Plotting B-H Curves
CONTENTS Experiments In Plotting Magnetization Curves ........ ................ ................. ................. ................ ................ ................. ................. ........... ... 2 Core Windings ............................................................................................................. 2 Circuit diagram ............................................................................................................ 2 Circuit Circui t Construction .................... ........................................... ............................................. ............................................ ..................................... ............... 3 Principle Of Operation ................................................................................................... 3 Finding H .................................................................................................................... 3 Finding B .................................................................................................................... 4 Plotting B against H ...................................................................................................... 4 Calculating The Permeability Of A Core ........................................................................... 5 Results Using Op-Amp Integrator And Test Core .................... ............................. ................. ................ ................ ................. ......... 7 Estimating Hysteresis Losses ......................................................................................... 8 B-H Curve Demonstration Unit ........................................................................................ 10 Triangle Wave Generator ............................................................................................ 10 Coil Driver ................... ......................................... ............................................ ............................................ ............................................ ............................ ...... 10 Integrator ................................................................................................................. 11 Core ......................................................................................................................... 11 Circuit Construction .................................................................................................... 12 Complete Circuit ........................................................................................................... 13 Issues ..................... ........................................... ............................................ ............................................ ............................................. ................................... ............ 14
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Plotting B-H Curves Experiments Experiments In Plotting Magnetization Curves The circuit discussed can be used to measure the B-H characteristics of certain ferromagnetic components. The core should be a toroid or other shape having a closed magnetic path. A “proof of concept” circuit was constructed and used to plot pl ot hysteresis loops with a rectangular core about 15mm by 10mm with a square cross section about 3mm by 3mm, giving a mean magnetic path length of about 50mm and a core volume of about 450mm 3
The required plot is of B against H (Flux Density vs Magnetic Field Strength) – typically of the form shown on the right
Core Windings Two windings on the core are required (a primary, and a secondary). secondary). The experimental core was wound with as many turns on the primary as would fit (to ensure the core could be taken to saturation with a reasonably small primary current). “Thick” wire (29 SWG – approx. 0.35 mm ∅) was used for the primary to ensure minimal primary resistance and so low I2R losses (the permeability for typi typical cal ferrites being significantly temperature dependent). The primary and secondary were bifilar wound to ensure good coupling (and for expedience…..).
Circuit diagram
R1 = 5 x 10 Ω resistors in parallel = 2 Ω, 1.25 W R2 = 100 KΩ C1 = 470 nF L1: Bifilar windings, 60T. Primary wound with thickest wire practical. Secondary – thickness not important, so use thin wire so the secondary occupies the minimal volume. Signal source: 5 KHz, variable amplitude
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Plotting B-H Curves Circuit Construction The test circuit was constructed on Vero b oard:
Core under test
R1
R2 C1
Principle Of Operation The core is energized through the primary using using an AC source. The wave shape is not important – a sinusoidal wave is most easily obtained, but a triangular wave is i s preferable for even display brightness (the waveform is used to drive the X axis of an oscilloscope in X-Y mode so a sine wave will dwell at the extremes of the sweep).
Finding H The current in the primary winding is monitored by measuring the voltage developed across R1: I p = V R1 / R1
Equation 1
The resulting magnetic field strength in the core is defined as: H = N p x I p / L
Eq. 2
where:
I p = Primary current N p = Number of turns L = Mean magnetic path length
From Eq. 1: V R1 ∝ I p So: V R1 ∝ H
Proportionality 1
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Plotting B-H Curves Finding B From Faraday’s Law: V s = N s × dΦ/dt dΦ/dt
Eq. 3
where:
dΦ/dt = Φ/dt = Rate of change of flux N s = Number of turns in secondary
Integrating w.r.t. time:
∫
V s dt = N s × Φ
Eq. 4
The RC network R2 C1 gives an approximate integration of the voltage on the secondary assuming the time constant R2 x C1 i s long compared with the signal frequency (so ΔV c is small resulting in the voltage across R2 being approximately constant and so making the charging current proportional to V s ). So the capacitor voltage is proportional to the flux: Vc ∝ Φ
Prop. 2
As flux density (B) is given by: B=Φ/A
Eq. 5
V c ∝ B
Prop. 3
where:
A = cross sectional area of core
So
Plotting B against H The key relationships are the proportionalities Prop. 1 and Prop. 3: V R1 ∝ H V c ∝ B So plotting V C against V R1 on an oscilloscope in X-Y mode will plot the B-H curve for the core.
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Plotting B-H Curves Below are plots produced by the test circuit:
Core not saturating
Core saturating
Both plots: X axis 50mV/div Y axis: 5mV/div Calculating The Permeability Of A Core To make accurate calculations of the permeability of the core, a more accurate integrator is is required. The standard op-amp integrator circuit can can be used as shown below, R3 R3 goes some way towards reducing drift, but offset null circuitry would be required to remove completely the tendency for the integrator to accumulate the effects of the op-amp input voltage and current offsets.
Op-Amp output, V 0 , is the inverse to the integral of the input voltage: V o = -
∫
V s /(C1 R2) dt
R1 = 5 x 10 Ω resistors in parallel = 2 Ω R2 = 1 KΩ R3 = 10 MΩ C1 = 0.47 µF To calculate the permeability of the core, the physical dimensions of the core and the turns on the secondary winding must be known. For the experimental core: N p = 60 N s = 60 Mean Magnetic Path Length, L = 50 mm = 0.05 m Core cross sectional area, A = 3 x 3 = 9 mm2 = 9x10-6 m2
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Plotting B-H Curves V c and V R1 are read directly from the oscilloscope and I p calculated from Eq 1 Referring to Eq. 2, values for N p , I p and L can be entered allowing a value for H to be calculated: Eq. 6 H = (N p V R1 ) / (L R1) Where V R1 , N P , L and R1 are all known
The output from the integrator is:
∫
V o = -
V s /(C1 R2) dt
Taking constants outside the integral: V o = - 1/(C1 R2) x
∫
V s dt
But from Eq. 4:
∫
V s dt = N s Φ
So V o = - (N s Φ) / (C1 R2)
Rearranging (and ignoring the minu s sign) gives: Φ = (V o C1 R2) / N s
Flux density is given by: B = Φ / A So B = (V o C1 R2) / (N s A)
Where V o , C1, R2, N s and A are all known
Eq. 7
With the two values provided by Eq. 6 and Eq. 7, the permeability can be calculated: µ=B/H
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Plotting B-H Curves The core losses can also be calculated (approximately) from the area enclosed by the h ysteresis loop. Draw a rectangle round the hysteresis loop – the energy represented by the rectangle (E R ) can be calculated from the area knowing the scale factor from Eq. 6 and Eq. 7. Measure / calculate the area of the hysteresis loop – the energy represented by the loop (E L ) is: E L = E R x (Area of loop) / (Area of rectangle)
Results Using Op-Amp Integrator And Test Core The op-amp integrator circuit was constructed constructed and the tests tests on the core repeated. The following image shows the results:
X axis: 20 mV / Div Y axis: 0.2 V / Div Source Frequency: 250 Hz
Core details (approx.):
Circuit values:
A = 9x10-6 m2 L = 0.05 m N P = 60 N S = 60
R1 = 2 Ω R2 = 1 KΩ C1 = 470 nF
Substituting the above values in to Eq. 6 and 7: H = (N p V R1 ) / (L R1) H = (60 x 0.12) / (0.05 x 2) H = 72 A/m
Eq. 8
B = (V o C1 R2) / (N s A) B = (0.96 x 470 x 10 -9 x 103) / (60 x 9 x 10 -6) B = 0.84 Tesla
Eq. 9
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Voltage measurements: X axis: 120 mV (V R1 ) Y axis: 0.96 V (V o )
Plotting B-H Curves Now the permeability can be calculated: µ=B/H µ = 0.84/72 H m-1 µ = 11.7 mH m-1
But
µ = µ 0 µ r
So
µ r = (11.7 x 10-3) / µ 0 µ r = (11.7 x 10-3) / (4 π x 10-7) µr
≈ 9300
Estimating Hysteresis Losses 1) The picture of the oscilloscope oscilloscope trace was imported to a graphics package.
2)
The "trace" selected and flood filled with white to give a clearly delineated outline.
3) The area enclosed in the outline flood filled white.
4)
The area outside the outline flood filled black and the image cropped to just enclose the curve.
Having achieved a clear, high contrast shape, a graphics analysis package will give the total number of pixels in the image, and and the total number of pixels coloured white. This task is outside the ability of most "painting" type graphics packages so something a little li ttle more specialised may be needed. ADI (Analyzing Digital Images) Images) is free and does the job very easily. easily. ADI returned a "white to whole picture" ratio of 11.7% for the captured image above.
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Plotting B-H Curves The axes of the plot have units B and H. In fundamental units: B
= (Newton x second) / (Coulomb x metre)
H
= Amps / metre = Coulomb / (second x metre)
So an area on the plot has units: B x H = (N x s x C) / (C x s x m 2) = N / m2 = Nm / m 3 B x H = J / m3
So substituting the values for H and B obtained in Eq. 8 and 9: Energy density for rectangle enclosing B-H curve = 0.84 x 72 J/m3 Energy density = 60.5 60.5 J/m3
Eq. 10
Of this rectangle, 11.7% was enclosed by the hysteresis curve, therefore: Energy density of losses per cycle = 0.117 x 60.5 J/m 3 Multiplying the energy loss density by the core volume gives the actual core loss per cycle: Core loss per cycle = 0.05 x 9 x 10--6 x 60.5 J Core loss per cycle = 27.2 x 10 --6
J
As frequency of signal was 250 Hz: Core loss per second = 250 x 27.2 x 10 --6 J/s Power loss = 6.8 mW
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Plotting B-H Curves B-H Curve Demonstration Unit Having shown the validity of the design d esign concept, a completely self-contained demonstration unit was designed with sufficiently robust construction to allow classroom use. The demonstration unit includes a signal generator (to create the primary current waveform) and power amplifier to drive the primary winding on the test test core. The primary current amplitude is adjustable to allow demonstration of the on-set of core saturation etc. The secondary winding output is fed to an op-amp integrator. The circuits run from a dual power power supply regulated at plus and minus five volts. The regulated supplies help to ensure minimum dri ft in the integrator circuits.
Triangle Wave Generator The primary current triangular waveform is generated using the standard inverting integrator circuit (U2, C1, R2) and a non-inverting Schmitt Trigger (U1, R1, R3). The output swing of the Schmitt trigger (the integration voltage) is not fully railto-rail which impacts on the output frequency. The frequency of of o oscillation scillation of the circuit as built was 485Hz.
Coil Driver The output of the triangle wave generator is fed to a small potentiometer (RV1) which allows adjustment of the signal level. The signal is AC coupled / DC restored by C2/R4 and fed to the power amplifier which has a voltage gain of 15 (determined by R5 and R6) and current gain is provided by transistors Q1 and Q2.
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Plotting B-H Curves Integrator The values of R8 and C3 were chosen to provide a suitably large integrated output (without driving the op-amp U4 in to saturation) when the core was driven in to saturation. RV2 and R12 provide off-set null and prevent drift of the integrator output.
Components R7, R8 and C3 are used in the calculation of B and H. Therefore these values need to be known accurately. accurately. The components used were measured and the following values recorded: R7 = 10.6 Ω R8 = 98.5 KΩ C3 = 11.6 nF
Core The core used had the following dimensions: OD: ID: Thickness: Height: Area: Mean Path Length: Primary turns: Secondary turns:
15.6 mm 9.8 mm 2.9 mm 6.78 mm 19.7 mm 40 mm 75 75
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Plotting B-H Curves Circuit Construction
B o u t p u t
H o u t p u t
0 V
O u t p u t s :
S c h m i t t t r i g g e r
T r i a n g l e w a v e
P o w e r I n p u t V o l t a g e R e g u l a t i o n
“ B ” I n t e g r a t o r
T r i a n g l e w a v e i n t e g r a t o r
“ B ” I n t e g r a t o r o f f s e t n u l l
C u r r e n t b o o s t e r s t a g e
T e s t c o r e
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P r i m a r y c u r r e n t l e v e l a d j u s t
Plotting B-H Curves Complete Circuit
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Plotting B-H Curves Issues The following issues will need to be considered: 1) The oscilloscope image was a photograph taken on a hand held, general purpose camera, and so some degree of keystone distortion was present – this will degrade the accuracy of area calculations. It would be preferable to use a proper oscilloscope camera or an oscilloscope that can provide a digital screen capture 2) The value of core power loss was quoted above to two significant figures, but it is difficult to read the oscilloscope to this l evel of accuracy. An oscilloscope with cursors to read values would reduce this problem 3) The thickness of the trace in the images introduces uncertainty in the area measurements. An A3 printout of the curve was produced and the areas measured with a planimeter:
This gave a curve/image ratio of 8.7% 8 .7%
4) The actual values / tolerance of R1, R2 and C1 will impact i mpact on the accuracy of the calculations. The true values of the critical components can be measured. 5) The core dimensions were estimated. The real core dimensions would be available through manufacturer’s data.
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