Let's break down the given expressions step by step to ensure clarity and accuracy:

First Equation:

$I_{12} = I_{23} + I_{24} + I_{25} - I_2$

Substituting the given values:

$I_{12} = (11,546 - j4,565) + (11,546 - j4,565) + (11,546 + 0j) - 50,037$

Now, let's simplify the real and imaginary components separately.

Real components: $11,546 + 11,546 + 11,546 - 50,037 = 34,638 - 50,037 = -15,399$

Imaginary components: $-j4,565 - j4,565 + 0 = -j9,130$

So, we have: $I_{12} = -15,399 - j9,130 \, A$ This matches the given result of $(-15,399 - j9,130) \, A$.

Second Equation:

$I_{01} = I_{12} + I_1$

Substitute the values: $I_{01} = (-15,399 - j9,130) + (7,696 + j5,772)$

Now, simplify again:

Real components: $-15,399 + 7,696 = -7,703$

Imaginary components: $-j9,130 + j5,772 = -j3,358$

Thus: $I_{01} = -7,703 - j3,358 \, A$ This matches the given result of $(-7,703 - j3,358) \, A$.

Conclusion:

The calculations have been verified as accurate.

Would you like any further explanations or details on any step?

Here are 5 related questions that might help expand on this:

1. How do we handle complex currents in AC circuit analysis?
2. What is the significance of the imaginary part of a current in electrical systems?
3. How can we interpret the real and imaginary components in terms of phase and magnitude?
4. What role do phasor representations play in simplifying AC circuit calculations?
5. What are the implications of complex current in relation to power dissipation in a circuit?

Tip: In AC circuit analysis, the imaginary part of a current or voltage represents the phase shift relative to the real (resistive) component.