Problem + Hard 53-1    Source:  Problem prepared by the author of the site.

Determine, based on the figure below, the values ​​of:

a) the current I;

b) the voltages V_b and V_c.

Solution of the Problem + Hard 53-1

Item a

One of the methods we can use to solve this problem is Kirchhoff's law for tensions. Let's base ourselves on the circuit shown in the figure below and build the system equations in the direction of the brown arrows.

For gmesh 1, we have:

For mesh 2, we have:

For mesh 3, we have:

Therefore, we managed to construct a system of three equations with three unknowns. For To solve this system the best way is to employ a graphing calculator such as HP, Texas, Sharp, etc ... or a software like Matlab, MathCad, or even a free, as is the case with Octave, which is 95% compatible with Matlab. we can access the Octave page and download at link to Octave HERE

Using Octave, the results for I₁, I₂ and I₃, are:

From the circuit we can conclude that:

With this result we conclude that the current I is ahead of 75.10° in relation to the voltage that the source supplies to the circuit, characterizing a predominance capacitive.

Item b

With these data in hand, we can calculate V_b and V_c. To do so, we must calculate the currents flowing through the capacitors of -j20 and -j5. In capacitor - j5 the current is I₃ itself. Then, applying Ohm's law, we have:

Notice that we changed -j5 into 5 ∠-90°. To determine the current that circulates through the capacitor -j20 we must look at the last circuit and conclude that I_j20 (let's call it that) equals    I_j20 = I₂ - I₃.   Making the calculation let's find the value of    I_j20  =   0.4987 + j1.3687  =   1.4567 ∠+70°.

Now, let's calculate the value of V_b by applying, once again, the Ohm's law.