# Impedances In Parallel

If Z1, which is a pure reactance of +10, is connected in parallel with Z2, which is a pure reactance of -10, this should equate to a combined infinite impedance (because this is an LC circuit at resonance).

$${Z}_{1}=a+bj=0+\mathrm{10}j$$ $$a=0$$ $$b=10$$ $${Z}_{2}=c+dj$$ $$c=0$$ $$d=-10$$The combined impedance when connected in parallel can be shown to be:

$$Z=\frac{{a}^{2}c+a{c}^{2}+a{d}^{2}+{b}^{2}c}{{\left(a+c\right)}^{2}+{\left(b+d\right)}^{2}}+\frac{\left({a}^{2}d+b{c}^{2}+{b}^{2}\left(-10\right)+b{d}^{2}\right)}{{\left(a+c\right)}^{2}+{\left(b+d\right)}^{2}}j$$ $$\Rightarrow Z=\frac{{0}^{2}0+0{0}^{2}+0{\left(-10\right)}^{2}+{10}^{2}0}{{\left(0+0\right)}^{2}+{\left(10-10\right)}^{2}}+\frac{\left({a}^{2}d+b{c}^{2}+{b}^{2}d+b{d}^{2}\right)}{{\left(0+0\right)}^{2}+{\left(10-10\right)}^{2}}j$$ $$\Rightarrow Z=\mathrm{infinity}$$