# Calculating Spatial Current in Resonant Inductors

• decaf14
In summary: The resonant frequency is given by ##f_res = \frac{1}{2\pi\epsilon_0 L^2}##.The self inductance is given by ##L = \frac{1}{2\pi\epsilon_0} \left(1 - e^{-j\omega t}\right)##.In summary, the paper talks about how to calculate the self inductance of an inductor, but does not go into detail about how to calculate the spatial current density or charge current density. They mention that these two quantities can be found using the time dependent current profile, but give no examples or instructions on how to do so.
decaf14
Hello,

I would like to replicate the results from the paper "wireless power transfer via strongly coupled magnetic resonances", but I'm having trouble interpreting their equations. I'm creating a MATLAB script to do so. In the paper, they get to a point where L and C are to be calculated in a self-resonating inductor. The equation is of the followng form:

Where J(r) is the spatial current density and p(r) is the charge current density. I'm confused on how to get spatial current density and charge current density. Previously, they mention that "the time dependent current profile has the form I0 * cos(pi * s / L) exp(i * w * t) where I0 is (I'm assuming) current input, s is a "parameterization coefficient" whatever that means (all i know is it varies in value from -l/2 to l/2 so it gives you the spatial location on the coil). "i" is imaginary number 1, L is total length, and t is time.

They jump right from this time dependent current profile into equations 3 and 4 I've listed above. Because they did not discuss methods for calculating J(r), I'm assuming the solution is trivial. However, I do not know where to even start for calculating J(r). Theoretical electromagnetics is not my strong suit. Any help would be greatly appreciated.

EDIT: to clarify a bit more, I understand that current density is simply current / area. However, the current varies at every point in space along the coil. The authors indicate that current is a function of radius, but previously they denote it as being a function of length or time, not radius. If it were to be a function of space, then time would need to be fixed. Even then, it should be a function of length, not radius, according to the equation they have given.

The paper can be found here: https://science.sciencemag.org/content/317/5834/83

Last edited:
Delta2
There is something wrong looking at those equations. I expect them to be volume integrals and not line integrals. The units of the LHS and RHS of the equations don't match . So are the differentials ##dr,dr'## that appear in those equations actually ##d^3r,d^3r'##?

Nope. There is no mention of what kind of integral. They don't even name what " r' "actually means, but I'm assuming it's a derivative.

They sort of just introduce the equation out of nowhere. It's a very highly cited (2.6k citations) paper in the field, so it's a reputable paper. I just can't seem to figure what they're doing.

No r' is not the derivative, its just the name of the second integration variable.

What does the paper say about L, is it the self inductance of the inductor and measured in units of Henry, or is it something else (self inductance per unit of length or per unit of volume perhaps?)

Delta2 said:
No r' is not the derivative, its just the name of the second integration variable.

What does the paper say about L, is it the self inductance of the inductor and measured in units of Henry, or is it something else (self inductance per unit of length or per unit of volume perhaps?)
L is referred to simply as the "effective inductance ... for each coil" . No units are listed, but based on the context, I think it's simply units of henries. Especially since they list this equation for resonant frequency that we know to be true (taken directly as a .jpeg from the text):

## 1. How do you calculate the spatial current in resonant inductors?

To calculate the spatial current in resonant inductors, you will need to know the inductance of the inductor, the frequency of the current passing through it, and the physical dimensions of the inductor. You can use the formula I = V/(2πfL) to calculate the spatial current, where I is the spatial current, V is the voltage, f is the frequency, and L is the inductance.

## 2. What is the significance of calculating spatial current in resonant inductors?

Calculating spatial current in resonant inductors is important because it helps us understand the behavior and performance of these components. It allows us to determine the amount of current flowing through the inductor and how it is affected by its physical dimensions and the frequency of the current. This information is crucial in designing and optimizing resonant inductor circuits for various applications.

## 3. How does the physical dimensions of an inductor affect the spatial current?

The physical dimensions of an inductor, such as its length, diameter, and number of turns, can significantly affect the spatial current. As the physical dimensions change, the inductance of the inductor also changes, which in turn affects the spatial current. A larger inductor with more turns will have a higher inductance and therefore, a higher spatial current compared to a smaller inductor with fewer turns.

## 4. Can the spatial current in a resonant inductor be controlled?

Yes, the spatial current in a resonant inductor can be controlled by adjusting the physical dimensions of the inductor or by changing the frequency of the current passing through it. By altering these parameters, you can increase or decrease the spatial current as needed for your specific application.

## 5. What are some common applications of resonant inductors?

Resonant inductors have a wide range of applications, including in power supplies, audio amplifiers, and radio frequency circuits. They are also commonly used in wireless charging systems, where they help transfer power wirelessly between a transmitter and a receiver. Additionally, resonant inductors are used in resonant circuits for tuning and filtering signals in electronic devices.

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