 ## Impedance, the Skin Effect, and their Implications in High Frequency Circuits

Many of the famous experiments Nikola Tesla performed involved high frequency alternating- or impulse currents. At these high frequencies, something interesting happens: these rapidly vibrating currents pass with great difficulty through a seemingly low resistance conductor. This effect is caused by impedance, or the opposition of a conductor to the flow of alternating current. This article explains impedance in detail, since it plays a crucial role in understanding high frequency circuits, and because the subject has more layers of complexity to it that one might expect.

Let’s start by defining some key terms:

Resistance R: friction against DC

Reactance X: inertia against AC

Impedance Z: combination of resistance and reactance

Of course these short definitions don’t tell you too much, but it does show you that these three concepts are intertwined, and so we will structure the rest of our exploration of impedance around them, starting with the easiest one: resistance.

## Resistance

Most people are familiar with resistance, which has the symbol R, is measured in ohm Ω, and is a way to state how difficult it is for a direct current to pass through a conductor. For example, copper has a very low resistance, so you don’t need a lot of pressure (voltage) to push a current through it, whereas lead, which is conductive, has about 12 times more resistance, and therefore requires 12 times more voltage to push the same amount of current through it.

This relation between resistance, voltage, and current is expressed in the formula for resistance: Besides the type of material that is being used, the shape of the conductor also has a large influence on resistance. For example, a thin, long wire has a higher resistance than a thick, short wire, just like it is harder to push water through a thin, long straw compared to a short, thick pipe.

It is very important to reiterate that resistance applies to DC currents only, since the opposition against alternating currents is called reactance, which we will cover next.

## Reactance

I just mentioned reactance is inertia against AC currents, but more precisely, it is the opposition of a circuit element to a change in voltage, and since AC changes its voltage constantly, reactance mainly applies to AC.

Changing currents, especially of high frequency, have a significant effect both on capacitors, as well as on inductors, so there are two parts to this puzzle:

1. Capacitive reactance
2. Inductive reactance

We will discuss both of these in detail, since each produces unique effects that are crucial in understanding some of Nikola Tesla’s experimental results.

### Capacitive reactance

When you apply a DC current to a capacitor, the capacitor will simply charge until it reaches the level of the supply voltage. In such a case there is no current at all flowing “through” the capacitor. The flow of current ends on one plate of the capacitor, never reaching the other side. However, when a high frequency current is applied, things change dramatically, as mentioned by Tesla:

“Another equally remarkable feature of high frequency impulses was found in the facility with which they are transmitted through condensers [capacitors], moderate electromotive forces and very small capacities being required to enable currents of considerable volume to pass.” 1

The reason high frequency currents pass through a capacitor with such ease, is because of something called capacitive reactance, which describes the opposition of a capacitor to a change in voltage. The amount of current passed through a capacitor is related to the capacitance and the rate of voltage change, according to the following formula: Since voltage does not change for a steady DC current, apart from when the current is first switched on, the current through the capacitor equals zero. However, as frequency increases, the rate of change of voltage over time increases, letting more current pass the higher the rate of change.

Some feel it is incorrect to say that currents pass through a capacitor, since electrons do not actually move from plate to plate. However, when I read up on what it is then that causes a current to appear on the other side of the capacitor when high frequency currents are applied, wildly different explanations were offered.

The explanation that made the most intuitive sense to me, was that one could view the dielectric between the capacitor plates as a flexible membrane, that, when hit with a large pressure, can oscillate. This, then, would not actually let any current through, but it would apply pressure to the other side of the capacitor by pushing into it and then retreating again, setting up a wave. This would also explain why a DC current does not create a current on the other side of the capacitor, since a steady pressure does not make the membrane oscillate.

Not sure if this explanation is wholly accurate of what actually happens, but it gives us a decent mental model to understand the behavior of capacitive reactance.

The formula for capacitive reactance itself is: If you fill in 0Hz in the above formula, which describes a DC current, capacitive reactance approaches infinity, therefore blocking DC. However, if you fill in higher and higher frequencies, capacitive reactance starts to approach zero, therefore acting as a short circuit, letting the alternating current pass.

#### High pass filters

In audio engineering, a high-pass filter is a circuit which blocks low frequency signals from the waveform, and only lets high frequency signals through. As you might have guessed, high pass filters work by making use of the effects of capacitive reactance. Only above a certain frequency, called the cut-off frequency, the reactance of the capacitor in the circuit becomes low enough for a signal to pass.

This cut-off frequency can be calculated as follows: In this sense, we could say that the capacitors in a Tesla Hairpin circuit act as high pass filters, blocking the 50 or 60Hz input signal coming from the transformer, allowing the capacitors to charge, but letting the high frequency pulses from the spark gap through with ease.

#### Frequency vs rate of change

So far I’ve used the terms “frequency” and “rate of change” interchangeably, but there is in fact a subtle distinction that should be made between the two, which becomes clear when Tesla describes his Hairpin circuit:

“These results, as I have pointed out previously, should not be considered to be due exactly to frequency but rather to the time rate of change which may be great, even with low frequencies.” 2

We already saw before that the “time rate of change” is calculated by: For example, if we assume we have a 10kV, 50Hz power supply, then each cycle takes 0.02 seconds (1 second / 50Hz). The rate of change is then 500kV/s (10kV / 0.02). Now, this calculation assumes we’re working with a sine wave, but if we instead disruptively discharge a condenser 50 times per second, where each discharge only takes 0.01 seconds, so half the time of one sine wave oscillation, the rate of change increases from 500kV/s to 1000kV/s, while the frequency is still 50Hz!

We can also achieve a 1000kV/s rate of change with a sine wave, but then we either have to double the voltage at 50Hz, or double the frequency to 100Hz, using the original 10kV. These calculations clearly show that, yes, a higher frequency leads to a higher rate of change, but that by using rapid discharges we can achieve a “time rate of change which may be great, even with low frequencies.” With discharges, frequency is the time between pulses, while the rate of change is determined by the duration of a single pulse.

This also has implications for the capacitive reactance formula mentioned before, which contained frequency in Hz as one of its denominators. This formula works as long as we’re dealing with sine waves, but for disruptive discharges we have to replace with , also known as angular velocity. This results in the following adjusted formula: We will see a similar theme when we describe the next component of reactance: inductive reactance.

As you can probably tell, I have a strong aversion to the traditional notation of mathematical formulas, especially variable names, since they trade off understandability for compactness, making them completely unreadable for the uninitiated. One is expected to know what a random Greek letter stands for, and also the unit used is often only implied (are we dealing with Hz or kHz, ohms or milliohms?), or, if one is lucky, the unit can be found in the text near the formula.

I develop software and so I work with variables all the time. HOWEVER, coding conventions state that one should “avoid ambiguous and small [variable] names which are hard to understand. Names should be descriptive such that it should tell what it is for” 3.  This is the reason why I use more verbose variables in my formulas. They are far from perfect as well, but at least it is clear what each variable stands for.

### Inductive reactance

Where capacitors let more current through the higher the frequency, conductors and inductors behave in exactly the opposite way, providing more opposition to current flowing as frequency and/or inductance increase. The reduction of current flow in a conductor due to induction is called inductive reactance 4.

So why does a conductor oppose the flow of AC current at all? The reason can be found in Lenz’s law, which states that “an induced current has a direction such that its magnetic field opposes the change in magnetic field that induced the current.” 4 In other words, when a current is passed through a conductor, a magnetic field is created, which in turn induces a current in the same conductor (self-induction) in the exact opposite direction, a so-called “back emf”, thereby opposing the initial current that generated the magnetic field in the first place.

Since a changing magnetic field induces a voltage that is directly proportional to the rate of change of the current producing it, this means that when the rate of change is doubled, the self-induced back emf is doubled, reducing the flow of current through the conductor accordingly. This is captured in the following formula for inductive reactance: As you can see from the formula, if frequency or inductance is increased, so is inductive reactance. Similar to the capacitive reactance formulas, the above formula can also be rewritten to apply better to pulse currents, like so: So the shorter the discharge time, the stronger the effect of inductive reactance.

Tesla once mentioned:

“One of the prominent characteristics of high frequency or, to be more general, of rapidly varying currents, is that they pass with difficulty through stout conductors of high self-induction.” 1

We already learned that high frequency currents have a high rate of change, and therefore set up a strong back emf, causing the current to “pass with great difficulty” through a conductor.

Tesla also mentions that this effect is present in “stout conductors of high self-induction”, and on another occasion he said that “the thicker the copper bar… the better it is for the success of the [impedance] experiments” 2. Why? Because the thicker the bar, the larger the surface area, the greater the self-inductance 3, and the more powerful the effect of inductive reactance, or inertia to AC.

### The skin effect Figure 2. Cross-section of a wire showing the Skin Effect, where the orange stands for the amount of current flow

We’ve seen how circuits behave differently to DC versus high frequency AC. One of the most curious effects is the so-called “skin effect”, which causes alternating currents to flow closer to the surface, rather than through the centre of a conductor, effectively increasing the resistance, since less volume is available for the current to flow through. In fact:

“For sufficiently high frequencies… the conductor might as well have a hollow core, as the central region of the conductor carries essentially none of the current.” 4 Figure 3. A 50kW radio transmitter, using hollow copper tubes coated with silver, to have great conductivity at the “skin”, where most of the current will be flowing.

As the image above shows, the skin effect is a well known phenomena which has practical implications. This is also why stranded Litz wire is often used to transmit radio frequency AC, since it maximizes surface area of the conductor(s) and therefore minimizes losses due to the skin effect.

The first to mention of the skin effect, sometimes called the “thick wire effect”, was mathematical genius Oliver Heaviside around 1883. Heaviside also championed the idea of induction, and is the one who transformed Maxwell’s original quaternion equations into their vector form still used today by engineers around the world, so he was a real heaviweight (pun intended). To understand Heaviside’s explanation of the skin effect, we must, however, first understand how he, like John Henry Poynting, believed that all energy propagation along the direction of the wire takes place outside of the wire instead of through it:

“[Energy transfer] takes place in the vicinity of the wire, very nearly parallel to it, with a slight slope towards the wire… It causes the convergence of energy into the wire.” 4

What Heaviside is saying is that the energy is induced into the wire from the dielectric surrounding it. This leads him to the following explanations of the skin effect:

“Since on starting a current the energy reaches the wire from the medium without, it may be expected that the electric current in the wire is first set up in the outer part, and takes time to penetrate to the middle. This I have verified by investigating special cases. Increase the conductivity of a wire enormously, still keeping it finite, however. Let it, for instance, take minutes to set up current at the axis. Thus, ordinary rapid signalling ‘through the wire’ would be accomplished by a surface current only, penetrating but a small depth.” 4

And:

“Having been, so far as I know, the first to correctly describe the way the current rises in a wire, viz, by diffusion from its boundary, and the consequent approximation, under certain circumstances, to mere surface conduction.” 4

So if we accept that energy is induced into a conductor from without, then Heaviside’s explanation makes sense: the energy reaches the outer skin of the conductor first, and does not have the time to penetrate deeply into it when oscillating at high frequencies.

Now let’s see what a modern textbook tells us about the skin effect:

“Skin effect: The tendency of alternating current (ac) to flow near the surface of a conductor, thereby (a) restricting the current to a small part of the total cross-sectional area and (b) increasing the resistance to the flow of current.” 5

A fair definition. Now what can the book tell us about the cause of the skin effect?

“Skin effect is caused by the inductance of the conductor, which causes an increase in the inductive reactance especially at high frequencies. The inner filaments of the conductor experience an inductive reactance with all the surrounding filaments, their reactance thus being higher than the outer filaments. Thus, the current tends toward the lower reactance filaments, i.e. the outside filaments. At high frequencies, the circumference is a better measure of resistance than the cross-sectional area. The depth of penetration of current at thigh frequencies can be very small compared to the diameter. Skin effect must be taken into account when designing antennas and metallic waveguides.” 5

So instead of Heaviside saying the current starts at the outside of the wire and does not have time to reach the centre, this explanation says that the skin effect is caused by a stronger inductive reactance at the centre of the conductor compared to the outside of the conductor, effectively pushing the current towards “the path of least reactance”, which is near the surface.

On Wikipedia I found yet another explanation, which highlighted eddy currents as the cause:

“The skin effect is due to opposing eddy currents induced by the changing magnetic field resulting from the alternating current.”

We know how to perform accurate calculations with electricity, but still don’t have a clue what electricity actually “is”. In the same vein, we can calculate the skin depth at any given frequency, but don’t seem to agree on what causes the phenomenon in the first place. This twilight zone seems to be the unfortunate fate of science in general, and physics in particular. While all explanations we just discussed of what causes the skin effect seem plausible, “the mere absence of nonsense may not be sufficient to make something true”, as Nassim Taleb once aptly said.

Thus, we are left with uncertainty about the exact cause of the skin effect, but at least everyone agrees that it increases resistance to current flow significantly at high frequencies. This is why I believe that the skin effect is a major cause for the unique Impedance Phenomena Nikola Tesla showcased in his Hairpin experiments.

## Impedance

Well, that was quite the journey to finally arrive at the overarching phenomenon of impedance, another term coined by Oliver Heaviside 6. We already learned that when applying direct current (DC) to a circuit, there is no distinction between impedance and resistance. However, when an alternating current (AC) is applied, reactance leads to an increase in opposition to current flow. I tried to capture the relationships of these concepts in the image below.

The formula for impedance provides another way to show these relationships: It is important to mention that for a series resonant circuit, like the Tesla Hairpin circuit, impedance is at its minimum and current at its maximum at the resonant frequency. Conversely, for a parallel resonant circuit, impedance is at its maximum and current at its minimum at the resonant frequency:

“The resonance of a series RLC circuit occurs when the inductive and capacitive reactances are equal in magnitude but cancel each other because they are 180 degrees apart in phase.” 7

To wrap things up, I would like to share with you an old but must-see video demonstration, titled Similarities of Wave Behavior. In this video, the presenter uses a physical wave device to show, amongst other highly interesting things, the way (electric) waves reflect when they encounter a conductor with a different impedance:

## Concluding words

We covered a lot of ground in this article, but I feel that a detailed understanding of impedance is a necessary prerequisite to understanding the behavior of high frequency circuits. It seems that we have now covered enough fundamental subject matter to take an informed shot at properly replicating Nikola Tesla’s Hairpin circuit and its curious effects. The next blog post will describe my own Hairpin replication, including a parts list, experimental results, and a detailed circuit analysis, so everyone will be able to follow along and duplicate the results. Feel free to leave some honest feedback in the comments below!