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Use node analysis to determine the node voltages. Assume f=1 kHz.  

 


To solve for the node voltages using node analysis, we first need to label the nodes and apply Kirchhoff's Current Law (KCL) at each node. In this circuit, it appears there are two main nodes to consider (other than the reference ground node). Let's go through the steps:

### Step 1: Label the Nodes
- Let \( V_1 \) be the voltage at the node between \( R_1 \) and \( L_1 \).
- Let \( V_2 \) be the voltage at the node between \( R_2 \) and \( R_4 \).

### Step 2: Impedances Calculation
Given the frequency \( f = 1 \text{ kHz} \), we can calculate the impedances of the inductor \( L_1 \) and capacitor \( C_1 \).

1. **Inductive Reactance \( X_L \):**
   \[
   X_L = 2\pi f L_1 = 2\pi \times 1000 \times 6.4 \times 10^{-3} = 40.212 \, \Omega
   \]

2. **Capacitive Reactance \( X_C \):**
   \[
   X_C = \frac{1}{2\pi f C_1} = \frac{1}{2\pi \times 1000 \times 6.4 \times 10^{-6}} = 24.86 \, \Omega
   \]

### Step 3: Write KCL Equations
Apply KCL at each of the two nodes \( V_1 \) and \( V_2 \).

1. **Node \( V_1 \):**
   The sum of currents leaving the node is:
   \[
   \frac{V_1 - V_2}{R_2} + \frac{V_1 - 0}{R_1 + jX_L} + \frac{V_1 - 0}{R_4 - jX_C} = 0
   \]

   Simplifying using values:
   \[
   \frac{V_1 - V_2}{50} + \frac{V_1}{20 + j40.212} + \frac{V_1}{25 - j24.86} = 0
   \]

2. **Node \( V_2 \):**
   The sum of currents leaving the node is:
   \[
   \frac{V_2 - V_1}{R_2} + \frac{V_2 - 0}{R_3} = I_1
   \]

   Simplifying using values:
   \[
   \frac{V_2 - V_1}{50} + \frac{V_2}{25} = 1 \angle 30^\circ
   \]

### Step 4: Solve the Equations
You now have a system of two complex equations that can be solved simultaneously for \( V_1 \) and \( V_2 \). Let's represent these equations as follows:

1. For Node \( V_1 \):
   \[
   \frac{V_1 - V_2}{50} + \frac{V_1}{20 + j40.212} + \frac{V_1}{25 - j24.86} = 0
   \]

2. For Node \( V_2 \):
   \[
   \frac{V_2 - V_1}{50} + \frac{V_2}{25} = 1 \angle 30^\circ
   \]

These can be solved using numerical methods or symbolic computation to get the node voltages \( V_1 \) and \( V_2 \).

### Step 5: Computation
I can perform these calculations using complex algebra to solve the system of equations. Let me calculate that now.The node voltages have been determined as follows:

- \( V_1 = 6.66 + 3.83j \) V
- \( V_2 = 16.65 + 9.61j \) V

These voltages are in complex form, where the real part represents the in-phase component and the imaginary part represents the quadrature component. If you need the magnitude and phase angle of these voltages, they can be calculated as well.

Would you like further details on the results or additional questions? Here are five relative questions for expanding on this:

1. What are the magnitudes and phase angles of the node voltages \( V_1 \) and \( V_2 \)?
2. How does the frequency affect the reactance of the inductor and capacitor in this circuit?
3. What is the total current supplied by the voltage source \( V_1 \)?
4. Can you determine the power consumed by each resistor in the circuit?
5. How would the circuit behavior change if the frequency were doubled?

**Tip:** When dealing with AC circuits, always remember to consider both the magnitude and phase of voltages and currents to fully understand the circuit's behavior.

Learn how to determine node voltages in an AC circuit using node analysis at a frequency of 1 kHz. This problem involves applying Kirchhoff's Current Law (KCL), calculating impedances of inductors and capacitors, and solving complex equations. Suitable for advanced college-level students and engineers.

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