Let's analyze the circuit using mesh analysis to find $V_3$.

Step 1: Identify Meshes

We have three loops (meshes) in this circuit:

• Mesh 1 (I1): Contains the $3V$ source, the $1 \Omega$ resistor, and the $4 \Omega$ resistor.
• Mesh 2 (I2): Contains the $4 \Omega$ resistor, the $2 \Omega$ resistor, the dependent current source $1.8 V_3$, and the $V_3$ voltage.
• Mesh 3 (I3): Contains the $5V$ voltage source, the dependent current source $1.8 V_3$, and the resistor $X = 15 \Omega$.

Step 2: Write Mesh Equations

We need to write mesh equations using Kirchhoff's Voltage Law (KVL) for each loop.

1. Mesh 1 (I1): The sum of voltages around Mesh 1 is: $3V - I1(1 \Omega) - (I1 - I2)(4 \Omega) = 0$ Simplifying: $3 = I1 + 4(I1 - I2)$ $3 = 5I1 - 4I2 \quad \text{(Equation 1)}$

2. Mesh 2 (I2): The sum of voltages around Mesh 2 includes the $V_3$ dependent source and the resistors: $(I2 - I1)(4 \Omega) + I2(2 \Omega) - V_3 = 0$ Simplifying: $4(I2 - I1) + 2I2 - V_3 = 0$ We also know that $V_3 = 15I3$ (from the value of $X = 15 \Omega$).

   Substituting this in: $4(I2 - I1) + 2I2 - 15I3 = 0$ $4I2 - 4I1 + 2I2 - 15I3 = 0$ $6I2 - 4I1 = 15I3 \quad \text{(Equation 2)}$

3. Mesh 3 (I3): The sum of voltages around Mesh 3 includes the $5V$ source, the $15 \Omega$ resistor, and the dependent source: $5 - 15I3 = 0$ Simplifying: $I3 = \frac{5}{15} = \frac{1}{3} \quad \text{(Equation 3)}$

Step 3: Solve the System of Equations

We now have the following system:

1. $3 = 5I1 - 4I2$
2. $6I2 - 4I1 = 15 \times \frac{1}{3} = 5$
3. $I3 = \frac{1}{3}$

Let's solve these equations to find $V_3$.

I will calculate it step by step.The solutions for the mesh currents are:

• $I_1 = \frac{19}{7} \, \text{A} \approx 2.71 \, \text{A}$
• $I_2 = \frac{37}{14} \, \text{A} \approx 2.64 \, \text{A}$

Additionally, the voltage $V_3$ is:

$V_3 = 5 \, \text{V}$

Would you like further details or clarifications on any part of this solution?

Here are 5 related questions to further expand on this:

1. How would the analysis change if we had a different value for the dependent current source?
2. What happens if the value of $X$ (the resistor) changes?
3. Can you explain the significance of dependent sources in mesh analysis?
4. How would you analyze this circuit using nodal analysis instead of mesh analysis?
5. How does the mesh current method compare with the superposition theorem in solving such circuits?

Tip: When dealing with dependent sources, always express the controlling variable (e.g., $V_3$) in terms of mesh or node variables to simplify the equations.