## Friday, January 23, 2015

### Significance of the RC time constant for super capacitors

When we think about an RC time constant, also known as tau, the thing that usually comes to mind is its relevance to filters. But when it comes to super capacitors it has a related, but somewhat different connotation. Let me explain.

One of the things we learn about early on in electrical engineering about the RC time constant is its relevance to the frequency response of first-order low pass RC filter, as depicted in Figure 1.

Figure 1: First order low pass RC filter

The cut off frequency, fc, is the point where the AC signal amplitude is down by 3 dB, as shown in Figure 2. Correspondingly the power is down by 6 dB, which is also referred to as the half-power point.

Figure 2: First order low pass RC filter response

Here the cutoff frequency is related to the RC time constant by the expression:

However, another aspect to consider about an RC time constant is its relevance to the time it takes for the capacitor to be charged and discharged.  This is significant for power and energy applications using capacitors for energy storage. The capacitor’s voltage response for charging is given by the expression:

For t = RC (one time constant) the capacitor is charged up to 63.2% of its final voltage. Similarly, when discharging, the capacitor is discharged to 36.8% of its final voltage.

So, how is this of significance with a super capacitor? The fastest limiting case for charging and discharging the capacitor is when any external resistance is set to zero. Then the only limiting resistance is the internal equivalent series resistance (ESR) of the capacitor. For conventional capacitors the RC time constant will typically be on the order of microseconds to tens of microseconds. However, super capacitors, with capacitance in Farads to hundreds of Farads (or greater) and ESR of milliohms (or less) have RC time constants on the order of roughly a second.  The fastest they can be charged and discharged for power applications is on the order of seconds or slower. This may sound slow compared to more conventional capacitors but this is not what you really want to compare them to. Because of their extremely large capacitance they are useful for energy storage applications. And when you compare them against other means of storing reasonable amounts of energy, like batteries for example, they are then extremely fast. This makes them ideal for an application needing to store and return quick surges of energy, such as is the case of regenerative braking.

So now when you see the RC time constant being used in reference to super capacitors you will know it’s meant as a figure of merit for how quickly they can be charged and discharged!