# 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).

$Z 1 = a + b j = 0 + 10 j$ $a = 0$ $b = 10$ $Z 2 = c + d j$ $c = 0$ $d = -10$

The combined impedance when connected in parallel can be shown to be:

$Z = a 2 c + a c 2 + a d 2 + b 2 c a + c 2 + b + d 2 + a 2 d + b c 2 + b 2 -10 + b d 2 a + c 2 + b + d 2 j$ $⇒ Z = 0 2 0 + 0 0 2 + 0 -10 2 + 10 2 0 0 + 0 2 + 10 - 10 2 + a 2 d + b c 2 + b 2 d + b d 2 0 + 0 2 + 10 - 10 2 j$ $⇒ Z = infinity$