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Circuits 01

If we think about circuits from the perspective of inputs and outputs, our first-level model will have to cover current, voltage, and resistance. We can use GeoGebra to help us look at these from different perspectives.

Linear Fuctions

Linear functions are probably the most familiar. They are just straight lines. The standard mantra in American education is y = mx + b. You may know this formula by name, but do you know what it means? When you move from the abstraction of a generalized line with some "slope" and "y-intercept" to the applications of a particular formula, the connection sometimes gets lost. And sometimes even the simple idea of a "linear function" becomes more complex. Let's take Ohm's law as an example. V = IR is the common way that this formula is stated and memorized. But we know that it can be rearranged into other forms:
All of these are equivalent expressions of Ohm's law, but they represent different ways of thinking about inputs and outputs. We usually think of the variable that is isolated (and usually put on the left) as the output. The stuff on the right side of the equals sign includes the input, but it often also includes constants (or coefficients). For now, let's work with just the first two versions of Ohm's law listed above.

Current as a function of voltage

If we treat voltage as the input, it will be plotted on the horizontal axis. If we treat current as the output, it will be plotted on the vertical. Resistance determines the relationship between the input and the output, and in the graph below you can change the resistance to see the different functions that result. Note that the resistance determines the slope of the line, but the value of the resistance does not equal the slope of the line according to the standard definition of slope. Why?

Voltage as a function of current

If we treat current as the input, it is plotted on the horizontal axis. Voltage, as the output, is plotted on the vertical axis. Resistance still connects the two, and this time it fits the standard definition of the slope of the line that traces the function.

In either (or both) of the plots above, set the resistance to 40 (which we take to mean 40 ). What current would flow if the voltage was a) 40 V, b) -80 V, or c) 60 V?

In either (or both) of the plots above, set the resistance to 100 . Estimate the current shown on the graph for voltages of a) -20 V, b) 40 V, and c) 60 V.

Non-linear functions

But there's another set of possibilities! What if we treat Ohm's law from the perspective of having a set voltage (which is reasonable if you always use the same type of battery), then we may want to think about current as the output given a particular resistance as the input. The relationship is still given by Ohm's law, and it can be thought of as coming from the second version of the formula given above: . But this time, we are treating R as the variable and V as the constant (or the coefficient). What does the graph of current as a function of resistance look like?

Current as a function of resistance

Here we treat resistance as the input, so it is on the horizontal axis. Current, as the output, is on the vertical. Voltage is now the quantity that connects the two, but current is inversely proportional to resistance. As resistance increases, current decreases; but the relationship is not linear!