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To solve this problem, we need to calculate the magnitude of the output voltage \( V_o \) phasor in the given network. The following information is provided:

- Source voltage \( V_s = 38 \, V \)
- Resistor \( R_1 = 4 \, \Omega \)
- Resistor \( R_2 = 9 \, \Omega \)
- Inductor impedance \( j2 \, \Omega \)
- Capacitor impedance \( -j1 \, \Omega \)

### Step 1: Circuit Analysis
The circuit contains a combination of resistors, inductors, and capacitors arranged in a parallel network, which requires us to use phasor analysis. To compute \( V_o \), we'll need to find the voltage division between the components in parallel.

The impedance of each branch is:

1. **First branch (left loop with \( R_1 \) and capacitor):**
   \[
   Z_{\text{left}} = R_1 - j1 = 4 - j1 \, \Omega
   \]

2. **Second branch (right loop with inductor and \( R_2 \)):**
   \[
   Z_{\text{right}} = R_2 + j2 = 9 + j2 \, \Omega
   \]

### Step 2: Parallel Impedance Calculation
The total impedance of two parallel branches is given by:
\[
\frac{1}{Z_{\text{total}}} = \frac{1}{Z_{\text{left}}} + \frac{1}{Z_{\text{right}}}
\]

Substituting the values:
\[
\frac{1}{Z_{\text{total}}} = \frac{1}{4 - j1} + \frac{1}{9 + j2}
\]

We will calculate the total impedance \( Z_{\text{total}} \), then use voltage division to find \( V_o \).

Let me calculate this for you.The total impedance \( Z_{\text{total}} \) of the parallel combination is approximately \( 2.9 - 0.3j \, \Omega \).

### Step 3: Voltage Division
To find the voltage \( V_o \), we use the voltage division rule. The output voltage \( V_o \) is across the right branch, so:

\[
V_o = V_s \times \frac{Z_{\text{right}}}{Z_{\text{total}}}
\]

Let me compute the magnitude of \( V_o \).The magnitude of the output voltage \( V_o \) phasor is approximately \( 120.2 \, \text{V} \) (rounded to the nearest tenth).

Would you like more details on any part of the process or have any questions?

Here are five related questions you might want to explore:
1. How does the phase angle of \( V_o \) compare to that of \( V_s \)?
2. What is the impedance of the circuit if only resistors were present?
3. How would the result change if the inductor's impedance doubled?
4. What is the power consumed by each resistor in this circuit?
5. Can the voltage phasor be calculated without using complex numbers?

**Tip:** When solving AC circuits, always convert all values to phasor form (with real and imaginary components) for correct voltage and current calculations.

Given the network in Fig. P8.52, determine the magnitude of the Vo phasor if Vs = 38V, R1 = 4 ohms, and R2 = 9 ohms.

This detailed guide solves an AC circuit problem involving resistors, inductors, and capacitors using phasor analysis and voltage division. Learn how to calculate the magnitude of the Vo phasor with Vs = 38V, R1 = 4 ohms, and R2 = 9 ohms.

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