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[
Producing wound components]
[ Air coils]
[ Power loss in wound components]
[The force produced by a magnetic field]
[ Faraday's law]
Quantity | Quantity symbol |
---|---|
coercivity | H_{c} |
effective area | A_{e} |
effective permeability | µ_{e} |
induced voltage | E |
inductance factor | A_{l} |
magnetic flux | |
magnetomotive force | F_{m} |
permeability of vacuum | µ_{0} |
reluctance | R_{m} |
Quantity | Quantity symbol |
---|---|
core factor | Σl/A |
effective length | l_{e} |
flux linkage | λ |
inductance | L |
magnetic field strength | H |
magnetic flux density | B |
permeability | µ |
relative permeability | µ_{r} |
remnance | B_{r} |
Quantity | CGS Unit | k | SI Unit |
---|---|---|---|
magnetomotive force | gilberts | 0.4π | ampere-turns |
magnetic field strength | oersteds | 79.6 | amps per metre |
magnetic flux density | gauss | 10^{-4} | tesla |
magnetic flux | maxwells | 10^{-8} | weber |
Converting formulae between the two systems requires care. You may also see flux being specified in 'lines' - this is synonymous with maxwells.
Parameter | Symbol | Value |
---|---|---|
Effective magnetic path length | l_{e} | 27.6×10^{-3} m |
Effective core area | A_{e} | 19.4×10^{-6} m^{2} |
Relative permeability | µ_{r} | 2490 |
Inductance factor | A_{l} | 2200 nH |
saturation flux density | B_{sat} | 360 mT |
Let's take a worked example to find the inductance for the winding shown with just two turns (N=2).
Σl/A = l_{e} / A_{e} = 27.6×10^{-3} / 19.4×10^{-6} = 1420 m^{-1}µ = µ_{0} × µ_{r} = 1.257×10^{-6} × 2490 = 3.13×10^{-3} Hm^{-1}
R_{m} = (Σl/A) / µ = 1420 / 3.13×10^{-3} = 4.55×10^{5} A-t Wb^{-1}
A_{l} = 10^{9} / R_{m} = 10^{9} / 4.55×10^{5} = 2200 nH per turn^{2}
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Quantity name | coercivity |
---|---|
Quantity symbol | H_{c} |
Unit name | amps per metre |
Unit abbreviation | A m^{-1} |
Fundamental units | A m^{-1} |
Coercivity is the field strength which must be applied in order to reduce (or coerce) a remnant flux to zero. Materials with high coercivity (such as those used for permanant magnets) are sometimes called hard. Conversely, materials with low coercivity (such as those used for transformers) are called soft. See the section on hysteresis for more details.
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Quantity name | core factor or geometric core constant |
---|---|
Quantity symbol | Σl/A |
Unit name | per metre |
Unit abbreviation | m^{-1} |
Fundamental units | m^{-1} |
The idea of core factor is, apart from adding to the jargon :-( , to encapsulate in one figure the contribution to core reluctance made by the size and shape of the core. It is usually quoted in the data sheet but it is calculated as -
Σl/A = l_{e} / A_{e} m^{-1}So for our example toroid we find -
Σl/A = 27.6×10^{-3} / 19.4×10^{-6} = 1420 m^{-1}If the core factor is specified in millimetres^{-1} then multiply by 1000 before using it in the formula for reluctance.
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Quantity name | effective Area |
---|---|
Quantity symbol | A_{e} |
Unit name | square metres |
Unit abbreviation | m^{2} |
Fundamental units | m^{2} |
The 'effective area' of a core represents the cross sectional area of one of its limbs. Usually this corresponds closely to the physical dimensions of the core but because flux may not be distributed completely evenly the manufacturer will specify a value for A_{e} which reflects this.
In the example toroid the area could be determined approximately as -
A_{e} = 6.3 × ((12.7 - 6.3) / 2) = 20.2 mm^{2}However, because the flux concentrates where the path length is shorter it is better to use the value stated by the manufacturer - 19.4 mm^{2}. For the simple toroidal shape A_{e} is calculated as
A_{e} = h×ln^{2}(R_{2}/R_{1}) / (1/R_{1}-1/R_{2}) m^{2}This assumes square edges to the toroid; real ones are often rounded.
Important: effective area is usually quoted in millimetres squared. Many formulae in data books implicitly assume that a numerical value in mm^{2} be used. Other books, and these notes, assume metres squared.
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Quantity name | effective length |
---|---|
Quantity symbol | l_{e} |
Unit name | metres |
Unit abbreviation | m |
Fundamental units | m |
The 'effective length' of a core is a measure of the distance which flux lines travel in making a complete circuit of it. Usually this corresponds closely to the physical dimensions of the core but because flux has a tendency to concentrate on the inside corners of the path the manufacturer will specify a value for l_{e} which reflects this.
In the toroid example the path length could be determined approximately as -
l_{e} = × (12.7 + 6.3) / 2 = 29.8 mmHowever, because the flux concentrates where the path length is shorter it is better to use the value stated by the manufacturer - 27.6 mm. For a simple toroidal shape l_{e} is calculated as
l_{e} = 2×ln(R_{2}/R_{1})/(1/R_{1}-1/R_{2}) mAnother common core type, the EE, is shown in Fig: EEE. The red line represents the shortest path which a flux line could take to go round the core. The green line is the longest. Shown in blue is a path whose length is that of the short path plus four sectors whose radius is sufficient to take the path mid-way down the limbs.
l_{e} = 2(3.8+1.2)+((2.63-1.2)/2) mmThis is all a bit approximate; but bear in mind that since manufacturing tolerances on permeability are often 25% there isn't much point in being more exact.
Important: effective length is usually quoted in millimetres. Many formulae in data books implicitly assume that a numerical value in mm be used. Other books, and these notes, assume metres.
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Quantity name | magnetomotive force |
---|---|
Quantity symbol | F_{m} or ℑ |
Unit name | amperes |
Unit abbreviation | A |
Fundamental units | A |
Quantity | Unit | Formula |
---|---|---|
Magnetomotive force | amperes | F_{m} = H × l_{e} |
Electromotive force | volts | E = D (Electric field strength) × l (distance) |
MMF can be thought of as the magnetic equivalent of electromotive force. Calculate MMF as the product of the current flowing in a coil and the number of turns it has-
F_{m} = I × N ampere turnsThe units of MMF are often stated as ampere turns (A-t) because of this. In the example toroid core-
F_{m} = 0.25 × 2 = 0.5 ampere turnsDon't confuse MMF with magnetic field strength. As an analogy think of the plates of a capacitor with a certain electromotive force (EMF) between them. How high the electric field strength is will depend on the distance between the plates. Similarly, the magnetic field strength in a transformer core depends not just on the MMF but also on the distance that the flux must travel round it.
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Quantity name | magnetic field strength |
---|---|
Quantity symbol | H |
Unit name | amperes per metre |
Unit abbreviation | A m^{-1} |
Fundamental units | A m^{-1} |
Quantity | Unit | Formula |
---|---|---|
Magnetic field strength | amperes per metre | H = F_{m}/l_{e} |
Electric field strength | volts per metre | = E/d |
Whenever current flows it is always accompanied by a magnetic field. The strength, or intensity, of this field is exactly proportional to the amount of current but inversely proportional to the distance from the conductor.
Magnetic field strength is analogous to electric field strength. Where an electric field is set up between two plates separated by a distance d and having a potential difference, E, between them the electric field is given by -
= E / d V m^{-1}Similarly, magnetic field strength is -
H = F_{m} / l_{e} A m^{-1}Where l_{e} is the length of the field line.
In the example the field strength is then -
H = 0.5 / 27.6×10^{-3} = 18.1 A m^{-1}
Do not confuse magnetic field strength with flux density, B. This is closely related to field strength but depends also on the material within the field.
H = B/µFlux also emerges from a permanent magnet even when there are no wires about to impose a field.
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Quantity name | magnetic flux |
---|---|
Quantity symbol | Φ |
Unit name | webers |
Unit abbreviation | Wb |
Fundamental units | kg m^{2} s^{-2} A^{-1} |
We talk of magnetism in terms of lines of force or flow or flux. Although the Latin fluxus, means 'flow' the English word is older and unrelated. Flux, then, is a measure of the number of these lines - the total amount of magnetism.
You can calculate flux as the time integral of the voltage on a winding divided by the number of turns, N.
= (1/N)V.dt webersThis is one form of Faraday's law. If a constant voltage is applied for a time T then this boils down to -
= V × T / N WbHow much simpler can the maths get? Because of this relationship flux is sometimes specified as volt seconds.
Quantity | Unit | Formula |
---|---|---|
Magnetic flux | volt seconds | = V × T |
Electric charge | amp seconds (= coulombs) | Q = I × T |
Although as shown above flux corresponds in physical terms most closely to electric charge, you may find it easiest to envisage flux flowing round a core in the way that current flows round a circuit. When a given voltage is applied across a component with a known resistance then a specific current will flow. Similarly, application of a given magnetomotive force across a ferromagnetic component with a known reluctance results in a specific amount of magnetic flux -
= F_{m} / R_{m} webersLastly, flux can also be derived by knowing the both the magnetic flux density and the area over which it applies:
= A_{e}×B webers
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Quantity name | Magnetic flux density |
---|---|
Quantity symbol | B |
Unit name | teslas |
Unit abbreviation | T |
Fundamental units | kg s^{-2} A^{-1} |
Quantity | Unit | Formula |
---|---|---|
Magnetic flux density | webers per metre^{2} | B = /Area |
Electric flux density | coulombs per metre^{2} | D = C/Area |
Flux density is simply the total flux divided by the cross sectional area of the part through which it flows -
B = / A_{e} teslasThus 1 weber per square metre = 1 tesla.
B = µ × H teslasSo for the example core -
B = 3.13×10^{-3} × 18.1 = 0.0567 teslas
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Quantity name | flux linkage |
---|---|
Quantity symbol | λ |
Unit name | weber-turns |
Unit abbreviation | Wb-t |
Fundamental units | kg m^{2} s^{-2} A^{-1} |
In an ideal inductor the flux generated by one of its turns would be encircle all the other other turns. Real coils come close to this ideal when the cross sectional dimensions of the winding are small compared with its diameter, or if a high permeability core guides the flux right the way round. In longer air-core coils the situation is likely to be nearer to that shown in Fig.TFK: Here we see that the flux density decreases towards the ends of the coil as some flux takes a 'short cut' bypassing the outer turns. Let's assume that the current into the coil is 5 amperes and that each flux line represents 7 mWb.
The central three turns all 'link' four lines of flux: 28 mWb.
The two outer turns link just two lines of flux: 14 mWb.
We can calculate the total 'flux linkage' for the coil as:
λ = 3×28 + 2×14 = 112 mWb-tThe usefulness of this result is that it enables us to calculate the total self inductance of the coil, L:
L = λ/I = 112/5 = 22.4 mHIn general, where an ideal coil is assumed, you see expressions involving N× or N×d/dt. For greater accuracy you substitute λ or dλ/dt.
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µ = B/H Hm^{-1}This is not the same as the slope of a tangent to the curve, unless it is 'incremental permeability' which is specifically being refered to.
The figure above is an initial magnetisation curve because it starts from an unmagnetised sample and shows how the flux increases as the field strength is increased. You can identify four distinct regions in most such curves. These can be explained in terms of changes to the material's magnetic 'domains':
There are two other common types of magnetization curve: the hysteresis loop and the normal magnetization curve. A circuit you can use to plot magnetization is described here.
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Quantity name | permeability |
---|---|
Quantity symbol | µ |
Unit name | henrys per metre |
Unit abbreviation | H m^{-1} |
Fundamental units | kg m s^{-2} A^{-2} |
Quantity | Unit | Formula |
---|---|---|
Permeability | henrys per metre | µ = L/d |
Permittivity | farads per metre | = C/d |
Permeability is defined as the ratio of flux density to field strength.
µ = B / H Hm^{-1}Permeability is determined by the type of material within the magnetic field. Think of permeability as a sort of 'resistance to magnetic flux'; just as those materials with high conductivity let electric current through easily so different materials allow flux through more easily than others. The analogy is not that good because for most materials (or no material at all, i.e. a vacuum - 'free space') the permeability is non-zero and is called µ_{0} -
µ_{0} = 4 ×10^{-7} = 1.257×10^{-6} henrys per metre.However, for iron, nickel, cobalt and manganese, or their compounds, the permeability may be significantly greater. These materials are classed as ferromagnetic. Using a ferromagnetic core will result in a higher value for µ. Iron has about 3.5 times the permeability of nickel.
It is quite unusual to see a straight or absolute permeability figure in data sheets. It's more common for permeability to be expressed as the ratio over µ_{0}. The factor by which permeability increases above µ_{0} is called the relative permeability, µ_{r}.
µ = µ_{0} × µ_{r} Hm^{-1}
So for the toroid example permeability is then:
µ = 1.257×10^{-6} × 2490 = 3.13×10^{-3} Hm^{-1}Many authors simply say "permeability" and leave you to infer that they mean relative permeability. If a figure greater than 1.0 is quoted then you can be almost certain it is µ_{r}. If you use a core with a high value of permeability then fewer turns will be required to produce a coil with a given value of inductance.
Material | µ_{r} | Application |
---|---|---|
Ferrite U 60 | 8 | UHF chokes |
Ferrite M33 | 750 | Resonant circuit RM cores |
Nickel | 2000 | - |
Ferrite N41 | 3000 | Power circuits |
Ferrite T38 | 10000 | Broadband transformers |
Silicon GO steel | 40000 | Dynamos, mains transformers |
supermalloy | 1000000 | Recording heads |
Note that, unlike µ_{0}, µ_{r} is not constant and changes with flux density. Also, if the temperature is increased from, say, 20 to 80 centigrade then a typical ferrite can suffer a 25% drop in permeability. This is a big problem in high-Q tuned circuits.
Another factor, with steel cores especially, is the microstructure, in particular grain orientation. Silicon steel is often made with the grains oriented along the laminations (rather than alowing them to lie randomly) giving increased µ. We call such material anisotropic.
Other types of permeability may be seen in data sheets. A common one is effective permeability, µ_{e}. This is often quoted when a core has an air gap which causes an apparent reduction in µ. Another is initial permeability, µ_{i} which is the relative permeability measured at low values of B (below 0.1T). The maximum value for µ in a material is frequently a factor 5 or more above its initial value.
Before you pull any value of µ from a data sheet ask yourself if it is appropriate for your material under the actual conditions under which you use it. Finally, if you do not know the permeabilty of your core then build a simple circuit to measure it.
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Quantity name | Inductance |
---|---|
Quantity symbol | L |
Unit name | henrys |
Unit abbreviation | H |
Fundamental units | kg m^{2} s^{-2} A^{-2} |
Quantity | Unit | Formula |
---|---|---|
Inductance | webers per amp | L = /I |
Capacitance | coulombs per volt | C = Q/V |
Inductance is the rate of change of flux with current -
L = N × d/dI henrys.Where I is the current flowing in the winding. If the material permeability is constant then the relation between flux and current is linear so:
L = N × /I henrys.In practice, where a high permeability core is used, inductance is usually determined knowing the number of turns in a coil, N, and the A_{l} value specified by the manufacturer for the core -
L = A_{l} × N^{2} nanohenrysInductance for the toroid example is then:
L = 2200 × 10^{-9} × 2^{2} = 8.8 μHIf there is no ferromagnetic core so µ_{r} is 1.0 (the coil is 'air cored') then a variety of formulae are available to estimate the inductance. The correct one to use depends upon
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Quantity name | inductance factor |
---|---|
Quantity symbol | A_{l} |
Unit name | nanohenrys |
Unit abbreviation | nH |
Fundamental units | kg m^{2} s^{-2} A^{-2} |
A_{l} is usually called the inductance factor. If you know the inductance factor then you can multiply by the square of the number of turns to find the inductance. In our example core A_{l} = 2200, so the inductance is -
L = 2200 × 10^{-9} × 2^{2} = 8800nH = 8.8 µ HThe core manufacturer may directly specify an A_{l} value, but frequently you must derive it via the reluctance, R_{m}. The advantage of this is that only one set of data need be provided to cover a range of cores having identical dimensions but fabricated using materials having different permeabilities.
A_{l} = 10^{9} / R_{m} nH per turn^{2}So, for our example toroid core -
A_{l} = 10^{9} / 4.55×10^{5} = 2200 nH per turn^{2}If you have no data on the core at all then wind ten turns of wire onto it and measure the inductance (in henrys) using an inductance meter. The A_{l} value will be 10^{7} times this reading.
A_{l} values are, like permeability, a non-linear function of flux. The quoted values are usually measured at low (<0.1mT) flux.
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Quantity name | reluctance |
---|---|
Quantity symbol | R_{m} or ℜ |
Unit name | ampere-turns per weber |
Unit abbreviation | A-t Wb^{-1} |
Fundamental units | A^{2} s^{2} kg^{-1} m^{-2} |
Reluctance is the ratio of mmf to flux -
R_{m} = F_{m} / ampere-turns per weberReluctance in a magnetic circuit corresponds to resistance in an electric circuit. It is proportional to the core factor, Σl/A, but inversely proportional to permeability -
R_{m} = (Σl/A) / µ A-t Wb^{-1}Take care to use the absolute rather than the relative permeability here. So for the toroid example reluctance is then:
R_{m} = 1420 / 3.13×10^{-3} = 4.55×10^{5} A-t Wb^{-1}Although it can be a useful concept when analysing series or parallel combinations of magnetic components reluctance is, like permeability, non-linear and must be used carefully.
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Quantity name | remnance |
---|---|
Quantity symbol | B_{r} |
Unit name | teslas |
Unit abbreviation | T |
Fundamental units | kg s^{-2} A^{-1} |
Remnance (or remanance) is the flux density which remains in a magnetic material when any externally applied field is removed (H = 0). For materials used in permanant magnets you usually need a high value of remnance. For transformers you need low remnance. See the section on hysteresis for more details.
Remnance is also what makes possible all magnetic recording technologies;
including the hard disk drive on which this text was stored until you
loaded it into your browser.
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Magnetic quantity | Electric quantity |
---|---|
magnetomotive force | electromotive force (voltage) |
magnetic field strength | electric field strength |
permeability | conductivity |
magnetic flux | current |
magnetic flux density | current density |
reluctance | resistance |
For example, if you have a transformer with a gapped core then imagine that the core and the gap form a series magnetic circuit with the same flux flowing through both reluctance components in an analogous fashion to a series electric circuit in which the same current flows through two resistors -
F_{m} = × (R_{m_gap} + R_{m_core}) ampere-turnscompare
V = I × (R1 + R2) voltsThere's an entire family of formulae which take similar forms in both the electric and magnetic worlds. Kraus lists most of them.
All analogies break down when pushed too far. This one falls rather quickly if you realise that curent flowing through a resistor dissipates energy while flux flowing through a reluctance does not. In fact you can ask whether flux is a real physical effect at all (in the way that electron flow is).
The current in a winding produces magneto-motive force -
F_{m} = I × N ampere-turns
The magneto-motive force produces magnetic field -
H = F_{m} / l_{e} ampere-turns per metre
The field produces magnetic flux density -
B = µ × H tesla
Summed over the cross-sectional area of the core this equates to a total flux -
= B × A_{e} webers
The flux produces induced voltage (EMF) -
E = N × d/dt volts
L = µ × A_{e} × N^{2} / l_{e} henrysI give the fundamental units for all the quantities in this equation; enabling thrill-seekers to make a dimensional analysis verifying that it is consistent. Right, so then our five step relationship between current and EMF boils down to:
E = L × dI/dt volts
You may be about to complain that you know the EMF on your winding but don't know the current in it. The answer is that the process then works in reverse - the current will build up until the induced voltage is sufficient to oppose the applied voltage. You can find out more by looking at Faraday's law.
How do you take into account the presence of the secondary windings in a transformer? One way is to take the first four steps of the sequence above and apply them separately to each winding (whether primary or secondary). The arithmetic sum over all windings gives total core flux. From the time rate of change of flux you then have the induced voltage in each winding (since you also know the number of turns for each). There are less tedious methods of analysing transformer operation which you would probably do better using. But they are another story.
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