Problem 23-1 Source:
Problem elaborated by the author of the site.
In the circuit show in the Figure 23-01.1, the S switch has been closed for a long time. For the time
t = 0 the switch has been opened. Calculate:
a) the current i to t ≥ 0.
b) the voltage v to t ≥ 0.
Solution of the Problem 23-1
As the S switch remained on for a long time, so in this case the inductor behaves like a short circuit. This way we
can calculate the current i that circulates through the inductor in time
t = O-.
From the circuit, it is noted that it is possible to calculate the equivalent resistance to the right of the inductor.
We have the resistors series 50 + 30 = 80 Ω in parallel with the resistor of
20 Ω. This results in a resistance of 16 Ω . Adding to the resistor of
4 Ω is the value of 20 Ω as equivalent resistance.
Analyzing the situation when opening the switch S, it is known that the inductor cannot abruptly change the value
of the current flowing through it. Then, it is concluded that the value of the initial current,
i (0+) = 24 A.
On the other hand, knowing the value of the inductor and the equivalent resistance (value calculated above) it is possible to calculate the time constant of the circuit.
Note that to t = 0+, The circuit comes down to the inductor and equivalent resistance. Therefore, it is easily concluded that the value of current i when
t → ∞ is equals zero. Knowing the value of the time constant, the start current, and the end current, the circuit response can be calculated using the eq, 23-03 studied in
item 4. Substituting for numeric values gives:
In the above equation we use the fact that 1 / 0.4 = 2.5.
To answer item b, we must calculate the current flowing through the resistance of 50 Ω. For this, a current divider is used, not forgetting that it will have the opposite sign to i. So:
We should pay attention to the fact that this value is only valid for t ≥ 0, because we know that in time t = 0- the value of i50 = 0. Performing the calculation:
So, to calculate the voltage on the 50 Ω resistor, just use the Ohm's law, or:
