1240315-1329 EDT
pegggu:
When playing with mathematical algebraic equations you need to ask some questions. But first realize that there are three fundamental components to an equation --- an equal sign, variables, and constants. Constants do not change while evaluating an equation, and variables change. The equal sign separates two parts of an equation that are equal.
Variables are many times written as lower case letters, and constants as upper case letters or actual numerical constants.
Consider the equation v = R*i . R is a constant, i is an independent variable, and v is a dependent variable. "v" is dependent upon on how "i" is changed. The slope of a curve is a measure of how rapidly the curve is changing. In this equation R is the slope. When plotted on a linear two dimensional graph the curve looks like a straight line with a slope of R. Double the value of R and the new slope is 2 times the original slope. In other words the voltage varies twice as rapidly with respect to current when resistance is 2*R compared to when resistance is R.
Rearrange this equation to read i = (1/R)*v. Now v is the independent variable and i is dependent. Here the slope is 1/R. This is still a straight line plot. Just a different slope.
Modify the first equation to v = V1 + R*i . Still a straight line, but offset from 0 V when i = 0. The slope is still R. Thus, the plot is a straight line parallel and offset from the first plot.
Change the equation to i = V/r . Now a plot of i vs r is not linear and the incremental slope of the curve at any point is constantly changing.
Next consider real world devices.
A 1/4 W 2000 ohms metal film resistor. Ohmmeter measurement 1996 ohms, very little current.
Voltage . Current . Resistance . Power
. Volts ..... mA ......... Ohms ..... mW
00.00 .... 00.000 .... ----- ...... 000.0
00.99 .... 00.494 .... 2004 ...... 000.5
04.99 .... 02.497 .... 1998 ...... 012.0
10.04 .... 05.042 .... 1991 ...... 050.6
15.02 .... 07.528 .... 1995 ...... 113.1
20.00 .... 10.023 .... 1995 ...... 200.5
25.05 .... 12.548 .... 1996 ...... 314.3
At the low voltage there was power supply instability, and reading two different meters caused some error. The 25 V level may imply some heating effect. By observation the resistance is constant over the rating range of the resistor.
An 1819 pilot lamp bulb rated 28 V at 35 mA. Ohmmeter measurement 93 ohms, very little current.
Voltage . Current . Resistance . Power
. Volts ..... mA ......... Ohms ..... mW
00.00 .... 00.00 ....... ---- ...... 000.0
01.01 .... 05.64 ....... 179 ...... 005.7
02.52 .... 08.80 ....... 286 ...... 012.0
05.00 .... 12.89 ....... 388 ...... 064.5
10.00 .... 19.31 ....... 518 ...... 193.1
10.52 .... 19.89 ....... 529 ...... 209.2 incremental resistance 0.52/0.58 = 897 ohms
14.98 .... 24.50 ....... 611 ...... 367.0
15.51 .... 24.98 ....... 621 ...... 387.4 incremental resistance 0.53/0.48 = 1104 ohms
20.02 .... 27.69 ....... 723 ...... 554.4
20.55 .... 28.12 ....... 731 ...... 577.9 incremental resistance 0.53/0.43 = 1233 ohms
25.00 .... 31.45 ....... 794 ...... 786.3
25.52 .... 31.86 ....... 801 ...... 815.6 incremental resistance 0.52/0.41 = 1268 ohms
You will notice that this does not obey Ohm's law because resistance is not a constant, but varies with current, or voltage. Current is the real factor because this heats the filament. Also note the incrementaal resistance (slope of the curve) varies with the filament temperature.
Note: an equation of the form y = K/x is not a hyperbola, but looks similar to one. It seems to be generally called a reciprocal curve. There is possibly a more specific name than just 1/x curve. A general equation for a hyperbola is ( (x-h)^2/a^2 ) - ( (y-k)^2/b^2 ) = 1 . A hyperbola is a conic section, obtained by cutting a cone with a plane surface.
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