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Example-Applying the Basic Laws

UNIVERSITY

ample 1.3 Consider a two-mesh network of Fig. 1.22. Express the currents in the three resistances and then write the KVL equation.

To solve Example 1.3, we need to apply the **Kirchhoff's Voltage Law (KVL)** to the two-mesh network shown in **Fig. 1.22**. Since you referred to a figure and I cannot see it, I will assume a basic two-mesh network with three resistors for explanation purposes. If your circuit has a different configuration, feel free to upload a picture.

### Step 1: Define the Mesh Currents
In a two-mesh network, there are two loops (meshes). Let's assign the following:
- Mesh 1 current: \( I_1 \) (in the clockwise direction)
- Mesh 2 current: \( I_2 \) (in the clockwise direction)

### Step 2: Label the Resistors
Assume the network has three resistances as follows:
- \( R_1 \) in the first mesh,
- \( R_2 \) shared by both meshes (between Mesh 1 and Mesh 2),
- \( R_3 \) in the second mesh.

Additionally, there may be voltage sources \( V_1 \) in Mesh 1 and \( V_2 \) in Mesh 2.

### Step 3: Express Currents in the Resistors
For each resistor, the current can be expressed based on the mesh currents:
- **Current in \( R_1 \)**: This resistor is only in Mesh 1, so the current is \( I_1 \).
- **Current in \( R_2 \)**: This resistor is shared by both meshes. The current through it is the difference between the two mesh currents, so the current is \( I_1 - I_2 \).
- **Current in \( R_3 \)**: This resistor is only in Mesh 2, so the current is \( I_2 \).

### Step 4: Apply Kirchhoff's Voltage Law (KVL)
We now apply KVL to both meshes.

#### Mesh 1 (Loop 1):
The voltage drops in Mesh 1 are across resistors \( R_1 \) and \( R_2 \). By KVL, the sum of voltage drops is equal to the voltage source \( V_1 \):
\[
V_1 = I_1 R_1 + (I_1 - I_2) R_2
\]

#### Mesh 2 (Loop 2):
The voltage drops in Mesh 2 are across resistors \( R_2 \) and \( R_3 \). By KVL, the sum of voltage drops is equal to the voltage source \( V_2 \):
\[
V_2 = (I_2 - I_1) R_2 + I_2 R_3
\]

### Final Equations
The two KVL equations for the system are:
1. \( V_1 = I_1 R_1 + (I_1 - I_2) R_2 \)
2. \( V_2 = (I_2 - I_1) R_2 + I_2 R_3 \)

You can solve these two simultaneous equations to find the values of \( I_1 \) and \( I_2 \). Once you have \( I_1 \) and \( I_2 \), you can calculate the current through each resistor using the expressions above.

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If you have any additional information about the circuit (like values for resistances or voltage sources), I can help you solve it fully. Do you want more details or clarifications?

### Related Questions:
1. How can Kirchhoff's Current Law (KCL) be applied to the same circuit?
2. What happens if one of the resistors in the two-mesh network is replaced with a capacitor?
3. Can the system be solved using matrix methods instead of solving it by hand?
4. How would the solution change if the current direction in Mesh 2 is counterclockwise?
5. How can superposition be used to simplify solving for currents in this circuit?

### Tip:
When writing KVL equations, always ensure that you maintain consistent sign conventions for voltage drops and rises.

Consider a two-mesh network of Fig. 1.22. Express the currents in the three resistances and then write the KVL equation.

Explore how Kirchhoff's Voltage Law (KVL) is applied to a two-mesh network with three resistors. This step-by-step solution involves calculating the currents in each resistor and writing the KVL equations. Ideal for undergraduate electrical engineering students.

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