day 6 notes mesh analysis[1]

8/3/2019 Day 6 Notes Mesh Analysis[1]

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MESH ANALYSIS

Mesh analysis is another general procedure for analyzing circuits, usingmesh currents as the circuit variables. Using mesh currents instead of

element currents as circuit variables reduces the number of equationsthat must be solved simultaneously.

Recall that a loop is a closed path with no node passed more than once.A mesh is a loop that does not contain any other loop within it.

Nodal analysis applies KCL to nd unknown voltages in a given circuit,while mesh analysis applies KVL to nd unknown currents. Meshanalysis is not quite as general as nodal analysis because it is onlyapplicable to a circuit that is planar.

Planar Circuit:A planar circuit is one that can be drawn in a plane with no branches crossing one another; otherwise it isnonplanar. A circuit may have crossing branches and still be planar if it can be redrawn such that it has no crossing branches. For example, the circuit in shown in a has two crossing branches, but itcan be redrawn as in b . Hence, the circuit is planar.

The circuit next is nonplanar, because there is no way to redraw it and avoid the branchescrossing. Nonplanar circuits can be handled using nodal analysis, but they will not beconsidered in this course.

A mesh is a loop which does not contain any other loops within it.

In the circuit, paths abefa and bcdeb are meshes, but path abcdefa is not a mesh. The current through a meshis known as mesh current. In mesh analysis, we areinterested in applying KVL to nd the mesh currents ina given circuit.


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We will start by applying mesh analysis to planar circuits that do not contain currentsources.

In the mesh analysis of a circuit with n meshes, we take the following three steps.

1. Assign mesh currents i 1, i2, . . . , i n to the n meshes.2. Apply KVL to each of the n meshes. Use Ohms law to express the voltages in termsof the mesh currents.3. Solve the resulting n simultaneous equations to get the mesh currents.

To illustrate the steps, consider the circuit shown.The rst step requires that mesh currents i 1 and i 2are assigned to meshes 1 and 2.Although a mesh current may be assigned to eachmesh in an arbitrary direction, it is conventionalto assume that each mesh current ows clockwise .

As the second step, we apply KVL to each mesh.Applying KVL to mesh 1, we obtain

V 1 + R 1i1 + R 3(i1 i 2) = 0or

(R 1 + R 3)i1 R 3i2 = V 1For mesh 2, applying KVL gives

R 2i2 + V 2 + R 3(i2 i 1) = 0or

R 3i1 + (R 2 + R 3)i2 = V 2 Note that in the first set of equations the coefcient of i 1 is the sum of the resistances inthe rst mesh, while the coefcient of i 2 is the negative of the resistance common tomeshes 1 and 2. Now observe that the same is true in the second set of equations.

The third step is to solve for the mesh currents. Putting the equations in matrix formyields

which can be solved to obtain the mesh currents i1 and i2. We can use any technique for solving the simultaneous equations. Previously we learned that if a circuit has n nodes, b

branches, and l independent loops or meshes, thenl = b n + 1.

Hence, l independent simultaneous equations are required to solve the circuit using meshanalysis.

Notice that the branch currents are different from the mesh currents unless the mesh isisolated. To distinguish between the two types of currents, we use i for a mesh currentand I for a branch current. The current elements I 1, I2, and I 3 are algebraic sums of themesh currents.It is evident from the circuit above thatI1 = i 1, I2 = i 2, I3 = i 1 i 2


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Given the circuit below, solve for the loop currents i 1 and i 2 indicated using meshanalysis.

Solution of Simultaneous Equations Using Cramers Rule

In circuit analysis, we often encounter a set of simultaneous equations having the form

where there are n unknown x1, x2, . . . , xn to be determined. These equations can bewritten in matrix form as


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This matrix equation can be put in a compact form asAX =B

where

A is square (n x n) matrix while X and B are column matrices.

There are several methods for solving these equations. These include substitution,Gaussian elimination, Cramers rule, and numerical analysis. In many cases, Cramersrule can be used to solve the simultaneous equations we encounter in circuit analysis.Cramers rule states that the solutions are

Where the Deltas are the determinants given by

Notice that is the determinant of the matrix A and k is the determinant of the matrixformed by replacing the kth column of A by B. Cramers rule applies only when != 0.When = 0, the set of equations has no unique solution, because the equations arelinearly dependent.


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The value of the determinant , for example, can be obtained by expanding along therst row:

where the minor Mij is an (n 1) (n 1) determinant of the matrix formed by strikingout the ith row and jth column. The value of may also be obtained by expanding alongthe rst column:

= a 11M11 a 21M21 + a 31M31 + + (1) n+1an1Mn1We now specically develop the formulas for calculating the determinants of 2 2 and 33 matrices, because of their frequent occurrence. For a 2 2 matrix,

For a 3 x 3 matrix:

An alternative method of obtaining the determinant of a 3 3 matrix is by repeating therst two rows and multiplying the terms diagonally as follows.

In summary:

The solution of linear simultaneous equations by Cramers rule boils down to ndingxk = k / , k = 1, 2, . . . , n

where is the determinant of matrix A and k is the determinant of the matrix formed by replacing the kth column of A by B.


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Apply Mesh Analysis to find in the circuit below:


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MESH ANALYSIS WITH CURRENT SOURCES

Applying mesh analysis to circuits containing current sources (dependent or independent)may appear complicated. But it is actually much easier than what we encountered in the

previous section, because the presence of the current sources reduces the number of equations. Consider the following two possible cases.

Case 1 . When a current source exists only in one mesh:Consider the circuit shown. We set i 2 = 5 A and write amesh equation for the other mesh in the usual way, that is,

10 + 4i 1 + 6(i 1 i 2) = 0 -> i 1 = 2 A

Case 2: When a current source exists between two meshes:Consider the circuit in (a), for example. We create a supermesh by excluding the currentsource and any elements connected in series with it,as shown in (b).


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A supermesh results when two meshes have a (dependent or independent)current source in common.

As shown in (b), we create a supermesh as the periphery of the two meshes and treat itdifferently. (If a circuit has two or more supermeshes that intersect, they should becombined to form a larger supermesh.) Why treat the supermesh differently? Becausemesh analysis applies KVLwhich requires that we know the voltage across each

branchand we do not know the voltage across a current source in advance. However, asupermesh must satisfy KVL like any other mesh.Therefore, applying KVL to the supermesh in (b) gives

20 + 6i 1 + 10i 2 + 4i 2 = 0or

6i1 + 14i 2 = 20We apply KCL to a node in the branch where the two meshes intersect.Applying KCL to node 0 in (a) gives

i2 = i 1 + 6Solving we get

i1 = 3.2 A, i 2 = 2.8 A (3.20) Note the following properties of a supermesh:1. The current source in the supermesh is not completely ignored; it provides theconstraint equation necessary to solve for the mesh currents.2. A supermesh has no current of its own.3. A supermesh requires the application of both KVL and KCL.

Use mesh analysis to obtain i 0 in the circuit below


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1. Identify the supermesh2. write a mesh equation for the super mesh.3. Write mesh equations for any addition meshes4. Write down any equations from KCL5. Solve


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Do Online Tutorial on Mesh Analysis

NODAL VS MESH ANALYSIS

Both nodal and mesh analyses provide a systematic way of analyzing a complex network.Someone may ask: Given a network to be analyzed, how do we know which method is

better or more efficient?Mesh analysis:

Networks that contain many series-connected elements, voltage sources, or supermeshes.a circuit with fewer meshes than nodes is better analyzed using mesh analysis.

Nodal Analysis

Networks with parallel-connected elements, current sources, or supernodes are moresuitable for nodal analysis. A circuit with fewer nodes than meshes is better analyzedusing nodal analysis

The key is to select the method that results in the smaller number of equations.

The second factor is the information required. If node voltages are required, it may beexpedient to apply nodal analysis. If branch or mesh currents are required, it may be

better to use mesh analysis.

It is helpful to be familiar with both methods of analysis, for at least two reasons. First,one method can be used to check the results from the other method, if possible. Second,since each method has its limitations, only one method may be suitable for a particular

problem. For example, mesh analysis is the only method to use in analyzing transistor circuits. Mesh analysis cannot easily be used to solve an op amp circuit, because there isno direct way to obtain the voltage across the op amp itself.For nonplanar networks, nodal analysis is the only option, because mesh analysis onlyapplies to planar networks. Also, nodal analysis is more amenable to solution bycomputer, as it is easy to program. This allows one to analyze complicated circuits thatdefy hand calculation.A computer software package based on nodal analysis will be introduced next class.

day 6 notes mesh analysis[1]

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