§2-2Mesh(网孔) analysis Mesh analysis is applicable only to those networks, which are planar. If it is possible to draw the diagram of a circuit on a plane surface in such a way that no branch passes over or under any other branch, then that circuit is said to be a planar circuit Non -planar-circuit (Planar-circuit
Mesh analysis is applicable only to those networks, which are planar. §2-2 Mesh(网孔) analysis If it is possible to draw the diagram of a circuit on a plane surface in such a way that no branch passes over or under any other branch, then that circuit is said to be a planar circuit. (Non -planar-circuit) (Planar -circuit)
Mesh analysis. L1 )=42 L2:=3(i1-i2)+4i2=10 42J 0( 6A =44 We define a mesh current as a current. which flows onl around the perimeter of a mesh. L1:6im+3(i )=42 (1)h2少+41210 42M 0 =64 十 i= 4A a mesh current may often be identified as a branch current,(支路中只有一个网孔电流通过时,有=uand (2)
Mesh analysis: L : 6i 3(i i ) 42 1 1 + 1 − 2 = L : 3(i i ) 4i 10 2 − 1 − 2 + 2 = = = i A i A 4 6 2 1 We define a mesh current as a current, which flows only around the perimeter of a mesh. L : 6i 3(i i ) 42 1 ( 1 ) + ( 1 ) − ( 2 ) = L : 3(i i ) 4i 10 2 − ( 1 ) − ( 2 ) + ( 2 ) = = = = = i i A i i A 4 6 2 (2) 1 (1) A mesh current may often be identified as a branch current,—(支路中只有一个网孔电流通过时，有i1=i (1) and i2=i (2)) − + 42V i 1 6 4 3 2 i 3 1 2 i = i − i + − 10V L1 L2 − + 42V i 1 6 4 3 2 i 3 1 2 i = i − i + − L1 10V L2 ( ) i 1 ( ) i 2
7+1(1-i2)+6+2(1-3)=0 (2-i1)+2i2+3(2-i3)=0 2(i3-i1)-6+3(i-i2)+1i3=0 )6V +6i,-3i,=0 i1-3i2+6i3=6 1-1 06 6-36 117 3A. 2A 3=34 39 2-36
− 7 +1(i 1 − i 2 )+ 6+ 2(i 1 − i 3 ) = 0 1(i 2 − i 1 )+ 2i 2 + 3(i 2 − i 3 ) = 0 2(i 3 − i 1 )− 6+ 3(i 3 − i 2 )+1i 3 = 0 − − + = − + − = − − = 2 3 6 6 6 3 0 3 2 1 1 2 3 1 2 3 1 2 3 i i i i i i i i i or 3 . 39 117 2 3 6 1 6 3 3 1 2 6 3 6 0 6 3 1 1 2 i 1 = = A − − − − − − − − − − = i 2 = 2A i 3 = 3A 4 − + 7V 3 1 1 2 − + 6V 2 2 i 1 i 3 i
When a current source is present in the network. We should assign an unknown voltage across the current source, apply KVL around each mesh. and then relate the source current to the assigned mesh currents +1(1-i2)+v+2(i1-i3)=0(1) l(i2-i1)+2i2+3(i2-i3)=0(2) 2(i3-i1)-+3(3-i2)+li3=0(3) 7(4) (1)+(3):-7+1(1-i2)+3(i-i2)+li3=0(5) +6i,-3i=0 OI i1-42+4i3 We have: i, =9A, i, =2.54, i2=2A
When a current source is present in the network .We should assign an unknown voltage across the current source, apply KVL around each mesh, and then relate the source current to the assigned mesh currents. (1) (3): 7 1( ) 3( ) 1 0 (5) + − + i 1 − i 2 + i 3 − i 2 + i 3 = 1( ) 2 3( ) 0 (2) i 2 − i 1 + i 2 + i 2 − i 3 = 2( ) 3( ) 1 0 (3) i 3 − i 1 − v + i 3 − i 2 + i 3 = 7 (4) i 1 − i 3 = 7 1( ) 2( ) 0 (1) − + i 1 − i 2 + v + i 1 − i 3 = − + = − = − + − = 4 4 7 7 6 3 0 1 2 3 1 3 1 2 3 i i i i i i i i or We have:i1=9A, i2=2.5A,i3=2A. − + − + 7V 3 1 1 2 2 2 i 1 i 3 i A7
Let us summarize the method by which we obtain a set of mesh equations for a resistive circui 1. Make certain that the network is a planar network If it is non-planar, mesh analysis is not applicable. 2. Make a neat, simple, circuit diagram. Indicate all element and source values. Resistance values are preferable to conductance values. Each source should have its reference symbol. 3. Assuming that the circuit has m meshes, assign a clockwise current in each mesh,i1,l2,……,iv
Let us summarize the method by which we obtain a set of mesh equations for a resistive circuit: 1.Make certain that the network is a planar network. If it is non-planar, mesh analysis is not applicable. 2.Make a neat, simple, circuit diagram. Indicate all element and source values. Resistance values are preferable to conductance values. Each source should have its reference symbol. 3. Assuming that the circuit has M meshes, assign a clockwise current in each mesh, i1 , i2 , …, iM
4. If the circuit contains only voltage sources, apply Kirchhoff 's voltage law around each mesh. To obtain the resistance matrix if a circuit has only independent voltage sources, equate the clockwise sum of all the resistor voltages to the counter-clockwise sum of all the source voltages, and order the terms, from in to iM. For each dependent voltage source present, relate the source voltage and the controlling quantity to the variables i1, i2,..., iM, if they are not already in that form
4. If the circuit contains only voltage sources, apply Kirchhoff 's voltage law around each mesh. To obtain the resistance matrix if a circuit has only independent voltage sources, equate the clockwise sum of all the resistor voltages to the counter-clockwise sum of all the source voltages, and order the terms, from i1 to iM. For each dependent voltage source present, relate the source voltage and the controlling quantity to the variables i1 , i2 , …, iM, if they are not already in that form
5. If the circuit contains current sources, temporarily modify the given circuit by replacing each source by a open circuit, thus reducing the number of meshes by one for each current source that is present. The assigned mesh currents should not be changed. Using these assigned mesh currents, apply kirchhoffs voltage law around each of the meshes or super-meshes in this modified circuit. relate each source current to the variables i1, i2,..., iM, if it is not already in that form
5. If the circuit contains current sources, temporarily modify the given circuit by replacing each source by a open circuit, thus reducing the number of meshes by one for each current source that is present. The assigned mesh currents should not be changed. Using these assigned mesh currents, apply Kirchhoff ' s voltage law around each of the meshes or super-meshes in this modified circuit. Relate each source current to the variables i1 , i2 , …, iM, if it is not already in that form