# Magnetic flux density meter

Several studies have shown that long time exposure to strong low-frequency magnetic fields can cause serious health problems. There are many devices that create powerful low-frequency magnetic fields such as transformers, electric motors, electric heaters, hair dryers, speakers, cathode ray tubes, and so on.

With a low frequency magnetic field meter, we could always detect strong magnetic field sources and take the necessary precautions. The magnetic field meter presented here is able to measure magnetic flux density from 100nT to 2.3uT and has a flat response across the frequency range from 40Hz to 10KHz. In the frequency band that operates the meter, it is able to measure fields coming from the power supply network (usually 50 or 60Hz) and harmonic frequencies of the power supply network that extend several times above the base frequency. Moreover, it is able to detect and measure all magnetic field sources up to 10KHz.

The circuit of the magnetic flux density (magetic field) meter is quite simple, and does not require any calibration. The device displays the measured field on a logarithmic 10 level scale with the aid of 10 LEDs, but it has also an analog output to which we can connect a voltmeter to obtain maximum resolution. The analog output voltage is directly propotional to the measured magnetic flux density at a rate of 0.1mV / nT.

#### Operating principle

According to the Law of Electromagnetic Induction (Faraday's law), the induced electromotive force in any closed circuit is equal to the negative time rate of the change of the magnetic flux enclosed by the circuit. Thus, when some magnetic flux passes through a coil it produces some voltage across it which depends on the field intensity time rate and on the total surface enclosed by the boundary of the coil. In mathematical terms:

Ε=-dΦ/dt=-A*N*dB/dt

(1)

Where Φ is the magnetic flux passing through the coil, A is the area enclosed by each turn of the coil, N is the total number of the turns of the coil, B is the magnetic flux density of the field (magnetic field) and t is the time. The rate of change of the magnetic flux density is essentially the time derivative of the magnetic flux density (derivative of the magnetic flux with respect of time).

Let us consider that in the space there is a harmonic (sinusoidal) magnetic field of the form:

B(t)=Bo*cos(2πft)

(2)

Where B(t) is the instantaneous value of the magnetic flux density (function of time), Bo is the maximum value of the oscillating field and f is the frequency of oscillation. From (1) and (2) we are able to tell that if a coil is placed into the harmonic field, then an harmonic voltage E(t) will be induced at its terminals which will be equal to:

(1)(2)-> Ε(t)=2πf*Α*Ν*Βο*sin(2πft)

(3)

We observe that the induced voltage is proportional to the geometric characteristics of the coil (A, N), proportional to the frequency f and proportional to the field strength Bo. There is of course the harmonic term sin(2πft) which shows that voltage is a harmonic function of time and has a phase difference of 90^{o} with respect to the field.

From (3), we observe that if we are able to measure the induced voltage E, then we will be able to calculate the B field, as long as we know the exact frequency f.

In practice, when we measure the magnetic field of an arbitrary (unknown) source we are not aware about its frequency. To solve the problem of the unknown frequency, we may multiply both sides of equation (3) with a G/2πf term (where G is a constant). Then we will have:

G*E(t)/2πf= G*Α*Ν*Βο*sin(2πft)

(4)

By setting V(t)= G*E (t)/2πf, we get:

V(t)= G*Α*Ν*Βο*sin(2πft)= G*Α*Ν*Β(t)

(5)

Speaking in RMS terms (Round Mean Square values) the RMS voltage V is proportional to the RMS magnetic field density B and independent of the frequency:

V_{RMS}=G*Α*Ν*Β_{RMS }

(6)

*Therefore, if we build a device that measures the voltage V, then we will be able to calculate the magnetic flux density B, ie we will build a magnetic flux density meter.*

Let's investigate now what it takes to build a real circuit that makes use of equation (6).

The block diagram of the magnetic flux density meter should be as in figure 1:

**Figure 1.**The block diagram of the magnetic flux density meter

We need a coil with well defined geometric characteristics, an amplifier which will have a voltage gain equal to G/2πf, an RMS detector and a display (figure 1)

From a technical point of view, winding a coil with a certain number of turns and specific dimensions and use it as a magnetic field sensor seems an easy task. It seems also quite reasonable to use a microcontroller, a VU (voltage unit) meter or a voltmeter to display the measured value of the magnetic field. For convenience, in our circuit we prefer to use a VU meter and a voltmeter.

Let’s now look at how to build the RMS detector and an amplifier with a G / 2πf gain:

**Let's start with the amplifier:**

Every electronic engineer knows that a circuit having an 1/f response is an integrator (a low pass filter with a linear amplitude response and an ideal cut-off frequency of 0 Hz). We may build an almost perfect integrator by using an operational amplifier and some passive components. If we wish to achieve some high gain (G to be a big number) we may use one high-gain integrator or several amplifying stages in series with a reasonable - gain integrator. In our circuit, we prefer the second solution and we use one amplifier unit before the integrator. High amplification is necessary in order for our detector to be sensitive without having to use any large coil (since the output voltage for a reasonable sized coil is relatively small, especially at low frequencies).

**Let's now explore the technical details of the RMS detector:**

We are interested to build a magnetic field meter which will be able to measure RMS field values rather than average or peak values. The output of the integrator is an AC signal. In order to measure the RMS value of this signal, we may use a rectifier and then extract the DC component of the rectified signal. Indeed, the DC component of a rectified signal is shown to be directly proportional to the RMS value of the harmonic input signal:

Let’s consider a half-wave rectified voltage, Us, during a full cycle (0 - 2π rad):

Us=Vm*sin(2πft) for 0<2πft<π and Us=0 for π<2πft<2π

Vm is the maximum value of the half-wave rectified voltage.

The Fourier series of the above function is:

Us=(Vm/π)+ (Vm/2) sin(2πft)-(2Vm/3π) cos(4πft)- )-(2Vm/15π) cos(8πft)+ …

The first term in the series represents the DC component of the semi-rectified signal, the second term is the 1st harmonic (base frequency) and the remaining terms are higher order harmonics.

We know that V_{RMS} = Vm/√2, so:

V_{DC} = √2*V_{RMS} / π ή V_{RMS} = π*V_{DC}/√2

(7)

*The RMS value of the input harmonic signal is equal to the π/√2 of the DC component of the semi-rectified signal.*

At rectifier’s output, apart from the DC component, there is also a large number of harmonics that should be rejected. The rejection - filtering of the harmonics can be done with a low pass filter. The cut-off frequency of the low pass filter should be as low as possible in order to obtain perfect harmonics rejection.

Therefore, the RMS detector can be implemented with a half - wave rectifier and a low pass filter.

The electronic circuit of the magnetic flux density meter

In figure 2, we present the electronic circuit of the magnetic field meter.

**Figure 2.**The electronic circuit of the magnetic flux density meter

IC1-A is used as a typical non-inverting amplifier with Ga gain:

Ga=1+R2/R18

(8)

Because we use a single supply voltage in our circuit (from battery), it is essential to have a virtual ground point for the op-amps at a positive voltage. So, we use virtual ground at about 3 volts, which is provided at point A from IC1-D. Virtual ground is used for DC bias. However, for AC signals, there must be a real path to earth (real ground at 0V) and this path is provided by capacitors C1 and C6. Since the op-amp has a very high input resistance, the R1 resistance does not affect the amplification. We use it only to stabilize the amplifier by preventing any oscillations that could arise from the parasitic capacitance at the ends of the L1 coil. R1, practically reduces the quality factor (Q) of the coil to prevent oscillations. R17 provides a discharge path for C1 because otherwise C1 could not be discharged via the coil due to the high input resistance of the op-amp.

The gain, Ga, is defined by R2 and R18 (see equation 8) and it is equal to 101 and does not depend on frequency. In practice, the gain of the amplifier would be less than 101 for frequencies lower than 40Hz due to capacitors C1 and C6. These capacitors will add some resistance to R18 at low frequencies (since they are in series with R18) and will lower the gain of the amplifier. At higher frequencies, the response of the amplifier is limited by the characteristics of the op-amp (the gain-bandwidth product). However, from 40Hz to 10KHz, the amplifier response is almost flat and the gain is approximately equal to the theoretical value (101) with a potential error of less than 1db if 1% accuracy resistors are used for R2 and R18.

The integrator follows up the amplifier. The integrator is made from IC1-B, R4, R3 and C2. The response of the integrator is inversely proportional to the frequency due to C2. Indeed, if we analyze the integrator as a typical inverse amplifier, we will find that its gain Gb, is equal to:

Gb=-R_{F}/R3

(9)

R_{F} is the impedance resulting from the parallel combination of resistor R4 and the Xc impedance (reactance) of C2. That is,

R_{F} = Xc//R4=R4*Xc/(R4+Xc)

(10)

We know that Xc reactance is equal to:

Χc=1/2πf*C2

(11)

By combining equations (9), (10) and (11) we find that:

Gb=(-R4/R3)/(1+2πfR4C2)

(12)

It is worth noting that the above relationship is not exactly at the form of 1/f, because we do not have a perfect integrator due to R4. However, with the components we use, the product 2πfR4C2 is much larger than 1 for frequencies above 40Hz and the “1” in the denominator can be omitted. Therefore, for frequencies above 40Hz, the gain of the integrator becomes equal to:

Gb=(-1/R3C2) *(1/2πf)

(13)

The total gain of the amplifier and integrator chain would be the product of the Ga and Gb, namely:

Gt=Ga*Gb

(14)

From equations (8), (13) and (14), we find that:

Gt=(1+R2/R18)*(-1/R3C2)*1/2π f

(15)

That is,

G=(1+R2/R18)*(-1/R3C2)

(16)

Now, taking into account equations (16) and (6), we find that at the output of the integrator, we will have an AC voltage, VRMS, equal to:

V_{RMS}=(1+R2/R18)*(-1/R3C2)*Α*Ν*Β_{RMS}

(17)

*From equation (17) we observe that we produced a voltage that is directly proportional to the field and does not depend on the frequency.*

**Figure 3.**The output voltage of the RMS detector as a function of the magnetic flux density (using a coil of 100 turns of 43.05x12mm rentagular cross section)

If we calculate the actual value of the G constant, we will find that:

G=(1+R2/R18)*(-1/R3C2)=101*10^{5 }

(18)

Suppose now that we want the V_{RMS} voltage to be 1.2V when the coil is placed inside a 2.3μT (RMS) magnetic field. From equation (17), we find that we should make:

Α*Ν=0.05166m^{2 }

(19)

If we choose our coil to have 100 turns, then each turn should include an area of 516.6 * 10^{-6} m^{2}. For a rectangular cross section coil, the desired area can be achieved if we choose each turn to be 43.05mm x 12mm.

*The meter coil should be of 100 turns of a 43.05mm x 12mm rectangular cross-section.*

After the integrator, follows up the rectifier circuit. The rectifier circuit is implemented from IC1-C. It is an uncommon circuit of rectification, since there is no diode anywhere. The rectification takes place because the negative half-period of the signal is simply cut off.

The IC1-C along with the resistors R5, R6, R7 and R8 is actually a classical circuit of subtraction (differential amplifier). The voltage applied to the left end of R5 is subtracted from the voltage applied to the left end of R6. However, the left edges of R5 and R6 are short-circuited. This means that we essentially subtract a voltage from itself. So, one would expect that at the output of the subtractor we will always have a voltage equal to zero. However, this is the case only for DC. For the AC signal, the capacitor C3 acts as a short circuit and grounds the non-inverting input of IC1-C. With this trick, the subtractor is converted to an inverting AC amplifier, with an amplification equal to R7 / R5 = 2.2.

The above analysis shows that IC1-C eliminates the DC component of the signal and amplifies the residual AC signal by 2.2 times. In addition to the above, IC1-C also performs half-rectification of the signal. Half-rectification occurs because the op-amp operates at a 0V reference level and there is no negative supply voltage. Therefore, it is able to only amplify the positive half-period of the signal and during the negative half-period its output becomes zero (that is, the half-phase rectification occurs).

At the output of IC1-C we have a semi-rectified signal. The semi-rectified signal contains a DC component and some harmonics (see the Fourier series mentioned above). The harmonics are filtered out from a simple low pass filter formed by R9 and C4. The cut-off frequency of this specific filter is equal to 1/2πR9C4 = 4.5Hz, which is a very low one and approximately the filter passes only the DC component.

Normally, the DC component of a half-rectified signal is equal to √2/π of the RMS value of the input signal of the rectifier. The term √2/π is equal to about 1/2.2. However, since we use amplification equal to 2.2 in the rectifier, we make the half-rectified signal to have a DC component equal to the exact RMS value of the input signal (by cancelling out the 1/2.2 factor). This DC component is passed to the up-following display section.

The display unit is a classic voltage unit level (VU) indicator based on LM3915. The display is made from 10 LEDs (D2 to D11). In order to save energy, the display operates in dot - mode rather than bar. LED D11 lights up when the DC voltage at the input of the LM3915 is equal to 1.2V, i.e. when the measured magnetic field is equal to or greater than 2.3μT. Each of the remaining LEDs illuminates at a lower field level. The VU meter operates in a logarithmic scale and the resolution step is equal to 3db. So, D2, D3, D4, D5, D6, D7, D8, D9, D10 and D11 turn on at field levels of 2.3uΤ, 1.6uΤ, 1.15uΤ, 810nT, 580nT, 409nT, 290nT, 205nT, 145nT and 103nT, respectively.

Magnetic field levels up to 300nT are considered within safe limits in most countries of the world, so we chose LEDs D2 to D5 to be of a green color. From 400nT to 1uT, the field is considered potentially dangerous and up from 1uT is considered definitely dangerous, so we chose LEDs D6 to D8 to be of a yellow color and D9 to D11 to be of a red color.

In addition to the 10-LED display, the circuit also has an analogue output for displaying the field at any other level indicator or even to a simple voltmeter. This output is provided on C5 terminals. C5, along with R14, R15 and R16, form a low pass filter that extracts the DC component from the output of the rectifier as well as the R9-C4 filter. The parallel combination of resistors R15 and R16, together with R14, form a voltage divider. The divider is designed so that the analog output provides a voltage equal to 0.1mV / nT. For instance, for a 2uT field, there will be a voltage of 200mV at the analog output.

**Figure 4.**The analog output voltage as a function of the magnetic flux density (using a coil of 100 turns of 43.05x12mm rentagular cross section)

The IC1-D is used as a simple non-inverting DC amplifier. At its input, there is a DC voltage of 1.2V which comes from the internal reference level circuit of LM3915. The DC gain of IC1-D is equal to

1+ R11/R12 = 2.471

Therefore, a reference voltage of

1.2* 2.471=3V

is produced at the output of IC1-D. This voltage is used to set a virtual ground level for IC1-A and IC1-B. C7 is used for noise filtering and R13 determines the driving current of the LEDs.

**Figure 5.**Printed circuit board of the magnetic flux density meter (copper side view)

How to assemble the electronic board

To make things easy, we have designed a printed circuit board. The printed circuit has copper only on one side and is shown in Figure 4:

All components should be soldered to this printed circuit board according to the assembly guide of Figure 5.

All the resistors we use in the circuit are of 1/4W. Some resistors are of normal 5% tolerance but there are several resistors that are of 1% tolerance. These specific low tolerance resistors are appropriately marked in the schematic.

The electrolytic capacitors we use are of low voltage (16V). The remaining non electrolytics are of polyester or ceramic type. C2 should be a low tolerance capacitor (5% or less - 1% is recommended for optimal accuracy).

**Figure 6.**How to assemble the electronic board of the magnetic flux density meter

The L1 coil should be winded carefully for optimal accuracy. To help you construct the coil, we have printed some guidelines on the circuit board, up on which you can wrap the coil. The guidelines are printed on two rectangular sectors of the board. Those sectors are marked as 43.05x12x100. The rectangle around the marks is the trace up on which the turns of the coil should be located. You should cut the one rectangular sector from the rest of the board and fit it on top of the other one by some thin wires. The wires should also act as guidelines and must be soldered to the 6 pads found on the rectangular guidelines. By this way you will have a "sandwich" in which you have to wrap the one hundred turns of the L1 coil. The coil turns can be wrapped along the printed guidelines of the “sandwich” and up to the thin guideline wires. For a better understanding of how to wrap the coil, please refer to the photos of the prototype.

The wire you will use for the coil should be thin enough to avoid any geometric deviations from the optimal dimensions and it is good to wrap the coil turns next to each other rather than in multiple layers. You may stabilize the coil with some glue after wrapping it. In the prototype we use a 0.2mm thick winding wire.

Once you solder the components, connect the S1 switch and the battery and the circuit will be ready for use. Make sure that you did not forget to solder a wire bridge which is required at the center of the circuit board and it's clearly denoted in the assembly guide.

You may test the magnetic field meter next to some magnetic field source such as a transformer, near a resistor of a heating device, etc.

The device does not need any calibration and there will be a negligible measurement error as long as you use the appropriate low tolerance components and you have correctly constructed the coil. If you want to test the accuracy of the device, you should create a magnetic field of a known value and test it on it (usually by using a standard coil for calibration) or you may compare the readings with a certified device or certify your own device in a specialized laboratory.

Once you have completed the assembly, you can place the device in a suitable plastic box to have a professional portable magnetic field meter.

In the prototype, we use the LM324 because of its relative low cost. You may alternatively use the most expensive and superior quality TL084 or TLC074. These op-amps provide greater stability and have less power consumption. You may also make some improvements or additions. For example, you may connect the analog input of a microcontroller to the analogue output of the circuit and display the measurements on a screen in a numerical or a graphic manner or you may get some measurements on a computer.

**Attachments:**

The printed circuit board of the magnetic flux density meter in pdf file.

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