Surprisingly, more often than not, a ceramic capacitor does not exhibit its specified value of capacitance in an actual powersupply application. So the 22µF capacitor you've selected won't look like 22 µF once its designed into your circuit. The underlying problem is parasitics. These nonideal characteristics are a chronic driver of performance and can be painful to deal with, especially when you don't realize they are impacting your circuit operation. A highfrequency dcdc converter provides a design example that illustrates the impact of various parasitics on capacitors when modeling the converter's performance in SPICE.
Worstcase circuit analysis (WCCA) is often invoked for space and aerospace hardware that cannot afford to suffer a performance outage or loss of any function. However, we are seeing other disciplines such as medical equipment, critical data backup and storage servers, and highvolume power systems use WCCA techniques to improve reliability much more frequently than in the past. Using WCCA techniques, we evaluate the impact of initial temperature aging and other environmental tolerances (radiation, vibration and moisture) on performance.^{[1]}
As designers are pushing their circuit designs to achieve higher performance, it is increasingly important to understand the consequences that secondorder effects such as capacitor dc bias and their associated tolerances, can have on circuit performance and simulation results.
SecondOrder Effects
To illustrate the significance of secondorder effects, consider a problem AEi Systems encountered when trying to model the performance of National Semiconductor's LM2831, a highfrequency 1.5A stepdown regulator. The process is one that AEi Systems regularly uses to develop SPICE models for National Semiconductor's WEBENCH online simulation system.
The normal process is to start by gathering benchtest data from an application circuit and then comparing the reallife performance with that of the SPICE model to provide concrete verification of the simulation results. In our experience, the models have consistently matched the bench data to within a few percentage points of phase margin and crossover frequency. However, in the case of the LM2831, we could not get the frequencyresponse bench data to line up with the SPICE model.
While it is a common reaction to blame the SPICE model for being the source of error, more often than not we find the test setup or measurement technique is at the heart of the discrepancy. Some of the anomalies are rather interesting.
In this case, as part of the model development process, we performed loadstep bench tests and recorded Bode plots. The loadstep response simulations agreed with the measured data almost perfectly, but the Bode responses did not. Interestingly, due to the transconductance nature of the control loop, the bandwidth of the regulator is inversely proportional to the capacitance and the closedloop impedance (stepload response) is related to 1/(capacitance × bandwidth). So the closedloop impedance is nearly independent of the output capacitance. Therefore, we conceived that the only way the stepload result could be correct and the Bode response incorrect would be if the capacitor value were the culprit (assuming the Bode measurement was correct).
The LM2831 test circuit is shown in Fig. 1 with the parts list indicating a 22µF ceramic X5R capacitor. Trying to understand the error, we calculated what we thought the capacitor would have to be to produce the bandwidth difference between the measured and simulated Bode plots. The answer was 16 µF, and so we might reasonably consider the possibility that a 15µF capacitor had been used in place of the 22µF capacitor.
It is generally assumed that ceramic capacitors can be modeled with a simple RLC or RCequivalent circuit (simple ESR representation), owing to the fact that the ESR is so low compared to the reactive impedance. This is usually an acceptable compromise in the fidelity of the capacitor model, and everything usually correlates reasonably well with this approach.
At this point, we requested and were provided with the manufacturer's capacitor datasheet. It quickly became clear that there was a severe dcbias effect (Fig. 2). The manufacturer's data indicated that at 1.8 V (the converter's output voltage) the capacitance decreased by 25%, which made the capacitance 16.5 µF. The resulting Bode data for the two different capacitor settings is shown in Fig. 3.
Development of a capacitor SPICE model with dcbias effects is fairly trivial.^{[23]} For a static solution based on the circuit's operating point, a capacitor with dc bias can be even more easily achieved with parameterpassing techniques common to most SPICE simulators, where the value of the capacitor is computed as {Cnom * Bias Effect * Temperature effect}. The Bias Effect parameter can be dynamically modeled with a table model that takes in voltage and outputs a percentage equal to that of the curve found in the manufacturer's datasheet, which then multiplies against the nominal capacitance of a voltagecontrolled capacitor. Programs such as Kemet SPICE^{[4]} output a SPICE model at a specified dc bias and temperature setting.
Parameters such as aging, frequency and ac bias (Fig. 4) are also often overlooked, as are variations in one of the most underspecified parameters with the greatest impact: ESR. One of the more common issues that we see concerns the use of tantalum and niobiumoxide capacitors. These general pitfalls also apply to aluminum electrolytic capacitors and wettantalum capacitors.
ESR is generally specified and discussed as a number, usually a maximum ESR at a specified frequency. It is important to remember that ESR is actually a strong function of frequency and temperature. But nominal and certainly statistical data can be hard to come by. Depending on the capacitor, the ESR curve can vary significantly.
What this means in your simulations is that the ESR you use to calculate the output ripple voltage at the switching frequency might be very different than the ESR you use to evaluate the phase margin at the bandwidth of the power supply. This same issue often appears in EMI filterdamping networks that have a relatively low resonant frequency. At these low resonant frequencies, the ESR is often high enough to significantly affect the damping effectiveness.
As the frequency increases, it is also common for the capacitance to decrease, which must be considered when assessing the effectiveness of the damping of the EMI filter and also in the stability assessment of the power supply. Fig. 5 shows ESR curves from two 22µF, 6.3V capacitors from the same manufacturer with clearly different lowfrequency ESR performance.
Another oftenoverlooked issue is cold temperature operation. The effects of cold temperature (55°C) on capacitor performance is significant. Cold temperature use of ceramic and solid tantalum capacitors generally results in a 10% to 20% reduction in capacitance with a 10% to 20% increase in ESR.
It is often assumed that this effect is similar for wettantalum and aluminum capacitors, which is not true. Very little information is published regarding cold temperature use of these capacitors. However, measurements we have made, and also measurements made by NASA, show that there is generally a 1500% (yes, 15 times) increase in ESR of wettantalum and aluminum capacitors when operated at 55°C (Fig. 6).^{[5]}
Fig. 7 and the table describe aging factors for various types of capacitors. Such variations are a concern for equipment that cannot be upgraded or replaced during its lifetime. Often life test data is not available. HALT and HASS testing can be expensive and, in any case, do not cover mission life in the 5 to 20year range. Arrhenius equations assume a linear time base and, therefore, are not useful in most cases.^{[6]} The MILHDBK1547A standard's estimate of 21% aging is based on such extrapolations and is considered overly pessimistic, especially with regard to today's manufacturing processes. It is not uncommon, however, to see 2.5% to 4% per decade hour factored into most designs. Using the equation, the aging rates presented in the table for various dielectrics may be used to calculate capacitance over time:
where C_{A} equals capacitance after t (hours), C_{1} equals cpacitance at time t_{0}, A equals dielectric aging constant (aging rate in percent) and t equals time from last heating (hours).
Beyond the IC Datasheet
It is important for the designer to have an understanding of the many parameters associated with the use of capacitors in powersupply applications, as well as an understanding of the tolerance effects that are associated with these parameters. As capacitor technology improves and switching frequencies increase, the power supplies become much more sensitive to these effects, which can, in some cases, dominate the powersupply performance. In some applications, more often in linear regulators than switching regulators, we see unstable control loops due to inordinately low capacitor ESR.
While the manufacturers of the linear regulators generally do not supply sufficient information to understand or calculate the stability of their devices, the capacitor ESR is often a significant zero in the regulator loop, meaning that a very low ESR will result in an unstable loop. This is even more common with the older regulators such as the LM117, since at the time these regulators were designed, there were no ultralow ESR capacitors, which means the effects of stability on these capacitors were never considered.
References

Sandler, Steve, and Hymowitz, Charles, “Worst Case Analysis of Three Terminal Regulators: Development of an HS117 SPICE Model,” PowerSystems World 2006.

Intusoft newsletters, November 1995/January 1996.

Prymak, John, “SPICE Modeling of Capacitors,” Kemet Electronics.

Kemet SPICE, www.kemet.com/kemet/web/homepage/kechome.nsf/Weben/kemsoft#spice, Kemet Electronics.

Holladay, A.M., “Guidelines for the Selection and Application of Tantalum Electrolytic Capacitors in Highly Reliable Equipment,” NASA Technical Memorandum TMx64755, Rev. A.

“Worst Case Circuit Analysis of Power Conversion Circuits,” APEC 2007.
Table. Aging rates of various dielectric types. *MILHDBK1547A is considered overly pessimistic.