153L BME - Spring 2009
Homework 5: Maple Introduction
5.1 Introduction
This homework focuses using Maple to find both the symbolic and the numeric solutions to the linear algebra equations involved with solving electric circuits. Specifically, this lab depends on the full set of equations generated from Ohm's law and Kirchhoff's laws.
5.2 Solving Equations
Once you have found the necessary equations to solve a circuit, there are multiple methods for solving them. If you choose to use matrix methods, you will need to organize the equations such that all the unknowns (and their coefficients) are on one side and the known items are on the other. You can then setup up and solve a matrix equation using whatever method you like (Cramer's Rule, Gaussian Elimination, etc.). If you have access to a computer, however, you may consider using a computational tool to do the algebra for you. One such tool is Maple, produced by Maplesoft. This program is written to perform many mathematical tasks, including symbolic algebra. Solving linear algebra problems with this program can be done in several steps.
5.2.1 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 graphics1 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.
5.2.2 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.
5.2.3 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."
5.2.4 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
1 See
the www.pundit.pratt.edu page on X-Win if you are a PC user or X11R6 if you are a Mac or Linux user.
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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: ax + by + cz = j dx + ey + f z = k gx + hy + iz = l 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. Also, you can put multiple equations in a single variable instead of defining each equation on its own; just separate them with commas: eqns := a * x + b * y + c * z =j , d * x + e * y + f * z =k , g * x + h * y + i * z = l Quick note on Maple variable names - use letters and numbers only. Subscripts are bad, bad ideas for Maple because the subscripted text is considered to be a new variable. Type in the following and note what happens as you try to type it and, once you figure out what is going on with that, when you get to the last line: p_abs := v_R_2 ^2/ R_2 R := 6 p_abs
5.2.5 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 ]) (or solve ({ eqns } , [x , y , z ]) if using the list of equations) 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: Copyright 2009, Gustafson Hwk 5 2
BME 153L - Spring 2009
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 solve line and pre-pend it with MySoln:= so it resembles: MySoln := solve ({ eqn1 , eqn2 , eqn3 } , [x , y , z ]) This will assign the solution list to a variable that we can use later.
5.2.6 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 equations for the known values, then tell Maple to substitute in the numerical values by using the built-in subs command. Add the following lines of code: 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 ) (note - the -1 for a is important; without it, the equations are not linearly independent). The list in MySoln will now be shown with numerical values instead of symbols. Note that you have 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 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.
5.2.7 Using Symbolic Representations
While the subs command was useful for putting numerical values into solutions, you can also use the equations generated with solve as part of the substitution list. What makes this a bit difficult is that MySoln is given as a single-row matrix and subs just wants the equations themselves. To extract only the equations, you can write: subs ( MySoln [1][] , Vals , T h i n g T o S u b I n F o r ) Go ahead and add the line alpha := x + y + z subs ( MySoln [1][] , Vals , alpha ) to the end of your worksheet. Note that the order is important - you need to first substitute in the equations for the circuit variables, then give values to the known quantities, then substitute all that into whatever is 1 in the final argument of subs. At this stage, will be 3 .
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BME 153L - Spring 2009
5.2.8 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 simplify(expand( )) compound function. The expand will take the expression and represent it using as many simple terms as necessary while simplify will recombine them in the most compact form. Finally, to get a decimal value, use the evalf[N]( ) function, where N represents the number of decimal digits to use. For example, s i m p l i f y( expand ( subs ( MySoln [1][] , alpha ))) will produce the most symbolically simplified version of (after substituting in the solutions found for x, y, and z) while evalf [8]( subs ( MySoln [1][] , Vals , alpha )) will produce a floating point result for . With practice, you will see how best to combine evalf, simplify, and expand to get the form of answer you want.
5.3 The Persistence of Memory2
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 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 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.
2 Salvador Dal The Persistence of Memory. 1931. Oil on canvas, 9 1/2 x 13" (24.1 x 33) c 2008 Salvador Dal Gala-Salvador i. i, Dal Foundation/Artists Rights Society (ARS), New York i
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BME 153L - Spring 2009
5.4 Assignment
In each case, you are going to be using Maple to solve for variables in homework problems you have already done. It is perfectly acceptable in this case to look at the solutions to get the equations and also to compare the solutions to your worksheet. (1) Write a Maple worksheet that will take the three KCL equations generated in Rizzoni 3.3 and solve for the node voltages. Call the top left node vx , the middle node vy , and the top right node vy . Show each symbolically and numerically. Then, use Maple to solve for the power delivered by the leftmost current source symbolically and numerically - this will require using subs). Be sure to simplify the symbolic forms for everything. (2) Write a Maple worksheet that will take the three KVL equations generated in Rizzoni 3.16 and solve for the mesh currents. Call the left mesh current i1 , the top right mesh i2 , and the bottom right mesh i3 . Show each symbolically and numerically. Then, use Maple to solve for the power delivered by the leftmost voltage source symbolically and numerically - this will require using subs). Be sure to simplify the symbolic forms for everything. In your Maple worksheet, be sure to include your name and the assignment name at the top. You should also clearly indicate where in your worksheet the answer for each section is by putting a comment just above it; for example, S y m b o l i c s o l u t i o n s for u n k n o w n v a r i a b l e s
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