To solve this circuit problem using mesh current analysis, let us break it down systematically. The goal is to find $i_1$, $i_2$, and $i_3$.

Step 1: Define Mesh Currents

• Let $i_1$, $i_2$, and $i_3$ represent the currents in the three loops as indicated in the diagram.

Step 2: Apply KVL (Kirchhoff's Voltage Law) in Each Loop

Loop 1: $i_1$

$24 - 4(i_1 - i_2) - 2i_1 = 0$ $24 = 6i_1 - 4i_2$ 6i_1 - 4i_2 = 24 \tag{1}

Loop 2: $i_2$

• The dependent voltage source is $3v_0$, where $v_0 = 4(i_3 - i_2)$. Hence: $3v_0 = 3[4(i_3 - i_2)] = 12(i_3 - i_2)$

Writing KVL for Loop 2: $-4(i_2 - i_1) + 2i_2 - 12(i_3 - i_2) = 0$ Simplify: $-4i_2 + 4i_1 + 2i_2 - 12i_3 + 12i_2 = 0$ $4i_1 - 14i_2 - 12i_3 = 0$ 4i_1 - 14i_2 + 12i_3 = 0 \tag{2}

Loop 3: $i_3$

$12(i_3 - i_2) + 8i_3 + 4i_3 = 0$ $12i_3 - 12i_2 + 8i_3 + 4i_3 = 0$ $24i_3 - 12i_2 = 0$ $12i_2 = 24i_3$ i_2 = 2i_3 \tag{3}

Step 3: Solve the System of Equations

We now have the following three equations:

1. $6i_1 - 4i_2 = 24$
2. $4i_1 - 14i_2 + 12i_3 = 0$
3. $i_2 = 2i_3$

Substitute $i_2 = 2i_3$ into (1) and (2):

From (1): $6i_1 - 4(2i_3) = 24$ $6i_1 - 8i_3 = 24$ $6i_1 = 24 + 8i_3$ i_1 = 4 + \frac{4i_3}{3} \tag{4}

From (2): $4i_1 - 14(2i_3) + 12i_3 = 0$ $4i_1 - 28i_3 + 12i_3 = 0$ $4i_1 - 16i_3 = 0$ i_1 = 4i_3 \tag{5}

Step 4: Solve for $i_3$

Equating (4) and (5): $4 + \frac{4i_3}{3} = 4i_3$ $4 = 4i_3 - \frac{4i_3}{3}$ $4 = \frac{12i_3 - 4i_3}{3}$ $4 = \frac{8i_3}{3}$ $i_3 = \frac{12}{8} = 1.5 \, \text{A}$

Step 5: Find $i_1$ and $i_2$

From (5): $i_1 = 4i_3 = 4(1.5) = 6 \, \text{A}$

From (3): $i_2 = 2i_3 = 2(1.5) = 3 \, \text{A}$

Final Answer:

• $i_1 = 6 \, \text{A}$
• $i_2 = 3 \, \text{A}$
• $i_3 = 1.5 \, \text{A}$

Would you like a more detailed step-by-step explanation or have any specific questions?

Follow-Up Questions:

1. What is the principle behind mesh current analysis?
2. How does the presence of a dependent source affect KVL equations?
3. Why do we substitute $i_2 = 2i_3$ to simplify the equations?
4. Can you solve this circuit using nodal analysis instead of mesh analysis?
5. How do you interpret the direction of the mesh currents physically?

Tip:

Always double-check dependent sources and their controlling variables when writing KVL equations, as errors there can propagate through your solution.