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ksio89
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I'm trying to solve item "c" of this question about the electrical circuit depicted on the attached picture.
The question is in Brazilian Portuguese, this would be a rough and not 100% gramatically correct translation:
"In the circuit shown below on Pic. 1 (labeled "Electrical Circuit"), the three resistors have values [itex]R_1[/itex] = 2 Ω, [itex]R_2[/itex] = 20 Ω and [itex]R_3[/itex] = 5 Ω. The battery B has constant tension 12 V. The current [itex]i_1[/itex] is considered positive in the indicated direction. Between the instants [itex]t[/itex] = 0 s and [itex]t[/itex] = 100 s, the generator G provides a variable tension [itex]V = 0.5 t[/itex] (V in volts and t in seconds).
a) Define the current value [itex]i_1[/itex] for [itex]t[/itex] = 0 s
b) Define the instant [itex]t_0[/itex] in which the current [itex]i_1[/itex] is null.
c) Plot the curve which depicts the current [itex]i_1[/itex] in function of the time [itex]t[/itex], in the interval from 0 to 100 s. Use the axis from the following picture (labeled "Graph") and clearly indicate the scale of current, in ampère (A).
d) Define the value of the power [itex]P[/itex] received or provided by the battery [itex]B[/itex] in the instant [itex]t[/itex] = 90 s."
I was able to solve items a, b and d using Kirchoff's Voltage Law and generating a linear system with two variables, [itex]i'_1[/itex] and [itex]i'_2[/itex], where [itex]i'_1 \pm i'_2 = i_1[/itex] and [itex]i'_1[/itex] is the current flowing around the left loop (loop I) and [itex]i'_2[/itex] is the current flowing around the right one (loop II). So:
a) 2 s;
b) [itex]t[/itex] = 30 s (in this case, [itex]i'_1 = -i'_2[/itex];
c) ?
d)[itex]P[/itex] = (I need to find the graph of [itex]i_1[/itex] vs. [itex]t[/itex] so I can calculate the absolute value of the power).
I wasn't able able to answer item c, because it asks to plot the graph of [itex]i_1[/itex] Vs [itex]t[/itex]. The issue is that [itex]i_1[/itex] is actually the algebric sum of [itex]i'_1[/itex] and [itex]i'_2[/itex], but I also have a third variable which is the tension [itex]V[/itex] generated by the voltage source G, which is a function of time [itex]t[/itex]. I don't have a third equation though, as Kirchoff's Voltage Law applied to the largest loop provides me an equation which is simply a linear combination of the equations associated with loops I and II.
According to the book answers section, the answer for item c) is a line of slope-intercept form [itex]f(t)= -\frac{t}{15}+2[/itex]. Then, |[itex]P[/itex]| would be 48 W. But how do you define [itex]i_1[/itex] as a function of [itex]V[/itex] and, consequently, [itex]t[/itex]?
The question is in Brazilian Portuguese, this would be a rough and not 100% gramatically correct translation:
"In the circuit shown below on Pic. 1 (labeled "Electrical Circuit"), the three resistors have values [itex]R_1[/itex] = 2 Ω, [itex]R_2[/itex] = 20 Ω and [itex]R_3[/itex] = 5 Ω. The battery B has constant tension 12 V. The current [itex]i_1[/itex] is considered positive in the indicated direction. Between the instants [itex]t[/itex] = 0 s and [itex]t[/itex] = 100 s, the generator G provides a variable tension [itex]V = 0.5 t[/itex] (V in volts and t in seconds).
a) Define the current value [itex]i_1[/itex] for [itex]t[/itex] = 0 s
b) Define the instant [itex]t_0[/itex] in which the current [itex]i_1[/itex] is null.
c) Plot the curve which depicts the current [itex]i_1[/itex] in function of the time [itex]t[/itex], in the interval from 0 to 100 s. Use the axis from the following picture (labeled "Graph") and clearly indicate the scale of current, in ampère (A).
d) Define the value of the power [itex]P[/itex] received or provided by the battery [itex]B[/itex] in the instant [itex]t[/itex] = 90 s."
I was able to solve items a, b and d using Kirchoff's Voltage Law and generating a linear system with two variables, [itex]i'_1[/itex] and [itex]i'_2[/itex], where [itex]i'_1 \pm i'_2 = i_1[/itex] and [itex]i'_1[/itex] is the current flowing around the left loop (loop I) and [itex]i'_2[/itex] is the current flowing around the right one (loop II). So:
a) 2 s;
b) [itex]t[/itex] = 30 s (in this case, [itex]i'_1 = -i'_2[/itex];
c) ?
d)[itex]P[/itex] = (I need to find the graph of [itex]i_1[/itex] vs. [itex]t[/itex] so I can calculate the absolute value of the power).
I wasn't able able to answer item c, because it asks to plot the graph of [itex]i_1[/itex] Vs [itex]t[/itex]. The issue is that [itex]i_1[/itex] is actually the algebric sum of [itex]i'_1[/itex] and [itex]i'_2[/itex], but I also have a third variable which is the tension [itex]V[/itex] generated by the voltage source G, which is a function of time [itex]t[/itex]. I don't have a third equation though, as Kirchoff's Voltage Law applied to the largest loop provides me an equation which is simply a linear combination of the equations associated with loops I and II.
According to the book answers section, the answer for item c) is a line of slope-intercept form [itex]f(t)= -\frac{t}{15}+2[/itex]. Then, |[itex]P[/itex]| would be 48 W. But how do you define [itex]i_1[/itex] as a function of [itex]V[/itex] and, consequently, [itex]t[/itex]?
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