This notes sheet covers capacitors, diodes & rectifiers, and the principles and processes behind these fundamental electronic devices.

Capacitance, C, is a measure of how much energy a capacitor stores. It is given in terms of charge
and voltage:

Since current is the rate of change of charge:

The energy stored in a capacitor is given as:

### Capacitors in Circuits

Capacitors can be seen as the opposite to a resistor when in networks:

in series

in parallel

### Charging Capacitors

A capacitor network consists of a capacitor and a resistor. This network has a time constant, :

Capacitors are charged in series with a resistor and a source:

When charging, the charge and potential difference both increase with the following exponsntial relationship:

Meanwhile the current decreases exponentially:

### Discharging Capacitors

When discharging, the source is disconnected, and the capacitor is connected to another resistor

When discharging, charge, potential difference and current all follow the decreasing exponential
relationship:

Note that current decreases in magnitude when charging and discharging, but the signs are reversed. This is because a capacitor is a load when charging, and a source when discharging.

The charging and discharging relationships can be plotted together to compare.

## Diodes

Diodes are semiconductors that work in one direction only. Ideally, they have no resistance in one
direction, but infinite resistance in the opposite direction.

In reality, however, there is a small threshold voltage that must be overcome before the resistance
increases dramatically. This is due to a slight internal resistance:

The threshold voltage is typically around 0.7 volts for silicone diodes.

To account for this, we need to model non-ideal diodes in series with a constant emf and resistor:

This linearises the graph:

As well as a threshold voltage, diodes have a maximum rated current.

Above this current, the diode will break.

### Light Emitting Diodes (LEDs)

These are some of the most common and useful diodes, as they emit light when used in the
forward direction. Their threshold voltage is slightly higher than normal didoes, at around 1.2 to
2V.

If a voltage is applied larger than the threshold voltage, the LED is destroyed. Therefore, they must
be connected to a high-resistance voltage source.

### Finding Operating Points

Since it is so important to use diodes at the right current and voltage, we need a method of finding

1. Thevenise the circuit into a single-port source and single-port load
2. Measure the current-voltage characteristics of both and plot on the same axes
3. The intersection of the load and source lines is the optimal operating point for the network

## Rectifiers & Rectification

Rectifiers are used to convert AC to DC. This is a three-stage process:

1. A transformer is used to reduce the AC voltage
2. A rectifier (a series of diodes) is used to convert the signal into DC pulses
3. A capacitor is used to smooth the pulses to a nearly constant voltage

### Half-Wave Rectifiers

A simple ‘half-wave’ rectifier consists of a single diode connected in series to the load. Since the
diode only works in one direction, the negative (or the positive) side of the AC voltage gets ignored:

Half-wave rectifiers are very inefficient, as half the power from the AC source is simply neglected.
However, they are so cheap that they are still used if the power is not required.

### Bridge Rectifiers

These are a network of four diodes, that preserve all the energy from the AC source. They do not
discard the negative half-waves, but reflect them in the x-axis:

Following the path around the circuit for a positive and negative source voltage shows that the
voltage will always run through the load in the same direction.

### Smoothing

Connecting a capacitor parallel to the load smooths the voltage to be nearly constant.

• When the voltage from the diode is increasing, the capacitor charges as a second load.
• As the voltage from the diode decreases lower than the voltage in the capacitor, the capacitor discharges, acting as a source to the load.
• Once the diode voltage matches and exceeds the voltage of the capacitor, it takes over and the cycle repeats

### Calculating Conversions

(See notes sheet on geometric waveforms)

Note that is the AC voltage.

This is because we assume the capacitor discharge to be a straight line, not exponential. We can
assume this because the time of discharge is so small (the typical AC frequency is 50 Hz, so a
bridge rectified frequency is 100 Hz!)

The difference between the maximum and minimum voltage is known as the ripple voltage:

This can also be written as:

If we assume that the maximum voltage roughly equals the DC output voltage:

Alternatively, the DC output voltage can be found as the maximum voltage minus half the ripple voltage:

When a bridge rectifier is used, the frequency doubles and the ripple voltage and peak supply current are halved.

• A capacitor is device that stores charge:
• The energy stored in a capacitor is:
• The time constant of a capacitor network is given as
• Capacitors are charged in series with a resistor and source, and discharged in series with a resistor and a load
• Real (non-ideal) didoes have a threshold voltage of around 0.7V (1.2-2V for LEDs)
• AC is converted into DC using a transformer, rectifier and capacitor
• The transformer reduced the voltage
• The rectifier ensures the signal is always positive using diodes
• The capacitor smooths the signal