To solve for \( I_o \) using the principle of superposition, we will follow these steps:

### Superposition Principle:
The superposition principle states that in a linear circuit with multiple sources, the response (in this case, the current \( I_o \)) can be found by considering the effect of each independent source separately, while turning off (setting to zero) all other sources. After analyzing each source separately, the total response is the algebraic sum of the individual responses.

### Steps:
1. **Turn off all sources except the \( 2 \text{ mA} \) current source**:
   - We will consider the effect of the \( 2 \text{ mA} \) current source alone by deactivating the independent voltage source (if there is any).
   - Calculate the current \( I_o^{(1)} \) due to the \( 2 \text{ mA} \) source.

2. **Turn off the \( 2 \text{ mA} \) source and activate the \( 1 \text{ mA} \) current source**:
   - We now deactivate the \( 2 \text{ mA} \) current source (replace it with an open circuit) and consider the \( 1 \text{ mA} \) source alone.
   - Calculate the current \( I_o^{(2)} \) due to the \( 1 \text{ mA} \) source.

3. **Sum the contributions**:
   - Add the contributions from the two independent sources to get the total current \( I_o \).

Let’s go ahead and solve this step by step.

### 1. Contribution from the \( 2 \text{ mA} \) Source Alone:
We'll deactivate the \( I_o \) source (open circuit it) and calculate the current through the resistor network.

Since this involves nodal or mesh analysis of the simplified circuit, I'll perform the calculations for this setup now.

from sympy import symbols, Eq, solve

# Define symbols for the unknown currents and resistances
Ix, Io = symbols('Ix Io')
R1 = 3e3  # 3 kOhms
R2 = 4e3  # 4 kOhms
R3 = 8e3  # 8 kOhms
I_source = 2e-3  # 2 mA

# Set up the nodal equation for the current source branch (Io = 0 assumption, open circuit):
# Current division between R1 and parallel combination of R2 and R3
# Use current division formula:
Req_parallel = R2 * R3 / (R2 + R3)  # Parallel combination of R2 and R3
Ix_1 = I_source * Req_parallel / (R1 + Req_parallel)  # Contribution from 2mA source

Ix_1The contribution to \( I_x \) from the \( 2 \text{ mA} \) current source alone is approximately \( 0.941 \text{ mA} \).

### 2. Contribution from the \( 1 \text{ mA} \) Source Alone:
Now, we deactivate the \( 2 \text{ mA} \) source (open circuit it) and calculate the contribution due to the \( I_o = 1 \text{ mA} \) current source. I'll calculate this next.# Now, consider the contribution from the 1 mA current source (Io):
I_o_source = 1e-3  # 1 mA

# For this scenario, the current directly flows through R2 and R3 in parallel.
Ix_2 = I_o_source * (R3 / (R2 + R3))  # Current division between R2 and R3

Ix_2The contribution to \( I_x \) from the \( 1 \text{ mA} \) current source alone is approximately \( 0.667 \text{ mA} \).

### 3. Total Current \( I_x \):
By superposition, we sum the contributions from both sources to find the total current \( I_x \):

\[
I_x = I_x^{(1)} + I_x^{(2)}
\]

Let me calculate that for you.# Sum of the contributions from both sources
Ix_total = Ix_1 + Ix_2
Ix_totalThe total current \( I_x \) found using the principle of superposition is approximately \( 1.61 \text{ mA} \).

If you have any questions or would like further details, feel free to ask!

### Additional Related Questions:
1. What is the principle of superposition in electrical circuits?
2. How do you deactivate a current source in superposition analysis?
3. How is current division applied in parallel resistor circuits?
4. Can the same approach be used for circuits with voltage sources?
5. What are the benefits of using superposition in circuit analysis?
6. How does nodal analysis help in solving more complex circuits?
7. What role does linearity play in the principle of superposition?
8. How would the result change if the resistor values were different?

### Tip:
When using superposition, always remember to deactivate sources properly: replace current sources with open circuits and voltage sources with short circuits.

Learn how to calculate Io (output current) using the principle of superposition in electrical circuits. Follow step-by-step instructions to find the total current by analyzing each independent source separately and summing their effects. Suitable for advanced high school students and electrical engineering enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation