We delve into these useful but often mysterious circuit building blocks.

Resonant circuits are dotted throughout the electronics landscape, performing tasks from filtering noise from an AC signal, to receiving radio waves. However, to many a Maker, particularly those who have come to the field without the historically typical background, they are a mysterious collection of magic. We set out to write a Fundamentals short primer on a specific resonant circuit, the tank circuit. In doing so, we found so much more that needed covering. As such, while the main aim of this article is still the tank circuit, we will explain different resonant circuits and their uses, as well as some of the maths around them.

WHAT IS RESONANCE?

Resonance is not unique to electronics. It is found throughout the physical world, and across different forms of energy. It is the name for a cycle of change of energy from one form to another and back again, over and over. In other words, it is an oscillation or vibration. It happens at a natural frequency that is different for different materials, and under different conditions.

One of the easiest ways to explain resonance to someone who is not physics-minded (which really means most of the population, myself included to a fair degree) is to illustrate it. A swing is one of the best examples of resonance. Without getting into how the swing starts, picture a swing that is already in motion with someone on it.

As the swing rises, it slows down. There is air resistance against the components of the swing and the load on it (the person), but the main force is gravity slowing the assembly down. This is negative acceleration, as there is no such thing as 'deceleration' in physics equations.

As the swing slows down near the top of its arc, the energy is not gone. It has changed from kinetic (moving) energy, into gravitational potential stored energy.

Any object at any height has a potential energy based on its height and the force of gravity. Gravity is constant and acceleration due to gravity is 9.8 metres per second per second, meaning that there will be a 9.8m/s increase in velocity for every second the object is moving, ignoring air resistance. That's not really true but it's close enough.

So, as the swing becomes stationary at the top of its arc, all of the energy in the system is now gravitational potential energy. Energy is not lost, just stored by the mass and height of the object against the force of gravity. When this transfer is complete, there is nothing holding the swing up.

Gravity immediately affects it in the way we are familiar with, and the swing falls downward in its arc. This is the transfer of the stored gravitational potential energy into kinetic energy again, and this transfer is complete only when the swing has reached the bottom of its arc.

Here, it has no gravitational potential energy left (technically it does because it is a weight suspended by a tether and would fall if the tether were cut, but that's not relevant to our situation) and all of the energy is kinetic. Now, the swing is travelling upward again, and the energy is transferring from kinetic to gravitational potential and the swing slows down. At stall, all of the energy is potential again, and the process repeats in the other direction.

While a swing is familiar to most of us, a pendulum is a better example of resonance, because they are designed to be affected as little as possible by air resistance and any other detracting factor.

Most pendulums in clocks take the form of a disc to present the least surface area to the direction of travel for the mass involved, but a ball on a string works for an everyday demonstration. One of the properties of a pendulum is that the time taken for a complete cycle is not dependent on the length of the arc of swing, but rather on the length of the pendulum. Note the numbers in the diagram are generalisations to illustrate the point. There is an entire set of maths regarding pendulums.

When the pendulum is swinging, more energy can be added to the system at precisely the right moment. Imagine trying to push a swing (which is a form of pendulum with far more friction and drag from many sources) when it is moving toward you: You slow the swing down and remove energy from the system.

However, if you wait until it reaches its peak and push it in the direction it is about to naturally go, or push in the direction it is moving, energy is added to the system. However, this increases the length of the arc, and the height of the swing (which is the amplitude in pendulum terms), but not its time period. This is because the speed increases with the amplitude, and so the time remains dependent only on the length of the pendulum. Remember this for later, because there is an electronic equivalent.

Actually, there is an electronic equivalent for everything so far, which is why we described these things. However, the forms are different. The energy is generally electrical current rather than kinetic energy, and can be stored as either a magnetic field or electric charge, rather than as gravitational potential energy. With that in mind, a recap on some of the components involved seems in order.

INDUCTOR RECAP

Any electric current flowing in a conductor produces a magnetic field around that conductor that is perpendicular (90°) to the current. Generally, the conductor will be a wire, and the direction of the magnetic field matches the direction of your fingers if you curl your right hand around the wire with your thumb pointing in the direction of the current. The magnetic field is dependent on the current. Its strength and direction are determined by the current, so if the current is constant, so is the magnetic field. If the current varies in strength and direction, then so does the magnetic field. Note that conductors could be plates or foil, too.

If a wire is wrapped around a core, two things happen: Firstly, the magnetic field of one turn influences the ones next to it. Secondly, the magnetic field becomes concentrated. This is an inductor, and they can be air cored, iron cored, or ferrite cored. The iron-bearing material Ferrite is not iron metal, but iron oxides sintered together and therefore it has different properties. In the case of iron and ferrite, the magnetic field is further concentrated and follows the core. This forms an electromagnet at the ends of the core, which can be both a help and a hindrance depending on the application.

More importantly, the magnetic field in one turn affects the turns next to it and vice versa. If the current is changing, then the magnetic field changes, too. Any changing magnetic field generates a current in any wire which is in the magnetic field, and that current is opposite to the one that created it in the first place. So, the current created flows in the opposite direction to the one that created it. However, these are not separate wires, but turns of the same wire. Therefore, the current created pushes against the one that created it. This is called 'Back-EMF', or backwards electromotive force.

That means that an inductor resists change in current. It will happen, but slowly. As the current first starts to flow, there is no magnetic field, and therefore a large current flows quickly. As it does, the magnetic field builds, and the back-emf builds with it. This means that the current lags behind the voltage. For DC, this depends on a variety of factors. For AC in a regular wave, this occurs at 90° out of phase.

However, as soon as a current is removed or begins to fall, the magnetic field collapses or begins to recede. The graph here with the flat sections is for the manual application and removal of DC current to an inductor. As power is removed, you can see a negative spike as the collapsing magnetic field produces a back-EMF in the coil that was containing the magnetic field.

With the right construction, a coil can be made into a dedicated inductor. Inductance is measured in Henries, given as 'H', and represented in equations by the letter 'L'. When buying inductors as components, some are ferrite cored, some are air cored, and some are iron cored. There are different types for different applications, such as ones designed for high currents in switchmode power supplies, and ones in packages a lot like a resistor, even marked with colour codes. Some of the ones we used in this article are wrapped on a ferrite core and rated for Radio Frequencies, which not all are.

INDUCTIVE REACTANCE

The effect of an inductor in opposing a current is called 'Inductive Reactance', designated in equations as XL, but is measured in ohms (Ω) like resistance. For an AC sine wave, it is determined by the frequency (ƒ) of the current in Hertz (Hz), the inductance (L) of the component in Henries (H), and a constant of two times Pi (2π).

X_L = f x 2π x L

Now, there are other equations for different waves and for DC, but it doesn't become particularly relevant yet. What is important is that you understand that inductive reactance exists and have an idea of how it behaves.

CAPACITOR RECAP

A capacitor is a different arrangement of conductors. In the capacitor, two conductors are present. When there is a potential difference (voltage) between the conductors, an electric charge builds. This could be two wires, and that happens, but capacitors are components made to exploit this property and as such, the conductors are generally plates. This is because surface area is important. The voltage present causes electrical charges in the plates to move, so that one side becomes positively charged, and one side negatively charged.

The conductors have to be separated by an insulator. In really simple capacitors, this can be air. In manufactured capacitors, this is usually a material like mylar, polyester, or some other insulator. This is called the 'dielectric'. Different dielectrics give different properties, like frequency tolerance or temperature stability. Some also only suit smaller values of capacitance, which is measured in Farads (F). More on that shortly. Ceramic capacitors, for example, are often used for only small values. Aluminium Electrolytic capacitors, however, are used generally for larger values. These have different constructions, but all involve multiple layers of the two conductors to maximise surface area.

The Farad is a big unit of measurement, too big for most purposes. We use some sort of division in multiples of one thousand: Generally, these are: MicroFarads, or one millionth of a Farad or x10-6, and using the Greek letter Mu (µF); NanoFarads, abbreviated as nF, are one billionth of a Farad, or x10-9. Then there is the PicoFarad, pF, which is one trillionth of a Farad, or x10-12.

We've covered the decimal multiplier system, if you're unfamiliar or out of practise with it, in Issue 51. Be aware, however, that sometimes MilliFarads are used. This will be abbreviated as mF, but can be confusing, Older fonts and typesets did not support Greek letters, and so sometimes mF actually means µF. In these cases, the abbreviation 'MFD" is often used to make the fact clear, but not always.

Operationally, the voltage difference causes electrons to move through the circuit and build up on one plate, leaving the other positively charged. Once all the available electrons have moved, no more current flows. All the energy is stored as a static charge. However, because the current has little resistance to flow at first, as the charge migrates, the voltage lags behind the current, instead of leading it like an inductor. This is by a phase angle of 90°.

Theoretically, a capacitor would hold a charge once the current was removed, as long as its terminals were not connected to anything. That sets it apart from an inductor. However, that's not true in the real world, although a large capacitor can store a charge for a long time, as one of the DIYODE staff once found when working on a camera flash that had been turned off for three weeks. In reality, there are current paths everywhere. Even the air between the terminals will leak a small amount of current, as air is not a perfect insulator and atmospheric air contains impurities. There is also leakage through the imperfect dielectric.

CAPACITIVE REACTANCE

Capacitors have reactance, too. However, the higher the frequency, the less reactance a capacitor has, which is the opposite to an inductor. Capacitive Reactance is designated as XC, also measured in Ohms, and can be found by:

where the Capacitance (C) in Farads, the frequency (ƒ) in Hertz, and two times Pi are all under the inverse, or one divided by the other factors.

The important point to note is that for a given capacitor, reactance decreases with frequency. For a given inductor, reactance increases with frequency.

THE BASIC PARALLEL RESONANT CIRCUIT

So, finally, on to resonant circuits! The absolute basic resonant circuit is a capacitor in parallel with an inductor. Much like the swing, current flows from the capacitor, where it was stored as a static charge rather than gravitational potential energy, into the inductor. At the place that would be the bottom of the arc in the swing or pendulum, the current is slowing the fastest between the two components. However, the back-EMF is building in the inductor, slowing down the current.

A magnetic field is building to store the energy, before finally, all the energy is stored in the magnetic field, which is opposing current flow completely, and the swing is now at the other end of its arc. The magnetic field now has nothing to generate it, so it collapses, causing a current in the reverse direction, back into the capacitor to start again. This process will go on forever! Remember, in the inductor, current lags voltage. In the capacitor, current leads voltage, meaning the current and voltage will always be in different places in the system.

NOTHING IS PERFECT

In reality, this does not occur forever. There are losses everywhere. There is straight-out resistance in the conductors, which causes losses due to heat, even minor. There is resistance in the conductors between the active elements. There are leakage paths for the capacitance and the magnetic field is not 100% efficient either. That means that energy must always be put into the system. The same can be said of a pendulum, too, although the losses are different there. A pendulum with a very big mass for its surface area, and a very thin supporting length such as a fishing line, will swing for a very long time with minimal loss.

The oscilloscope screenshot here is the voltage measured at the junction of the capacitor and the inductor in a basic resonator circuit. The circuit involved a 1mH inductor and a 470nF capacitor in parallel. These values were chosen arbitrarily and not calculated. They were lying on the workbench. Power was applied manually to start the oscillation, by touching a jumper lead to one side of the circuit, where the oscilloscope probe was attached. The other side is grounded, as is the probe. 6V was the power source, and the manual touch was as quick as practical but intimed.

You can see from the scope shot that the voltage achieved was much higher than the 6V input. You can see the build-up, too, as small oscillations start before a sudden increase, then power is removed and the oscillations fade away due to losses. This was a poorly controlled experiment and the input signal, a human hand touching a wire to the top of the circuit, was longer than one oscillation cycle. It was longer than many, many cycles and so this is an indication only and does not display all of the correct properties that will be detailed further on.

RESONANT FREQUENCY

All resonant circuits are resonant only at one frequency. There will be some resonance either side of this, forming a bandwidth, but some resonant circuits have a narrower bandwidth that others. This is because capacitors block lower frequencies, while inductors block higher frequencies. The point where these two intersect is the resonant frequency. More accurately, the resonant frequency is where the capacitive reactance and inductive reactance have the same magnitude in Ohms. Most of the maths presented works only for sine waves, but the value for other waves can be calculated. It's pretty complex and in most cases, calculating for a sine wave works just fine. A square wave, for example, contains a sine wave. Keep that in mind for later.

The resonant Frequency can be found by:

USES

If an Alternating Current (AC) source is applied to a resonant circuit, and the frequency of this source matches the resonant frequency of the circuit, then current keeps applying energy just at the right time, like pushing a swing when it reaches the top of its arc. This means that an LC circuit can be used to amplify small signals. This is not magic of course, and the law of conservation of energy must be maintained. The amplification still turns current into voltage, so while stronger signals may be generated, the current must be there in the first place.

The reality is that in most cases, an LC circuit will be part of a larger whole if used to amplify a weak signal, being driven by an active element. This is what happens in switch mode inverters, where a high current is turned into a high voltage. This is the compressed version of course, as we'll detail soon.

Other uses for LC circuits are for filtering signals. This is probably the most extensive use for LC circuits and is the basis for radio receivers, where the antenna is either connected to an inductor, or itself forms the inductor. Then, a variable capacitor can be used to change the specific resonant frequency. Other frequencies are therefore blocked, and the resonant frequency passes through.

LC resonant circuits can also be used as signal shapers. If fed a square wave signal, for example, a correctly-tuned LC circuit will resonate and generate a sine wave. However, there is a lot more to using an LC circuit for this because the reality is quite a bit more complicated than that. It is more correct to say that an LC circuit can be a component of a signal generator.

PARALLEL LC CIRCUITS

While we have mainly discussed parallel LC circuits so far, LC circuits can be series-connected as well. Before we discuss them, however, there is a bit more to say about the parallel version. Because the components are in parallel, the voltage at the terminal of each, when current is being applied, is equal. However, the current flowing is the sum of the inductor current and capacitor current. Remember, only at a certain point in the cycle is all the energy stored in one or the other, and even then only in the theoretical realm.

However, at resonance, when the inductive reactance and capacitive reactance are equal, the current in the capacitive branch of the circuit is equal in magnitude but opposite in polarity to that in the inductive branch. The currents are 180° out of phase with each other. Therefore, the capacitive and inductive currents cancel each other out, giving a very high impedance. In fact, mathematically it is infinite, but we know in real life that there are imperfections.

It is possible to calculate the impedance of a parallel LC circuit, but we don't expect many Classroom readers are going to want to try. If you are just learning about LC circuits, we find it unlikely that you are ready to attempt an equation like this:

where j = the square root of negative one, ω (Ancient Greek lower case Omega) is the angular frequency in Radians, ZLC is the impedance of the LC parallel circuit. We weren't feeling like it either, especially because that is only needed in reasonably advanced applications. The take-home point is that the current applied to a parallel LC circuit is minimal.

This has effects on a load connected to the circuit. If a parallel LC circuit is connected in series with a load, a band-stop filter is formed. At the resonant frequency, almost no current passes. However, current at any other frequency does pass. This is very useful to filter, say, an induced noise signal off another signal, or a DC supply. If the circuit is connected in parallel, however, a bandpass filter is formed, allowing that frequency to pass while shorting and therefore blocking all others before the load.

SERIES LC CIRCUITS

It is also possible to connect an inductor and capacitor in series. Because the current must flow through one before the other, there will still be an interaction. The voltage will still lead or lag the current depending on which component is first, before entering the other. In this case, the voltage across the whole circuit is divided between the capacitor and the inductor, and it is the current which must remain the same through the circuit.

V_T = V_C + V_L, while I_T = I_C = I L

The main calculations and concepts are the same: Capacitive reactance decreases in magnitude for an increase in frequency, while inductive reactance increases in magnitude for an increase in frequency. The main difference is that in the series arrangement, it is the voltage that is minimised and the current maximised. The voltages are 180° out of phase rather than the currents as in a parallel arrangement. The capacitor can be connected before or after the inductor, and different applications may demand one over the other. However, the inductor before the capacitor seems to be the most common.

FILTERING

Knowing this, we can think about filters. As discussed above, LC circuits can be used to either block or pass only a desired signal, and this is either a bandstop or bandpass filter, respectively. Note that some of these circuits feature a resistor. Sometimes this is part of the operation of the filter. At other times, however, it is there because the filter presents a short circuit at resonance and current must be limited to avoid damage.

BANDPASS FILTERS

A bandpass filter passes a certain band of frequencies through it. The width of this band depends on circuit design but in general will have a narrow peak at a target frequency and taper steeply away either side. This is why radio stations have gaps between the operating frequencies: There is always some spill-over on either side of the frequency in question.

If an inductor and capacitor are connected in series, we know from earlier that the combination presents almost no impedance to the target frequency but almost total impedance to other frequencies. In this way, a series LC filter can be used to select a frequency from out of a cluster of frequencies, and pass through only the desired frequency.

The load is still important here. The resonant frequency is not dependent on the resistance in any way, but the load resistance does have an effect on the width of the pass band, or the selectivity of the filter.

The other way to make a bandpass filter is to use a parallel LC circuit, which as we know, has almost total impedance to the target signal but almost none to other frequencies. Therefore, the parallel LC filter is connected in parallel to the load to filter out a frequency: All others are shorted to ground by the filter. This is why the resistor R1 is present: It will limit the current of the shorted signals, whereas in the series circuit, the current of the target frequency is limited by the load.

This is really just the tank circuit, with a load resistance on one side, and a series resistor to limit current when needed. This is the circuit found in most radios, where the signal mix comes from the radio antenna picking up everything that is in the air, and the tank circuit filters out the desired frequency for further processing.

In both the series and parallel bandpass filters, the load resistance may be pure resistance or an impedance made up of capacitors, inductors, or both. In most cases, it is either pure (or almost pure) such as to drive the base of an amplifying transistor; or capacitive, through a decoupling capacitor.

BANDSTOP FILTERS

Bandstop filters are the opposite of bandpass filters. They stop the target frequency and let other frequencies through. They are less commonly used but make good audio or noise filters when a frequency or range of frequencies is a problem in a given situation. And, they look rather familiar!

A parallel bandstop filter sees the tank circuit connected in series with the load. Remember, the 'parallel' bit refers to the LC elements, not how the LC block is placed in the broader circuit. Because the parallel LC circuit presents almost total impedance to the resonant frequency, this frequency is blocked, and others allowed to pass. In reality there is a band of frequencies, with the resonant frequency at the centre of the peak and a curve of varying steepness either side.

Not having a series resistor means the signals passing through are not impeded by any more than the very minor impedance of the LC tank circuit at non-band frequencies. This is particularly important if this circuit is being used to filter, say, a switching noise from a DC power supply line, and where all of the current for the supply rail flows through this filter. It is also a very selective form of filter, not suitable for a broad general noise filter.

On the other hand, a series LC filter can also be used as a bandstop, but by being connected in parallel with the rest of the circuit and shorting the resonant frequency to ground, necessitating a series resistor. The presence of this resistor limits the overall current that can be drawn through to the load, as it has to pass though the limiting resistor as well. Why not just put the resistor in series

with the filter? Because that would leave the RLC combination in parallel with the load. In that case, whether or not this matters depends on the load impedance: If the load is low impedance, the resistor could cause too much impedance in the filter and thus leave too much current flowing straight through the load to render the filter effective. With a high-impedance load, this is unlikely to happen.

ADDING RESISTANCE

Unfortunately, no LC circuit is free of resistance. There are resistances in conductors, connections, and through impurities in material. Capacitors have an Equivalent Series Resistance as well, although some types more than others. For aluminium electrolytic capacitors, it may be a serious consideration. This resistance can change the resonant frequency of an LC circuit away from the calculated frequency, a property called 'antiresonance'.

For parallel LC circuits, resistance (either directly like long conductors or poor joints, or as equivalent series resistance inside the component) in series with the inductor lowers the resonant frequency. Resistance in series with the capacitance increases the resonant frequency. For series LC circuits, resistance in parallel with the inductor increases the resonant frequency, while resistance in parallel with the capacitor reduces the frequency.

We have no hope of explaining the maths of it here, and how to calculate it. We're not sure we understand it ourselves! However, it is worth being aware of. Choosing quality components, reducing the length of current paths, making good solid connections, and other general quality building practices will reduce the effects.

THE RLC CIRCUIT

RLC circuits involve adding a resistor to either series or parallel LC circuits. The resistor can be added in various combinations and these all produce different effects for different purposes. We were going to try to fit that information here, but really, there is enough to say to make it too big for this space, and worthy of a future Fundamentals short lesson specifically on RLC circuits.

DAMPING

Something else that needs considering is the concept of damping. Damping is the term applied to the restriction of any resonance. You can dampen a pendulum by adding drag, for example. Tall structures often have sway dampers to stop resonance building to the point where the building is structurally compromised.

The same applies to electrical resonance. Earlier in the article, we showed a screenshot from the oscilloscope of our very rough, random RC tank circuit when DC power had been fed in then disconnected. The circuit oscillated but each oscillation was less than the one before.

This was partly because the combination was poorly tuned, and partly because of the very high resistances of a breadboard connection with over-used component legs. This resistance slowed the current each time, converting some of the energy to heat (and on a breadboard, probably a dozen other sources of loss), and therefore reducing the oscillations.

These losses can also be eddy currents in an iron-cored inductor, something that should be absent from ferrite-cored inductors which do not electrically conduct. Electromagnetic radiation is also a path of loss. If the losses exceed the energy input, no resonance will ever occur. If the losses are high, the resonance will fade quickly.

Q FACTOR

Q factor is a way to measure
how effective a resonant circuit is in terms of the accuracy of its bandwidth, and is often said to stand for 'Quality Factor'. However, we cannot verify that this is not a backronym, like the old TWAIN scanner driver story. Regardless of the name, the higher the Q-factor, the narrower the bandwidth. While it might seem logical at face value to design all resonant circuits with the highest Q-factor possible, the reality is that sometimes a broader bandwidth is desirable.

This is made up of circuit reactance and the circuit resistance:

where X is the inductive or capacitive reactance (remember, they're the same at resonance), and R is the series resistance. In a series LC circuit, the formula applies as is. For parallel LC circuits, generally it is the inductor we are interested in.

BANDWIDTH

We've introduced the idea of bandwidth, but now it is time to calculate it. Bandwidth is the range of frequencies above 70.7% of the amplitude of the peak frequency.

For a series or parallel LC circuit, where ƒ = the resonant frequency and Q = the Q factor:

In the real world, you're unlikely to be calculating either bandwidth or Q factor in such a way. The more usual methods involve a lot of frequency, current, voltage, and resistance measurement and more data than presented here. However, it is very useful to understand the concept and be able to recognise it when you see its effects.

THE SLAYER EXCITER

One of the more exciting uses of an LC circuit in our recent memory was the Slayer Exciter that we built in Issue 41. In this build, we made a tank circuit without an actual capacitor. This device involves two coils. One is a few turns of high-current wire, the primary, wrapped loosely around a great many more turns of fine wire, the secondary.

The circuit is connected in such a way that when current is applied to the primary, a voltage is induced in the secondary. The whole assembly functions a bit differently to the traditional inductor, but the important point is, the secondary is the inductor in one half of a tank circuit. The capacitor is there too, but you can't see it: It is the air gap between the top of the secondary, and the ground plane.

This arrangement permitted a wild increase in voltage generated, and was able to produce sparks and corona discharge, as well as lighting neon globes, fluorescent tubes, and compact fluorescent bulbs as well. The circuit theory on pages 65 to 67 of Issue 44 is well worth a read. You can access the whole article, called 'Sparking Ideas', for free online. The circuit has some quirks unlike any other application of either LC circuits or transformers that we have ever seen, and does seem counterintuitive at times. The results, however, are undeniable and visually spectacular.

SUMMARY

We hope this answers some of the questions you may have had over LC circuits as you progress on your electronics journey. There was a lot of pretty dry information here, and we had to pick and choose what to say and what not to say. We're never exactly sure who the audience is for Classroom, as makers come from all backgrounds with all levels of knowledge. Some makers and even plenty of experienced engineers or tradespeople, have gaps in their knowledge. However, we pitch to those who are just exploring, which is why we chose to gloss over or not cover some things that
an engineer may think is necessary for a complete explanation.

They're probably right, but there is only so much information someone who is learning can absorb, and someone who has come to the hobby from another background may not think the same as those who traditionally did so through tinkering with electronics.

However, we hope you have been able to take home some points to consider in your future endeavours:

• Inductive reactance increases as frequency increases. Capacitive reactance decreases as frequency increases.
• Resonance occurs when inductive and capacitive reactance have the same value.
• Parallel LC circuits have maximum impedance at their resonant frequency, while series LC circuits have their minimum impedance at the resonant frequency.
• You can pass or block a certain frequency with an LC filter, depending on whether it is series or parallel, and how you connect that block to the rest of the circuit.
• Resistance is everywhere, and leads to damping of resonance if not controlled well
• Bandwidth and Q-factor are related concepts, but calculating them is really beyond the level of this article. Just remain aware that bandwidth can vary, and the calculated frequency is never the only one passed or blocked down to the single Hz chosen.
• The Slayer Exciter is a really fun use of an LC circuit and you should totally build one and show us!