- What is electricity?
- Resistance, Conductance & Ohms Law
- Practical Resistors
- Power and Joules Law
- Maximum Power Transfer Theorem
- Series Resistors and Voltage Dividers
- Kirchhoff’s Voltage Law (KVL)
- Parallel Resistors and Current Dividers
- Kirchhoff’s Current Law (KCL)
- Δ to Y Network Conversion
- Y to Δ Network Conversion
- Voltage and Current Sources
- Thevenin’s Theorem
- Norton’s Theorem
- Millman’s Theorem
- Superposition Theorem
- Mesh Current Analysis
- Nodal Analysis
- Capacitance
- Series & Parallel Capacitors
- Practical Capacitors
- Inductors
- Series & Parallel Inductors
- Practical Inductors
Inductors
Electromagnetism – A Quick Introduction
An electromagnetic field is the combination of both an electric field and a magnetic field. Both affect each other and both always exist whenever there is a moving electrical charge. However, both can also exist independently of each other (think static electricity and a bar magnet).
Electricity to Magnetism
Whenever electrical current flows through a conductor, a magnetic field forms around that conductor. This occurs for all current flow no matter the shape or size of the conductor. Magnetic fields, like electric fields, have polarity, but instead of being positive and negative, it is known as North and South. The field is considered to leave via the north pole and return via to the south pole. Like current flow this is just a convention, but stick to it anyway.
The diagram on the left demonstrates something called the Right-Hand Rule in action. Take your right hand and point your thumb in the direction of conventional current flow and curl your fingers around the conductor. Your fingers now point in the direction of the magnetic field.
Note that the magnetic fields are concentric flowing around the conductor. In theory these fields reach out to infinity, but in practice they are attenuated more and more as distance from the source increases.
Note also that the strength of the field is proportional to the current flow.
However. If the conductor is wound in to a coil then the fields align in the same direction and the field goes around the coil end-to-end. Try it. Use the right-hand rule at any point of the coil. You will find that (in the case of the following drawing), the magnetic field always flows upwards through the centre of the coil.
The more loops on the coil, the stronger the magnetic field produced. You may have noticed that the lines of force between adjacent loops are in opposition. To avoid this, the coil should be wound as tightly as possible. Thus assuring that the field merges more cleanly as the direction on the inside and outside of the winding are the same for all loops of the coil.
Further. If a core made from a low magnetic reluctance material, such as iron or ferrite is inserted, the magnetic field is concentrated even more.
To Summarise
- No current = No magnetic field
- More current = stronger magnetic field
- More loops (and /or longer conductor) = stronger magnet
- Lower the magnetic reluctance of the core in the coil = stronger magnet
- The direction of the magnetic field (polarity) can be reversed by reversing the direction of electrical current flow.
Magnetism to Electricity
Faraday’s law of induction states, in a roundabout way, that:
The voltage across a conductor in a magnetic field is proportional to the rate of change of the magnetic field.
The important thing to note here is that it is proportional to the rate of change of the magnetic field. This means that if the field is static, then there is no voltage generated. No matter how strong the field, zero times anything is still zero.
There are four ways that the magnetic field can be changing (from the point of view of the conductor):
- The magnetic field could be static, but the conductor could be moved rapidly through the lines of the field. The faster the conductor is moving, the higher the induced voltage. Which is the basis for one form of generator.
- The conductor could be static, but the magnet could be swept across the conductor. Again, the faster the movement the greater the induced voltage. Which is the basis for another form of generator.
- Actually vary the strength of the magnetic field. This can be achieved by having two stationary coils in close magnetic proximity. One of the coils is driven by a changing electrical current which causes an expanding and contracting magnetic field. The other coil experiences the changing magnetic field and a corresponding, but reversed, electrical current is induced as a result. The faster that the current changes, the greater the induced voltage. This is the basis of both transformers and induction motors (both of which are topics for later).
- Self induction: Consider a single coil driven by a rapidly changing current.
- As the current increases the magnetic field increases.
- As the magnetic field increases it effectively cuts across the very conductor that is creating it generating a corresponding, but opposing, voltage in that conductor. Thus opposing the increase (as predicted by both Lenz’s law and the negative sign in Faraday’s Induction law). This induced, reversed voltage is called back E.M.F.
- Similarly as the driving current decreases, so the magnetic field decreases.
- Again the conductor experiences this resulting change in magnetic field as an induced voltage in opposition to the collapsing field (again as predicted by both Lenz’s law and the negative sign in Faraday’s Induction law).
Inductors
Inductors get their name from the phenomena known as inductance, which Wikipedia defines as:
The tendency of an electrical conductor to oppose a change in the electric current flowing through it.
Physically, inductors are generally a tightly wound single coil of wire which may, or may not, have a low reluctance core (usually iron or ferrite) inserted into the centre of the coil. If a core is present then it may, or may not, be adjustable as to the depth of the insertion.
Schematic symbols similar to these are often used:
Inductors are designed to take advantage of the phenomena known as self inductance. Consider the following simple circuit with a voltage source in series with a 1k resistor and an inductive coil. For the purposes of this explanation we are assuming a perfect inductor (no internal resistance):
Assuming that we start with a steady state where the voltage is a constant 10V and the current 10mA. At this point there is no changing magnetic field and so no induced voltage.
Now the voltage is suddenly doubled and hence the current tries to double as well. At this point the magnetic field starts to expand rapidly.
As the magnetic field expands it induces a back E.M.F. in to the coil, thus slowing the rise in current.
At some point the current reaches the target value and the magnetic field is once again static with no voltage being induced.
This may be easier to visualise as a graph.
- Between t0 and t1 the input is steady resulting in a steady output of the same value.
- At t1 the input rapidly increases. However, the back e.m.f. slows the rise in the output.
- By t2 the output has reached the input value and then remains steady again.
The effect of the inductor on the circuit has been to resist any changes to it’s input and smooth out those changes over time. Note that the change function imposed by the inductor is that of an exponential curve, similar to what we saw with capacitors. However, there are some very important differences:
Let us compare the two by looking at the three stages of operation. For the purposes of this analysis we will assume a perfect capacitor and a perfect inductor (which clearly would not be the case, but we will discuss the impact of that later).
Comparison
The circuit is switched on. The capacitor is initially fully discharged and the inductor has no magnetic field around it.
Capacitor
Maximum current starts to flow, there is zero volts across the device.
Inductor
Maximum voltage appears across the device, zero current is flowing.
As charge builds up on the plates the voltage across the device increases and the current decreases.
As current starts to change the magnetic field starts to grow rapidly inducing back e.m.f. which reduces the voltage across the device and chokes the rise in current.
The plates finally hold all the charge that they can and the voltage across the device reaches maximum with zero current flowing.
The current through the device finally reaches it’s target. Maximum current is flowing and the voltage across the device is zero.
They are basically mirror images of each other. The capacitor is a voltage driven device which stores it’s energy as physical particles of charge and is only happy while things are changing. Meanwhile, the inductor is a current driven device that has no physical storage. The energy is stored as a magnetic field which can only exist while there is current flowing. It is only happy when nothing is changing and does all it can to oppose any change forced upon it.
Real World Practicalities
So far we have only talked about a hypothetical perfect inductor which has only inductance. In practice an inductor is made from coils of wire. Wire has resistance. Therefore it is impossible to have a zero ohm inductor. All real world inductors MUST be modelled with an internal series resistance.
Sizing an Inductor
Inductors have the designation L (I have no idea why) and the unit of measure is the henry (H), which is an absolutely enormous size so you will more commonly see millihenry (mH) or microhenry (µH). The henry is defined as:
\(L~{(1H)} = \frac{{1V}}{1 \frac{{A}}{s}} \)
One henry is the amount of inductance that causes one volt to be generated by a one ampere per second change in current.
The formula which defines how to build an inductor is:
\(L = \frac{N^2 . {Area} . \mu_r . \mu_0}{{length}} \\\) \(\text{Where:} \\\) \(L = \text{the inductance in henrys}~(H) \\\) \(N = \text{the number of turns of the coil} \\\) \({Area} = \text{the cross sectional area of the coil in }~m^2 \\\) \(\mu_r = \text{the relative permeability of the core material – a unit less number} \\\) \(\mu_0 = \text{the permeability of free space} = 4 \pi \times 10^{-7}~ \frac{W_b}{A.m} \\\) \({length} = \text{the length of the coil (not the wire that makes the coil) in}~m \\\)
Example 1 – 6mm x 25mm with 100 turns on an Air Core
\(L = \frac{N^2 . {Area} . \mu_r . \mu_0}{{length}} \\\) \({Area} = \pi . r^2 = \pi . (3 \times 10^{-3})^2 \approx 28.3 \times 10^{-6}~m^2 \\\) \(\mu_r = 1~ \text{as it is an air core} \\\) \({length} = 25 \times 10^{-3}~m \\\) \(L = \frac{100^2 . (28.3 \times 10^{-6}) . (4 \pi \times 10^{-7})}{25 \times 10^-{3}} = 14.2 \mu H \)
Example 2 – 6mm x 25mm with 100 turns on an Iron Core
\(L = \frac{N^2 . {Area} . \mu_r . \mu_0}{{length}} \\\) \({Area} = \pi . r^2 = \pi . (3 \times 10^{-3})^2 \approx 28.3 \times 10^{-6}~m^2 \\\) \(\mu_r = 5000~ \text{according to Wikipedia assuming 99.8% pure iron} \\\) \({length} = 25 \times 10^{-3}~m \\\) \(L = \frac{100^2 . (28.3 \times 10^{-6}) . 5000 . (4 \pi \times 10^{-7})}{25 \times 10^-{3}} = 71.1mH \)