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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Analyzing an Efficient Wireless Power Transmission System}





\markboth{Efficient Wireless Power transmission}{Efficient Wireless Power Transmission}

\author{

\hspace*{.5 true cm} \begin{minipage}[t]{7.5 true cm}
Maila Brucal-Hallare\\
Norfolk State University\\
Norfolk  VA \teacher{23504} USA  \\
\teacher{mcbrucal-hallare@nsu.edu}
\end{minipage}
\ \quad \
\begin{minipage}[t]{7.5 true cm}
Iordanka Panayotova\\
Christopher Newport University\\
Newport News VA \teacher{23606} USA\\
\teacher{iordanka.panayotova@cnu.edu}
\end{minipage}

}

\makeStitlePDFLaTex


\begin{abstract}
This activity presents an engineering application that is modelled by a coupled system of two linear second-order differential equations with constant coefficients.  One of the equations is homogeneous while the other one is non-homogeneous.  The application is called wireless power transmission (WPT), which belongs to a bigger class of engineering technology that does not utilize wires or cables to transfer power across space.  The students will be guided in modelling WPT and the resulting system is most efficiently solved using Laplace transforms method.
\end{abstract}





\section*{The Story}
You woke up this morning with a terrible headache. You have been studying for your final unit exam in your Differential Equations class where you have been learning about Laplace transforms.  The theory is fine and neat and mumbled to yourself ``I am sure glad I listened to my Calculus II professor when we talked about improper integrals\ldots'' but you still feel uneasy because they all seem so \ldots  abstract.  ``Well, my professor said that we will talk about an interesting engineering application today so I better get ready!'' You hopped off your bed because you get excited every time your professor talks about electric circuits.  As a future engineer, you like applications that involve tinkering with the components of a circuit or designing circuit diagrams and simulating them with online circuit simulators.  You are secretly hoping that your professor will somehow make connections between Laplace transforms and electric circuits. ``Hmmm \ldots maybe I can make an Honors project about differential equations and circuits \ldots'' You went to the bathroom and grabbed your electric toothbrush that has been sitting overnight in its charging base...



\section*{SCENARIO DESCRIPTION}

Wireless power transmission (WPT) happens when electrical energy is transmitted without the use of cables or wires.  A wireless charger, the most popular application of WPT, allows one to charge cellphones or toothbrushes without the use of any cable or plug.  Do you know that your training in differential equations up to this point gives you the power to model the current flowing from the power source to your device, through the charging base?  See Figure 1 for a simplified diagram of an electric toothbrush, viewing it with modelling in mind.  

\begin{figure}
\centering
\includegraphics[width=10cm]{electric toothbrush diagram.PNG}
\caption{A schematic diagram of an electric toothbrush as a WPT engineering application.}
\end{figure}

In this learning activity, you will model and explore a WPT system by coupling two second-order differential equations.  This activity is interesting because it gives you a chance to solve an engineering application via the Laplace transform method.  Although current research analyzes systems of differential equations using sophisticated computer simulations, it is still important to analyze such systems of equations step-by-step.  As a bonus, this application is something that you actually see in your home. 

        
\begin{figure}
\centering
\includegraphics[width=10cm]{WPTequivcircuit.PNG}
\caption{An equivalent circuit of a WPT system}
\end{figure}

An equivalent circuit for a WPT system is illustrated in Figure 2.  An equivalent circuit is a theoretical, highly idealized circuit that captures and retains the most important electrical components and characteristics of a circuit.  Actual circuits for an electric toothbrush and its charging base are more complicated.  The electric circuit on the left-hand side of Figure 2 corresponds to the electric circuit of the transmitting coil (source) of Figure 1 while the electric circuit on the right-hand side of Figure 2 corresponds to the electric circuit on the receiving coil (drain) Figure 1.     

\subsubsection*{Part 1.  Creating the mathematical model.}  
The diagrams in Figure 1 and Figure 2 should help in writing the systems of differential equations for WPT.  Observe that these two pictures have visually led you from the physical realm (the electric toothbrush) to the electrical world (the equivalent circuit) and now it is time to go the mathematical world!  Towards the goal of writing the model, let $i_{1}, i_{2}$ be the currents in the first and second loops; $L_{1}, L_{2}$ be the inductors in the first and second loops; $R_{1}, R_{2}$ be the resistors in the first and second loops; $C_{1}, C_{2}$ be the capacitors in the first and second loops; and $V_{1}$ be the voltage power source.  In the meantime, do not worry about the broken arrow in Figure 2 which is labelled $M$.

Assume that the transmitting coil is an RLC-series electrical circuit. Recall from your modeling experience (for example, Chapter 5 in textbook [6]), that an RLC-series electric circuit may be written as a linear second-order differential equation with constant coefficients.  Then the left circuit in Figure 2  gives us the equation
\begin{equation}\label{firstEQN}
L_{1}\dfrac{d^2i_{1}(t)}{dt^2} + R_{1}\dfrac{di_{1}(t)}{dt} + \frac{1}{C_{1}}i_{1}(t) = \dfrac{dV_{1}(t)}{dt}.
\end{equation}
Now, since $V_{1}$ is an electric power source (also called alternating-current (AC) source), $V_{1}$ is typically of the form $A \cos(mt) + B\sin(mt)$ and so $\dfrac{dV_{1}(t)}{dt}$ is nonzero and hence (\ref{firstEQN}) is a non-homogeneous differential equation.   

Next, let us look at the right circuit in Figure 2.  Analogously, we have
\begin{equation}\label{secondEQN}
L_{2}\dfrac{d^2i_{2}(t)}{dt^2} + R_{2}\dfrac{di_{2}(t)}{dt} + \frac{1}{C_{2}}i_{2}(t) = 0
\end{equation}
and unlike the first equation, this is a homogeneous linear second-order differential equation with constant coefficients.  In other words, the transmitting coil (source) is non-homogeneous while the receiving coil (drain) is homogeneous.  

It is time to couple our two equations to form our system.  Observe that the two electrical circuits are linked by the mutual inductance $M.$  By Ampere's Law, the flow of the electric current in the left circuit will create a magnetic field.  Since the two electric circuits are linked by $M$ and that the field is time-varying, Faraday's Law predicts the creation of a voltage within the second circuit.  Although this occurs to some extent in all circuits, the effect is magnified in coils because the circuit geometry amplifies the linkage effect.  

Mathematically, we capture the effect of magnetic inductance into the two circuits by adding the term $M \dfrac{d^2 i_{2}(t)}{dt^2}$ to the left-hand side of (\ref{firstEQN}) and adding the term $M \dfrac{d^2 i_{1}(t)}{dt^2}$ to the left-hand side of (\ref{secondEQN}).  This magnetic inductance is the key to the engineering and mathematics behind wireless power transmission.   

 The wireless power transmission system can be modelled by: 
\begin{equation}\label{System}
\left\{
\begin{array}{lll}
L_{1}\dfrac{d^2i_{1}(t)}{dt^2} + M 
\dfrac{d^2i_{2}(t)}{dt^2} + R_{1}\dfrac{di_{1}(t)}{dt} + \frac{1}{C_{1}}i_{1}(t) = \dfrac{dV_{1}(t)}{dt}\\ 
L_{2}\dfrac{d^2i_{2}(t)}{dt^2} + M 
\dfrac{d^2i_{1}(t)}{dt^2} + R_{2}\dfrac{di_{2}(t)}{dt} + \frac{1}{C_{2}}i_{2}(t) = 0
\end{array}
\right.
\end{equation} subject to the initial conditions $i_{1}(0), i_{2}(0), i_{1}'(0), i_{2}'(0).$  What we have presented here is a slightly different perspective of modelling the WPT circuit; please refer to [1] for another way to derive  the system (\ref{System}) via first principles, using Kirchoff's Voltage Law.    

You have learned how to solve homogeneous and non-homogeneous, linear differential equations with constant coefficients such as (\ref{firstEQN}) and (\ref{secondEQN}), respectively.  For example, you may want to review the method of undetermined coefficients and variations of parameters [6].  Also, refer to [2] for a review on how to analyze a second-order differential equation arising from an RLC-series electric circuit.

Let us consider a simpler version of the system by assuming that we have a loss-less circuit, that is, let us assume that $R_{1} = R_{2} = 0$.  This is a simple way to analyze the efficiency of a circuit because there is no resistance component.  For example, the discussion in [5] starts with a similar analysis and investigates the efficiency of a WPT via transitionless quantum driving.
\begin{equation}\label{SystemLossLess}
\left\{
\begin{array}{lll}
L_{1}\dfrac{d^2i_{1}(t)}{dt^2} + M 
\dfrac{d^2i_{2}(t)}{dt^2} + \frac{1}{C_{1}}i_{1}(t) = \dfrac{dV_{1}(t)}{dt}\\ 
L_{2}\dfrac{d^2i_{2}(t)}{dt^2} + M 
\dfrac{d^2i_{1}(t)}{dt^2} + \frac{1}{C_{2}}i_{2}(t) = 0
\end{array}
\right.
\end{equation}

The voltage power source function $V_1(t)$ is usually assumed to be of the form $A \cos(mt)+B \sin(mt)$ to account for the alternating current AC voltage source (connected to a power outlet).  Note that for the sake of practice, one of the exercises below assumes that $V_{1}(t)$ is a constant.      

Upon taking the Laplace transforms of the system (\ref{SystemLossLess}), we get
\begin{equation}\label{Laplace}
\left(
\begin{array}{cc}
L_{1}s^2 + \frac{1}{C_{1}} & Ms^2  \\
Ms^2 & L_{2}s^2 +\frac{1}{C_{2}} 
\end{array}
\right)
\left(
\begin{array}{l}
I_{1}(s)\\I_{2}(s)
\end{array}
\right)
=
\left(
\begin{array}{l}
\mathcal{L}\{V_1'(t)\}\\0
\end{array}
\right).
\end{equation}

Then, using Cramer's Rule on (\ref{Laplace}) to solve for $I_{1}(s)$ and $I_{2}(s)$, we have
\begin{equation}\label{System3rdSolutions}
\begin{array}{lll}
I_{1}(s) &=& \dfrac{(L_{2}s^2 +  \frac{1}{C_{2}})\mathcal{L}\{V_{1}'(t)\}}{(L_{1}s^2 +  \frac{1}{C_{1}})(L_{2}s^2 + \frac{1}{C_{2}}) - M^2s^4},\\
I_{2}(s) &=&\dfrac{-Ms^2\mathcal{L}\{V_{1}'(t)\}}{(L_{1}s^2 + \frac{1}{C_{1}})(L_{2}s^2 +  \frac{1}{C_{2}}) - M^2s^4}
\end{array}
\end{equation}
In one of the exercises below, you will be asked to provide the details in deriving the system of linear equations (\ref{System3rdSolutions}) using Cramer's Rule. Finally, the solutions are given by the inverse Laplace transforms of these functions.   

      


%The value of the mutual inductance $M$ cannot exceed %the geometric mean of the inductors $L_{1}, L_{2}$, %that is:
%$$
%0 \leq M \leq \sqrt{L_{1}L_{2}}.
%$$
%The proof of this requires some power and energy analysis [3].  Furthermore, in mathematics, the coefficient $M$ is called a coupling coefficient between the two equations; in electrical engineering, however, the coupling coefficient between two circuits joined by a mutual inductance is given by
%\begin{equation}\label{couplingcoefficient}
%K = \dfrac{M}{\sqrt{L_{1}L_{2}}}, 
%\end{equation}
%and hence, the coupling coefficient is a quantity %between 0 and 1.  The coupling coefficient is a way to %measure the strength of the connection between the two %circuits.



%27s_triangle% write a narrative about the partner discipline background that an instructor needs to know in order to implement this activity.  This can be up to 1 or 1.5 pages in length

\section*{Analyzing an Efficient Wireless Power transmission: Activity} 

The goal of the first two exercises below is to provide you with some opportunities to learn more about WPT.  The next three exercises require that you analyze WPT using your knowledge of Laplace transforms.  To solve the problems, make sure that you have a table of Laplace transforms next to you.  Suppose $I_{k}(s) = \mathcal{L}\{i_{k}(t)\}$ for $k = 1, 2$ and $s > 0$.  In engineering courses, the functions $i_{k}(t)$ and $I_{k}(s)$ are usually called currents expressed in the time and frequency domains, respectively.  Assume that $i_{1}(0)=i_{2}(0)=i_{1}'(0)= i_{2}'(0) = 0$.  
\begin{enumerate}
\item \textit{Relating WPT to another electromagnetic induction application.} Use the internet to learn about electricity transformers.  In your exploration, you will see that a transformer is based on a very simple fact about electricity: when a time-varying power source enters a circuit, it generates an oscillating magnetic field which, in turn, generates an electric current that enters another circuit.  Observe that this concept is similar to the basic principle of a WPT.  Find out what the difference is between a WPT and an electricity transformer.  What are the equations that govern an electricity transformer?

\item \textit{Relating WPT to TET.}  Another engineering wonder that uses the same principle as WPTs and transformers is transcutaneous energy transmission (TET). This concept is used to recharge biomedical prosthetic devices that are implanted in the human body, for example, artificial hearts, pacemakers, and insulin pumps.  Read the paper \cite{Niu} to see that TETs are governed by electromagnetic induction, too.  Moreover, based on your reading, what are some practical issues regarding TETs when used by patients?


\item \textit{Analytical solution of the WPT system.}  Provide the details to show how the matrix equation (\ref{Laplace}) is obtained from (\ref{SystemLossLess}).  Then, write out the details to show how (\ref{System3rdSolutions}) is obtained from (\ref{Laplace}).
\item \textit{Thinking about a mathematical WPT system.} For a first experience in analyzing the system (\ref{SystemLossLess}), assume that  $V_{1}(t)$ is a constant value $K$.  In electrical engineering, this assumption happens when the voltage power or input is a DC source.  (Note, however, that a real WPT system requires a time-varying source by virtue of Faraday's Law.)    
\begin{enumerate}
\item Assuming $V_{1}(t) = K$ for some constant $K$, write the two equations in the WPT system.   
\item Write the solutions to the system (\ref{SystemLossLess}) using the equations in (\ref{System3rdSolutions}) by taking the inverse Laplace transforms.
\item Observe that the inverse Laplace Transform of zero does not appear in your table of Laplace transforms. What property of the inverse Laplace transforms guarantees that the inverse Laplace of zero is also zero?  
\end{enumerate}

\item \textit{Computing and graphing solutions to a WPT example.}  Analyze the wireless power transmission system whose component values are given in Table \ref{Tab:WPT}.
\begin{table}
\center
\caption{Table of values for Exercise 5.}
\begin{tabular}{|l|l|}
\hline
Voltage Source $V_{1}(t)$ & $\sin(t) V$\\
\hline
Inductor $L_{1}$ & $1 H$\\
\hline 
Capacitor $C_{1}$ & $1F$\\
\hline
Inductor $L_{2}$ & $1 H$\\
\hline
Capacitor $C_{2}$ & $1F$\\
\hline
Mutual Inductor $M$ & $1 H$\\
\hline
\end{tabular}
\label{Tab:WPT}
\center
\end{table}


\begin{enumerate}
%\item Compute the coupling coefficient $K$ using (\ref{couplingcoefficient}). 
\item Sketch a circuit diagram that corresponds to this WPT system.  Write the corresponding system of differential equations.
\item Write the solutions of the system you developed in (a).  You may find \cite{symbolab} useful in checking your work.
\item Sketch the solutions to the system on one coordinate plane.  You may use \cite{desmos} or other graphing software.
\item Analyze the graph of the solutions that you obtained in (c) using the following guide questions.  For exploration purposes, choose any closed interval that you see in your graph.  For example, the interval $[0, 4\pi]$.
\begin{enumerate}
\item What is the maximum amplitude of $i_{1}(t)$ and $i_{2}(t)$ in this interval ?    
\item Find the points of intersection between $i_{1}(t)$ and $i_{2}(t)$ for $t \in [0, 4\pi]$.
\item What are the zeros of $i_{1}(t)$ and $i_{2}(t)$?
\item From the electrical circuit perspective, why does it make sense that the current $i_{1}(t)$ is a linear combination of two cosines while the the current $i_{2}(t)$ is a single cosine function?
\end{enumerate}
\end{enumerate}


\end{enumerate}


Now that you have learned about wireless power transmission system, you realize that mathematics is indeed everywhere, even in your bathroom!  Every time you use your electric toothbrush or put your cellphone on its charging base, you now know that the underlying principle for such technology is a system of second-order differential equations that you can solve using Laplace transforms methods.  Back to your terrible headache this morning, be sure to visit simiode.org to find that activity that lets you model how a medicinal pill is dissolved into your bloodstream.  Yes, there are applications of differential equations in your first-aid box! 

\teacher{
\section*{NOTES FOR THE TEACHER}

In a typical one-semester undergraduate course in differential equations, the topic of Laplace Transforms is usually covered in the final few weeks of the semester.  In our experience, we scramble to finish up the semester by wrapping up the topic on Laplace Transforms with a few examples of solving initial-value problems.  This activity allows the instructor to present an interesting engineering application that hopefully helps students visualize a coupled system of linear second-order differential equations with constant coefficients.  Some of our students have said that Figures (1) and (2) finally helped them make connections between electric circuits and differential equations before the semester ended.

Moreover, looking at the textbook [6] as one illustration of what modern texts may offer, the example on a linearized double pendulum is the only example whose system involves second-order derivatives in both equations. Unfortunately, the double pendulum example is intimidating to many students and takes a significant amount of time in creating the mathematical model.  Moreover, such a system needs to be linearized with small angle substitution of $x$ for $\sin x$.  We believe that our engineering application example is more interesting and less intimidating to students.

In the Student Version, there is a {\it Story} to motivate the fact that this activity is best presented at the latter part of the semester.  In particular, the activity requires that the following topics have already been covered:
\begin{enumerate}
\item computing Laplace and inverse Laplace transforms,
\item using Laplace transforms to solve initial-value problems, and
\item using systems of second-order linear differential equations to model RLC-series electrical circuits.
\end{enumerate}
The activity also requires that students are still familiar with some skills from Precalculus and Calculus 1, namely, 
\begin{enumerate}
\item analyzing trigonometric functions, 
\item solving trigonometric equations, and
\item solving optimization problems on a closed interval.
\end{enumerate}
Although these skills are not required to model a WPT, we believe that this is a great opportunity to review these elementary skills at this point in a student's mathematics training.
     
More details on the derivation of (\ref{System}) that uses first principles (Kirchoff's Voltage Law)  can be found in the paper [1].  The modelling process as described in this activity is slightly different but it is more intuitive.  With sufficient practice on the prerequisite topic RLC-series electrical circuits, the instructor may be able to use less discussion time by presenting the model  (\ref{System}) directly and emphasizing that it consists of two RLC-series second-order equations with the mutual inductance $M$ as the coupling coefficient.  

Next, let us look at the exercises in the {\it Activity}.  The first two exercises aim to engage student's curiosity about electromagnetic induction by relating WPT to transformers and TETs.  By asking the students to learn more about related engineering applications, they are able to see that these real-life technological wonders are governed by the same electricity principles.

The third exercise is a standard pen-and-paper analysis of solving systems of equations using Laplace transforms, Cramer's Rule, inverse Laplace transforms, and partial fraction decomposition.  The fourth exercise is a great opportunity to re-visit the definition of the Laplace transform as an integral and its linearity property. The final exercise provides an opportunity for students to visualize the solutions to (\ref{SystemLossLess}).  We suggest that an online graphing software be used in graphing the solutions.  

}

\begin{thebibliography}{99}

\bibitem{MAANotes2021WPT} Brucal-Hallare, M. {\it et al.} 2021. {\it Modelling Wireless Power Transfer.}  Submitted to MAA Notes.  

\bibitem{MAANotes2021Second} Deo, M, {\it et al.} 2021 {\it Studying the Response of Second-Order Electrical Circuits.}  Submitted to MAA Notes.    

\bibitem{Irwin} Irwin, J. D. and R.M.~Nelms. 2015. {\it Basic Engineering Circuit Analysis, 11th Edition}. New York: Wiley.

\bibitem{desmos} DESMOS. 2021. DESMOS, an online graphing software and calculator. \href{https://desmos.com}{https://desmos.com}.


\bibitem{Kurs} Kurs, A. {\it et al.}  2007.  Wireless Power Transfer via Strongly Coupled Magnetic Resonances. {\it Science\/}. 317(5834):83–86.

\bibitem{Niu} Niu, C., {\it et al.}  2006. The Transcutaneous Charger for Implanted Nerve Stimulation Device. {\it Proceedings of the 28th IEEE EMBS Annual International Conference, New York City, USA.}

\bibitem{Paul} Paul, K and A.K.~Sarma.  2018 Fast and efficient wireless power transfer via transitionless quantum driving. \href{https://www.nature.com/articles/s41598-018-22562-9.pdf}{https://www.nature.com/articles/s41598-018-22562-9.pdf}.      

\bibitem{symbolab} SYMBOLAB. 2021 SYMBOLAB - an online mathematics software that provides help on most precalculus, calculus, and differential equations topics. \href{https://symbolab.com}{https://symbolab.com}


\bibitem{Zill} Zill, D.  2013.  {\it A First Course in Differential Equations with Modelling Applications, 10th Edition}. Pacific Grove CA: Brooks/Cole.
  
\end{thebibliography}



\end{document}
