Coils and Inductance

Make a ferrite-core inductor and find its Q; also, observe resonance and self-resonance and study air-core coils

Energy Storage in Fields

Capacitors store energy in the electric field between their plates, giving it up when the capacitor is discharged and the electric field disappears. By making the field large through thin dielectric layers of large dielectric constant, considerable energy can be stored in a small volume. Capacitors are very useful circuit elements, as we have seen. Values from a few pF to 0.01F are easily obtained at low cost. A capacitor, basically, tries to maintain the voltage across it, supplying current to this end.

An inductor, on the other hand, stores energy in the magnetic field produced by the current flowing through it. When the magnetic field collapses, its energy is given up, sometimes appearing as a spark when a circuit is broken. An inductor tries to keep a current flowing, providing the voltage necessary. The relation between voltage and current in an inductor is V = L(dI/dt), or, in frequency domain, V = ωLI. If V is in volts, I in amperes and t in seconds, L is in henries, H. Magnetic fields in air (space) created by ordinary currents are very small, so it is difficult to store much energy this way, and inductors are not common circuit elements, at least at low frequencies.

An inductance in a DC circuit that is switched should have a diode across it to carry the surge when the current through it is suddenly switched off. If it is in an AC circuit, a snubber consisting of a capacitor (say 0.1 μF) and a small resistor (say 100Ω) in series is the best that can be done. If this is not arranged, whatever is doing the switching will soon be dead.

At radio frequencies, say 100 kHz and above, even a small inductance will have a reasonable reactance, so even air core inductances on the order of microhenries are useful, especially in LC tuned circuits that offer the advantage of frequency selectivity and resonance.

Magnetic Fields

However, the great utility of magnetic fields at low frequencies is well-known, in transformers and rotating machinery, all of which depends on magnetic fields, and on the wonderful effect of iron in strengthening the magnetic fields that otherwise would be feeble. The iron is actually another source of magnetic field that we call into action by the weak fields produced by current-carrying windings. There is only one magnetic field, but it is usual to consider it in two parts: first, the fields produced mainly by currents that is called H, and the total field called B. B is usually a function of H. In space, they are proportional (since they are the same thing physically), but in magnetic materials like iron the relation may be more complex.

For concreteness, let us consider a uniform ring or toroid of magnetic material. The beauty of the toroid is that it is uniform and has no end, which means that the field we call H is due entirely to the currents in the windings that link it. We wind N turns around the toroid, and if we pass a current I through them, we get NI ampere-turns, which are what get things moving. The field H is ampere turns per unit length. Meters, centimeters and inches have all been used. It's best if the turns are distributed evenly around the toroid, but does not matter a lot. The important thing is the ampere-turns per length. The ampere-turn per meter is the MKS unit of H, but the unit commonly seen is the cgs unit called the oersted, which is 0.4π times ampere-turns per centimeter. An oersted is 79.58 At/m or 2.02 At/inch, so we can handle any unit for H that we may find.

The permeability μ of a material is the ratio of the magnetic flux density B to the intensity (as it is usually called) H. If we use the cgs unit for B, gauss, then the permeability of space is 1. (the funny factors are in the 0.4π defining H). If we use the MKS unit for B, the tesla, then the permeability of space is 4π x 10^-7 (the funny factors are now in the permeability). It's all the same, of course, and 10,000 gauss = 1 tesla. The gauss is often seen, the tesla rarely. If the material of the toroid is 950 times more permeable than space, it is obvious that most of the field is going to be within the toroid, just as most of the current in the wire goes through the copper, not the plastic insulation. Therefore, B is uniform within the toroid, and is equal to μH. Magnetic flux is the product of B and the area normal to it, Φ = BA. The gauss is a line or maxwell per square centimeter, while the tesla is weber per square meter. Since a tesla is 10,000 lines per sq. cm., a weber is 10,000 x 10,000 = 1 x 10⁸ lines. The MKS units are rather gloriously inconvenient for practical work, which is why they are still seldom seen. However, it is nice to have a consistent system of units to fall back on in times of doubt.

Finally we come to the interesting bit: when the magnetic flux inside a turn of wire changes, there is a voltage induced in the turn equal to the time derivative of the flux. One weber per second gives us one volt, as do 1 x 10⁸ lines per second. There must be an inconveniently large number somewhere in this relation, which is related to the speed of light. Let's find the voltage induced in the winding of N turns around the toroid from all of this when the current in it changes. V = -NdΦ/dt = -N(d/dt)BA = -(μAN²/l)(dI/dt). The minus sign is to show that the voltage opposes the change in current. The inductance is then μAN²/l, proportional to the square of the number of turns.

A voltage is induced in any winding that links the flux in the toroid. As long as the circuit is not closed, it is just a voltage, and no energy is transferred. However, when we close the circuit, current flows, energy is transferred, and there is a reaction due to the current in this secondary winding. This is, of course, a transformer, a device that is treated in more detail elsewhere. Transformers are much better than capacitors for coupling amplifier stages. There is absolutely no disturbance to the bias, and impedance can be transformed by choosing the number of turns properly, a joy that capacitors do not supply.

The changing flux also induces voltages in any conductors that happen to be around, not just in windings. The result is eddy currents in a magnetic material that also happens to be electrically conducting, such as iron. In power frequency machines, eddy currents are greatly reduced by laminating the magnetic material, and insulating the lamina from each other. This becomes useless at radio frequencies, so different core materials must be sought. One is powdered iron, formed together in an insulating matrix, and another is ferrite, a magnetic material like iron, but one with very low electrical conductivity. These materials are not as good magnetically as iron, but have reasonably low losses even at much higher frequencies, and extend the range of utility of high permeability materials to radio frequencies.

Ferrites have the composition XFe₂O₃, where X is Mn, Zn, Co or Ni, and a spinel crystal structure. Most are mixed MnZn or NiZn. The powders are sintered in the desired form. The properties are controlled by the composition of the mixture and its heat treatment. Their principal advantage is high resistivity, which means low eddy current loss. They are close to saturation at about 3 Oe, and some have a very narrow hysteresis loop.

Making an Inductor

I selected a ferrite core for experiment at random. The main requirement was that it was large enough to be handled easily. I used an Amidon FT-82-43 ferrite core. The FT-82 is the size of the toroid, and the 43 gives the ferrite mixture used. More details on core sizes and mixtures can be found in the ARRL Handbook, together with much practical information on making inductors. The 43 ferrite is useful from 0.1 to 1 MHz, and has an initial permeability of 950 with a saturation flux density of 2750 gauss. The FT-82 toroid has in inner diameter of 0.520", an external diameter of 0.825" and a height of 0.250". Its cross-sectional area is 0.0381 square inches, and its circumferential average length is 2.070 inches.

I put two windings on the core, one of 65 turns, and the other of 30 turns, using #26 enameled wire. To make the winding less annoying, you need to make a bobbin that will fit through the toroid. I made one of flat wood, 3/8" wide and about 2" long, with notches in each end to hold the wire. I then wound about 5 ft. of wire on the bobbin (each turn will take a little more than 0.8" of wire). Wind the wire as evenly as possible around the toroid, which would hold about 100 turns if the wire were carefully placed. You do not have to use the same number of turns--put as many on as you have the stomach for. When you are finished, twist the leads around each other to secure the winding. With a piece of fine sandpaper, remove the enamel from the end of each lead. It is convenient to solder the leads to a header that will fit into the breadboard like an IC. The #26 wire is a little fiddly to use directly in the breadboard.

We now have an inductor of two possible values, or a transformer with turns ratio of 65/30 = 2.17. If we calculate the inductance of the 65 turn coil by the formula given above, we find 2.4 mH. The 30 turn coil would give 0.51 mH, while both coils in series aiding would furnish 5.1 mH. To test the inductor, connect the 65 turn coil in series with a 1k resistor, and feed it with a sine wave from the function generator at 100 kHz. Look at the voltages across the LR combination and across R alone with the oscilloscope. I found peak-to-peak voltages of 4.0 and 2.1 V, respectively, so Z = 4.0 V/2.1 mA = 1.905k. The inductive reactance (square root of Z² - R²) was, therefore, 1.621k, or L = 2.6 mH, in very close agreement with the calculated value. This was only due to luck in using a good value for μ, but shows we are on the right track. The phase angle of 58° was also obtained from the Lissajous figure in the X-Y display.

Now remove the 1k resistor, and connect a 100Ω resistor across the 30 turn winding as a load. By the voltage drop at the output of the function generator (600Ω output resistance), and the voltage across the 100Ω resistor, you can estimate the voltages and currents at the primary and the secondary. I found a voltage ratio of 2.08 and a current ratio of 2.4, both close to the turns ratio of 2.2. The current measurement was rough, and with some care better agreement could be obtained. The 100Ω resistor looks like (2.17)² x 100Ω = 471Ω to the function generator. Recall that the impedance transformation ratio of an ideal transformer is the square of the turns ratio.

The ARRL Handbook gives a formula for the number of turns required for a certain inductance, which is N = 1000 √(mH/A), where A is a constant for the core size and ferrite mix. For the FT82-43, A = 577, so for 2.4 mH, we need 1000 √(2.4/557) = 65.6 turns. Again we have agreement. You can now easily construct an inductor of any reasonable inductance that you require.

One further consideration must be mentioned. All ferromagnetic materials, such as iron and ferrite, have a saturation flux density. This means that as you increase H, B eventually starts increasing less quickly, and eventually levels off at some value. For 43-mix ferrite, this is B = 2750 gauss. If we assume a permeability of 950, then at the value of H that would give this flux density if the relation continued linear, B is already leveling off, and there is no point in continuing. The core must be large enough that the total flux you desire can be reached without saturating the core. In the present example, if I = 186 mA, at which H = 2.9 Oe, the saturation flux density would be reached for a linear relation, and this is probably a good estimate of the highest instantaneous current that should flow in the windings. The saturation flux density is actually measured at 13 Oe, but by this time the B curve is nearly horizontal.

A dc component in the winding current can also bring the core into saturation, at which point the inductance largely disappears. A direct current should never pass through a transformer winding or an inductor that is not designed for it, since it will seriously affect the inductance. An RF choke is an inductance designed to allow for a dc component. Any old inductor will not do its job. Similar inductors are required in DC power supplies that use them.

Resonance

Capacitance and inductance have, in a sense, opposite effects, so they tend to cancel each other. The phenomenon of resonance occurs when the capacitive and inductive reactances are equal in a series or parallel combination of C and L, which happens at a frequency f_o= 1/2π√(LC), called the resonant frequency. Resonance can be defined as zero phase angle, or as minimum or maximum impedance. For L and C in series, the zero phase angle resonant frequency is given by the formula above. When L and C are in parallel, the frequency of zero phase is slightly different, but very close to the value given by the formula if Q is large.

A resistance R must always be included in series with the inductance L of a coil to represent the losses in the core material as well as the DC resistance. For an air-core inductor, this resistance is small, but it is considerable for a ferrite-core inductor. This R is larger than the DC resistance of the winding, and is roughly proportional to frequency (since core losses are proportional to frequency, roughly). The ratio ωL/R is, therefore, roughly constant with frequency and is a useful parameter, denoted by Q. In a resonant, or tuned, circuit, this turns out to be the Q you are familiar with.

When L and C are connected in series, the impedance at resonance is due to R alone. Since the impedance is purely resistive, the phase angle between the voltage applied to the branch and the current through it is zero. At lower frequencies, the capacitor obstructs the current and soon dominates with its impedance of 1/jωC. At higher frequencies, the inductor obstructs the current and soon dominates with its impedance of jωL. The magnitude of the impedance is lowest at resonance, and is proportional or inversely proportional to the frequency away from resonance.

When L and C are connected in series, the L lets the current through at low frequencies, and the C lets it through at high frequencies. If it were not for R, at the resonant frequency the currents let through by each would be in antiphase, and would exactly cancel: the impedance would be infinite. This does not happen, because R spoils the exact antiphase relation, and some current gets through. If you analyze the circuit, you find that exactly at (zero phase) resonance the impedance R_P = Q²R. This will be large if Q is large, and is one of the most important properties of a parallel tuned circuit. At resonance, a current is oscillating in the closed loop of L and C. When the exciting voltage is removed, this current decays at a rate determined by R. All these things are mentioned in a circuits course, but it is very instructive to observe them in practice.

A circuit for studying the resonance properties of the coil we made above is shown at the right. Any similar coil can be used for these experiments, but it is fun to use one that you have made. The op-amp is used in a differential amplifer to read the current I while allowing the circuit under test to be grounded, so that V can be measured similtaneously. The sensitivity of the differential amplifier is 1 V/mA, so the voltage on Ch 2 gives the current directly. Adjust the function generator to supply a few volts of sine wave. I use a frequency counter to read the frequency accurately, not depending on the calibration of the function generator. Peak-to-peak voltages are measured on the scope to an accuracy of about 5%.

Study the series circuit first, if you like. Vary the frequency and note the resonant frequency when the voltage across the circuit will be a minimum, or the current a maximum. This is not very hard to do, and gives the resistance R. Take enough points (about 7-10) to be able to sketch the impedance as a function of frequency. Plot log |Z| against log f, as for a Bode plot, which will give a nice shape and make the skirts linear. It's quite satisfying to see the curve appear. For my coil, I found a value of R = 100Ω at the resonant frequency. Although the voltage across the whole circuit is a minimum, the separate voltages across the capacitor and inductor are large, Q times the voltage across the circuit. A series resonant circuit can act as a voltage amplifier (in the same way a transformer acts; there is no power gain).

Now study the parallel circuit, in the same way. The plot of the impedance versus frequency should resemble the diagram on the left. Find the resonant frequency, which will give a minimum current or a maximum voltage. This should be the same as for the series circuit (to this approximation). The maximum impedance gives the resistance R_P. I found a value of about 2500Ω, so that Q = √(2500/100) = 5.0. I also estimated Q from the widths of the resonance curves, at the points where Z = √2 R or Z = R_P/√2, getting 3.9 and 4.4. These values are very inaccurate since I did not take many points for the plots, but at least are in the same region. The value of 5.0 is probably closest to the truth.

In the parallel circuit, the current through the whole circuit is a minimum, but the current through the inductance and capacitance is Q times as large. The parallel resonant circuit amplifies current, as the series resonant circuit amplifies voltage. This large circulating current stores energy alternately in electric and magnetic fields. A parallel-resonant circuit is called a tank circuit for this reason.

The measurements on the resonant circuit can be repeated with a different resonant frequency by changing the capacitor, say to 0.001 μF. These should show that the Q is roughly the same, in spite of the tenfold change in frequency. It is also easy to show that the transformer transforms impedances with the square of the turns ration with this arrangement, by measuring V and I at the primary for various resistances in the secondary. Try the transformer as step-up or step-down. You will notice that for the transformer to act as an ideal transformer, the reactance of the windings must be much greater than the load resistances. At 300 kHz, the 30-turn secondary has a reactance of about 940Ω, so a load of 100Ω will be reflected as 469Ω. There is a lot more that could be said about transformers, but experimenting with this one will show some interesting properties.

It's possible to display a resonance curve on the oscilloscope; the circuit is shown at the right. The load resistor is used to sense the current, which is what is displayed on the scope. Because the function generator has an internal impedance, the voltage applied to the resonant circuit is not quite constant, so the method is approximate, but good enough in practice. The X-Y display of the oscilloscope is employed. This is one good use of the triangle output of a function generator. If you only have one function generator, it would not be hard to make a sweep oscillator (based on any one of the relaxation oscillators presented in these pages) that generates a ramp or triangle wave of about 20 Hz and 4 V peak to peak. The parallel resonant circuit shown is made from a 1.0 mH rf choke and a 0.01 μF capacitor, which should resonate at about 50 kHz. A 620Ω resistor is used as a load. None of the values are critical, of course--these simply gave good results. By making static measurements of the frequency output of the signal generator as a function of the VCO input voltage, the horizontal axis can be calibrated in frequency, so that quantitative measurements are possible. Even without this, the shape of the response can be seen. This procedure is useful in the alignment of a receiver, and other jobs where resonant circuits are adjusted.

I have an air-core coil weighing 313 g, ID 15 mm, OD 35 mm and 58 mm long, wound with 0.35 mm enameled wire, with a DC resistance of 37.5Ω. The wire would seem to be #26, which is 1300 ft/lb and 41.62 Ω/1000'. On a weight basis, there would be 896 ft of wire, and on a resistance basis 901 ft. With an average turn length of 3.77" this means about 2800 turns. The coil was sold for the wire it contains, but I kept it to use as an air-core inductor. Inductance formulas predict about 92 mH for the coil, but this is only a rough estimate, since we do not know exactly how many turns there are. To determine the inductance, the best way is to resonate the coil with a capacitor, and use the resonance formula to determine L.

There are approximate formulas for the inductance of air-core coils. For a single-layer coil of diameter d inches, length l inches and N turns, Wheeler's formula is L = d²N²/(18d + 40l) μH. For a coil of rectangular cross-section, of thickness t inches, length l inches and mean diameter (average of inside and outside) d inches, Hazletine's formula is L = 0.8d²N²/(12d + 36l + 40t) μH. These formulas are useful for estimation, but do not apply well to extreme cases. Grover (see references) gives more exact semianalytical formulas.

I used an 0.1 μF capacitor, expecting a resonance at around 1600 Hz. This gives an L/C ratio of 0.9 x 10⁶, a proper value. The inductor and capacitor were connected in parallel, in series with a 1k resistor so that the current could be measured. The lower end of the resistor was grounded, and the signal generator supplied the top of the tuned circuit. The applied voltage to the combination, and the voltage across the 1k resistor, were displayed on the oscilloscope, while the frequency was counted. The signal generator was tuned for the minimum current, which occurred at 1800 Hz. The current waveform had a small bump on it that was symmetrically placed at the highest and lowest parts of the sine wave, allowing very precise setting of the frequency. From this, the inductance was determined to be 78.2 mH, not far from the estimate. The Q can be found from the ratio of input and output voltages at resonance. 3.0 V peak to peak gave 0.2 V output, so the ratio of resistances was 15, making the resistance of the tuned circuit at resonance 14k. This should be Q²R, which gives Q = 19. From Q = ωL/R, the result is 24. These are close enough, showing that the Q of the coil is about 20.

Now remove the capacitor, so that the inductor and the 1k resistor make a simple RL circuit. As you increase the frequency, the current becomes smaller and smaller, and lags the voltage as it should for an RL circuit (remember ELI). As you increase the frequency more, the current becomes quite small--and then begins increasing! If you examine its phase relation, it will be seen to lead the voltage, a shocking observation (remember ICE). The inductor has, somehow, become a capacitor! If you look carefully at the current waveform near its minimum, you will see that it is in phase with the voltage, so that what we are observing is a parallel resonance. For my coil, this happened at 50 kHz. This is called self-resonance, and is an effect of the capacitance between the windings of the coil. An excellent lesson comes with this observation: a coil of wire is an inductor only over a certain frequency domain. Outside it, it may well become an inductor, and at self-resonance it is a resistor. All circuit components share an analogous behavior--capacitors can also be self-resonant, and act like inductors at high frequencies. My coil is excellent for demonstrating self-resonance, since it occurs at a reasonable frequency because the coil is simply wound with parallel turns. Radio-frequency chokes are usually given a honeycomb winding to reduce interturn capacitance. In a honeycomb winding, the windings cross at an angle, which reduces the capacitance significantly.

Another effect that should at least be mentioned here is the skin effect. At high frequencies, even the inductance of a straight wire is important, and fields within the wire expel current from this region, so that it is carried mainly on the surface. This raises the AC resistance of wire above its DC value. The skin effect already is noticeable at power frequencies, but is very significant at radio frequencies. The wire called litz wire, from its German name of Litzendraht, is made of fine insulated #28 wires intertwined so that no wire is always on the surface or interior, reducing the skin effect. Litz wire is useful for RF conductors, besides being quite flexible.

Reference

F. W. Grover, Inductance Calculations, Working Formulas and Tables (New York: Dover, 1962; reprint of 1946 edition)

Anonymous, The Impoverished Radio Experimenter, No. 2