该领域的经典。Middlebrook指出，振荡可以通过保证源（滤波）输出阻抗的大小保持低于电源（measured beyond the filter）的输入阻抗大小来有效阻止。
In 1992, Dr. Fred Lee proposed a less stringent criterion based on consideration of the phase of the impedance as well as the magnitude. It is true that while switching power supplies have negative input impedance at DC, both the phase and amplitude of this impedance are functions of frequency. At frequencies above the bandwidth of the open loop gain of the voltage feedback loop, the phase and amplitude both shift dramatically. It is possible to have the magnitude of the filter output impedance exceed the magnitude of the power supply input impedance and still have a stable system, but from a practical standpoint the stability margins will probably not be satisfactory for high-reliability applications.
1992年，Dr. Fred Lee提出一种较为宽松的判据，基于考虑阻抗的相位以及大小。确实，虽然开关电源有直流负输入阻抗，其相位和幅值都是关于频率的函数。在电压反馈回路开环增益带宽频率之上，相位和振幅都发生了戏剧性的转变。滤波器输出阻抗的振幅超过电源输入阻抗的大小，而且仍然保持一个稳定的系统，这是有可能的。但从实际的角度去看，这个稳定的裕度可能无法满足高可靠性的应用。
This paper starts with a simple L-C filter, then progresses through damping to stability criteria and gives practical examples of real-life impedance plots in both magnitude and phase. Both magnitude and phase guidelines are discussed as the examples are presented.
本文从一个简单的LC滤波器讲起，然后通过阻尼进展到稳定性判据而且给出生活中存在的实例的阻抗大小与相位的图谱并进行了讨论。
Simple L-C Filter
Figure 1 shows a simple L-C filter. L-C filters are common in power supplies. They are used to attenuate AC signals generated inside the power supply and prevent these signals from being conducted either back into the source or out into the load. One characteristic of a simple L-C filter such as the one shown is that, if perfect L's and C's could be constructed, the output impedance of the filter approaches infinity at the resonance of the filter. (Resonance is the frequency where the inductive reactance of the inductor is equal to the capacitive reactance of the capacitor.)
图1介绍了一个简单的LC滤波器。LC滤波器在电源中是常见的，它们被用来在电源内削弱交流信号的产生并且防止这些信号回源或者进入负载。对于如图所示简单的LC滤波器有一个特性，滤波器的输出阻抗在滤波器谐振情况下是趋近于无穷大，前提是要有完美的电感和电容用来构建电路。
To control the impedance peaking, damping is normally employed in L-C filters. There is some natural damping because of series losses in the inductor and the capacitor. It is frequently the case that additional damping is necessary. Figure 2 shows an L-C filter with damping. Parallel damping of the capacitor is shown in lieu of series damping of the inductor because, in general, series losses are to be avoided. Parallel damping of the sort shown would cause excessive losses also, but the next section of the paper discusses a method of controlling the losses from parallel damping.
为了控制阻抗突升，阻尼在LC滤波器是常见器件。因为有电感和电容串联的损失，所以存在自然阻尼。通常情况下，附加的阻尼是有必要的。图2显示了一个带阻尼的LC滤波器。跟电容并联的阻尼在图中由跟电感串联的阻尼代替，因为通常情况下，串联的损失是可以避免的。所示如此类似的并联阻尼也会导致过多的损耗，但是本文的下一部分将会讨论一种方法来控制减少并联阻尼的损耗。
Figure 3 shows a plot of impedance versus frequency of the inductor and the capacitor together with plots of the filter output impedance in the under-damped and over-damped cases. The characteristic impedance of the L-C circuit is given by Zo= the square root of (L/C). A value of damping resistance roughly equal to this value is normally
（没有图3）图3显示了在阻尼情况下和过阻尼情况下电感与电容相对于频率的阻抗图和滤波器输出阻抗图。LC电路的特性阻抗Zo=(L/C)的平方根。该阻尼电阻通常大约等于这个值。
L-C Filter with Practical Damping
带有实际阻尼的LC滤波器
Just as it is not practical to place a resistor in series with the inductor (series damping) because of the conduction losses, it is not practical to simply place a resistor in parallel with the capacitor. This is because the value of this resistor is usually low, on the order of 1 ohm, and the shunt loss would be excessive. Figure 4 shows an L-C filter with a more practical damping scheme. A capacitor is placed in series with the damping resistor to block the DC voltage and prevent DC current from flowing through the damping resistor. The value of this series capacitor has to be larger than the value of the filter output capacitor, a factor of 4 being a common rule of thumb.
正如将一个电阻和一个电感串联在一起是不现实的，因为存在传导损耗，简单的将一个电阻与一个电容并联同样是不现实的。因为这个电阻值通常是比较低的，大于近似于一欧姆，分流损耗会比较严重。图4显示了一个带有更加实用阻尼方案的LC滤波器。电容与阻尼电阻串联以阻挡直流电压而且放置直流电流流经阻尼电阻。串联电容的值必须大于滤波器输出电容，这是4的一个基本的经验法则。
Figure 5 shows an impedance versus frequency plot of the three branches that make up the output impedance of the damped filter. Remember in calculating output impedance the input (input side of the inductor) is grounded. The output impedance is the lower of the curves on the graph. The impedance still follows the inductor impedance up to resonance, the impedance at resonance is still controlled by the resistor, and the curve follows the capacitor impedance down at frequencies above resonance. The damping capacitor impedance is always higher than the inductor impedance and its effect is relatively insignificant.
图5显示了3支路组成的阻尼滤波器输出阻抗的阻抗频率图。记住在计算输出阻抗时候输入（电感的输入端）要接地。输出阻抗是较低的曲线图。阻抗依旧随着电感阻抗达到共振，共振阻抗依然受控于电阻，而且在共振频率之上，电容阻的曲线抗随着频率上升而下降。阻尼电容阻抗始终高于电感阻抗，而且影响相对不明显。
The use of a damping network including a large capacitor may seem like an excessive burden in size, weight, and cost but the benefits are many and the burden is not as great as it first appears. The output capacitor needs to be of good quality with low internal resistance (internal resistance is called Equivalent Series Resistance or ESR). The damping capacitor does not need to be of this high quality, since a resistor will be placed in series with it anyway in the application. For example, in a space application, the filter output capacitor may be a low ESR solid tantalum while the series damping capacitor could be a lossy but much smaller wet-slug tantalum
包含了大型电容的阻尼网络的使用可能看起来在规格重量和造价上过度负担，但是相比带来的许多好处，这些负担并没有一开始体现的那么大。输出电容需要低内阻质量好（内阻称为等效串联电阻或者ESR）。阻尼电容不需要这么高的质量，因为电阻无论如何都会串联进应用中。例如，在空间应用中，滤波器输出电容可能是一个低ESR的固态钽电容器，而串联阻尼电容可能是一个有损耗的但是更小的wet-slug tantalum
L-C Filter With Damping And Negative Resistance Load
带有阻尼和负电阻负载的LC滤波器
When an L-C filter is driving a switching power supply, a different and fundamentally dangerous situation occurs. Switching power supplies are for all practical purposes constant power devices. If the output power is constant, the input power will be also. If the input voltage increases, the input current will decrease so that the volt-amp product will remain the same. This characteristic of decreasing current with increasing voltage is effectively a negative resistance. Figure 6 shows schematically what happens when a negative load is connected to a damped L-C filter. If the magnitude of the effective negative resistance is lower than the magnitude of the effective damping resistor, the L-C filter will have a net negative damping resistance and the filter will oscillate.
当一个LC滤波器驱动一个开关电源时，一个不同的而且根本的危险情况会发生。开关电源实际上是恒定的功率器件。如果输出功率是恒定的，则输入功率也会如此。如果输入电压升高，则输入电流会下降，所以volt-amp产品会保持一样。这种随着电压上升而电流下降的特性实际上是一种负电阻。图6示意表明当一个负的负载连接到一个阻尼LC滤波器的情况。如果有效的负电阻的大小小于有效的阻尼电阻，LC滤波器将会有一个纯负阻尼电阻而且滤波器将会振荡。
This phenomenon is obvious at DC, but what may not be so obvious is that the negative resistance is relatively constant up to the bandwidth of the internal voltage feedback loop of the power supply regulation circuitry. If the filter resonance frequency is lower than the power supply voltage feedback bandwidth, then it is necessary to control the peaking of the output impedance of the input filter so that the peak value remains below the magnitude of the negative resistance of the power supply input.
这种现象在直流中是很明显的，但是可能不会那么明显的是，负电阻是相对恒定的由电源供电管理电路的内部电压反馈回路带宽所决定。如果滤波器共振频率低于供电电压反馈带宽，那么就有必要控制输入滤波器的输出阻抗的峰值，所以峰值要保持低于电源供应输入电路的负电阻大小。
Relative Placement of Filter Resonance and Power Supply Bandwidth
滤波器共振和电源带宽的相关位置
Problems of filter resonance occur primarily when the resonance of the filter is lower than the bandwidth of the power supply voltage feedback loop. This is rare in commercial power supplies since the input filter is usually placed there for prevention of electromagnetic interference (EMI) and has a resonant frequency on the order of 300 kHz, well above the feedback loop bandwidth. In military and aerospace power supplies, constraints of conducted emissions and conducted susceptibility testing require that the filter resonance always be below the bandwidth of the power supply voltage feedback loop. This situation comes about because the conducted susceptibility requirement does not allow signals to pass through the power supply from input to output at any frequency. At low frequency, noise on the input is rejected by the regulating action of the feedback loop. At high frequency, noise on the input is rejected by the input filter. There is necessarily an overlap between these two frequency ranges, since to leave a gap would create a band of frequencies where noise would pass through the power supply unimpeded by either the loop or the input filter. This means that in military and aerospace power supplies there is always the potential of an input filter oscillation and tests and/or analysis should be used to verify sufficient margin between the filter output impedance at resonance and the power supply input impedance. Watch out for cables. The resonance and oscillation problem is not restricted to overt filters. Long cables terminated by capacitors can create the same conditions if the load is a switching regulator.
当滤波器共振带宽低于电源电压反馈回路时，滤波器主要的问题是发生共振。这在商业电源中是罕见的因为输入滤波器通常置于那儿是预防EMI（电磁干扰）而且共振的频率接近300KHz，显著高于反馈回路带宽。在军事和航空航天电源里，因为排放管理的约束和传导敏感度实验的要求，滤波器谐振始终低于电源电压反馈回路的带宽。这种情况是由于传导敏感度要求不允许信号通过电源从输入到输出在任意频率。在低频，输入的噪声由反馈回路调节作用所拒绝。在高频，输入的噪声由输入滤波器所消除。所以说必然地在两个频率范围会发生重叠，因为要留个缺口来创建一个频带，使噪声可以无阻碍地通过电源，无论是回路或者输入滤波器。这意味着在军队或者航空航天电源里，总要有一个输入滤波器振荡、测试或者分析的可能，以用来核实滤波器输出阻抗在谐振时和电源输入阻抗之间有足够的裕度。要注意电缆。谐振和振荡问题并不局限于这些滤波器，长电缆终端的电容也会产生相同的情况，如果负载是一个开关调节器。
Power Supply With Input Filter
电源带输入滤波器
Figure 7 shows a power supply with input filter. Notice that there is not only an input L-C filter, but an output L-C filter as well. The power supply is partitioned in a way that is perhaps unusual but is more representative of the actual functional blocks at work. The portion labeled "Power Supply" is the actual switching regulator stripped of the input and output filters. The power supply draws pulses of current from the input filter and puts out pulses of voltage to the output filter.
图7显示了一个带输入滤波器的电源。注意到这里不仅是只有一个输入LC滤波器，而且还有一个输出LC滤波器。电源被划分在某种程度上或许是不寻常的但是对于实际工作中功能模块也是非常具有代表性的。部分标有“电源”是从输入输出滤波器里剥离出来的实际的开关稳压器。电源从输入滤波器拉动电流脉冲然后熄灭电压脉冲到输出滤波器
Dual Nature of Filters
滤波器的二重性
One interesting characteristic of all passive filters is a duality whereby the voltage transfer function in the forward direction is identical to the current transfer function in the reverse direction. Filters which are designed to attenuate voltage in the forward direction are equally adept at filtering current in the reverse direction.
所有的无源滤波器都有一个有趣的特性就是二象性，正向电压传递函数与反向的电流传递函数相同。这样设计滤波器目的是降低正向电压，同样地反向滤波电流也如此。
A major purpose of the input filter is to attenuate the AC components of the pulses of current drawn by the power supply so that only the DC average value flows from the source. A major purpose of the output filter is to attenuate the AC portion of the pulses of voltage at the output of the power supply so that only the DC average value of the voltage is applied to the load.
输入滤波器的一个主要的目的是降低电源电流消耗的脉冲的交流分量，这样来源就只有直流的平均值。输出滤波器的一个主要目的是降低电源的输出电压脉冲的交流部分，这样就只有直流平均值得电压被施加到负载。
In the circuit of Figure 7, a control voltage Vc is compared to a ramp to generate a pulse width modulated drive signal for the transistor. This circuit is not a complete power supply, since the compensation amplifier is not shown. This portion of the circuit is called the "plant" or "modulator". The transfer function from control voltage Vc to the output normally has constant gain and negligible phase shift at low frequency, and then falls at a 2 slope (-40 dB/decade) beyond the corner frequency of the output filter. A typical plant transfer function is shown in Figure 8.
在图7的电路中，控制电压Vc与斜线相比对晶体管生成一个PWM驱动信号。这不是一个完整的电源电路，因为补偿放大器是不显示的。电路的这个部分叫做"plant"或者调幅器。控制电压Vc到输出的传递函数在低频通常具有恒定的增益和微不足道的相移，然后在超过输出滤波器转角频率后突然开始以2 slope (-40 dB/decade)下降。图8所示是一个典型的plant传递函数。
Figure 8 does not include any effects of the input filter. If the input filter is well damped and the output impedance peak of the input filter is well below the negative input impedance of the power supply, this is what a typical plant transfer function looks like.
图8不包括任何输入滤波器的影响，如果输入滤波器具有良好的阻尼性而且输入滤波器的输出阻抗峰值低于电源的负输入阻抗，这就看起来想一个典型的装置传递函数。
Effect of Input Filter on Transfer Function
输入滤波器传递函数的影响
In the case of a power supply with poorly damped input filter, the gain portion of the plant transfer function dips significantly at the resonant frequency of the input filter. Figure 9 shows the plant transfer function of a circuit with little damping. The input filter resonant frequency in this particular example is 225 Hz and the output filter resonates at 503 Hz. Note the large perturbations in both gain and phase at approximately 200 Hz. These are due to the input filter and are characteristic of systems with a potential input filter problem.
在电源带有低阻尼滤波器的情况下，在输入滤波器共振频率，装置传递函数的增益部分电压跌落非常值得注意。图9显示了带有低阻尼电路的装置传递函数。输入滤波器谐振频率在这个例子中是225HZ而且输出滤波器谐振频为503hz。注意到大的扰动在每个增益都出现，相位大约是200HZ。这是由于输入滤波器和一个潜在的输入滤波器所特有系统的问题。
Effect Of Filter On Feedback Loop Transfer Function
反馈回路传递函数滤波效果
In order to complete the feedback loop, a compensation amplifier is needed to sense the power supply output voltage, compare it to a reference, and output an appropriate control voltage to the Vc input of the plant. Figure 10 shows a complete power supply with plant and compensation amplifier interconnected. Gain of the compensation amplifier is set by the ratio of the feedback impedance Zf to the input impedance Zin. This ratio can be freely adjusted to achieve a wide range of overall loop bandwidth and phase margin. When selecting amplifier gain, it is usually best to design for loop bandwidth greater than the resonant frequency of the output L-C filter. The reason is that the loop provides active damping for the output filter if the loop bandwidth is greater than the filter resonant frequency. If the loop bandwidth is less than the output filter resonant frequency, then the loop cannot damp the filter and the filter Q determines the amount of output impedance peaking with no help from the loop. There is no essential difference between an input filter and power supply interface and the interface between one switching power supply driving another. As in the examples just discussed, it is necessary to keep the output impedance of a filter or an entire power supply lower than the negative impedance of the next unit in series.
为了完成反馈回路，补偿放大器需要监测电源输出电压，对照参考进行比较，输出一个适当的控制电压到装置Vc输入。图10一个完整的带装置和补偿放大器连接的电源。补偿放大器的增益是反馈阻抗Zf和输入阻抗Zin的比。这个比值可以自由调节以实现更广泛的环路带宽与相角裕度。当选择好放大增益，它通常很好是设计成环路带宽大于输出LC滤波器的谐振频率。因为如果环路带宽大于滤波器谐振频率，则回路为输出滤波器提供了主动阻尼。如果环路带宽小于输出滤波器谐振频率，那么环路无法使滤波器产生阻尼性而且滤波器Q决定输出阻抗尖峰的数量且无法从环路中得到帮助。输入滤波器和电源接口之间没有本质的区别，而且对于接口，开关电源与驱动另一个也是一样。就如刚讨论的例子，有必要保持滤波器或者一个完整的电源的输出阻抗低于串联的下一个单元的负阻抗。
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Figure 11 shows the open loop transfer function of the complete power supply including compensation amplifier and under-damped filter. This loop was designed for a bandwidth of 3 kHz and phase margin of 60 degrees. The effect of the input filter resonance at 200 Hz can be clearly seen. Loop gain actually dips below unity (0 dB) at the filter resonance, but the loop is stable because the phase dip did not quite reach zero degrees. (When discussing a loop, zero phase is really 360 degrees, and the condition to avoid is the one where the gain is unity (0 dB) and the total phase shift is 360 degrees or one whole cycle. This is the condition for oscillation.) In this circuit, the loop crosses over well above the output filter resonance (3 kHz vs. 500 Hz) and the phase margin near optimum at 60 degrees. The output impedance peaking will be well controlled by the feedback loop, but the input filter resonance is severe and will have a significant effect on the output impedance near the 200 Hz resonant frequency.
图11显示了一个包含了补偿放大器和低阻尼滤波器完整的电源开环传递函数。这个环路设计为一个3KHZ带宽和60°的相位裕度。输入滤波器在200HZ的谐振效果可以清晰看到。环路增益在滤波器谐振时实际上降至低于统一（0分贝），但是环路是稳定的，因为相位下降不完全到0°（当讨论一个环路，0相位实际上也是360°，而且避免的条件是增益是统一的而且总相移是360°或者一个完整的周期，这是振荡的条件）。在电路中，环路交叉点远高于输出滤波器共振（3KHZvs500HZ）而且相位裕度接近60°。输出阻抗峰值将由反馈回路控制，但是诸如滤波器的谐振很严重，会对附近200HZ谐振频率的输出阻抗产生影响。
Testing Impedance：测试阻抗
Figure 12 shows a test set-up for measuring both output impedance and input impedance of the next unit using a Venable Model 350 System or equivalent frequency response analysis system. Other equipment also required is a power amplifier, injection transformer capable of carrying high DC current, and a current probe for measuring the current. The injection transformer should be located near the source and the current probe should be near the injection transformer. The idea is to make the measurement at a "slice" in the power transfer cables so that the output impedance can be measured as well as the input impedance of the next unit in conjunction with the connecting cables. It is of course possible to measure the input impedance of the load without the cable effects simply by moving the Channel 2 voltage measurement point to the input terminals of the load, but that measurement in itself would not be sufficient to assure system stability since the cables are in fact a part of the input impedance of the load from the standpoint of the source. Source output impedance is CH3/CH1. Load input impedance is CH2/CH1. The scale factor of the current probe and the direction of the current probe have to account for in scaling the measurement data. The measurement system chosen must be capable of plotting user-selected channels, scaling the data to compensate for the scale factor of the current probe, and inverting the data on one measurement to compensate for the direction of the current probe. The data for the unit the current probe is pointing to will be correct. The data for the unit the current probe is pointing away from needs to be inverted to be accurate.
图12显示了一个用于测量下一个单元输出阻抗和输入阻抗的测试装置——Venable 350型号系统或者等效的频响分析系统。其他设备还需要功率放大器，注入变压器可以承载高直流电流而且一个电流探头用于测量电流。注入变压器应当靠近电源而且电流探头应该靠近注入变压器。这个想法是使测量处于电力传输电缆的“slice”以至于输出阻抗可以被测量到，以及下一个单元连同连接电缆的输入阻抗。不考虑电缆效果而将通道2的电压测量点移动到负载的输入终端来测量负载的输入阻抗当然是有可能的，但是测量本身不能足以保证系统的稳定性，因为从源的角度来看，电缆实际上是负载输入阻抗的一部分。源输出阻抗是CH3/CH1。负载输入阻抗是CH2/CH1。电流探头的比例系数和方向必须考虑拓展的测量数据。测量系统的选择必须能够绘制用户选择通道，拓展数据以补充电流探头的比例系数。数据单元的电流探头的指向都是正确的。数据单元的电流探针指向偏离需求需要精确地调整。
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Figure 13 shows a plot of impedance versus frequency of the input filter output impedance over-plotted with input impedance of the power supply in the circuit of Figure 7 and for the underdamped filter case. In this example the impedance magnitude of the input filter output impedance does actually touch the magnitude of the power supply input impedance, but the phase of the power supply input impedance is a few degrees from the 180° required to have a pure negative resistance and the circuit does not oscillate. This circuit is on the verge of oscillation and would be unacceptable in any high-reliability application. In a transient loading situation, this circuit will ring at 200 Hz because of the input filter resonance. There will be no appreciable ringing at 3 kHz, the true loop bandwidth and the frequency that would normally be associated with a potential oscillation.
图13显示了如图7中电路中带有电源输入阻抗的输入滤波器和欠阻尼情况的输出阻抗的阻抗对应频率图。在这个例子中，输入滤波器的输出阻抗的阻抗值大小实际上接近电源输入阻抗的大小，但是电源输入阻抗的相位需要从180°降到很低使得有一个纯粹的负电阻而且电路不会发生振荡。这个电路在振荡的边缘而且不能承受任何高可靠性需求的应用。在瞬态情况下，该电路因为输入滤波器的谐振将振铃在200HZ。3000hz不会有明显的振铃，真正的环路带宽与频率通常会与一个潜在的振荡相关。
Effect of Feedback Loop on Power Supply Output Impedance
反馈回路对电源输出阻抗的影响
In addition to the design and damping of the input filter, the voltage feedback loop bandwidth and phase margin have dramatic effects on the overall output impedance of the power supply. This impedance is important because it becomes the source impedance for the next switching regulator down the line. If the feedback loop bandwidth is less than the output filter resonant frequency, then the peaking will come mostly from the filter and the loop will have relatively little effect. If, as should be the case, the voltage feedback loop bandwidth is higher than the output filter resonance, then the loop characteristics dominate. The peaking frequency is determined by the loop bandwidth and the amplitude of peaking is determined by the phase margin.
除了输入滤波器的设计和阻尼，电压反馈环路带宽和相位裕度在对于整个电源输出阻抗上有巨大的影响。阻抗重要是因为对于下一个开关稳压器它是源阻抗。如果反馈环路带宽低于输出谐振频率，然后峰值会主要来源于滤波器而且环路会相对较小的影响。如果，应该也是如此，电压反馈环路带宽高于输出滤波器谐振频率，那么环路特性将占主导地位。峰值频率由环路带宽决定，峰值的大小由相位裕度决定。
Crossover Below Output L-C Corner
输出LC角以下的交叉
Figure 14 is a plot of the overall power supply output impedance when the loop bandwidth is less than the filter resonance. Note that the absolute magnitude of the impedance is higher than in the following example, even though the circuit was identical except for the loop bandwidth and the phase margin was actually higher than in the following case. The input filter was disabled for the purposes of this test.
图14是一个总电源当环路带宽小于滤波器谐振频率时候的输出阻抗图。注意到阻抗的绝对大小是高于下文的例子，虽然下文中的电路除了环路带宽和相位裕度较高外，其他都是相同的。输入滤波器无法达到实验目的。
Crossover Above Output L-C Corner
输出LC角以上的交叉
Figure 15 shows a plot of overall power supply output impedance with 3 kHz loop bandwidth and 60 degrees of phase margin. The input filter was disabled for this test. Note that the peaking occurs at the loop bandwidth, not at the resonant frequency of the output filter.
图15显示了在3KHZ环路带宽和60°相位裕度情况下总电源输出阻抗图。输入滤波器不能用于这个实验。注意到环路中出现了峰值，但并不是出现在输出滤波器的谐振频率。
Crossover Above Output L-C Corner with Under-Damped Filter
带有低阻尼滤波器时输出LC角以上的交叉
Figure 16 is a plot of the overall output impedance of the power supply with 3 kHz bandwidth but also with the under-damped input filter. Note the impedance peak at 200 Hz due to the filter. It is actually higher than the principal peaking at the loop bandwidth.
图16是3Khz带宽的电源总输出阻抗图，而且带有低阻尼输入滤波器。注意到因为存在滤波器，阻抗峰值出现在200HZ。它实际上是高于环路带宽的主峰值的。
One Supply Driving Another：驱动情况
When one power supply is driving another, exactly the same principles apply as in the case of a power supply driven by an input filter. If the power supply feedback loop does not have sufficient bandwidth or phase margin, the output impedance of the source supply can be excessive. If this impedance exceeds the input impedance of the supply, which is acting as a load, and the phase of that impedance is close to 180 degrees, the system will oscillate or be dangerously close to doing so. Also, the cables have to be taken into account. If the circuit is "cut" for measurement purposes near the source, then the cables count as part of the load. If, on the other hand, the circuit is "cut" near the load, the cables become part of the source impedance and again have to be accounted for. I have personally experienced a case where an off-line supply had a large electrolytic filter capacitor as a line frequency filter, then cables over to a bank of high frequency bypass capacitors near the input to a switching power supply. The cable from the electrolytic capacitor resonated with the bank of bypass capacitors and caused a system oscillation, even though there was no overt input filter.
当一个电源驱动另一个电源时，同样的原则也适用于由滤波器驱动的电源情况。如果电源反馈回路没有足够的带宽或者相位裕度，电源的输出阻抗就过多了。如果这个阻抗超过了电源的输入阻抗而扮演一个负载的角色，而且阻抗的相位接近180°，那么系统就会振荡或者变得不安全。同时，电缆也应当被考虑在内。如果电路从电源看进去测量，那么电缆是负载的一部分。如果从另一方面，从负载方向看进去测量，电缆就成了源阻抗的一部分。我曾亲身经历过的情况，一个离线电源有一个大的电解滤波电容作为一个线频滤波器，然后电缆越过一堆靠近输入的高频旁路电容连接到开关电源。虽然没有明显的输入滤波器，但是带有旁路电容堆的电解电容电缆会产生共振而且引起系统振荡。
One Power Supply Driving Several Others
电源驱动多个
In the case of a distributed power system where one power supply is driving many others, the same principles still apply although the analysis becomes very difficult. Measurement remains easy and is the recommended way to assure system stability and reliability. If analysis is absolutely required, the easiest way to do it is to convert each impedance into an admittance for each branch of interest at the "cut" where the system is to be analyzed. At any one frequency, each admittance can be converted to an equivalent inductive or capacitive admittance and a parallel resistor. The resistor may be positive or negative. All of these quantities will be functions of frequency. At the frequencies where the inductive and capacitive admittances add to zero (inductive and capacitive admittances cancel each other out if the amplitudes are equal), if the sum of the resistive admittances is negative then the system has a stability problem. This calculation is not trivial since in many cases loads are combinations of series and parallel elements, which have to be accurately combined, and calculation of essential elements such a cable impedance is no trivial task in itself. Fortunately, measurement remains easy and the measurement equipment automatically gives the results of these combinations without having to calculate and combine each one independently.
在分布式电源系统的情况下，一个电源会到驱动多个部件，同样的原则依然适用，虽然分析变得非常困难。测量很容易而且是一个比较靠谱的方式来确保系统的稳定性和可靠性。如果分析是绝对必要的，简单的方式是将每个阻抗转换成每个相关的支路中的导纳at the "cut"。在任意频率，每个导纳可以被转化为等效的电感或者电容性导纳和一个并联的电阻。电阻可能是正的也可以是负的。所有的这些数量都是频率的函数。在频率下，电感和容性导纳和为零（电感和电容性导纳互相抵消如果振幅相等），如果电阻性导纳的和为负则系统具有稳定性问题。这种计算时不重要的因为在多种情况下负载是一系列元件的串联和并联组合，必须精确的组合，而且对于其本身例如电缆阻抗这样必要元素的计算都不是必要的任务。幸运的是，测量仍然是容易的而且测量设备自动给出结果，就不用必须独立计算组合每一个元部件。
Conclusions
Criteria for stable combinations of sources and loads were developed starting with simple examples of L-C filters. Easily understood principles were derived and expanded to cover increasingly complicated situations, which occur in real-life power and power distribution systems. The culprit was shown to be the negative input impedance of switching power supplies and the ability of this negative impedance to reduce damping of L-C filters and even feed energy into them to cause oscillation. An easy and simple measurement technique was presented for measuring these difficult to model quantities and assuring system stability and reliability.
2017-02-11
