concertina: How to : solve resistor warming case mathematically

A magnetic coil, which made from copper have constant voltage. In operation the temperature is going higher from 12 degrees Celsius to 60 degrees Celsius. How much percentage has the electric current changed by the warming of the coil?

bonus. how much is the percentage of the resistor?

What we know?

voltage is constant
This function about how to search resistor from a certain temperature

$\120dpi \large R{}_{\theta}=R{}_{\theta_{0}}[1+\alpha(\theta-\theta_{0})]$

$\120dpi \large R{}_{\theta}$ = Resistor from temperature theta
$\120dpi \large R{}_{\theta_{0}}$ = Resistor from referenced temperature
$\120dpi \large \alpha$ = coefficient from temperature which depend on the material
$\120dpi \large \theta_{0}$ = referenced temperature (in most case 20 degrees Celsius)
$\120dpi \large \vartheta_{1}=+12^{o}C\rightarrow\vartheta_{2}=+60^{o}C$
$\120dpi \large \alpha_{20}=3,93\times10^{-3}\frac{1}{K}$

what do we need to calculate
$\120dpi \large ?\frac{\bigtriangleup R}{R_{12}} and \frac{\bigtriangleup I}{I_{12}}$

okay here is the answer
$\inline \120dpi \large \frac{R_{60}-R_{12}}{R_{12}}=\\\frac{R_{20}[1+\alpha_{20}(60-20)]-R_{20}[1+\alpha_{20}(12-20)]}{R_{20}[1+(12-20)]}=\\\frac{R_{20}[\alpha_{20}(60-12)]}{R_{20}[1+(12-20)]}=\\\frac{\alpha_{20}(48)}{1+\alpha_{20}(-8)}=0,194$

that is the way to search the resistor's percentage

then move to electric current

$\120dpi \large \bigtriangleup I=I_{60}-I_{12}=\\\frac{U}{R_{60}}-\frac{U}{R_{12}}=\frac{-U\times(R_{12}-R_{60})}{R_{60}\times R_{12}}$

$\120dpi \large I_{12}=\frac{U}{R_{12}}$

with that we can search for the percentage
$\120dpi \large \frac{\bigtriangleup I}{I_{12}}=\frac{\frac{-U\times(R_{12}\times R_{60})}{R_{60}\times R_{12}}}{\frac{U}{R_{12}}}=\frac{-R_{12}\times R_{60}}{R_{12}}=\\\frac{-R_{20}[\alpha_{20}(12-60)]}{R_{20}[1+\alpha_{20}(60-12)]}=\\\frac{-\alpha_{20}(12-60)}{1+\alpha_{20}(60-12)}=0,16$
thats all. 19% change on resistor and 16% on Electric current.

headfake from this post.

use Latex for this mathematic calculation
thanks to codecogs for letting me write latex on blogspot
i use lyx to get better view of what i calculate
it also give me another project about how to install latex plugin on Ubuntu

concertina

Sunday, October 31, 2010

How to : solve resistor warming case mathematically

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