Maple/Simultaneous Equations
Contents
Introduction
This page focuses on using Maple to find both the symbolic and the numeric solutions to equations obtained from electric circuits.
Starting the Program
Maple is free to Duke students and resides on the OIT system in the same way that MATLAB does. To start Maple, make sure your terminal is set up to receive graphics and type
xmaple &
at the prompt. Maple will start up. It may have a window at startup containing hints or tips - go ahead and close that window. There will most likely be some initial blank document in the main window - go ahead and close it as well by selecting File-Close Document. Then, open a new blank worksheet with File-New-Worksheet Mode.
Documenting Your Work
When Maple starts a worksheet, it expects everything to be an input. To document your work with the title of the assignment, your name and NET ID, and any kind of explanation you would like to add, you need to tell Maple to switch to paragraph mode. Go to Insert-Paragraph-Before Cursor and you will notice that a blank line opens up above the red cursor mark. You can type text in here and Maple will know not to try to process it. Go ahead and call this assignment Introductory Maple Assignment, hit return, put in your name followed by your NET ID in parenthesis, hit return, and put in today's date.
Clearing the Worksheet
When Maple runs, it "remembers" everything that it has done in the worksheet, regardless of what order you ran lines of code. For that reason, it is good programming practice to have Maple "restart" itself at the beginning of each worksheet. To give Maple a command, first tell Maple you are ready to issue commands by selecting Insert-Execution Group-After Cursor. This will start a new bracket (black lines at the left of the worksheet) and give you a prompt (red >). At the prompt, type restart. When you hit return, if you quickly look at the bottom left of the Maple window, you will see that Maple evaluates the command then then tells you that it is Ready. The restart command clears out any variables Maple was taught and also clears out any packages that were loaded. It is a good way to make sure you have a "fresh start."
Defining Variables and Equations
In Maple, the way you define a variable is by typing the name of the variable, followed by the symbols :=, followed by whatever items you want to store in the variable. Note the importance of the colon directly in front of the equals sign - without it, Maple will not assign a value to a variable but will merely print out the equation you typed. One benefit of this is you can define variables to hold on to equations and then use those variables later, in concert with Maple's solver, to get answers for the unknowns. Let us assume that we want to solve the following equations:
where x, y, and z are unknowns, a through i are known coefficients, and j through l are known variables. To teach Maple about these equations, you would create three variables, each holding on to one of the equations. At the prompt, type:
eqn1:=a*x+b*y+c*z=j
eqn2:=d*x+e*y+f*z=k
eqn3:=g*x+h*y+i*z=l
Note that each time you hit return to go to the next line, Maple processes your input and reports back what it has done. It will also number the outputs for you so you can refer to them later. At this point, Maple now has three variables, each of which defined as an equation. It is perfectly happy having undefined items in the equations.
Solving Equations With Maple
To solve the equations, all you need to do is use Maple's built in solve function. One of the best ways to use the solve function is to give it a list of the equations and an array of items for which to solve. In the equations above, for example, there are three equations with a total of fifteen symbols - we need to tell Maple which ones are unknown and it will assume that the others are known. Add the line:
solve({eqn1, eqn2, eqn3}, [x, y, z])
and note that the equations are bracketed with curly braces while the unknowns are in a list set off with square brackets. Hit return, and you will note that Maple produces a list - set off with double brackets - containing the answers for x, y, and z in terms of the other variables. If we had not included the variable list and instead had asked
solve({eqn1, eqn2, eqn3})
Maple would have given all possible combinations of all 15 symbols that would satisfy the equations. Conversely, if we had given Maple only $x$ to work with as an unknown by typing:
solve({eqn1, eqn2, eqn3}, [x])
the answer would come back as empty because no value of $x$ satisfies the three equations for arbitrary values of the other 14 variables.
In order to use these solutions, you should give them a name. Click at the start of the {\tt solve} line and pre-pend it with {\tt
MySoln:=} so it resembles:
\begin{lstlisting}[frame=shadowbox] MySoln:=solve({eqn1, eqn2, eqn3}, [x, y, z]) \end{lstlisting} This will assign the solution list to a variable that we can use later.
\subsection{Substituting Values} Now that you have the symbolic answers to the variables $x$, $y$, and $z$, you may want to substitute the actual coefficient values to obtain a numerical solution. One way to do this is to generate a list of the known values, then tell Maple to substitute in the numerical values by using the built-in {\tt subs} command. Add the following lines of code: \begin{lstlisting}[frame=shadowbox] Vals := {a=1, b=2, c=3, d=4, e=5, f=6, g=7, h=8, i=9, j=10, k=11, l=12} subs(Vals, MySoln) \end{lstlisting} The list in {\tt MySoln} will now be shown with numerical values instead of symbols. Note that you have {\it not} made any actual changes to any of the variables - you have just asked Maple to show you what they would look like given the particular substitutions presented in {\tt Vals}. This is a very powerful tool, since you can substitute in a variety of values to see how one or more parameters influence a particular variable or variables.
\subsection{Assigning Representations} There will be many times you actually want to assign the solutions found by {\tt solve} - that is, you want to take the equations out of the list and have Maple process them as if the = were := so that Maple could use those expressions later. Maple has a command called {\tt
assign} that does exactly that. Add the commands:
\begin{lstlisting}[frame=shadowbox] assign(MySoln) x y z \end{lstlisting} You will see that while the {\tt assign} command does not report anything back to you, when you ask Maple to tell you what $x$, $y$, and $z$ are, it responds with the symbolic representation produced in {\tt MySoln}. This is very useful if, for example, the answer you are looking for is some function of the variables $x$, $y$, and $z$. Assuming that you have determined the variable you are looking for, $alpha$, is \begin{align*} \alpha&=x+y+z \end{align*} you can now use the symbolic representations in Maple to generate a symbolic representation for $\alpha$: \begin{lstlisting}[frame=shadowbox] alpha:=x+y+z \end{lstlisting} Note, among other things, that Maple represents the variable named {\tt alpha} as its symbol, $\alpha$. If you want a numerical value, you can again use the {\tt subs} command and the value list from before: \begin{lstlisting}[frame=shadowbox] subs(Vals, alpha) \end{lstlisting}
\subsection{Cleaning Things Up} Many times, Maple will produce an expression that is more complicated than it needs to be. To get what it considers to be the simplest form, use the {\tt simplify(expand( ))} compound function. The {\tt
expand} will take the expression and represent it using as many
simple terms as necessary while {\tt simplify} will recombine them in the most compact form. Finally, to get a decimal value, use the {\tt
evalf[N]( )} function, where {\tt N} represents the number of
decimal digits to use. For example, \begin{lstlisting}[frame=shadowbox] simplify(expand(alpha)) \end{lstlisting} will produce the most symbolically simplified version of $\alpha$ while \begin{lstlisting}[frame=shadowbox] evalf[8](subs(Vals, alpha)) \end{lstlisting} will produce a floating point result for $\alpha$. With practice, you will see how best to combine {\tt evalf}, {\tt simplify}, and {\tt
expand} to get the form of answer you want.
\section{Memory Issues} A major issue to consider with Maple is its memory. At the end of this worksheet, there are several variables that are defined, including $x$, $y$, and $z$. If you go back near the beginning, click in the line where {\tt
eqn1} is defined, and hit return, you will notice that where $x$,
$y$, and $z$ were before, their symbolic solutions from much further down the worksheet are being used. This is why the {\tt restart} command is so helpful - if you need to to run a worksheet again, it is best to always start from scratch. A shortcut for running an entire worksheet is the !!!~button at the top of the window.
\section{Sample Circuit} Each of the samples presented below will involve solving for the power absorbed by resistor $\E{R}{4}$ in the circuit: \begin{center} \epsfig{file=./XMAPLE/SampleStarter.ps, scale=0.5} \end{center} by using a different method. The circuit will be presented several times along the way to demonstrate how to label it properly for the given method.
\subsection{Node Voltage Method} First, choose a ground node. In this case, the large bottom node is as good as any: \begin{center} \epsfig{file=./XMAPLE/SampleNVM1.ps, scale=0.5} \end{center}
Next, circle or otherwise mark each node: \begin{center} \epsfig{file=./XMAPLE/SampleNVMnodes.ps, scale=0.5} \end{center} At this point, you can see that there are five total nodes. One of them is the ground node. From ground, you can determine the node voltage of the node at the positive terminal of $\E{v}{s}$ since the voltage source determines the different in voltage between ground and that node. Otherwise, you will need to assign labels to the remaining three nodes. In this case, use $\E{v}{x}$, $\E{v}{y}$, and $\E{v}{z}$ as node voltages: \begin{center} \epsfig{file=./XMAPLE/SampleNVMnodeslab.ps, scale=0.8} \end{center}
Note that other unknowns could have been selected so long as they gave equations for each of the nodes.
Finally, you can write the KCL equations. In this case, there are three unknowns so you need to pick three of the nodes. To help choose, remember that voltage sources are bad for KCL equations because {\it any} amount of current can go through a voltage source. Conveniently enough, that eliminates two of the five nodes from consideration ($n_1$ and the ground node) leaving three available. The KCL equations, written by determining the amount of current {\it
leaving} the node through each branch that passes through the node
boundary, are: \begin{align*} \mbox{KCL, n}_2&: & \frac{\E{v}{x}-\E{v}{y}}{\E{R}{1}}+ \frac{\E{v}{x}-\E{v}{s}}{\E{R}{2}}+ \frac{\E{v}{x}-\E{v}{z}}{\E{R}{3}}+ \frac{\E{v}{x}-0}{\E{R}{4}}&=0\\ \mbox{KCL, n}_3&: & \frac{\E{v}{y}-\E{v}{x}}{\E{R}{1}}- \E{i}{t}&=0\\ \mbox{KCL, n}_4&: & \frac{\E{v}{z}-\E{v}{x}}{\E{R}{3}}+ \frac{\E{v}{z}-0}{\E{R}{5}}+ \E{i}{t}&=0 \end{align*} These three equations can be put into Maple {\it as is} - again, no need to set them up as a matrix if you are using Maple. Maple can solve for the three unknowns, and the power absorbed by $\E{R}{4}$ will be the voltage across the resistor ($\E{v}{x}-0$) squared divided by the resistance. \pagebreak
\subsection{Branch Current Method} For the BCM, start be determining the number of branches. In this particular case, there are five branches. One of them - the very top one - has an independent current source so its current is known. The other four are yet to be determined: \begin{center} \epsfig{file=./XMAPLE/SampleBCMbranches.ps, scale=0.5} \end{center} Without more information, no more currents can be labeled so you will need to create an unknown that will help determine other branch currents. Labeling the current through $\E{R}{5}$ as $\E{i}{y}$, for example, will not only allow you to note what the current through $\E{R}{5}$ is but also allow you to write an equation for the current through $\E{R}{3}$ using KCL at the node common to $\E{R}{3}$, $\E{R}{5}$, and $\E{i}{t}$: \begin{center} \epsfig{file=./XMAPLE/SampleBCMbranchesy.ps, scale=0.5} \end{center} This still leave two unknown currents, and there is as yet no way to solve for either of them. Since we eventually want to know the power absorbed by $\E{R}{4}$, it makes sense to label its current so call that $\E{i}{x}$. Once that is done, you can use KCL at the ground node to determine the current through the branch with voltage source $\E{v}{s}$: \begin{center} \epsfig{file=./XMAPLE/SampleBCMbranchesxy.ps, scale=0.5} \end{center} You may also want to do a quick check at the node connecting all four resistors to make sure KCL holds there. Calculating the current leaving that node yields: \begin{align*} -(\E{i}{t})-(\E{i}{x}+\E{i}{y})+(\E{i}{t}+\E{i}{y})+(\E{i}{x})&=0 \end{align*}
At this point, you have two unknowns and six loops to choose from. For the current methods, you must avoid current sources which knocks out three loops (the top right primary loop, the right superloop, and the overall superdooperloop\footnote{Well...what would {\it you} call
it?}). That leaves three loops and thus two loop equations. In
this case, each of the primary loops will have four elements while the bottom superloop has four, so you may choose the primary loops: \begin{center} \epsfig{file=./XMAPLE/SampleBCMloops.ps, scale=0.8} \end{center} \begin{align*} \mbox{KVL, l}_1&: & -\E{v}{s}+ \E{R}{2}\left(\E{i}{x}+\E{i}{y}\right)+ \E{R}{4}\left(\E{i}{x}\right)&=0\\ \mbox{KVL, l}_2&: & \E{R}{4}\left(-\E{i}{x}\right)+ \E{R}{3}\left(\E{i}{t}+\E{i}{y}\right)+ \E{R}{5}\left(\E{i}{y}\right)&=0 \end{align*} These two equations can be put into Maple and Maple can solve for the two unknowns. In this case, the power absorbed by $\E{R}{4}$ will be the current through the resistor ($\E{i}{x}$) squared multiplied by the resistance. \pagebreak
\subsection{Mesh Current Method} For the MCM, start by labeling the mesh currents in the primary loops. In this particular case, there are three primary loops. As to avoid confusion with the unknowns in the previous examples, they are labeled with subscripts $a$, $b$, and $c$: \begin{center} \epsfig{file=./XMAPLE/SampleMCMcurrents.ps, scale=0.8} \end{center} Meshes $b$ and $c$ are ready for KVL. Mesh $a$ has a problem in that there is a current source. This can be solved by using an auxiliary equation based on the current source - looking at that mesh, it should be clear that: \begin{align*} \E{i}{a}&=-\E{i}{t} \end{align*} so you may use that as one of your equations. If a current source happens to be between loops, you will end up using two auxiliary equations - one will be the equation to define the current source in terms of two mesh currents, and the other will be a superloop equation that combines the two meshes touched by the current source.
At this point, you have two meshes left for which to write KVL equations: \begin{align*} \mbox{KVL, l}_b&: & -\E{v}{s}+ \E{R}{2}\left(\E{i}{b}\right)+ \E{R}{4}\left(\E{i}{b}-\E{i}{c}\right)&=0\\ \mbox{KCL, l}_c&: & \E{R}{4}\left(-\E{i}{b}+\E{i}{c}\right)+ \E{R}{3}\left(-\E{i}{a}+\E{i}{c}\right)+ \E{R}{5}\left(\E{i}{c}\right)&=0 \end{align*} The three equations - the auxiliary equation plus the two KVLs - can be put into Maple and Maple can solve for the three unknowns. In this case, the power absorbed by $\E{R}{4}$ will be the current through the resistor (now, $\E{i}{b}-\E{i}{c}$) squared multiplied by the resistance.
\section{Assignment} Create three worksheets - one each to solve the equations generated by the circuit above with the three methods. At the end of each worksheet, there should be a variable named {\tt pabsR4XXX} where {\tt XXX} is the method used to generate the equations. Use the {\tt simplify} and {\tt
expand} commands on this final symbolic representation for the power
absorbed and make sure each of the three methods produce the same answer. Then use Maple's {\tt subs} command to determine the power absorbed assuming $\E{v}{s}$=5 V, $\E{i}{t}$=3 mA, $\E{R}{1}$=1.2 k$\Omega$, $\E{R}{2}$=2.2 k$\Omega$, $\E{R}{3}$=3.6 k$\Omega$, $\E{R}{4}$=4.7 k$\Omega$, and $\E{R}{5}$=5.6 k$\Omega$. Display your final answer with 4 significant figures (i.e. use the {\tt evalf[4]} command).
% LocalWords: NVM BCM MCM KCL KVL knowns supernode supernodes Vals MySoln OIT
% LocalWords: evalf superloop Cramer's Maplesoft xmaple startup eqn
