# Transient Energy Storage

## Devices which Store Energy Transiently

In engineering, a great deal of use is made of devices which can store energy in modest amounts and often for shortish periods of time. For example, a spring stores potential energy, while a rotating wheel can store kinetic energy.

Almost any engineered device that depends upon events happening with controlled timing will make use of such transient energy stores, directly or indirectly. A common arrangement is one in which energy is transferred between two devices A and B in such a way that the energy is entirely present in A, then entirely in B, then entirely in A ... at precisely regular intervals. For example a mechanical watch depends on the regular tranfer of energy between a wheel and a spring.

Another common arrangement is one in which energy is transferred at a controlled rate into a device until an event is triggered which releases the energy. Alternatively energy may be transferred out of a device until an event is triggered. For example in an hour glass sand is given potential energy in the upper half container. It trickles through the narrow neck. When the human using the glass observes that the upper section is empty he takes an appropriate action.

Transient energy stores are also useful in helping to keep the value of a physical variable from getting too large or small. For example the springs on a vehicle serve to prevent forces arising from uneven ground being transmitted to the occupants - that is the force exerted on the occupants by the vehicle is prevented from getting too large.

Other kinds of energy stores can be used to produce very high values of a physical variable by exploiting the sudden release of stored energy. For example, by storing energy in a hammer, a blacksmith is able to exert much higher forces on a workpiece than he could just by pressing directly on it.

It's worth noting that such sudden release of energy can happen unintentionally, as when the blacksmith drops his hammer on his toe. Whenever you use an energy storage device in electronics, you should be aware of possible damage (or even danger) that can arise from unintended release of that energy.

## Capacitors

A capacitor is a two terminal device which stores energy in the form of an electric charge according to the equation:

Q = CV

Here C is the capacitance of the capacitor, measured in farads (after Michael Faraday). A one-farad capacitor is very large, so the most common units of capacitance are the micro farad µ F and the pico-farad pF. For some reason the nano-farad and the milli-farad are not used as units.

Differentiating, we obtain the current:

I = dQ/dt = C(dV/dt)

### Capacitors in Series and Parallel

When capacitors are connected in parallel we obtain:

C = C1 + C2 + C3...

And when two are connected in series:

C = C1 C2/(C1 + C2)

A capacitor is formed of two conductors separated by a thin layer of insulator, called the dielectric. For a capacitor of any significant size, the conductors are in the form of plates or sheets of metal. The larger the area of the conductors and the thinner the dielectric the larger the value of the capacitance.

There is a capacitance formed between any two conductors. This means that a signal can pass from one to the other. This is usually not a problem for digital electronics except in cases such as we find in a ribbon cable, where two adjacent wires are close together for a long distance. In this case we can get ``crosstalk'' between wires. To mitigate this problem, it is common to have every alternate conductor in a ribbon cable grounded.

## Kinds of Capacitor

Small capacitors are relatively easy to make. In the pico-farad range, a capacitor can be made out of two quite small metal plates separated by a thin sheet of insulator such as mica or polythene. Larger value capacitors may make use of metal foil to provide a larger surface area in a small volume. Small value capacitors will take all the voltage we are likely to put across them without breakdown.

The problem in making big capacitors is how to get the conductors near enough and of big enough area. This problem is usually solved by making use of electrolytic capacitors in which the insulator is formed by electrolytic action forming a very thin film of insulator on a metal electrode. Beware! These are polarised, that is to say, you must always keep one terminal more positive than the other, or you will knacker the device, possibly messily. The polarity of the terminals is indicated.

Tantalum capacitors are a superior kind of electrolytic capacitor, which are none-the-less polarised.

Horowitz and Hill have a nice table listing the vices and virtues of various kinds of capacitor. Horowitz and Hill rate electrolytic capacitors as ``terrible, ghastly and awful'' for their accuracy, temperature stability and leakage respectively. They also have a relatively short lifetime. If a piece of consumer electronics starts to emit a loud buzzing sound, it is more likely that an electrolytic capacitor has died somewhere in its innards than it is that a swarm of bees has moved in.

Nuclear bombs are triggered by the precisely timed discharge of capacitors which cause the detonation of shaped charges of conventional explosive thereby assembling a critical mass of fissile material and maintaining it long enough for the nuclear reaction to achieve near completion. For this reason the export of certain high performance capacitors is discouraged by Uncle Sam, [and by Frau Battenburg's Government too].

### RC Circuits

These are the electronic equivalent of the hour glass. The capacitor C is charged up to V0 volts.

C(dV/dt) = I = V/R

This is a first order differential equation with constant coefficients. It has the solution

V = V0 exp(-t / (RC))

The product RC is called the time constant of the circuit. The voltage will fall to 37% of its initial value in time RC.

According to the equation the voltage V never gets to zero, but a given circuit should be designed to trigger an event in a small multiple of RC seconds if reasonably accurate timing is required.

You can also use a charging-up circuit rather than a discharging circuit.

Here the equation is:

I = C(dV/dt) = (Vin-V)/R

with the solution

V = Vin(1 - exp(-t/(RC))

In designing these circuits we have to allow for the imperfections of the components, especially leakage of the capacitor. Also, we can only trigger an event by sensing the state of the circuit by some additional component, and this may perturb the circuit significantly, by consuming current for example.

## RC Circuits as Smoothing Filters

Well then,

We can use a RC circuit to compute the weighted average of a signal. We'll talk about how to analyse this using the Laplace Transform later in the course.

We will use this kind of smoothing filter as part of a circuit for detecting choos. As tiny choos run along the track they draw current from it. The presence of this current thus indicates the existence of a choo in a particular piece of track. However a choo actually bounces along the track (not unlike Tigger) so the current is intermittent. By using an RC circuit with a time constant of a few seconds we can generate a signal for the computer reliably indicating the presence or absence of a choo.

Question: we could also write a simple algorithm for the computer which performed a near-equivalent computation. What is it?

## Extending our Analogue Computation Repetoire

### Differentiators

I = C (d/dt)(Vin-V) = V/R

choose R and C small, so that V is small, we have approximately:

V(t) = RC(d/dt)Vin(t)

so that the output is an attenuated form of the derivative of the input. Note - this can happen by accident, so if what should be a beautifully smooth sine wave is crawling with horrid little ticks, you may suspect that some capacitative coupling is occurring somewhere - perhaps a digital line has been placed too close to an analog signal.

### Integrators

I = C(dV/dt) = (Vin - V)/R

If we keep V small compared with Vin by having RC large, then approximately:

```C(dV/dt) = Vin/R
```
```V(t) = (1/RC) integral Vin(t)dt + constant
```

## Ramp generators

If we replace the resistance in an RC circuit with a current source we get a ramp generator, which provides a signal in which voltage is proportional to time.

V(t) = (I/C)t