B vs H Curve Tracer Notes ------------------------------ Initial Rev. 10-Dec-2021 Current Rev. 11-Jan-2022 Calculate the Magnetizing Force H: ---------------------------------- From the display on the oscilloscope it is easy to calculate the magnetizing force H. Assume the scope is setup so that zero primary current is in the center of the scope and you want to know the value of H at +- Volts on the X axis. - The current in the primary winding is just the value of the scope X axis voltage at the point of interest divided by the value of the current measurement shunt resistor which is typically 1 Ohm. I pri = Volts X Axis / Shunt Resistor Ohms - The magnetizing force, expressed in Amp-Turns is just this current times the number of turns in the primary winding which is typically 10 turns. H in Amp-Turns pri = I pri x Number Turns pri or H in Amp-Turns pri = (Volts X Axis / Shunt Resistance) x Num Turns pri - When working with a given core, expressing the magnetizing force in Amp-Turns is often exactly what you want for engineering work. The SI value for the magnetizing force is the total net primary current (i.e. the Amp-Turns) divided by the effective length of the magnetic path. The effective magnetic path length for a given core will be given in its data sheet specification and is frequent called the le value for the core. If you don't have a data sheet for your core it's simple to estimate the effective magnetic path length for toroids as the average circumference of the toroid, i.e. pi times the average of its inner and outer diameters. To keep all units SI - express the effective magnetic path length in meters. - The SI value of the magnetizing force in Amps per meter is then calculated by: H in Amps per meter = Amp-Turns / effective magnetic path length or H in Amps per meter = Volts X Axis x Number of Turns pri --------------------------------------------------- Shunt Resistance x effective magnetic path length - Many of the data sheets for commercial cores express the magnetizing force H in the Gaussian cgs unit Oersted. 1 Amp per meter = 4 pi 10E-3 Oersted 1 Amp per meter = 0.012566 Oersted Calculate the Resulting Magnetic Flux Density B in the Core: ------------------------------------------------------------- From the display on the oscilloscope we can also calculate the B field (the magnetic flux density) that is generated in the core by the H field that comes from current flowing in the primary winding. Assume the scope is setup so that the zero of the vertical display is in the center of the scope and you want to know the value of B at +- Volts on the Y axis. 1. Faraday's Law gives us: Vsec = Number of Turns sec x dFlux/dt Vsec is the voltage induced in the N turn secondary by dFlux/dt which is the time rate of change of the total flux in the core. Note that Vsec is proportional to the rate of change of the total flux in the core. Vsec is not proportional to the total flux in the core which is what we want to know. We will integrate over time to recover the value of the total flux in the core. Note that we must Not put a significant load on the secondary winding, i.e. cause a current to flow in the secondary winding, because that current would modify the situation that we are trying to measure. In our case the secondary is working into a 1k Ohm load at the input to the Integrator. In some situations an even higher, e.g. 10k Ohm, load may be more appropriate. 2. The time rate of change of the output of the Integrator is related to its input signal by: dVout/dt = Vin / RC Here R is the input resistor to the opamp summing node (in our case 1k Ohm) and C is the feedback capacitor from the opamp output to its summing node ( 1 uFd in our case). This gives an RC value of 0.001 second. Note that I'm skipping the minus sign that is used to indicate that this circuit is "voltage inverting". 3. With the secondary winding connected to the input of the integrator this gives us an equation with a time derivative on each side: Number of Turns sec dVout/dt = ------------------- x dFlux/dt RC or after integration: Number of Turns sec Vout = ------------------- x Flux RC or: Vout x RC Flux = ------------------- Number of Turns sec We are using SI units for everything going into this equation so the total Flux will be expressed in Webbers. 4. From here the Flux Density B is just the total Flux divided by the cross section area of the core. In many core data sheets this is often called the Ae value of the core, aka the effective cross section area of the core. If you don't have a published specification for the Ae of your core then just estimate it by measuring the core's width and depth with a caliper. Total Flux B = ------------------ Cross Section Area Express the cross section area in meters squared to keep everything in SI which will give you B in Webers per meter sqrd. One Weber per meter sqrd is one Tesla. Volts Y Axis x RC B = ------------------------------------------ Number of Turns sec x Cross Section Area If you need to express the Flux Density in the Gaussian cgs system recall that: 1 Tesla = 10,000 Gauss Calculate the Relative Permeability of the Core: ------------------------------------------------ Finding the Relative Permeability is straight forward except you need to remember that what most the data sheets for magnetic cores call Permeability is actually the Relative Permeability of the core. Relative Permeability is the ratio of the permeability of the core to the permeability of free space. ur x u0 = B/H The relative permeability ur is unit-less because the permeability of free space u0 carries the units Henrys per meter. The permeability of free space is: u0 = 4 pi 10E-7 Henry / meter or u0 = 4 pi 10E-7 Weber / Amp-meter B ur = -------- H x u0 B is in Webers per meter sqrd or Tesla H is in Amps per meter Be smart - keep everything in the SI system. Expect the relative permeability of your core to be a function of the magnetizing force H and for it to fall off sharply once the core starts into saturate. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= One of the important properties of an inductor, either a coil on a magnetic core or an air-core coil, is its inductance. The inductance of a coil is its fundamental property in such relationships as: Energy stored = 1/2 x I_sqrd x L in the inductor Voltage across = L x dI/dt an inductor Time-Constant = L / R Current through = V/L x t (ignoring its DC resistance) an inductor LC circuit = 1 / ( 2 pi x f x sqrt( LC )) resonant frequency L = (N x Flux) / I L is the inductance in Henry I is the current in Amps R is the resistance in Ohms V is the voltage in Volts t is the time in seconds f is frequency in Hz N is the number of turns in the coil Flux is the total magnetic flux generated by the current in the coil In the first two section of this note the data was collected that allows us to calculate the inductance of the inductor using the final equation in the set of 6 equations above. N x Flux in the final equation just above is a quantity that is commonly called the "number of flux linkages". Having an N in the numerator of this equation makes sense. If other parameters are held constant (e.g. the geometry, magnetic core, and current through a coil are held constant) then doubling the number of turns will double the magnetizing force H (and thus double the total magnetic Flux passing through the coil) and it will also double the voltage generated across the coil when there is any change in the Flux passing through the coil, e.g. as caused by a change in the current through the coil. Thus doubling the number of turns will cause a 4x increase in the coil's inductance. Picking up with the equation for total Flux in section two from above: Vout x RC Flux = ------------------- Number of Turns sec and keeping things simple by using the same Number of Turns on both the primary and secondary of the measurement setup we end up with: Vout x RC L = ------------- I pri Look at the dimensions of this equations and things make sense. One Henry is 1 Volt-sec per Amp. This is as expected because Flux has the dimensions of Volt-sec ( 1 Weber is 1 Volt-sec ) and inductance just tells us how much total Flux is generated per Amp of current flowing through the coil. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= How close does a magnetic path length or magnetic cross section calculated based on the specified Outside Diameter, Inside Diameter, and Thickness come to the specified Effective Magnetic Path Length and the specified Effective Magnetic Cross Section ? Here are the results for three cores that I have used and have data sheets for. Specified Effective Specified Effective Magnetic Path Length Magnetic Cross Section --------------------------- --------------------------- Core Calculated from OD,ID,Thick Calculated from OD,ID,Thick ZJ-43813-TC 92.5 % 89.9 % 77894A7 96.7 % 80.6 % 846t250_3e2a 98.4 % 101 % The ZJ-43813-TC and 77894A7 cores are coated. The 846t250_3e2a core is un-coated. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Some interesting background, especially that on units, can be found at: http://info.ee.surrey.ac.uk/Workshop/advice/coils/ http://info.ee.surrey.ac.uk/Workshop/advice/coils/BHCkt/ http://info.ee.surrey.ac.uk/Workshop/advice/coils/unit_systems/ =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The standard "B-H" curve has the magnetizing force H on the X axis and the resulting magnetic flux density B on the Y axis. Thus the name "H-B" curve may make more sense.