Problem

# A series RCL circuit has R = 30 Ω, Xc = 50 Ω, and XL = 90 Ω. The impedance of the circuit...

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

#### Step-by-Step Solution

Solution 1

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 $${(b)}$$.