Maple Worksheet 1 - Introduction to Maple and DC circuits

[notation, labeling, sets, solve(),subs(),assign()]

We will be using Maple, a computer algebra system, to solve equations, do numerical analysis, and explore physical models in this course. This introductory worksheet explores some of the basic concepts and reviews the basics of DC circuit analysis.

restart:

Part I - Maple Basics - based on Intro to Maple for physics students by Ross L. Spencer (http://www.mapleapps.com/powertools/physics/Physics.shtml)

Page setup - The window space marked off by the the square bracket is called an execution group and everything happens inside these groups. Here's what you have to know: everything before the > sign is text, or typeset mathematics, and the red stuff that comes after is code that Maple will execute to solve your problems. After the expression is keyed in, it is entered into Maple using the enter key on your keyboard. Maple results print in blue, like this:

sin(Pi/4);

The purpose of the text area is to provide you with space to document your code - use it! Note that we have terminated the Maple code with a semi-colon; Maple requires every executable statement to end with a semicolon or a colon. Use of a colon will suppress output.

Online help - Like all programs, you can find online help by clicking on the Help icon at the top of the page. But there is a quicker way to get help directly at the Maple prompt: just use the question mark. For example, suppose you are pretty sure that Maple knows how to find the prime factors of big numbers, but you don't know the command. Try this:

?factor;

When you do you will find that this is the wrong command because it does algebra on polynomials. But if you read a bit, and look at the helpful See Also links at the bottom of the page you will see that the command you want is ifactor, so you either go there or use

?ifactor;

Now help will tell you that this is just what you want, and you can try

ifactor(5040);

Special Constants - There are several constants pre-loaded in Maple that are useful for physics. Some of them are:

Pi;I;infinity;

Pi is the number Pi to about as many significant figures as you want. To see its digits, use the Maple command evalf to convert a number to floating point form

evalf(Pi);

You can increase the number of digits by changing the Maple constant Digits (default value is 10);

Digits:=100;evalf(Pi);

and you should be impressed. I is the imaginary unit, i.e., NiMvJSJJRy0lJXNxcnRHNiMsJCIiIiEiIg== ; try executing

sqrt(-5);

Infinity is, of course, not a number, but an idea, and only makes sense in limits. And Maple does the right thing in calculations involving limits, like

limit(exp(-x),x=infinity);

Variable names - Like all other modern computer languages, Maple allows you to name a variable just about anything you want, with some exceptions. Use names that are as close as possible to the names you would usually use when writing out equations. For example, the equation expressing Ohm's Law could be

V=i*R;

Many greek letters are also possible

alpha=beta/gamma;

Maple will let you know if a variable can't be used. Also avoid putting subscripts on variables, even though Maple's typesetting ability will let you do it. For instance you might want to work with the variables NiMmJSJpRzYjIiIi and NiMmJSJpRzYjIiIj which you get by doing this in Maple

i[1];i[2];

They look great, but now watch what happens if you give NiMlImlH a value, then display NiMmJSJpRzYjIiIi

i:=3;

i[1];

I think we can all agree that this is not going to help us do mathematics. So if you want to code NiMmJSJpRzYjIiIi in Maple, call it i1. And as you choose variables, experiment a little with the names and try for elegance and simplicity.

Assigning value to variables - in Maple the statement

R=5;

doesn't mean that the variable a is equal to 5. Instead, to assign to variable NiMlIlJH the value 5 the Maple statement would be

R:=5;

Now we should be able to calculate the voltage in a circuit where R=5 ohms and i=3 A.

This doesn't quite work as we have not assigned the value of V to the product i*R

V:=i*R;

Note that Maple has remembered the values we assigned to i and R earlier - Maple remembers forever. The quickest way to make sure this doesn't mess up your worksheet is to restart Maple,

restart;

We can assign all sorts of things to our variables, including floating point numbers, complex numbers, equations etc.

f1:=4./3.;

We can assign labels to our equations to make it easier to refer to them later

OL:=V=i*R;

Expressions and functions - An expression looks like what you would probably call a function, but there is a subtle difference. An expression is a combination of variables and Maple functions like this

Fn1:=cosh(x^3)/(1+x^2);

The reason it isn't a function is because of what you have to do to evaluate it for a specific value of x: Fn1(x) doesn't work! Instead you have to assign x a value like this

x:=2.;

then evaluate Fn1

Fn1;

A Maple function is defined with this syntax

Fn2:=x->cosh(x^3)/(1+x^2);

This does behave like a function because you give it an argument and it gives you an answer back

Fn2(2.);

Or you can define a function with multiple arguments like this

Fn3:=(x,y,z)->x*y^2*z^3;

Fn3(1,2,3);

Hence, the -> function notation makes functions like you are used to from math classes. So why would you even want to mess with expressions? Because Maple often has an easier time doing algebra with them, integrating them, differentiating them, plotting them, and using them to build differential equations. So you have to get used to both. Try to remember it this way: a Maple expression looks like a function in a math book; a Maple function is a machine that takes input and gives back output.

Part II - Review of DC circuit analysis

Analysis of the voltages and currents in a linear DC (direct current, i.e. not oscillating) circuit requires only

1. Ohm's law V=iR

2. Kirchoff's laws. These express the concepts of conservation of charge and energy

a. The algebraic sum of the currents at any junction in a circuit is zero (conservation of charge)

b. The algebraic sum of the potential differences around any closed loop in a network is zero (conservation of energy).

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The circuit on the right is a basic voltage divider. The ratio of the voltages across the two resistors is equal to the ratio of the values of the resistances. Furthermore, the voltage at the junction between R1 and R2 is a fraction of the supply voltage. By controlling R1 and R2 we can obtain any fraction of the supply voltage to use in another circuit. We can derive these relations using Kirchhoff's laws and Ohm's law. We assume that V, R1, and R2 are known, use Kirchhoff's laws and Ohm's law to write down a set of coupled equations relating the current through the resistors and the voltage drop across each resistor, and then use Maple to solve the set of coupled equations that we obtain to find equations for the unknown variables.

First restart:

restart;

Consider the junction point between R1 and R2. Using Kirchhoff's first law we can say that all the current passing through R1 must pass through R2 as well. Lets call the first equation KL1

KL1:=i1=i2;

The voltages across the two resistors are V1 and V2. Kirchhoff's second law states that the voltage rise across the battery V is the same as the voltage drop across both resistors, V1+V2.

KL2:=V=V1+V2;

We need two more equations to solve for the four unknowns. These equations can be the current-voltage characteristic of the resistors. Since Ohm's law applies to the resistors, we can write:

OL1:=V1=i1*R1;

OL2:=V2=i2*R2;

We can define these four equations as a set.

{KL1,KL2,OL1,OL2};

Let's given the label "eqns" to our set of four equations.

eqns:={KL1,KL2,OL1,OL2};

Once the set is defined, its elements can be referred to.

eqns[2];

Use the function "solve" to solve this set of equations for the desired quantities: V1,V2, i1, and i2. "Solve" looks for a closed form solution. For information about usage of this function, check the Maple Language Help.

soln:=solve(eqns,{V1,V2,i1,i2});

You can pick off a particular solution using the "subs" command, e.g., the solution for i1 is obtained as

subs(soln,i1);

Let's check that the voltage is divided properly using the elements of soln.

vdiv:=V1/V2=subs(soln,V1)/subs(soln,V2);

It is!

[Note: Using the "subs" command is the recommeded method to pick off various elements out of a set of solutions. You can also get the solution for i1 by looking at the right-hand side of the fourth element of soln.

rhs(soln[4]);

However, sometimes the order of the solutions in soln is reversed. In particular, if you use Maple on a machine with a different operating system, you cannot be sure that the solution for i1 is still the fourth element of soln! Therefore, never use something like rhs(soln[4]) or lhs(soln[1]), but always use the subs command. ]

If you do not want to use the subs command for the rest of the worksheet, it is possible to assign the solutions permanently to the variables and then use their names as placeholders:

assign(soln);

i1;i2;V1;V2;

How is the power divided between the two resistors?

p1:=i1*V1;

p2:=i2*V2;

Then the ratio of power dissipated in the two resistors is given by:

pdiv:= p1/p2;

Is this surprising?

Equivalent Circuits

It is sometimes easier to analyze DC circuits by replacing all or part of a network with an equivalent circuit that has the same characteristics as the original. There are two types of Equivalent Circuits:

1. Thevenin Equivalent Circuit - Thevenin's theorem states that any black box containing a complicated network of resistors, batteries, or current sources can bereplaced by a series combination of an output resistor Req and an ideal voltage source Veq and a pair of output terminals. By definition, an ideal voltage source has zero resistance. A good electronically voltage-stabilized power supply is close to ideal, up to a specified current, beyond which the voltage falls. Thevenin's theorem therefore provides us with a simple model for describing the operation of a real source. Consider the circuit below (notice that this is just another voltage divider circuit, with a load resistor attached).

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The voltage Veq is the voltage between the terminals of the Thevenin's equivalent circuit when the resistor R is removed (open circuit voltage); so if the two circuits behave identically, itmust also be the open-circuit voltage of the orignal circuit.

Veq:=R2/(R1+R2)*V;

We can obtain Req by noting that the current of the equivalent circuit when the resistor R is shorted (short-circuit current) is Veq/Req.

ieq:=V/R1;

Then the resistance Req is just

Req:=Veq/ieq;

2. Norton Equivalent Circuit - here a black box with two output terminals containing a complicated network of resistors, batteries, and current sources is replaced by a parallel combination of an output resistor Req and an ideal current source ieq.

Problem 1: Connect two resistors in parallel across a current source, as in the circuit below. Set up Kirchhoff's laws for this case and solve for the current through each resistor, the voltage across the resistors, and the power dissipated by each resistor.

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Problem 2: Find the direction of the current through R3 and the power dissipated in R3 as in the circuit below. Assume that V1 = 10V, V2 = 15V and R1=10kW, R2 = 15kW, R3 = 12kW.

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

Problem 3: For the unbalanced bridge circuit shown below, carry out the full network analysis using the Kirchoff method. Calculate the current through a load RL connected across A and B, as well as the voltage across RL. Do the calculation for RL = 10kW and RL = 2kW and compare with the answers obtained by the Thevenin method.

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