A series RCL circuit has R = 30 Ω, Xc = 50 Ω, and XL = 90 Ω. The impedance of the circuit is:
(a) 30 + j140 Ω
(b) 30 + j40Ω
(c) 30 – j40 Ω
(d) –30 – j40 Ω
(e) –30 + j40 Ω
Calculate the impedance of the resistor.
$$ \begin{aligned} \mathbf{Z}_{R} &=R \\ &=30 \Omega \end{aligned} $$
Calculate the impedance of the inductor.
$$ \begin{aligned} \mathbf{Z}_{L} &=j X_{L} \\ &=j 90 \Omega \end{aligned} $$
Calculate the impedance of the capacitor.
$$ \begin{aligned} \mathbf{Z}_{c} &=-j X_{c} \\ &=-j 50 \Omega \end{aligned} $$
Calculate the impedance of the series RLC circuit.
$$ \mathbf{Z}=\mathbf{Z}_{R}+\mathbf{Z}_{L}+\mathbf{Z}_{C} $$
Substitute \(30 \Omega\) for \(\mathbf{Z}_{R}, j 90 \Omega\) for \(\mathbf{Z}_{L}\) and \(-j 50 \Omega\) for \(\mathbf{Z}_{C}\).
$$ \begin{aligned} \mathbf{Z} &=30+j 90-j 50 \\ &=30+j 40 \Omega \end{aligned} $$
Thus, the correct answer is \({(b)}\).