Weird Science

Ohm’s Law in Action: Measuring Resistance Without an Ohmmeter

Polish ver­sion is here

What Is Resi­stance?

The que­stion may seem very sim­ple, and every phy­si­cist and elec­tro­nics enthu­siast sho­uld know the answer. Howe­ver, in prac­tice, it often poses chal­len­ges. Let’s cla­rify it now.

Elec­tri­cal resi­stance is a mea­sure of the oppo­si­tion a com­po­nent pre­sents to the flow of elec­tric cur­rent. It is deno­ted by the upper­case let­ter R. The SI unit of resi­stance is the ohm, sym­bo­li­zed by the Greek let­ter Ω.

For many con­duc­tive mate­rials, the cur­rent flo­wing thro­ugh a resi­stor is line­arly pro­por­tio­nal to the vol­tage across it—this is true for metals. Such con­duc­tors are cal­led linear, as oppo­sed to non­li­near ones (e.g., semi­con­duc­tors), where the cur­rent is a more com­plex func­tion of vol­tage. Today, we will focus on linear pas­sive resi­stance. In this case, the rela­tion­ship between cur­rent and vol­tage is descri­bed by Ohm’s law, expres­sed by the equ­a­tion:

Ilustracja

Where:

I - elec­tric cur­rent (A)

U - vol­tage (V)

R - resi­stance (Ω)

By rear­ran­ging the equ­a­tion to solve for resi­stance, we get R = U/I. This is one of the fun­da­men­tal defi­ni­tions of elec­tri­cal resi­stance. The ratio of the vol­tage across a con­duc­tor to the cur­rent flo­wing thro­ugh it is con­stant (in the case of linear con­duc­tion) and is cal­led resi­stance. This para­me­ter cha­rac­te­ri­zes pas­sive com­po­nents com­monly used by elec­tro­nics enthu­sia­st­s—re­si­stors.

Eno­ugh The­ory: The Mystery Resi­stor

Let’s play detec­tive using our know­ledge of elec­tri­cal laws.

The sub­ject of our inve­sti­ga­tion is a resi­stor. Below is a cera­mic high-power resi­stor:

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Unfor­tu­na­tely, we don’t know its resi­stance. Sure, one could sim­ply use an ohm­me­te­r—but where’s the fun in that? Instead, we’ll use the method pro­vi­ded by Mr. Georg Simon Ohm. Accor­ding to the equ­a­tion above, we need to mea­sure both the vol­tage across and the cur­rent thro­ugh the resi­stor to deter­mine its resi­stance. Let’s build a sim­ple cir­cuit:

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R is the unk­nown resi­stance. The volt­me­ter mea­su­res the vol­tage across R, while the amme­ter mea­su­res the cur­rent flo­wing thro­ugh it. We will cal­cu­late the ratio U/I for seve­ral dif­fe­rent sup­ply vol­ta­ges. By using four AA bat­te­ries (R6 cells) con­nec­ted in series, we can gene­rate mul­ti­ples of 1.5V (~1.5V, ~3.0V, ~4.5V, ~6.0V).

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For each vol­tage, we record both the vol­tage and cur­rent rea­dings. The mea­su­re­ment setup is shown below:

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The digi­tal mul­ti­me­ter's display appe­ars unc­lear because the LCD has a sli­ght delay, and the pic­ture was taken during a value change.

The recor­ded values are ente­red into a spre­ad­sheet (ava­i­la­ble here). My results are shown below:

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The table shows the vol­tage and cur­rent mea­su­re­ments. The third column lists the cal­cu­la­ted ratio U/I, which is the resi­stance. As we can see, the value rema­ins con­si­stent for each mea­su­re­ment, allo­wing us to take the ave­rage as the final result. When using stan­dard SI units (V, A), the cal­cu­la­ted resi­stance is auto­ma­ti­cally in ohms.

Now, let’s plot the cur­rent-ver­sus-vol­tage graph to see why this type of con­duc­tor is cal­led linear:

Ilustracja

The cur­rent incre­a­ses pro­por­tio­nally to the vol­tage applied across the resi­stor. Gra­phi­cally, this rela­tion­ship is repre­sen­ted by a stra­i­ght line.

Our detec­tive work has yiel­ded a resi­stance value of 9.871Ω for the resi­stor. Let’s see how close this is to rea­lity. Here’s the resi­stor’s label:

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The label indi­ca­tes a resi­stance of 10Ω—very close to our mea­su­red result. The mea­su­red value falls well within the resi­stor’s tole­rance range.

Enjoy expe­ri­men­ting! :)

Fur­ther rea­dings:

Marek Ples

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