Problem Solving Techniques

Introduction

Students (and professionals!) in science, engineering and technology often experience a challenge in "getting started" on a problem. If the solution to a scientific problem is not immediately obvious, it may help to adopt one of several techniques for seeking the answers. One such technique involves these five steps:

DRAW a picture of the problem, LABEL everything you know on the picture, DEFINE the unknown quantity, EXPRESS the unknown as a function of labeled known quantities, and SOLVE the resulting arithmetic problem.

Draw

Drawing a picture of the problem is often the best first step to take. By drawing a picture, one is forced begin stating the problem in scientific terms, to begin identifying the components or processes involved. Sometimes as the picture is being drawn, previously hidden relationships between different elements of the problem are uncovered. At this stage missing information may be identified and the search for it begun. It is only very rarely that creating a picture of a problem does not result in a more logical approach to a solution.

Label

Labeling every known on the drawing is an important step towards finding the parameters which will be used in formulating the definition. If labeling is done in a generic sense; that is, the name of the quantity rather than its present value, a general solution can be developed which will also serve to solve future problems of the same configuration. It is important that label names be chosen according to standard industry practice, or "debugging" the solution later may be cumbersome.

Define

To define a quantity is not to predict it, nor to estimate it, nor to express it in numbers. The best definitions go as far back to the basics as possible: define velocity as ds/dt, or acceleration as dv/dt, or resistance as dv/di. The gain of an amplifier is defined as V_out /V_in - equations involving resistor values or transconductance are predictions. It will obviously be of great benefit if a definition in terms of previously labeled parameters can be found.

Now is the time to exercise those algebra skills. Manipulate the labeled factors so they can express the definition. Label some new factors on the drawing if the definition requires them, because they might be derivable from quantities already labeled. Perhaps, for example, V_out is not known - but load power and load resistance are. Finding V_out now involves just one simple equation! It is still a good idea to create this expression using named variables instead of numeric quantities, so the solution will be general and applicable to future problems of the same nature.

Solve

Now use whatever arithmetic and algebra skills are at your disposal to simplify the solution. Find variables which cancel, factors which can be gathered, mathematical operations which will help express the solution in the simplest possible form. At this stage of the solution it may be appropriate to introduce approximations which will result in an answer which is correct within a predictable degree of error which is generally agreed to be acceptable. The acceptance of such approximations often leads to an extremely simple answer: for example it may be found that the voltage gain of an amplifier is predicted simply by R_f /R_i, the velocity of an object simply by distance/time.

An example

As an example of this problem solving method, let us examine the solution to the problem of predicting the voltage gain of a simple common-emitter transistor amplifier. Given will be the h_fe and h_ie of the transistor, and a collector load resistor R_L.

Draw & label

The first drawing of this problem results in a picture like that of Figure 1, a conventional schematic drawing. This will prove useful, but it is quickly seen that there is no way to label h_fe and h_ie on this drawing. Therefore, a supplementary drawing using the popular h-parameter model of a transistor, incorporated into the small-signal model of the entire amplifier, will be added - see Figure 2. Now we can label the h-parameters and, thinking ahead, label the input and output voltage and base and collector current as well. Theoretically h_oe and h_re should be considered also, but in most cases we are willing to accept the small error caused by leaving them out.

Define

We have been asked to predict the voltage gain of this amplifier, but have no numerical values for the variables. This is just as well, since it forces us to express a general solution into which we later insert any sets of values we wish. Since the ultimate purpose of a voltage amplifier is accept some small input voltage and present it, in greater amplitude but otherwise unchanged, at the output; the simplest definition of voltage gain is output voltage divided by input voltage, or A_v = V_out /V_in. Had we neglected to label V_out and V_in earlier, we could go back and do it now.

Express

Just having the above definition does not complete the solution - if we wished the measured gain, we could certainly go to the lab and measure V_out and V_in, but that would not be a prediction! So, we must discover a way to express variables in terms of parameters whose values might be given, or found on a data sheet.

First look at V_out. Notice that it is the potential difference across the load resistor, and also that all of the collector current must flow through that same resistor. Thus we may exercise Ohm's Law and state that V_out = I_c*R_L. If these terms were not labeled before, go back and do it now.

Next notice that the input voltage similarly obeys Ohm's Law, so that V_in = I_b*h_ie (we will neglect the current flowing in the bias resistors, as it should be much smaller than I_b). Then we can write:

But is there a simpler way to express this?

Solve

Sometimes solving the last step is just simple algebra. Since we have no numbers to plug in, simplifying the results of the "Express" step may be all that is required. In any case, even if numbers are available, working with the simplest expression produces the fewest errors.

Notice that I_c may also be expressed as h_fe*I_b. Write out the new equation for yourself and you will see that this substitution permits a further simplification. The I_b term appears in both the numerator and denominator, and may thus be divided out. The final result, if you canceled properly, should look like this:

How did you do?

At this point, if the h-parameters are available from a data sheet, and the load resistor known from the specified design, the numerical gain is easy to find. Since we solved the problem using only variable names, the solution is general and will apply to all amplifiers having the same configuration. It is also easy to see what steps could be taken to get more gain: get a transistor with higher h_fe or lower h_ie, or increase the load resistance.

Conclusion

All recipes require some interpretation on the part of the cook. To successfully use this five-step problem solving technique, you must be prepared to interpret the requirements of each step as they apply to the problem at hand. If a required variable was not labeled in a previous step, go back and label it as soon as the need is recognized. Inspect all equations carefully to determine what variable substitutions and simplifications can be performed. Carefully consider the subtle differences between words like "definition", "approximation", "formula" and "prediction". As an exercise, study a problem in which approximations were made to determine what percent error resulted - was the approximation acceptable? And by all means, return to the lab occasionally to confirm that real-world measurements on circuits you build support the predictions you made on paper.

---

Visit Dan@Work – Comments to dan@stlcc.info