Worked Example of Impedances in Parallel - LC resonant circuit

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 = \frac{a^{2} c + a c^{2} + a d^{2} + b^{2} c}{{(a + c)}^{2} + {(b + d)}^{2}} + \frac{(a^{2} d + b c^{2} + b^{2} (-10) + b d^{2})}{{(a + c)}^{2} + {(b + d)}^{2}} j

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

\Rightarrow Z = infinity