Problem 10-19    Source:   Problem 2.24 - page 66 -    IRWIN, J. David -    Book: Análise de Circuitos em Engenharia - 4ª edição - Ed. Pearson Education do Brasil - 2013.

The circuit shown in Figure 10-19.1 is used in digital-to-analog converters. Determine expressions for I₁, I₃ and I₅ in terms of I_o.

Solution of the Problem 10-19

Note that from point c flows the current I₆ and circulates through two resistors connected in series of values equal to R each. This association results in a resistor of value 2 R and is in parallel with the other resistos of 2 R where it circulates I₅. As a result of this parallel we obtain a resistor of value R. Between point b and point c we have a resistor of value R that added to the equivalent resistance of point c , a equivalent resistance of value 2 R results. Therefore, we realize that there will be a repetition of the situation described above and each point always "sees" the value R. Thus, we easily conclude that the equivalent resistance that the circuit offers to the voltage source V has the value 2 R . Therefore, the value of I_o it is given by:

In relation to point a, we have an equivalent resistance of 2 R where it circulates I₂. Therefore, it is evident that I₁ = I₂. So we can write the value of I₁ in terms of I_o as:

Following the same line of reasoning, we easily concluded that I₃ = I₄ = I₁/ 2. So, writing I₃ in terms of I_o, we have:

So, without difficulties, we have: