 Desirae Washington

2022-07-09

Technique to solve this equation of 2 unkowns
I was solving a problem of single phase eletrical circuits where I had to find the inductor $L$ and resistance $R$. I managed to get two equations containing the two unknowns.
$\frac{R}{{R}^{2}+\left(w\ast L{\right)}^{2}}={c}_{1}$
and
$\frac{wL}{{R}^{2}+\left(w\ast L{\right)}^{2}}={c}_{2}$
where are known.How do I solve this? Ronald Hickman

Expert

Squaring both equations and adding them you get
$\frac{{R}^{2}}{\left({R}^{2}+\left(w\ast L{\right)}^{2}{\right)}^{2}}+\frac{\left(w\ast L{\right)}^{2}}{\left({R}^{2}+\left(w\ast L{\right)}^{2}{\right)}^{2}}={c}_{1}^{2}+{c}_{2}^{2}$
or
$\frac{1}{{R}^{2}+\left(w\ast L{\right)}^{2}}={c}_{1}^{2}+{c}_{2}^{2}$
This yields:
${R}^{2}+\left(w\ast L{\right)}^{2}=\frac{1}{{c}_{1}^{2}+{c}_{2}^{2}}$
Now replace the denominators in both equations. Holetaug

Expert

Alternate solution
Dividing the two equations you get
$\frac{wL}{R}=\frac{{c}_{2}}{{c}_{1}}$
Thus
$wL=\frac{{c}_{2}R}{{c}_{1}}\phantom{\rule{thinmathspace}{0ex}}.$
Replacing in either equation you get an equation in $R$.

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