Construct an electrical circuit to find the values of Xa, Xb, and Xc in this system of equations:
21Xa – 10Xb – 10Xc = 1
–10Xa + 22Xb – 10Xc = –2
–10Xa – 10Xb + 20Xc = 10
Your circuit should include only the following elements:
one 1-Ω resistor
one 2-Ω resistor
three 10-Ω resistors
three ideal constant voltage sources
three ideal ammeters
The circuit should be designed so that each ammeter displays one of the values Xa, Xb, or Xc. Given that the Xa, Xb, and Xc values represent currents, what kind of circuit analysis yields equations in this form?
You get equations in this form when you do mesh analysis of a circuit. Each equation represents the sum of the voltages around one loop in the mesh.
What do the coefficients on the left side of the equations represent? What about the constants on the right side?
The coefficients on the left side of each equation represent resistances. Resistance multiplied by current (the unknown Xa, Xb, and Xc values) yields voltage.
The “bare” numbers on the right side of each equation represent voltages directly (i.e., independent voltage sources).
What is the numerical solution for the equations?
To solve the equations directly, start by solving the third equation for Xc and substituting it into the other two equations:
Xc = 1/2 Xa + 1/2 Xb + 1/2
21Xa – 10Xb – 5Xa – 5Xb – 5 = 1
–10Xa + 22Xb – 5Xa – 5Xb – 5 = –2
16Xa – 15Xb = 6
–15Xa + 17Xb = 3
Solve for Xa by multiplying the first equation by 17 and the second equation by 15 and then adding them:
272Xa – 255Xb = 102
–225Xa + 255Xb = 45
47Xa = 147 → Xa = 147/47
Solve for Xb by multiplying the first equation by 15 and the second equation by 16 and then adding them:
240Xa – 225Xb = 90
–240Xa + 272Xb = 48
47Xb = 138 → Xb = 138/47
Finally, substitute those two results into the equation for Xc:
Xc = 147/94 + 138/94 + 47/94 = 332/94 = 166/47
Finally, what is the actual circuit? Draw a diagram of the circuit and indicate the required value of each voltage source.
The circuit is a mesh comprising three loops, each with a voltage source. The common elements of the three loops are the three 10-Ω resistors, connected in a Y configuration (see the figure below).
The values of the voltage sources in each loop are given directly by the equations, as shown. To verify the numeric solution calculated previously, you can calculate all of the node voltages around the outer loop, plus the voltage at the center of the Y, and ensure they’re self-consistent.
We’ll start by naming Va as ground, or 0 V:
Vb = Va + 2 V = 2 V
Vc = Vb + 2 Ω × Xb = 2V + 2 Ω × 138/47 A = 370/47 V = 7.87234 V
Vd = Vc + 1 Ω × Xa = 370/47 V + 1 Ω × 147/47A = 517/47 V = 11.000 V
Ve = Vd – 1 V = 11.000 V – 1.000 V = 10.000 V
Va = Ve – 10 V = 0 V
which is where we started.
The center node, Vf, should be at the average of the three voltages Va, Vc, and Ve:
0 V + 370/47 V + 10 V/3 = 840/141 V = 5.95745 V
We should also be able to get this value by calculating the voltage drops across each of the three 10-Ω resistors:
Va + (Xc – Xb) × 10 Ω = 0 V + (166 – 138)/47A × 10 Ω = 280/47 V = 5.95745 V
Vc + (Xb – Xa) × 10 Ω = 370/47V + (138-147)/47A × 10 Ω = 280/47 V = 5.95745 V
Ve + (Xa – Xc) × 10 Ω = 10 V + (147-166)/47 A × 10 Ω = 280/47 V = 5.95745 V