Heaviside's Equations
Symbol Quantity Name Units (S.I.) Electric field V/m E B Magnetic field often called "magnetic flux density" Tesla (often use gauss This is the field that measures/does the work in and 1 T = 110 4 gauss) magnetically-based systems H H field or "magnetic intensity" - a convenient A/m ( we sometimes function that can sometimes be computed easily use oersted and 1 A/m and used to find B = .0126 oe.) D Electric displacement field. For most systems DE 2 J Free current, that is, the kind of current you amp/m measure with a meter, density. Integrated over an area, say the cross section of a wire, gives the current I in the wire.
0 Permeability of free space - a defined quantity Henry/m 410 7
0 Permittivity of free space or dielectric constant of farad/m free space, a derived quantity 8.85 1012 F Force on a charge or segment of current carrying Newton wire
Ampere's Law - the key to motors and transformer design: zero D HJ+ HdlJdSI ∇ enc t CS Faraday's Law - the key to generating electricity:
B ∇EEdlBdS ttCS
Flux conservation - there is no magnetic equivalent to the charge of an electron: ∇BBdS 0 0 S Conservation of charge - Kirchoff's current law:
∇JJdS0 0 t S
Gauss's Law - DE = DdSdV ∇∇rfree0 SV Lorentz force equation - FqEqvB on a charged particle and FdiB where di idl and i is the current in an infinitesimal length of wire dl.
We are in the business of generating, distributing and using electrical energy so it may be worthwhile to see how energy and the fundamental electromagnetic equations are related to one another.
Poynting's vector (Heaviside independently derived the same result but Poynting published first by a matter of weeks.) This vector is the flow of energy per unit area in watts per square meter in the direction of the vector. To get the total flow, you have to integrate over all the area of the fields. We are primarily interested in this is terms of where the flow takes place and how dense it is for things like high voltage transmission lines and lamp cords. Notice that E has units of Volts/m and H has units of Amps/m so the units of P are Volts x Amps/meter^2 as advertised. PEH Stored energy - If energy can flow in a set of fields, it can probably be stored there too. In non-magnetic materials or materials with linear permeability, the energy per cubic 112 meter stored in the magnetic field is UBHBM 22 and the energy per cubic meter
1 r 0 2 in the electric fields of a material with linear dielectric constant is UDEEE 22. The total energy is 11 UU ME U22 BHDE However, for most "magnetic" materials, the stuff that is used to get large magnetic fields from small currents at room temperature, the relation of B to H is very complicated and leads to significant losses. We will frequently pretend they are linearly related, writing B r 0 H where r is the dimensionless relative permeability. That permeability is often so large (it ranges from 200 to 200,000 depending on the alloy) that the exact relation for a properly designed system is not important as long as we correct for losses. I will discuss what is going on in these materials in class.
B For the exact non-linear case, the stored energy per cubic meter is UHdB. M 0
Magnetic Circuits Relationships
Magnetic Circuit Variables Electrical Circuit Equivalents Symbol Name Units Symbol Name Flux weber (volt-sec) I Current F Magnetomotive Force ampere-turns V Voltage R Reluctance 1/henries R Resistance P Permeance henries G Conductance 2 lAmC, , Mean magnetic path meters, m , lA, C , length, cross length, cross sectional henries/m sectional area and area and permeability conductivity
Ampere's Law in integral form is behind all the simple magnetic circuit equations and ideas. Also, we generally simplify life by assuming linear permeabilities. The value of the linear relative permeability to use in a calculation is based on the mean of the permeability over the operating range of the device. Ampere's Law: Hdl JdSNI CS
For a very simple magnetic circuit with uniform cross section, the integral reduces to: NI HdlHl NI and the flux density or magnetic field is B r 0 . m C lm The flux is defined as the product of B with the cross sectional area, so
rC0 A BANIC lm Flux is important because voltage is induced in a wire loop cut by a time varying flux as d VN . dt l Define the magnetomotive force as F=NI and the reluctance of a circuit as R m . rC0 A With those definitions, Ampere's law takes on the same algebraic form as Ohm's law: F=ΦR with the correspondence between Ohm's law quantities and the magnetic variables as shown in the table above. Note that the usual relationship of the resistance of l a uniform bar: R exactly matches the expression for reluctance in magnetic AC circuits with the correspondence of conductivity with permeability . Finally the utility of breaking magnetic systems into "circuits" lies in the ways in which such circuits can be analyzed using the equivalent of Kirchoff's laws. The conservation of flux expressed in ∇B 0 gives us an analogy to Kirchoff's current law. As a result, the reluctance of two circuits in series add while two circuits in parallel combine reluctance the same way as for parallel resistors. Magnetic circuits in parallel have permeances that add.
Customarily the people who make and use magnetic materials characterize them in different units than the SI system. The historical roots for this practice are not of interest to us but we do need to know how to go back and forth. The practical result of the use of the Gauss/Oersted unit set is that the slope on the Gauss/Oersted plot is the unitless relative permeability of the material. On datasheets for magnetic steels, the magnetic field B is given in Gauss and the required H field is in units of Oersteds. The SI units are Tesla and ampere-turns/meter respectively. The table below shows the conversion factors between the two sets of units.
Converting Units Related to Magnetic Fields Example From To Multiply By From Result 16,000 Tesla Gauss 10,000 1.6 T Gauss Gauss Tesla 1.00E‐04 5 Kgauss 0.5 T 43 Ampere‐ 0 10 4 10 turns/meter Oersted = 0.0126 100 A‐N/m 1.26 Oe Ampere‐ Oersted turns/meter 79.6 3 Oe 240 A‐N/m