ELECTRICAL CIRCUIT AND MACHINE ENT 249/3. Methods of Analysis

Transcription

1 ELECTRICAL CIRCUIT AND MACHINE ENT 49/ Methods of Analysis

2 Introduction Nodal Analysis KCL & Ohm s Law Nodal Analysis with Voltage Sources (Supernode) KCL, Ohm s & KVL Law Mesh Analysis - KVL & Ohm s Law Mesh Analysis with Current Sources (Supermesh) - KVL, Ohm s & KCL Law

3 INTRODUCTION Another circuit analysis method is:. Nodal analysis ( application of KCL) known as Node oltage Method in some reference book.. Mesh analysis ( application of KVL) known as loop current Method in some reference book. Useful in analyzed multiple loop circuits haing two or more oltage or current sources.

4 NODAL ANALYSIS. In nodal analysis, we are interested in finding the node oltages.. Steps to determine node oltage:. Select reference node. n. Assign oltage,,..., to the remaining n- nodes (with respect to reference node). Apply KCL to each nonreference nodes. Use Ohm s Law to express the branch currents. 4. Sole the resulting simultaneously equations to sole for node oltage

5 NODAL ANALYSIS The first step is selecting a node as the reference node. The reference node is commonly called the ground since it is assumed to hae zero potential.

6 NODAL ANALYSIS Consider the aboe figure as an example:. Ground has been chosen as the reference node. Assign and as node and respectiely. (node oltage = oltage of node with respect to the reference node).. Apply KCL. Use Ohm s Law to express the branch currents. Node: I Node : I = I + i + i = i + i 4. Sole the equations to find node oltages

7 NODAL ANALYSIS Apply Ohm s Law, where current flows from a higher potential to a lower potential in a resistor: 0,, 0 R i R i R i = = = R i lower higher = R R I = + R R I I + + = We obtain: Substitute into equation: Sole for and using elimination technique/ Cramer s rule.

8 NODAL ANALYSIS Example.: Calculate the node oltages in the circuit shown in Figure.. Nodes Voltage, & V V.Applying KCL (refer next slide) Add references node

9 NODAL ANALYSIS Solution: Eq. (..) Using the elimination technique

10 NODAL ANALYSIS

11 NODAL ANALYSIS Example.: Determine the oltages at the nodes in Figure shown. Solution:

12 NODAL ANALYSIS We hae three simultaneous equation to sole to get the node oltages,,. Answer : We can get the answer by Using approaches: i. elimination technique ii. Cramer s rule

13 4 7 From(5) Substitute (6) in(4) = + 7 () () () + () + 4 = 4 9 = =.4V + =...(4) = 0...(5)...(6) = =...() = 0...() = 0...() Answer : = 4.8V Continue with &..!!!!!!! =.4V = -.4V How to sole using cramer s rule??

14 NODAL ANALYSIS WITH VOLTAGE SOURCES Possibilities:. If a oltage source is connected between the reference node and a nonrefence node, simply set the oltage at the nonreference node equal to the oltage of the oltage source. If the oltage is connected between two nonreference nodes, the two nonreference nodes form a generalized node or supernode; apply both KCL and KVL to determine the node oltages. Supernode: formed by enclosing oltage source connected between two nonreference nodes and any elements connected in parallel with it.

15 From the figure: = 0V Nodes and form a supernode KCL at supernode: = = = + + KVL at supernode: 4 i i i i + = + NODAL ANALYSIS WITH VOLTAGE SOURCES

16 NODAL ANALYSIS WITH VOLTAGE SOURCES Example.: For the circuit shown in Figure.9, find the node oltages. Answer: = -7.V, = -5.V

17 NODAL ANALYSIS WITH VOLTAGE SOURCES Practice Problem: Find and i in the circuit in Fig.: i 4 At node = 7V...() i i At node i + = i + i i 4...() i From () Node & node form super node + = = = +...() 0...(4) Answer: -0.V,.4A To be continued.

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19 MESH ANALYSIS. Mesh analysis is also known as loop analysis or the mesh-current method.. Mesh is a loop which does not contain any other loops within it.. Application: to find unknown currents 4. Only capable to a planar circuit 5. Planar circuit: can be drawn in a plane with no branches crossing one another.

20 MESH ANALYSIS Fig..5 a) a Planar circuit with crossing branches. b)the same circuit redrawn with no crossing branches b)

21 MESH ANALYSIS Steps in determining node oltage:. Assign mesh currents i,i, in to the n meshes Assign mesh current in the clockwise direction around each non redundant closed loop. Apply KVL to each of the n meshes. Use Ohm s Law to express the oltages in terms of the mesh currents.. Sole the resulting n simultaneously equations to get the mesh currents by using substitution or determinants.

22 MESH ANALYSIS Consider the figure below:. Assign i and i as meshes and.. Apply KVL to each mesh: Mesh: V Mesh : R i + R i + V + + R R ( i ( i i i ) = 0 ) = 0. Sole for mesh currents i and i I in = I out I = i - i

23 MESH ANALYSIS Example: For the circuit in Figure, find the branch currents I, I and I using mesh analysis. I in = I out I=i-i or I = -(i-i)

24 MESH ANALYSIS Practice Problem: Calculate the mesh currents i and i in the circuit of Figure shown. Answer: i= / A, i=0a

25 MESH ANALYSIS WITH CURRENT SOURCES Possibilities:. When a current source exists only in one mesh set the current as equal to the source. CASE set i = -5A Mesh equation: 0+ 4i + 6( i i) = 0, i = A

26 MESH ANALYSIS CURRENT SOURCES CASE. When the current source exists between two meshes create a supermesh (by excluding the current source and any elements connected in series with it). A supermesh results when two meshes hae a (dependent or independent) current source in common. Fig: a) Two meshes haing a current source in common, b) a supermesh, created by excluding the current source

27 MESH ANALYSIS CURRENT SOURCES From the aboe figure:. Apply KVL to supermesh:. Applying KCL to node 0:. Soling: i 0+ 6i + 0i + 4i = i + 6 = 0 i =.A, i. 8A =

28 Example: For the circuit in figure below, find i to i4 using mesh analysis. Answer: i = 7.5A,i =.5A,i =.9A,i 4 =.4A

29 FURTHER READING. Fundamentals of Electric Circuits, 6 th Edition,McGrawhill Alexander, C. K. and Sadiku, M. N. O.. Electric Circuit, 8 th Edition, Pearson, Nillson and Riedel