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:

    \[Q=CV\]

Since current is the rate of change of charge:

    \[i=\frac{dQ}{dt}=C\frac{dv}{dt} \quad v=\int\frac i C dt\]

The energy stored in a capacitor is given as:

    \[E=W=\frac 1 2 QV = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}V^2 C\]

Capacitors in Circuits

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

\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+... in series

C=C_1+C_2+... in parallel

Charging Capacitors

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

    \[\tau=CR\]

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

charging a capacitor, capacitor charging network, charging capacitor circuit

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

    \[x=x_0(1-e^{-\frac t \tau})\]

Meanwhile the current decreases exponentially:

    \[I=I_0 e^{-\frac t \tau}\]

Discharging Capacitors

When discharging, the source is disconnected, and the capacitor is connected to another resistor
(this is the load):

discharging a capacitor, capacitor discharge network, discharge capacitor circuit

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

    \[x=x_0 e^{-\frac t \tau}\]

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.

capacitor graphs, charging vs discharging capacitors

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:

Diode graph, diode IV graph, real vs ideal diodes, threshold voltage

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:

non-ideal diode, diode model, diode threshold voltage, diode internal resistance

This linearises the graph:

linear diode, linearised diode 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
this: the load-line method.

loand-line method fo rfinding operating point of a diode and circuit, finding operating points electronics
  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

half-wave rectifier circuit, rectification network

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 rectification graph, rectifier graph

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

bridge rectifier, bridge rectification, rectification circuit, bridge rectifier circuit diagram

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:

bridge rectifier graph, bridge rectification graph, full rectification

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.

smoothing AC current, converting AC to digital, rectification smoothing graph
  • 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

calculating dc conversions, ac to dc conversion,

    \[V_{max}=\sqrt 2 \times V_{rms, sine wave}\]

(See notes sheet on geometric waveforms)

Note that

    V_{rms,sine wave} is the AC voltage. <!-- /wp:paragraph --> <!-- wp:paragraph --> <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="http://engineeringnotes.net/wp-content/ql-cache/quicklatex.com-a1b744c5976199cba4a93e966481ec8c_l3.svg" height="41" width="184" class="ql-img-displayed-equation " alt="\[V_{min}=V_{max}(1-\frac t \tau)\]" title="Rendered by QuickLaTeX.com"/> <!-- /wp:paragraph --> <!-- wp:paragraph --> This is because we assume the capacitor discharge to be a straight line, not exponential. We canassume this because the time of discharge is so small (the typical AC frequency is 50 Hz, so abridge rectified frequency is 100 Hz!) <!-- /wp:paragraph --> <!-- wp:paragraph --> The difference between the maximum and minimum voltage is known as the <strong>ripple voltage</strong>: <!-- /wp:paragraph --> <!-- wp:paragraph --> <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="http://engineeringnotes.net/wp-content/ql-cache/quicklatex.com-664b719020af6f4aa7a5400c6a92ff85_l3.svg" height="21" width="178" class="ql-img-displayed-equation " alt="\[V_{rpp}=V_{max}-V_{min}\]" title="Rendered by QuickLaTeX.com"/> <!-- /wp:paragraph --> <!-- wp:paragraph --> This can also be written as: <!-- /wp:paragraph --> <!-- wp:paragraph --> <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="http://engineeringnotes.net/wp-content/ql-cache/quicklatex.com-3be7e71e9d9c4e8fec080dc17592857e_l3.svg" height="41" width="125" class="ql-img-displayed-equation " alt="\[V_{rpp}=V_{max}\frac t \tau\]" title="Rendered by QuickLaTeX.com"/> <!-- /wp:paragraph --> <!-- wp:paragraph --> If we assume that the maximum voltage roughly equals the DC output voltage: <!-- /wp:paragraph --> <!-- wp:paragraph --> <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="http://engineeringnotes.net/wp-content/ql-cache/quicklatex.com-84926f14b52dea5cd567f3f2a3f693bc_l3.svg" height="41" width="119" class="ql-img-displayed-equation " alt="\[V_{rpp}=V_{DC}\frac t \tau\]" title="Rendered by QuickLaTeX.com"/> <!-- /wp:paragraph --> <!-- wp:paragraph --> Alternatively, the DC output voltage can be found as the maximum voltage minus half the ripple voltage: <!-- /wp:paragraph --> <!-- wp:paragraph --> <span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="http://engineeringnotes.net/wp-content/ql-cache/quicklatex.com-2a157b337cf92a2c8d224a580a111d06_l3.svg" height="43" width="190" class="ql-img-displayed-equation " alt="\[V_{DC}=V_{max}-\frac{1}{2} V_{rpp}\]" title="Rendered by QuickLaTeX.com"/> <!-- /wp:paragraph --> <!-- wp:paragraph {"className":"redbox"} --> When a bridge rectifier is used, the frequency doubles and the ripple voltage and peak supply current are halved. <!-- /wp:paragraph --> <!-- wp:list {"className":"summary"} --> <ul class="summary"><li>A capacitor is device that stores charge:

Q=CV</li><li>The energy stored in a capacitor is:E=\frac{1}{2}QV=\frac{Q^2}{C}=\frac{1}{2}V^2C</li><li>The time constant of a capacitor network is given as\tau=CR$
  • 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