When Kirchhoff's Laws are applied to the electrical network...

When Kirchhoff's Laws are applied to the electrical network in the figure, the currents $I_{1}, I_{2}$, and $I_{3}$ are the solution of the system
$\left\{\begin{aligned} I_{1}-I_{2}+I_{3} &=0 \\ 3 I_{1}+2 I_{2} &=7 \\ 2 I_{2}+4 I_{3} &=8 \end{aligned}\right.$
Find the currents.

Key Concepts

Circuit Analysis

Circuit analysis encompasses the methods and procedures used to understand and predict the behavior of electrical circuits. By applying fundamental concepts like Kirchhoff's Laws and solving the resulting linear equations, one can determine the currents, voltages, and other important characteristics of a circuit, facilitating both the design and analysis of electrical systems.

Solution Techniques for Linear Systems

Various mathematical methods, including substitution, elimination, and matrix-based methods like Gaussian elimination, can be employed to solve systems of linear equations. These techniques provide reliable procedures to determine the numerical values of the unknown electrical parameters in the circuit.

Kirchhoff's Laws

Kirchhoff's Laws consist of the Current Law (KCL) and the Voltage Law (KVL), which are essential in analyzing electrical circuits. KCL asserts that the total current entering a junction equals the total current leaving it, reflecting the conservation of charge. KVL states that the total voltage around any closed loop in a circuit must sum to zero, encapsulating the conservation of energy. These laws allow us to formulate equations that represent the behavior of complex circuitry.

Systems of Linear Equations

When applying Kirchhoff's Laws, the relationships between currents and voltages in various parts of the circuit are expressed as a system of linear equations. Such systems neatly capture the linear dependencies between the electrical quantities, allowing techniques from linear algebra to be used to compute the unknown values.

Transcript

00:01 So in this problem, we're giving us electrical circuit, electrical network, and using kirchhoff's law, you can come up with three equations with three unknowns to figure out the currents through each of the different circuits.

00:16 So we get these equations, right? i -1 minus i -2 plus i -3 -i -3 -1 plus -2 -2 -i -3 -2 -2 -i -2 -2 -i -3.

00:33 2 equals 7 and 2 i2 plus 4 i 3 is 8 i'm going to draw a line here just to keep our work kind of neat call this equation 1 this equation 2 and this equation 3 now then 3 is not very neat there let me clean that up now then let's see we need to combine these so that we end up with two equations with two unknowns right so equation two doesn't have an i3 so i can combine equation one and three together and i have an i three then i would be where i want to be so let's take minus four times equation one so that's minus four i one plus four i two minus four i three is zero let's add equation three to this.

01:42 So i'll just write equation 3 down here.

01:44 2 i2 plus 4 i3 equals 8.

01:49 Add these together.

01:51 And the i3s drop out by design.

01:56 So i'm left with minus 4 i1.

02:00 4 i2 plus 2 i2, that's plus 6 i2 equals 8.

02:07 I'm going to call this equation 4.

02:11 So now what can we see? between equation 2, an equation four.

02:18 Well, if i multiply equation two by minus three, then the i -2s will drop out, won't they? so let's do minus three times equation two.

02:29 So minus three times three -i, that's minus nine, i -1, minus six i -2, equals seven times minus three, that's minus 21.

02:43 Okay, add these together...