Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Solutions to various electrical engineering problems using nodal analysis. The problems involve finding voltages, currents, and power dissipation in circuits. The solutions are presented step-by-step, including the use of MATLAB to solve certain problems.
Typology: Study notes
1 / 19
This page cannot be seen from the preview
Don't miss anything!
1. For the circuit in Fig. 1, obtain v 1 and v 2.
Fig. 1. For Prob. 1. Solution: At node 1, โv 1 10 โ^
v 1 5 = 6 +^
v 1 โv 2 2 โ 60 = โ8v^1 + 5v^2 (1) At node 2, v 2 4 = 3 + 6 +^
v 1 โv 2 2 โ 36 = โ2v^1 + 3v^2 (2) Solving (1) and (2), v 1 = 0 V, v 2 = 12 V
2. Given the circuit in Fig. 2, calculate the currents i 1 through i 4. (4%)
Fig. 2. For Prob. 2. Solution:
At node 1,
โ6 โ 3 + v 201 + 10 v^1 = 0 โ v 1 = 9 โ (^203 ) = 60 V (1)
At node 2,
3 โ 2 + v 402 + v 402 = 0 โ v 2 = โ1 โ (^160080 ) = โ20 V (2)
i 1 =
v 1 20 = 3 A, i^2 =
v 1 10 = 6 A, i 3 = v 2 40 = โ500 mA, i^4 =
v 2 40 = โ500 mA.
3. Solve for V 1 in the circuit of Fig. 3 using nodal analysis.
Fig. 3. For Prob. 3. Solution: Set the bottom of the circuit as the reference node. At node 1: (V 1 โ10) 5 +^
(V 1 โ10) 10 +^
(V 1 โ0) 10 +^
(V 1 โ20) 4 = 0, (0.2 + 0.1 + 0.1 + 0.25)V 1 = 2 + 1 + 5, V 1 = 8/0.65= 160/13=12.308 V.
4. Using nodal analysis, find vo in the circuit of Fig. 4.
Fig. 4. For Prob. 4.
Solution:
6. Determine voltages v 1 through v 3 in the circuit of Fig. 6 using nodal analysis.
Fig. 6. For Prob. 6. Solution:
At the supernode,
2 = v 1 + 2 (v 1 - v 3 ) + 8(v 2 โ v 3 ) + 4v 2 , which leads to 2 = 3v 1 + 12v 2 - 10v 3 (1)
But v 1 = v 2 + 2v 0 and v 0 = v 2.
Hence v 1 = 3v 2 (2)
v 3 = 13 V (3)
Substituting (2) and (3) with (1) gives, 2 = 9 v 2 + 12v 2 - 130
v 1 =132/7 V= 18.858 V , v 2 =44/7 V= 6.286 V v 3 = 13 V
7. Use nodal analysis to find v 1 , v 2 , v 3 and in the circuit of Fig. 7.
Fig. 7. For Prob. 7. Solution: At node 1,
5 = 3 + v^1 โv 2 3 + v^1 โv 8 2 + v 41 โ 16 = 7v 1 โ v 2 โ 4v 3 (1)
At node 2, v 1 โv 2 8 =^
v 2 โv 3 4 +^
v 2 2 โ 0 = โv^1 + 7v^2 โ 2v^3 (2) At node 3,
3 + 12โv 8 3 + v^1 โv 2 3 + v^2 โv 4 3 = 0 โ โ36 = 4v 1 + 2v 2 โ 7v 3 (3)
Form (1) to (3),
Using MATLAB,
8. For the circuit in Fig. 8, find v 1 , v 2 and v 3 using nodal analysis.
Fig. 9. For Prob. 9. Solution: Consider the circuit below.
Use nodal analysis at nodes 1~4,
Now we can use MATLAB to solve for the unknown node voltage.
10. Calculate the node voltages v 1 , v 2 , and v 3 in the circuit of Fig. 10.( ML )
Fig. 10. For Prob. 10. Solution: At node 1,
At node 2,
Fig. 11. For Prob. 11. Solution: (a) This is a planar circuit because it can be redraw as shown below,
(b) This is a planar circuit because it can be redraw as shown below,
12. Apply mesh analysis to the circuit in Fig. 12 and obtain Io. ( ML )
Fig. 12. For Prob. 12. Solution: Consider the circuit below with the mesh currents.
But, we need one more equation, so we use the constraint equation โ I 3 +I 4 = 10. This now gives us three equations with three unknows.
We can now use MATLAB to solve the problem.
For mesh 2,
For mesh 3,
Solving (1), (2), and (3) using MATLAB, we obtain,
14. Use mesh analysis to obtain i 0 in the circuit of Fig.14.
Fig. 14. For Prob. 14. Solution:
Loop 1 and 2 form a supermesh. For the supermesh,
For loop 3,
Also,
Solving (1) to (3),
15. Use mesh analysis to find the current i 0 in the circuit of Fig. 15. ( ML )
Fig. 15. For Prob. 1 5. Solution:
For loop 1, 16i 1 โ 10i 2 โ 2i 3 = 0 which leads to 8i 1 โ 5i 2 โ i 3 = 0 (1)
For the supermesh, โ 35 + 10i 2 โ 10i 1 + 10i 3 โ 2i 1 = 0
Or โ 6i 1 + 5i 2 + 5i 3 = 17.5 (2)
Also, 3i 0 = i 3 โ i 2 and i 0 = i 1 which leads to 3i 1 = i 3 โ i 2 (3)
Solving (1), (2), and (3), we obtain i 1 = 1.0098 and
i 0 = i 1 = 1.0098 A
16. Find the mesh currents in the circuit of Fig. 16. ( ML )
We obtain,
I = inv(Z)*V
I= 1.6196 mA
Fig. 17. For Prob. 17. Solution:
For mesh 2,
But at node A, i 0 = i 1 โ i 2 so that (1) becomes i 1 = (16/6)i 2 (2) For the supermesh, or (3) At node B, i 3 + 0.2v 0 = 2 + i 1 (4) But, v 0 = 10i 2 ,so that (4) becomes i 3 = 5 + (2/3)i 2 (5)
Solving (1) to (5) , i 2 = 0.2941 A,
18. Write the node-voltage equations by inspection and then determine values of V 1 and V 2 in the circuit of Fig. 18.
Fig. 18. For Prob. 18. Solution:
With two equations and three unknowns, we need a constraint equation,
Ix = 2V 1 , thus the matrix equation becomes,
This results in V 1 = 20/(โ5) = โ 4 V and V 2 = [โ8(โ4) โ 7]/5 = [32 โ 7]/5 = 5 V.
19. By inspection, obtain the mesh-current equations for the circuit in Fig. 19.
Fig. 19. For Prob. 19. Solution:
For loop 1, -0.75 + 1.5kIB + VBE + 400IE = 0 = -0.75 + 0.7 + 1500IB + 400(1 +ฮฒ)IB
IB = 0.05/81,900 = 0.61 uA
v 0 = 400IE = 400(1 + ฮฒ)IB = 49 mV
