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In an animal cell, especially neuron and in particular its axon, while there is electrical resistance and capacitance mechanism in the cell, which play essential roles in the cable theory model of neuronal action potential transmission, is there a prominent self inductance mechanism in the sense of electromagnetism?

What one thinks, no matter how intuitive it may appear is not particularly relevant in science. The inductance associated with a neural axon has been well documented since Cole (1966). Its role in the propagation of neural signals is developed extensively in http://neuronresearch.net/hearing/pdf/7Projection.pdf#page=39 . The actual development begins earlier in Section 7.4 on page 322 of that document.

Failure to consider the inductance associated with any alternating electrical signal passed along a coaxial cable leads to disaster. The first undersea cable based on the ideas of William Thompson,Lord Kelvin, and described as an RC cable by Hermann (page 322 in the above document) was a technical and financial disaster. Two years later, a more sophisticated RLC cable based on Maxwell's Equations for a coaxial structure was laid with great success. No RC cable has ever been used in practice since that time. For unknown reasons, the biological community keeps trying to ignore the inductance of the coaxial myelinated axon (leading to ridiculous modeling data). This appears to be the result of introductory courses in electricity for non-engineers trying to avoid the necessary mathematics to understand electromagnetic signal propagation through space and along various types of cables and waveguides.

There surely is inductance in neurons. This inductance is introduced by two different mechanisms. 1. The coil structure of myelin sheaths can introduce a real electrical inductance. The solid evidence for this is the opposite spiraling directions between the adjacent myelin sheaths.

Here I quote the description on Wikipedia:*At the junction of two Schwann cells along an axon, the directions of the lamellar overhang of the myelin endings are of opposite sense.* You can also check the details in this paper: Uzmman B. G.; Nogueira-Graf G. (1957). "Electron microscope studies of the formation of nodes of Ranvier in mouse sciatic nerves". Journal of Biophysical and Biochemical Cytology. 3 (4): 589-597. doi:10.1083/jcb.3.4.589

The opposite spiraling directions can introduce a positive mutual inductance between adjacent myelin sheaths then further enhance the propagation speed of the neural signal. Meanwhile, it is easy to predict that the myelinated nerve can be stimulated by a magnetic field because of this coil inductor. Because of this opposite spiraling direction, the stimulation result is determined by the spatial gradient of the magnetic field. This phenomenon was validated by years and can be easily understood now.

- The piezoelectric effect of the plasma membrane. If you check the molecular structure of the lipid bilayer of the plasma membrane, you will find that it is naturally a piezoelectric layer. The definition of the piezoelectric layer is a layer consists of two crystal layer with opposite polarities, which is exactly the same as the lipid bilayer. The equivalent circuit of this piezoelectric layer will be a RLC circuit, which contains an equivalent inductor. Since the inductor is not a real one, the value is only used to match with the mechanical resonance frequency. When the mechanical resonance frequency is very low, which is the case for a soft and thin plasma membrane, this inductance will be huge. This is why in Cole's paper for measurement of the squid axon, this inductance is 0.2H. Then as a direct prediction, there should be a mechanical wave accompany with the electric signal of the action potential. This mechanical wave has been measured in this paper: Gonzalez-Perez, A., Mosgaard, L.D., Budvytyte, R., Villagran-Vargas, E., Jackson, A.D. and Heimburg, T., 2016. Solitary electromechanical pulses in lobster neurons. Biophysical chemistry, 216, pp.51-59.

I think here I already give a comprehensive answer to this question. You can check all the details in this paper on bioRix:https://www.biorxiv.org/content/early/2018/10/22/343905

Here I may talk something more, but these things will make most of the people in neuroscience unhappy. If this inductance introduced by the coil structure of myelin and the piezoelectric effect is true, then the whole neuroscience is wrong from the Day 1. The H-H model is built based on a RC circuit and so many people have developed their theories and models based on this H-H model. But ridiculously, everyone claims his model or theory is correct and can explain the data when tha basis is wrong. I have seen so many absurd explanations to bypass this inductance, such as the frequency-dependent capacitor, virtual cathode, negative resistor, and even negative capacitor. And indeed, more and more people begin to realize the so-called neuroscience is a complete failure. You can see in the homepage of Neuralink(https://neuralink.com/), they officially claim that they do not need any neuroscience experience, quote here:*No neuroscience experience is required* Also, there are many groups now only use deep learning or machine learning to study neurons.

## Inductance in cell - Biology

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## Inductance in cell - Biology

In electricity there are two forms of inductance. Mutual inductance and self inductance.

Mutual inductance is what happens in a transformer. When changing current in one oil causes a current in another coil. The unit of mutual inductance is Henry (H)

Change in Magnetic field is proportional to current.

That is φ / I is a constant (M) Therefore φ = m X i.

We know that Volts = change in flux / time. That is V = N x Δ φ / Δ t .

Substituting the value of φ in the above we get :- V = M . Δ I / Δ t .

Inductors are used to tune radio circuits, filter out unwanted noise and in various other electronic circuits.

The rate of change in magnetic flux is proportional to voltage of the induced current

The direction of the induced current causes a force to oppose The change that caused it.

The current in primary increases from zero to 0.8A in 2.5 s. The secondary coil Voltage was 1.2 v.

Rate of increasing current.I/t

The time taken for the current to drop to zero at the rate of 0.2A/s.

The induced voltage during the drop.

Inductor and self induction. .

An inductor can simply be a coil of wire with a several attached to an electrical circuit. When a current is passed it develops a strong magnetic field. According to Faraday’s law , when the current is building up , as there is a change in magnetic flux, a current is induced. According to Lenz’s law the induced current will be in the opposite direction to the current that caused the change. So this is sometimes referred to as the back emf.

When the inductor is charging the equation V= ir has to be modified as

Supplied voltage - Inductor voltage = ir.

This can be written as V s - V L = ir

Charging and Discharging an Inductor.

When the circuit is switched on, the bulb glows brightly and becomes dim. Then again when switched off the bulb becomes bright and goes off.

Let us use the following data for a calculation:-

Normal resistance of circuit = 0.3 Ω.

Time taken for current to be steady = 0.2 seconds.

Calculating the steady current in the circuit

Using V - ir we get 1.5 = i x 0.3

∴ i = 1.5 / 0.3 = 5 .0 A . (amperes)

Induced voltage in resistor

In the absence of supply current this can give a current of I= v/r

This is the reason for the bulb to be bright on switching on and off.

Energy stored in an inductor

The basic unit of measurement for inductance is the Henry, ( H )

It can also be given as Webers per Ampere ( 1 H = 1 Wb/A ).

a.Find the induced voltage when a current of 5A in a inductor of 3 H. is reduced to zero in 0.2 seconds.

b. Find the energy stored i the capacitor.

b) Energy = ½ L x 5 x 5 = 0 .5 x 3 x 5x5 = 37.5

Her the inductor and the resistance symbol are enclosed inside a rectangle . That shows it is the resistance of the coil.

## Lift-off Invariant Inductance of Steels in Multi-Frequency Eddy-Current Testing

How to cite: Lu, M. Meng, X. Huang, R. Chen, L. Peyton, A. Yin, W. Lift-off Invariant Inductance of Steels in Multi-Frequency Eddy-Current Testing. *Preprints* **2021**, 2021040387 (doi: 10.20944/preprints202104.0387.v2). Lu, M. Meng, X. Huang, R. Chen, L. Peyton, A. Yin, W. Lift-off Invariant Inductance of Steels in Multi-Frequency Eddy-Current Testing. Preprints 2021, 2021040387 (doi: 10.20944/preprints202104.0387.v2). Copy

### Cite as:

Lu, M. Meng, X. Huang, R. Chen, L. Peyton, A. Yin, W. Lift-off Invariant Inductance of Steels in Multi-Frequency Eddy-Current Testing. *Preprints* **2021**, 2021040387 (doi: 10.20944/preprints202104.0387.v2). Lu, M. Meng, X. Huang, R. Chen, L. Peyton, A. Yin, W. Lift-off Invariant Inductance of Steels in Multi-Frequency Eddy-Current Testing. Preprints 2021, 2021040387 (doi: 10.20944/preprints202104.0387.v2). Copy

## Chapter 1. How to Understand the Inductance?

The primary purpose of this section is to provide an in-depth understanding of the nature of inductance to biological researchers, who typically do not have a substantial physical background. It will be beneficial for the illustration and understanding of the whole theory proposed in this study.

### The Nature of an Inductor

In an actual circuit, an inductor is an electronic component for storing energy in the form of a magnetic field. However, in most cases, biological researchers are not studying an actual circuit but an equivalent circuit that is modeled from some biological tissue or organism, for instance, an equivalent neural circuit. In this kind of equivalent circuit, an inductor is not an actual unit but a symbol for reproducing the voltage oscillation and resonance frequency measured in electrophysiological tests. Since the voltage oscillation and resonance frequency are the typical characteristics of an RLC circuit, adding an inductor in the equivalent circuit becomes inevitable.

However, the actual phenomena to be observed in tests are the voltage oscillation and resonance frequency, which is not directly associated with the existence of inductance in the equivalent circuit. There are a lot of cases which can generate oscillation and resonance frequency without the presence of inductance. One example is a simple pendulum, as shown in Figure 1A . Another example is the one-dimensional harmonic oscillator, as shown in Figure 1B . In these two cases, there is no presence of inductor, but the oscillation and resonance frequency exists. A system with simple harmonic motion can always be modeled as an RLC circuit, as shown in Figure 1C . Here we need to emphasize two points, which are critical to the theory in this study:

Illustration of the nature of inductance. **(A)** The case of a simple pendulum **(B)** The case of the one-dimensional harmonic oscillator **(C)** A system with simple harmonic motion can always be modeled as an RLC circuit **(D)** The observed voltage oscillation and resonance frequency in all systems with simple harmonic motion.

The reason for the oscillation and resonance frequency is that the total energy of the whole system has a conversion between different energy forms. In the case of the simple pendulum, the energy conversion is between the gravitational potential energy and kinetic energy. In the case of the one-dimensional harmonic oscillator, the energy conversion is between the elastic potential energy of the spring and kinetic energy of the oscillator. In the case of an actual RLC circuit, the energy conversion is between the electric field in the capacitor and the magnetic field in the coil inductor. Therefore, the inductor in an equivalent circuit means there is an energy conversion between two forms.

Adding an inductor in the equivalent circuit is to reproduce the oscillation and resonance frequency, as shown in Figure 1D . The inductor itself does not necessarily have a physical meaning, and its value can be unrealistic compared with the one in an actual circuit. In the case of the pendulum, the swinging frequency can be very low, which is about 1 Hz. Based on the equation to calculate the resonance frequency, f = 1 2 π L C , a huge value of the inductance can be obtained. This huge inductance can never happen in an actual circuit but is quite normal in an equivalent circuit.

### The Potential Fallacy of Conventional Neuroscience

By understanding the two points mentioned above, it is quite clear to see what is theoretically inadequate with the conventional neuroscience. There is a large inductance in neural systems, which has been reported in a lot of studies (Cole and Baker, 1941 Curtis and Cole, 1942 Hodgkin and Huxley, 1952 Sjodin and Mullins, 1958 Araki et al., 1961 Freeman, 1961 Huxley, 1963 Ranck, 1963 Guttman, 1969 Mauro et al., 1970 Scott, 1971 Homblé and Jenard, 1984 Hutcheon and Yarom, 2000 Dwyer et al., 2012 Thomas, 2013 Mosgaard et al., 2015 Kumai, 2017 Rossi and Griffith, 2017). The evidence of this inductance, as explained above, is the voltage oscillation and resonance frequency measured in experiments. The first study of this large inductance is the paper proposing the H-H model (Hodgkin and Huxley, 1952). The measured inductance can be about 0.21 to 0.39 H, which is much higher than a reasonable value of a physical coil inductor. Based on their proposal, this large inductance is induced by the impedance change of the ion channels. This bizarre phenomenon also aroused lots of other theoretical guesses in later studies, such as frequency-dependent membrane capacitance (Howell et al., 2015), negative resistance (Rissman, 1977), and negative capacitance (Takashima and Schwan, 1974).

Nevertheless, all of them were misled by two points emphasized here:

They considered the voltage oscillation and resonance frequency as the evidence of the inductance.

They believed that a coil is an exclusive origin accounting for the inductance in an equivalent circuit.

With the clarification of these two points, a better theoretical hypothesis can be taken into consideration. The inductance in the neural circuit means there is a kind of biological structure that can store the energy in a non-electrical form. Since the cell membrane is typically modeled as a capacitor, the energy conversion happens between the electrical field stored in the cell membrane and some unknown form stored in an unknown biological structure.

The same idea was first proposed by Cole (1941). It was said in the paper that the measured large inductance in neurons could be raised from the piezoelectric effect of the cell membrane. The energy conversion happens between the electrical field in the cell membrane and the surface tension by the piezoelectric effect. Since the inductance is calculated from the resonance frequency f = 1 2 π L C , its value can be quite large if the resonance frequency is very low. However, at the time of 1941, the lipid bilayer structure of the cell membrane, which is naturally piezoelectric, remains unknown to Kenneth S. Cole, he proposed this idea as a hypothesis. It is a pity that his opinion drew no attention in later research. We will make a detailed discussion of this point below.

In recent years, the use of induction-heating systems has increased and wireless power transmission (WPT) systems have been discussed. These applications are installed close to a human body. Therefore, it is important to discuss the effects of alternating magnetic fields and to evaluate electromagnetic interference. This paper discusses the design procedure of a magnetic field generator to evaluate the electromagnetic interference at 85 kHz that is being studied in WPT systems for EV and HEV. The magnetic field generator presented in this paper consists of a single-phase inverter circuit that uses SiC-MOSFETs and an air–core inductor that is used as the coil for generating a magnetic field. In particular, this paper shows that the coil used for generating magnetic field needs to reduce the winding voltage to generate higher magnetic flux. In addition, this paper presents the design procedure of the proposed coil structure that can satisfy some limited conditions. The experimental results of the proposed system rated at 82 kHz and 100 A are presented.

In the recent years, induction heating devices using magnetic field of 20 kHz to 100 kHz have spread rapidly in both industrial 1 and home 2, 3 applications. In addition, 85-kHz wireless power transmission (WPT) systems were proposed for electric vehicles, and extensive research and development (R&D) is conducted on power supplies, power transmission coils, and other components. 4-7 Transmission coils of WPT systems for electric vehicles are supposed to be used in the vicinity of other electronic devices and human bodies, and there are concerns about electromagnetic interference and biological effects due to alternating magnetic fields generated by these coils. Therefore, biological effects of 85-kHz alternating magnetic fields must be evaluated through experiments. 8-10

Some systems were developed to evaluate biological effects of alternating magnetic fields there are reports about systems operated at 200 Hz to 100 kHz. 11-14 We studied experimental systems to evaluate biological effects of 85-kHz magnetic field, and explored structure of magnetic field generators using air–core solenoid coils. 15 However, with double-layer solenoid coils offering optimal magnetic flux density, voltage of the kilovolt order may occur between coil windings, and possibility of dielectric breakdown must be taken into account in design of magnetic field generation coils.

In this paper, we aim at development of a magnetic field generator in 85-kHz band intended for evaluation of biological effects particularly, we propose a double-layer solenoid coil with reduced maximum voltage between windings. First, in order to combine winding voltage reduction with magnetic flux density output comparable to conventional design, we propose an improved structure of double-layer solenoid coil, and demonstrate usefulness of this coil in magnetic field generation for evaluation of biological effects. Then, we derive a practical coil structure by numerical analysis, and manufacture a prototype of double-layer solenoid coil with water-cooling copper pipes. In addition, we show through experiments that winding voltage can be reduced in the prototype coil. Finally, we conduct experiments at 82 kHz and 100 A with a circuit in which a resonant capacitor and the improved solenoid coil are connected in series to single-phase output using MOSFET to confirm that the proposed coil can be used for evaluation of biological effects.

## Inductance Energy, Dennis Danzik's Earth Engine

I have noticed that some lawsuits have been filed, apparently for breach of contract, but like most, they're continually dragged out with no resolution.

Byron Wyoming was welcoming them, I also see a National Training Center in Scottsdale AZ.

Website claims to have produced actual generators, but I can't find any info on who has them, where they are and how they're producing.

### Earth Engine - Inductance Energy

ie.energyThe vast majority of internet news pertains to it all being another Dennis Danzik scam, because it's very nature and premise defies the known laws of physics.

Many have assumed it's demonstrator is either powered or nothing more than a ruse and a kinetic flywheel.

For anybody not familiar, here's a link to a Youtube video -- there's more if you search Dennis Danzik, or Inductance Energy, or Earth Engine.

I've also not read of any Patents being issued.

I'm just curious if anybody has kept up with this, no real updates or recent internet chatter, yet seems to be the perfect solution for CA's power problem, in fact the entire world's.

### Earth Engine Live Demo | Steorn's Orbo

dispatchesfromthefuture.com#### Spectras only

##### Well-Known Member

#### Chili Palmer

##### Master of My Domian

#### Spectras only

##### Well-Known Member

#### Monkeyswrench

##### Well-Known Member

#### Taboma

##### Well-Known Member

Well I have been waiting to see, for more than two years, I'm sure his investors are as well. I see a fancy website with big claims, youtube vids of the same, info showing a national training center and satellite office in Nevada. Claims to have produced working generators, yet I can find nothing to substantiate any of his or the company's claims.

By all appearances this is another carburetor that claims to get 100 mpg, or some other miracle that the oil and energy company's have bought out, or killed off, or ?

What I find odd is, the internet chatter regarding this apparent hoax, has grown extremely quiet since the 2019.

I brought this up because I was going back through old bookmarks and it popped up, when I tried to follow it for more recent updates, I came up crickets. So was curious if anybody on RDP had kept up, or was aware of any recent revelations.

## Keywords

**Siyang Zheng** received the BS degree in biological sciences and biotechnologies from Tsinghua University, Beijing, China, in 1996. He received his MS in electrical engineering from Pennsylvania State University, University Park, PA, USA, in 2000. He worked at Lucent Technology, Holmdel, NJ, USA for one year before he was awarded Atwood Fellowship and joined the Micromachining group at California Institute of Technology (Caltech), Pasadena, CA, USA as a graduate student in 2002. He obtained his PhD in electrical engineering from Caltech in 2007 under the guidance of Dr. Yu-Chong Tai. Currently he is working as a postdoc researcher at the same institute. His broad research experiences covered lab-on-a-chip system for cell separation and analysis, electrical impedance sensing, laser-induced fluorescence detection, MEMS sensors/actuators, micro/nano fabrication, on-chip molecular self-assembly, microfluidic integration technology. He has published more than 20 peer reviewed journal/conference papers and filed several patents over the years.

**Mandheerej Singh Nandra** earned his BASc degree from the University of Toronto, specializing in Nanoengineering. He joined the Caltech Micromachining Lab in 2005 where he received his MS in electrical engineering and is currently working towards his PhD. His research interests include MEMS for retinal implants and integrated electronics for bioMEMS.

**Chi-Yuan Shih** received his BS and MS degrees in electrical engineering from National Tsing Hua University (NTHU) in 1996 and 1998 in Taiwan. In June of 2001, he was awarded the Killgore Fellowship and joined the Department of Electrical Engineering at California Institute of Technology (Caltech). He then got the MS and PhD degree in electrical engineering in 2002 and 2006. Currently he works for Taiwan Semiconductor Manufacturing Company (TSMC). Between 2001 and 2006, Mr. Shih worked as a graduate researcher in the Caltech Micromachining Laboratory under the advice of Prof. Yu-Chong Tai. His broad research experiences covered on-chip separation technology, laser-induced fluorescence detection, MEMS sensors/actuators, micro/nano fabrication, on-chip molecular self-assembly, microfluidic integration technology and ferroelectric material flash memory devices. He has published more than 10 journal/conference papers and filed several industrial patents over the years. His PhD thesis is focused on chip-based High Performance Liquid Chromatography (HPLC) systems. Recently, Mr. Shih was awarded the distinguished Study Abroad Fellowship by Ministry of Education in Taiwan for his outstanding academic and research achievements. He was also awarded the 2004 Phi Tau Phi Honor Scholastic Scholarship in America. Mr. Shih is a member of IEEE, California Separation Science Society (CaSSS) and The Chinese Institute of Engineers.

**Wei Li** was born in China in 1981. He will receive his BS degree in electrical engineering from California Institute of Technology (Caltech), Pasadena, USA, in 2007. He joined Dr. Yu-Chong Tai's Caltech MEMs group as an undergraduate researcher in 2005. The research he participated in was On-Chip Temperature Gradient Liquid Chromatography. In 2006, he joined Dr. Axel Sherer's NanoFabrication Laboratory as a summer undergraduate research fellow. His project was Nano-Structures Enhanced InGaN/GaN Light Emitters. In addition, he did his senior thesis on Micro Power Generator and High Efficiency Battery Circuits in Caltech MEMs group again. He is current at MIT to pursue his PhD degree in electrical engineering.

**Yu-Chong Tai** received his BS degree from National Taiwan University, and the MS and PhD degrees in electrical engineering from the University of California at Berkeley. After Berkeley, he joined the faculty of electrical engineering at the California Institute of Technology and built the Caltech MEMS Lab. Not long ago, he joined the Bioengineering department and he is currently a professor of electrical engineering and bioengineering at Caltech. His current research interests include flexible MEMS, bioMEMS, MEMS for retinal implants, parylene-based integrated microfluidics, neuroprobes/neurochips, HPLC-based labs-on-a-chip. He has received several awards such as the IBM fellowship, the Best Thesis Award, the Presidential Young Investigator (PYI) Award and the David and Lucile Packard Fellowship. He co-chaired the 2002 IEEE MEMS Conference in Las Vegas. He is currently a subject editor of the *Journal of Microelectromechanical Systems*.

## Pluripotency induction in HEK293T cells by concurrent expression of STELLA, OCT4 and NANOS2

Germline stem cells (GSCs) are attractive biological models because of their strict control on pluripotency gene expression, and their potential for huge epigenetic changes in a short period of time. Few data exists on the cooperative impact of GSC-specific genes on differentiated cells. In this study, we over-expressed 3 GSC-specific markers, STELLA, OCT4 and NANOS2, collectively designated as (SON), using the novel polycistronic lentiviral gene construct FUM-FD, in HEK293T cells and evaluated promoter activity of the Stra8 GSC marker gene We could show that HEK293T cells expressed pluripotency and GSC markers following ectopic expression of the SON genes. We also found induction of pluripotency markers after serum starvation in non-transduced HEK293T cells. Expression profiling of SON-expressing and serum-starved cells at mRNA and protein level showed the potential of SON factors and serum starvation in the induction of ESRRB, NANOG, OCT4 and REX1 expression. Additionally, the data indicated that the mouse Stra8 promoter could only be activated in a subpopulation of HEK293T cells, regardless of SON gene expression. We conclude that heterogeneous population of the HEK293T cells might be easily shifted towards expression of the pluripotency markers by ectopic expression of the SON factors or by growth in serum depleted media.

**Keywords:** NANOS2 OCT4 Reprogramming STELLA Starvation Stra8.

## AC Circuit Theory and Representation of Complex Impedance Values

### Impedance Definition: Concept of Complex Impedance

Almost everyone knows about the concept of electrical resistance. It is the ability of a circuit element to resist the flow of electrical current. Ohm's law (Equation 1) defines resistance in terms of the ratio between voltage, E, and current, I.

While this is a well known relationship, its use is limited to only one circuit element -- the ideal resistor. An ideal resistor has several simplifying properties:

- It follows Ohm's Law at all current and voltage levels.
- Its resistance value is independent of frequency.
- AC current and voltage signals though a resistor are in phase with each other.

However, the real world contains circuit elements that exhibit much more complex behavior. These elements force us to abandon the simple concept of resistance, and in its place we use impedance, a more general circuit parameter. Like resistance, impedance is a measure of the ability of a circuit to resist the flow of electrical current, but unlike resistance, it is not limited by the simplifying properties listed above.

Electrochemical impedance is usually measured by applying an AC potential to an electrochemical cell and then measuring the current through the cell. Assume that we apply a sinusoidal potential excitation. The response to this potential is an AC current signal. This current signal can be analyzed as a sum of sinusoidal functions (a Fourier series).

Electrochemical impedance is normally measured using a small excitation signal. This is done so that the cell's response is pseudo-linear. In a linear (or pseudo-linear) system, the current response to a sinusoidal potential will be a sinusoid at the same frequency but shifted in phase (see Figure 1). Linearity is described in more detail in the following section.

**Figure 1**. Sinusoidal Current Response in a Linear System

The excitation signal, expressed as a function of time, has the form

where E_{t} is the potential at time t, E_{0} is the amplitude of the signal, and ω is the radial frequency. The relationship between radial frequency ω (expressed in radians/second) and frequency f (expressed in hertz) is:

(3)

In a linear system, the response signal, I_{t}, is shifted in phase (Φ) and has a different amplitude than I_{0}.

(4)

An expression analogous to Ohm's Law allows us to calculate the impedance of the system as:

(5)

The impedance is therefore expressed in terms of a magnitude, Z_{o}, and a phase shift, Φ.

If we plot the applied sinusoidal signal E(t) on the X-axis of a graph and the sinusoidal response signal I(t) on the Y-axis, the result is an oval (see Figure 2). This oval is known as a "Lissajous Figure". Analysis of Lissajous Figures on oscilloscope screens was the accepted method of impedance measurement prior to the availability of modern EIS instrumentation.

**Figure 2.** Origin of Lissajous Figure

(6)

it is possible to express the impedance as a complex function. The potential is described as,

and the current response as,

(8)

The impedance is then represented as a complex number,

(9)

**Data Presentation**

Look at Equation 9 in the previous section. The expression for Z(ω) is composed of a real and an imaginary part. If the real part is plotted on the X-axis and the imaginary part is plotted on the Y-axis of a chart, we get a "Nyquist Plot" (see Figure 3). Notice that in this plot the Y-axis is negative and that each point on the Nyquist Plot is the impedance at one frequency. Figure 3 has been annotated to show that low frequency data are on the right side of the plot and higher frequencies are on the left.

On the Nyquist Plot the impedance can be represented as a vector (arrow) of length |Z|. The angle between this vector and the X-axis, commonly called the “phase angle”, is f (=arg Z).

**Figure 3.** Nyquist Plot with Impedance Vector

Nyquist Plots have one major shortcoming. When you look at any data point on the plot, you cannot tell what frequency was used to record that point.

The Nyquist Plot in Figure 3 results from the electrical circuit of Figure 4. The semicircle is characteristic of a single "time constant". Electrochemical impedance plots often contain several semicircles. Often only a portion of a semicircle is seen.

**Figure 4.** Simple Equivalent Circuit with One Time Constant

Another popular presentation method is the Bode Plot. The impedance is plotted with log frequency on the X-axis and both the absolute values of the impedance (|Z|=Z_{0}) and the phase-shift on the Y-axis.

The Bode Plot for the electric circuit of Figure 4 is shown in Figure 5. Unlike the Nyquist Plot, the Bode Plot does show frequency information.

**Figure 5.** Bode Plot with One Time Constant

**Linearity of Electrochemistry Systems**

Electrical circuit theory distinguishes between linear and non-linear systems (circuits). Impedance analysis of linear circuits is much easier than analysis of non-linear ones.

The following definition of a linear system is taken from *Signals and Systems* by Oppenheim and Willsky:

A linear system . is one that possesses the important property of superposition: If the input consists of the weighted sum of several signals, then the output is simply the superposition, that is, the weighted sum, of the responses of the system to each of the signals. Mathematically, let y_{1}(t) be the response of a continuous time system to x_{1}(t) ant let y_{2}(t) be the output corresponding to the input x_{2}(t). Then the system is linear if:

2) The response to ax_{1}(t) is ay_{1}(t) .

For a potentiostated electrochemical cell, the input is the potential and the output is the current. Electrochemical cells are not linear! Doubling the voltage will not necessarily double the current.

However, Figure 6 shows how electrochemical systems can be pseudo-linear. If you look at a small enough portion of a cell's current versus voltage curve, it appears to be linear.

**Figure 6.** Current versus Voltage Curve Showing Pseudo-Linearity

In normal EIS practice, a small (1 to 10 mV) AC signal is applied to the cell. With such a small potential signal, the system is pseudo-linear. We don't see the cell's large nonlinear response to the DC potential because we only measure the cell current at the excitation frequency.

If the system is non-linear, the current response will contain harmonics of the excitation frequency. A harmonic is a frequency equal to an integer multipled by the fundamental frequency. For example, the “second harmonic” is a frequency equal to two times the fundamental frequency.

Some researchers have made use of this phenomenon. Linear systems should not generate harmonics, so the presence or absence of significant harmonic response allows one to determine the systems linearity. Other researchers have intentionally used larger amplitude excitation potentials. They use the harmonic response to estimate the curvature in the cell's current voltage curve.

**Steady State Systems**

Measuring an EIS spectrum takes time (often up to many hours). The system being measured must be at a steady state throughout the time required to measure the EIS spectrum. A common cause of problems in EIS measurements and analysis is drift in the system being measured.

In practice a steady state can be difficult to achieve. The cell can change through adsorption of solution impurities, growth of an oxide layer, build up of reaction products in solution, coating degradation, or temperature changes, to list just a few factors.

Standard EIS analysis tools may give you wildly inaccurate results on a system that is not at steady state.

**Time and Frequency Domains and Transforms**

Signal processing theory refers to data representation domains. The same data can be represented in different domains. In EIS, we use two of these domains, the time domain and the frequency domain.

In the time domain, signals are represented as signal amplitude versus time. Figure 7 demonstrates this for a signal consisting of two superimposed sine waves.

**Figure 7.** Two Sine Waves in the Time Domain

Figure 8 shows the same data in the frequency domain. The data is plotted as amplitude versus frequency.

**Figure 8.** Two Sine Waves in the Frequency Domain

You use a transform to switch between the domains. The Fourier Transform takes time domain data and generates the equivalent frequency domain data. The common term, FFT, refers to a fast, computerized implementation of the Fourier transform. The inverse Fourier transform changes frequency domain data into time domain data.

In modern EIS systems, lower frequency data are usually measured in the time domain. The controlling computer applies a digital approximation to a sine wave to the cell by means of a digital-to-analog converter. The current response is measured using an analog-to-digital converter. The FFT is used to convert the current signal into the frequency domain.

Details of these transforms are beyond the scope of this Application Note.

### Electrical Circuit Elements

EIS data are commonly analyzed by fitting to an equivalent electrical circuit model. Most of the circuit elements in the model are common electrical elements such as resistors, capacitors, and inductors. To be useful, the elements in the model should have a basis in the physical electrochemistry of the system. As an example, most models contain a resistor that models the cell's solution resistance.

Some knowledge of the impedance of the standard circuit components is therefore quite useful. Table 1 lists the common circuit elements, the equation for their current versus voltage relationship, and their impedance.

**Table 1.** Common Electrical Elements

Notice that the impedance of a resistor is independent of frequency and has no imaginary component. With only a real impedance component, the current through a resistor stays in phase with the voltage across the resistor.

The impedance of an inductor increases as frequency increases. Inductors have only an imaginary impedance component. As a result, the current through an inductor is phase-shifted -90 degrees with respect to the voltage.

The impedance versus frequency behavior of a capacitor is opposite to that of an inductor. A capacitor's impedance decreases as the frequency is raised. Capacitors also have only an imaginary impedance component. The current through an capacitor is phase shifted 90 degrees with respect to the voltage.

**Serial and Parallel Combinations of Circuit Elements**

Very few electrochemical cells can be modeled using a single equivalent circuit element. Instead, EIS models usually consist of a number of elements in a network. Both serial (Figure 9) and parallel (Figure 10) combinations of elements occur.

Fortunately, there are simple formulas that describe the impedance of circuit elements in both parallel and series combination.

**Figure 9.** Impedances in Series

For linear impedance elements in series you calculate the equivalent impedance from:

**Figure 10.** Impedances in Parallel

For linear impedance elements in parallel you calculate the equivalent impedance from:

(11)

We will calculate two examples to illustrate a point about combining circuit elements. Suppose we have a 1 Ω and a 4 Ω resistor in series. The impedance of a resistor is the same as its resistance (see Table 1). We thus calculate the total impedance as:

Resistance and impedance both go up when resistors are combined in series.

Now suppose that we connect two 2 μF capacitors in series. The total capacitance of the combined capacitors is 1 μF.

Impedance goes up, but capacitance goes down when capacitors are connected in series. This is a consequence of the inverse relationship between capacitance and impedance.

### Physical Electrochemistry and Equivalent Circuit Elements

**Electrolyte Resistance**

Solution resistance is often a significant factor in the impedance of an electrochemical cell. A modern three electrode potentiostat compensates for the solution resistance between the counter and reference electrodes. However, any solution resistance between the reference electrode and the working electrode must be considered when you model your cell.

The resistance of an ionic solution depends on the ionic concentration, type of ions, temperature, and the geometry of the area in which current is carried. In a bounded area with area, A, and length, l, carrying a uniform current, the resistance is defined as,

ρ is the solution resistivity. The reciprocal of ρ (κ) is more commonly used. κ is called the conductivity of the solution and its relationship with solution resistance is:

(15)

Standard chemical handbooks will often list κ values for specific solutions. For other solutions, you can calculate κ from specific ion conductances. The units of κ is Siemens per meter (S/m). The Siemen is the reciprocal of the ohm, so 1 S = 1/ohm.

Unfortunately, most electrochemical cells do not have uniform current distribution through a definite electrolyte area. The major problem in calculating solution resistance therefore concerns determination of the current flow path and the geometry of the electrolyte that carries the current. A comprehensive discussion of the approaches used to calculate practical resistances from ionic conductances is well beyond the scope of this application note.

Fortunately, you usually don't calculate solution resistance from ionic conductances. Instead, you calculate it when you fit experimental EIS data to a model.

**Double Layer Capacitance**

An electrical double layer exists on the interface between an electrode and its surrounding electrolyte. This double layer is formed as ions from the solution adsorb onto the electrode surface. The charged electrode is separated from the charged ions by an insulating space, often on the order of angstroms. Charges separated by an insulator form a capacitor so a bare metal immersed in an electrolyte will be have like a capacitor. You can estimate that there will be 20 to 60 μF of capacitance for every 1 cm 2 of electrode area though the value of the double layer capacitance depends on many variables. Electrode potential, temperature, ionic concentrations, types of ions, oxide layers, electrode roughness, impurity adsorption, etc. are all factors.

**Polarization Resistance**

Whenever the potential of an electrode is forced away from its value at open-circuit, that is referred to as “polarizing” the electrode. When an electrode is polarized, it can cause current to flow through electrochemical reactions that occur at the electrode surface. The amount of current is controlled by the kinetics of the reactions and the diffusion of reactants both towards and away from the electrode.

In cells where an electrode undergoes uniform corrosion at open circuit, the open circuit potential is controlled by the equilibrium between two different electrochemical reactions. One of the reactions generates cathodic current and the other generates anodic current. The open circuit potential equilibrates at the potential where the cathodic and anodic currents are equal. It is referred to as a mixed potential. If the electrode is actively corroding, the value of the current for either of the reactions is known as the corrosion current.

Mixed potential control also occurs in cells where the electrode is not corroding. While this section discusses corrosion reactions, modification of the terminology makes it applicable in non-corrosion cases as well as seen in the next section.

When there are two, simple, kinetically-controlled reactions occurring, the potential of the cell is related to the current by the following equation.

- I = the measured cell current in amps,
- I
_{corr}= the corrosion current in amps, - E
_{oc}= the open circuit potential in volts, - β
_{a}= the anodic Beta coefficient in volts/decade, - β
_{c}= the cathodic Beta coefficient in volts/decade.

If we apply a small signal approximation to equation 16, we get the following:

which introduces a new parameter, R_{p}, the polarization resistance. As you might guess from its name, the polarization resistance behaves like a resistor.

If the Beta coefficients, also known as Tafel constants, are known you can calculate the I_{corr} from R_{p} using equation 17. I_{corr} in turn can be used to calculate a corrosion rate.

We will discuss the R_{p} parameter in more detail when we discuss cell models.

**Charge Transfer Resistance**

A similar resistance is formed by a single, kinetically-controlled electrochemical reaction. In this case we do not have a mixed potential, but rather a single reaction at equilibrium.

Consider a metal substrate in contact with an electrolyte. The metal can electrolytically dissolve into the electrolyte, according to,

In the forward reaction in the first equation, electrons enter the metal and metal ions diffuse into the electrolyte. Charge is being transferred.

This charge transfer reaction has a certain speed. The speed depends on the kind of reaction, the temperature, the concentration of the reaction products and the potential.

The general relation between the potential and the current (which is directly related with the amount of electrons and so the charge transfer via Faradays law) is: