Final answer:
To find currents i1 to i5 in a complex circuit, apply Kirchhoff's rules to formulate and solve equations based on the junction and loop rules, allowing for the determination of each individual current.
Step-by-step explanation:
Calculating Current: Using Kirchhoff's Rules
To determine the currents i1 to i5 in a complex circuit, it is essential to apply Kirchhoff's rules. This involves formulating and solving a set of simultaneous equations based on the given circuit configuration and the specified voltage and resistance values. The calculations begin by applying Kirchhoff's junction rule to determine the relationship between the different currents flowing into and out of a junction. Then, Kirchhoff's loop rule is applied to each independent loop in the circuit to formulate equations that account for the voltage drops and gains around each loop. By solving these equations, the values of currents i1, i2, i3, i4, and i5 can be determined.
For example, if you have the equation I2 = 6 - 2i1 indicating that current i2 is dependent on current i1, and another equation defining i3 in terms of i1, you would substitute the value of i1 to find the respective currents. Finally, you can verify the consistency of the solutions by checking the power supplied by the voltage sources against the power dissipated by the resistors, ensuring they are equal, as per the principle of conservation of energy.