1

Problems on Chapter Three Loop and Cut-set Analysis

1. Consider the circuit with 6 branches shown in Fig.1.a and its graph with the chosen tree (4,5,6) shown in Fig.1.b:

(a).Write KVL equations for the fundamental loops in the matrix form of Bv=0, in which v is the branch voltage vector.

(b).Write KCL equations in the matrix form of j = BTi , in which j and i are , respectively, branch current vector and loop current vector.

(c).Write the branch equations in the matrix form of v = Rj – Rjs + vs .(R is a 6x6 asymmetric matrix due to the dependent sources in the circuit).

(d).Eliminating j and v in the above equations, find the loop equation Zei=es. (e).Find the equations in part (d), without using the fundamental loop matrix B. To

do so, first, transform the current sources (dependant and independent) into voltage sources, and use the short-cut method to get the loop equations. Solve these equations, calculate the current vector i, and finally determine va, vb, and vc.

av51

−

cv45

bv2−

+b

v−

+

av

Fig. 1.b Fig. 1.a

2. Consider the circuit with 5 braches shown in Fig.2.a and its graph with the chosen tree shown in Fig.2.b.

(a).Write KCL equations for the fundamental cut-sets in the matrix form of Qj = 0. (b).Write KVL equations in the matrix form of v = QTe , in which e is, tree

branch-voltage vector . (c).Write the branch equations in the matrix form of j= Gv – Gvs + js. (d).Eliminating j and v in the above equations, Find the equations Yq e=is. (e).Find the node equations in part (d),without using the fundamental cut-set matrix Q.

Solve this equation to calculate the vector e , and finally determine v1 and vx .

Fig. 2.a Fig. 2.b

3. Which of the “Loop” or “Cut-set” analysis is more convenient to analyze the following circuit? And why? Write the integro differential equations Zl(D)i=es for the specified tree. Determine the required initial conditions.

2

3Li3L

1R

xi

sv

1Li

1L

2L

R

2Li

M

xiα

−

+cv •

•

4. Which of the “Loop” or “Cut-set” analysis is more convenient for the circuit? And why? Does your answer depend on the chosen tree? Write the integro differential equations Yqe=is for the tree consist of the capacitor and the 2Ω resistor. Determine the required initial conditions for solving the equations. Using this equation, write the cut-set equations in the sinusoidal steady-state from Yq (jω) E=Is. Use the input is=4cos2t

1Li

H3

−+ cv

Ω3 Ω2H4si

2Li

F2

HM 2=

5. By using the equations Q =( E ¦ 1n ) and B= (1l ¦ F), prove that if the fundamental the loop of a link contains a tree branch, then the fundamental cut-set of that tree branch, contains that link.

6. (a) Why the node voltages with respect to a datum node are independent of each other ?

(b)Why the mesh currents are independent of each other ? (c) Use parts (a) and (b) to prove MAT=0 and AMT=0. (d)Can one find matrix A, knowing matrix M and vise versa? Has the equation a

unique answer? For example, find M1 for A1 specified below and find A2 for M2 specified below.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

100100111000101

1A

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

110000111000011

2M

7. (a) Find a relation between the incidence matrix A and the fundamental loop matrix B. (b)Find a relation between the mesh matrix M, and fundamental cut-set matrix Q and

for the following graph verify this relation.

3

(c)Number of the trees in a graph is given by Det [AAT], in which A is the incidence matrix of the graph .For the graph of problem 1, verify this statement by specifying all trees of the graph.

8. In the following graph:

(a)Choose a tree such that (1,3,4,6) and (4,5,6,8) are two of its fundamental cut-sets and (1,6,2,5,7) is one of its fundamental loops. Then, determine all other fundamental loops and fundamental cut-sets of the tree.

(b)Determine a tree that has branches 1, 4 and 6 as its links (find all possible answer). (c)Determine a tree that has branches 1, 4 and 6 as its tree branches (find all possible

answer).

9.(a)In the following graph ,find a tree that (e,c,d) and (f,c,d,g) are two of its fundamental loops and (d,e,f) is one of its fundamental cut-sets. Give all possible answers.

(b)For the tree (a,b,c,g,k) write KVL equations of all fundamental loops and KCL equation of all cut-sets.

(c)Write the loop impedance matrix Zl, when all the circuit branches are 3-ohm resistors.

10. (a) In the following graphs find a tree (or trees) that its fundamental cut-sets are branches connected to a node.

(b)In the same graphs, find a tree (or trees) that its fundamental loops are the meshes of the graph?

4

11. In the following graph determine a tree (or trees) which (d,c,b) is one of its fundamental loops and (h,e,b,c,k) is one of its fundamental cut-sets.