Nodal Analysis:

• 3 variables, thus 3 equations needed.
• 240 V source directly connects V₃ to the reference node, thus we immediately know V₃, therefore we really only have 2 equations to solve.
• As usual with nodal analysis, we have the extra algebraic work of finding least common denominators, etc.
• When defining the dependent source parameter i_d, we will need to use KCL expressed in terms of node voltages since i_d is not directly associated with a resistor and we cannot use Ohm's Law.

Mesh Analysis:

• 4 variables, thus 4 equations required.
• I₁ is the only mesh current through the19 A source, thus we immediately know I₁, therefore we really only have 3 equations to solve.
• As usual with mesh analysis, the algebraic simplification is a bit easier than with nodal analysis.
• The dependent source parameter i_d is easy to define since it equals one of the mesh currents.

Verdict:

Since the equations are not to be solved manually but with a calculator or computer, mesh analysis would probably be my choice despite the extra equation to avoid the additional algebra and the slight difficulty with i_d, but it is a close call. If I had to solve the equations manually, I would probably choose nodal analysis.

I have worked this problem by both methods. Look in the homework index under mesh analysis for the mesh solution or click here.

i_b = (V₂ - V₁) / 5

As mentioned above, we will have to use KCL to determine an expression for i_d. Note that trying to write KCL at node 3 would be difficult since there is no easy way to express the current through the dependent voltage source in terms of the node voltages, thus I will write KCL at the reference node. This might seem to violate the rule about writing KCL at the reference node, but we are not trying to write our final set of equations, but merely defining a parameter used by a dependent source. It is, by the way, possible to write KCL at node 3 for this purpose; can you determine how?

i_d = 2 i_b - V₁ / 40 + 19
inserting i_b from above gives
i_d = 2 (V₂ - V₁) / 5 - V₁ / 40 + 19

Cleaning this up a bit for later:

multiply by 40
40 i_d = 16 (V₂ - V₁) - V₁ + 760
collect terms
40 i_d = 16 V₂ - 17 V₁ + 760
solve for i_d

i_d = (16 V₂ - 17 V₁) / 40 + 19

Independent source:

V₃ = - 240 (equation 1)

Dependent source:

4 i_d = V₃ - V₁
Substituting for i_d from above
4 ( (16 V₂ - 17 V₁) / 40 + 19) = V₃ - V₁
Multiplying through by 4 on the left
(16 V₂ - 17 V₁) / 10 + 76 = V₃ - V₁
Multiplying through by 10
16 V₂ - 17 V₁) + 760 = 10 V₃ - 10 V₁
Collecting terms
7 V₁ - 16 V₂) + 10 V₃ = 760
Substituting for V₃ from above
7 V₁ - 16 V₂) - 2400 = 760

7 V₁ - 16 V₂ = 3160 (equation 2)

Marking the supernode on the circuit diagram

Summing the currents into Node 2:

2 i_b + (V₁ - V₂) / 5 + (V₃ - V₂) / 10 = 0
Substituting for i_b from above
2 ((V₂ - V₁) / 5) + (V₁ - V₂) / 5 + (V₃ - V₂) / 10 = 0
Multiplying through by 10
4 (V₂ - V₁) + 2 (V₁ - V₂)+ V₃ - V₂ = 0
Collecting terms
- 2 V₁ + V₂ + V₃ = 0
Substituting for V₃ from above
- 2 V₁ + V₂ = 240 (equation 3)

V₁ = - 280
V₂ = - 320
V₃ = - 240

ia = V₁ / 40 = - 280 / 40
ia = - 7 A

ib = (V₂ - V₁) / 5 = (- 320 + 280) / 5
ib = - 8 A

ic = (V₃ - V₂) / 10 = (- 240 + 320)
ic = 8 A

id = (16 V₂ - 17 V₁) / 40 + 19 = (16 (-320) - 17 (-280)) / 40 + 19
id = 10 A

ie = ic + id = 8 + 10
id = 18 A

Also, I calculated total power on the mesh analysis solution which also checked.