plotting b-h curves - in - 2

8/16/2019 Plotting B-H Curves - In - 2

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Plotting B-H Curves

CONTENTS

Experiments In Plotting Magnetization Curves ..................................................................... 2

Core Windings ............................................................................................................. 2

Circuit diagram ............................................................................................................ 2

Circuit 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 In Plotting Magnetization Curves

The circuit discussed can be used to measure the B-H characteristics of certain ferromagneticcomponents.

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 hysteresis loops with a rectangular core about 15mmby 10mm with a square cross section about 3mm by 3mm, giving a mean magnetic path lengthof about 50mm and a core volume of about 450mm3

The required plot is of B against H (Flux Density vsMagnetic Field Strength) – typically of the form shown onthe right

Core Windings

Two windings on the core are required (a primary, and a secondary). The experimental core

was wound with as many turns on the primary as would fit (to ensure the core could be takento 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 typical ferrites being significantlytemperature 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 theminimal volume.

Signal source: 5 KHz, variable amplitude

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Plotting B-H Curves

Circuit Construction

The test circuit was constructed on Vero board:

Principle Of Operation

The core is energized through the primary using an AC source. The wave shape is not

important – a sinusoidal wave is most easily obtained, but a triangular wave is preferable foreven display brightness (the waveform is used to drive the X axis of an oscilloscope in X-Ymode 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:

Ip = VR1 / R1

The resulting magnetic field strength in the core is defined as:

H = Np x Ip / L where: Ip = Primary currentNp = Number of turns

L = Mean magnetic path length

From Eq. 1:

VR1 ∝ Ip

So:

VR1 ∝ H

Equation 1

Eq. 2

Proportionality 1

R1

C1

Core under test

R2

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Plotting B-H Curves

Finding B

From Faraday’s Law:

Vs = Ns × dΦ/dt where: dΦ/dt = Rate of change of flux

Ns = Number of turns in secondary

Integrating w.r.t. time:

Vs dt = Ns × Φ

The RC network R2 C1 gives an approximate integration of the voltage on the secondary

assuming the time constant R2 x C1 is long compared with the signal frequency (so ΔVc is small

resulting in the voltage across R2 being approximately constant and so making the charging

current proportional to Vs).

So the capacitor voltage is proportional to the flux:

Vc ∝ Φ

As flux density (B) is given by:

B = Φ / A where: A = cross sectional area of core

So

Vc ∝ B

Plotting B against H

The key relationships are the proportionalities Prop. 1 and Prop. 3:

VR1 ∝ H

Vc ∝ B

So plotting VC against VR1 on an oscilloscope in X-Y mode will plot the B-H curve for the core.

Eq. 3

Eq. 4

Prop. 2

Prop. 3

Eq. 5

∫

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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 isrequired. The standard op-amp integrator circuit can be used as shown below, R3 goes someway towards reducing drift, but offset null circuitry would be required to remove completely thetendency for the integrator to accumulate the effects of the op-amp input voltage and currentoffsets.

Op-Amp output, V0, is the inverse to the integral ofthe input voltage:

Vo = - Vs /(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 onthe secondary winding must be known.

For the experimental core:

Np = 60

Ns = 60Mean 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

Vc and VR1 are read directly from the oscilloscope and Ip calculated from Eq 1

Referring to Eq. 2, values for Np, Ip and L can be entered allowing a value for H to becalculated:

H = (Np V

R1) / (L R1) Where V

R1, N

P, L and R1 are all known

The output from the integrator is:

Vo = - Vs /(C1 R2) dt

Taking constants outside the integral:

Vo = - 1/(C1 R2) x Vs dt

But from Eq. 4:

Vs dt = Ns Φ

So

Vo = - (Ns Φ) / (C1 R2)

Rearranging (and ignoring the minus sign) gives:

Φ = (Vo C1 R2) / Ns

Flux density is given by:

B = Φ / A

So

B = (Vo C1 R2) / (Ns A) Where Vo, C1, R2, Ns and A are all known

With the two values provided by Eq. 6 and Eq. 7, the permeability can be calculated:

µ = B / H

∫

∫

∫

Eq. 7

Eq. 6

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Plotting B-H Curves

The core losses can also be calculated (approximately) from the area enclosed by the hysteresis

loop.

Draw a rectangle round the hysteresis loop – the energy represented by the rectangle (ER) 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 (EL) is:

EL = ER x (Area of loop) / (Area of rectangle)

Results Using Op-Amp Integrator And Test Core

The op-amp integrator circuit was constructed and the tests on the core repeated. Thefollowing image shows the results:

X axis: 20 mV / Div

Y axis: 0.2 V / Div

Source Frequency: 250 Hz

Core details (approx.):

A = 9x10-6 m2

L = 0.05 m

NP = 60NS = 60

Circuit values:

R1 = 2 Ω

R2 = 1 KΩ

C1 = 470 nF

Voltage measurements:

X axis: 120 mV (VR1)Y axis: 0.96 V (Vo)

Substituting the above values in to Eq. 6 and 7:

H = (Np VR1) / (L R1)

H = (60 x 0.12) / (0.05 x 2)

H = 72 A/m

B = (Vo C1 R2) / (Ns A)

B = (0.96 x 470 x 10-9 x 103) / (60 x 9 x 10-6)

B = 0.84 Tesla

Eq. 8

Eq. 9

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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 m2)= N / m2

= Nm / m3 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/m

3

Energy density = 60.5 J/m3

Of this rectangle, 11.7% was enclosed by the hysteresis curve, therefore:

Energy density of losses per cycle = 0.117 x 60.5 J/m3

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

Eq. 10

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Plotting B-H Curves

B-H Curve Demonstration Unit

Having shown the validity of the design concept, a completely self-contained demonstration unitwas 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 core. The primary currentamplitude 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 supply regulated at plus and minus five volts. The regulatedsupplies help to ensure minimum drift in the integrator circuits.

Triangle Wave Generator

The primary current triangular waveformis 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 rail-

to-rail which impacts on the outputfrequency. The frequency of oscillationof the circuit as built was 485Hz.

Coil Driver

The output of the triangle wave generator is fed to a small potentiometer (RV1) which allowsadjustment of the signal level.

The signal is AC coupled / DC restored by C2/R4 and fed to the power amplifier which has avoltage gain of 15 (determined by R5 and R6) and current gain is provided by transistors Q1and 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 (withoutdriving 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 needto be known accurately. The components used were measured and the following valuesrecorded:

R7 = 10.6 Ω

R8 = 98.5 KΩ

C3 = 11.6 nF

Core

The core used had the following dimensions:

OD: 15.6 mmID: 9.8 mm

Thickness: 2.9 mmHeight: 6.78 mm

Area: 19.7 mmMean Path Length: 40 mm

Primary turns: 75Secondary turns: 75

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Plotting B-H Curves

Circuit Construction

V o l t a g e R e g u l a t i o n

C u r r e n t b o o s t e r s t a g e

“ B ” I n t e g r a t o r

P o w e r I n p u t

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

T r i a n g l e w a v e i n t e g r a t o r

T e s t c o r e

“ B ” I n t e g r a t o r o f f - s e t n u l l

T r i a n g l e w a v e

S c h m i t t t r i g g e r

O u t p u t s :

0 V

H o u t p u t

B o u t p u t

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Plotting B-H Curves

Complete Circuit

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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 theaccuracy of area calculations.

It would be preferable to use a proper oscilloscope camera or an oscilloscope that canprovide a digital screen capture

2) The value of core power loss was quoted above to two significant figures, but it isdifficult to read the oscilloscope to this level 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 areameasurements.

An A3 printout of the curve was produced and the areas measured with a planimeter:

This gave a curve/image ratio of 8.7%

4) The actual values / tolerance of R1, R2 and C1 will impact on the accuracy of thecalculations.

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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