Question 2.13: Solve for the current in each element of the circuit shown i...
Solve for the current in each element of the circuit shown in Figure 2.34.
First, we select the mesh currents. Following our standard pattern, we define the mesh currents to flow clockwise around each mesh of the circuit. Then, we write a KVL equation around mesh 1:
20(i_1 − i_3) + 10(i_1 − i_2) − 70 = 0
For meshes 2 and 3, we have:
10(i_2 − i_1) + 12(i_2 − i_3) + 42 = 0 (2.57)
20(i_3 − i_1) + 14i_3 + 12(i_3 − i_2) = 0 (2.58)
Putting the equations into standard form, we have:
30i_1 − 10i_2 − 20i_3 = 70 (2.59)
− 10i_1 + 22i_2 − 12i_3 = −42 (2.60)
− 20i_1 − 12i_2 + 46i_3 = 0 (2.61)
In matrix form, the equations become:
\left[\begin{matrix} 30 & -10 & -20 \\ -10& 22& -12\\ -20 & -12 & 46 \end{matrix} \right] \left[\begin{matrix} i_1 \\ i_2 \\ i_3 \end{matrix} \right] = \left[\begin{matrix} 70 \\ -42 \\ 0 \end{matrix} \right]These equations can be solved in a variety of ways. We will demonstrate using MATLAB. (The same results can be obtained by using these same commands in LabVIEW MathScript.)We use R for the coefficient matrix, because the coefficients often are resistances. Similarly, we use V for the column vector for the right-hand side of the equations and I for the column vector of the mesh currents. The commands and results are:
>> R = [30 -10 -20; -10 22 -12; -20 -12 46];
>> V = [70; -42; 0];
>> I = R\V % Try to avoid using i, which represents the square root of
>> % -1 in MATLAB.
I =
4.0000
1.0000
2.0000
Thus, the values of the mesh currents are i_1 = 4 A, i_2 = 1 A, and i_3 = 2 A. Next, we can find the current in any element. For example, the current flowing downward in the 10-Ω resistance is i_1 − i_2 = 3 A.