Introduction: What is Mesh Analysis? Mesh Analysis is a technique for the rigourous solution of many electrical circuits. With this method, the user can systematically find sufficient and necessary equations for the solution of the problem. In conjunction with the Node Voltage Method, Mesh Analysis provides a second analysis technique for the verification of the solution. A mesh is the simplest form of closed loop that passes through circuit elements without there being any elements enclosed within the Kirchhoff's Voltage Law is satisfied around each mesh in the circuit with the resulting equations written in terms of the mesh currents which are the variables for the solution. The current flow in a circuit element will be determined from the appropriate mesh currents which flow through that element. In the beginning, don't anticipate signs associated with individual voltages or currents. Be methodical! Assign each mesh current as flowing clockwise around its mesh. A detailed analysis procedure is described on the right. In the next three sections, the procedure will be implemented by way of three examples. The first example describes a circuit which only has resistors and an independent voltage source; the second example introduces a current source with a supermesh (the current source will be inside the supermesh) and the third example includes a dependent source. The Unit concludes with mini-tests and exercises. Procedure: For a circuit which may be drawn on a plane surface without the need for any crossing paths: Draw the circuit without any crossing paths. Label all the mesh currents flowing DON'T label any voltages on the circuit, except those associated with voltage sources. Draw up a matrix for n equations in terms of the n mesh current variables. Express all dependent sources in terms of the circuit's mesh current variables. For every current source (both independent and dependent sources), equate the current value of the source to the adjacent mesh currents which flow through the source. Enter the equations so that the first non-zero voltage coefficient lies on the diagonal of the matrix. As each equation is entered into the matrix, note in words alongside it the reason for the equation. (This has not been done in some of the examples here 1
because of space limitations.) Form supermeshes around all each current sources, condensing them into larger supermeshes if two or more of them interlock in the circuit. Supermeshes are formed because it is impossible to write a meaningful equation in terms of the mesh currents for the voltage across a current source, while it will be possible to express all the voltages around a supermesh. For every mesh which is not an internal part of a supermesh, apply Kirchhoff's Voltage Law in a clockwise direction around the mesh, expressing the sum of the voltage drops across passive elements = the sum of all the voltage rises for the voltage sources. Only mesh currents should appear as circuit variables in the resulting equations. If the mesh currents are labelled in a clockwise direction and Kirchhoff's Voltage Law is being applied in a clockwise direction around the mesh then, for passive components, the coefficient for the current of the mesh being considered is positive while the coefficients of adjacent mesh currents are negative. For each supermesh, likewise apply Kirchhoff's Voltage Law as in the previous point. Remember that supermeshes have different mesh currents at various points around their length. At this stage, the matrix should be exactly filled with sufficient and necessary equations. Solve the matrix equation and check the result by applying Kirchhoff's Voltage Law either around the circumference of the circuit or along the longest closed path which does not pass through a current source. 2
The Basic Circuit with Independent Voltage Sources and Resistors Step 1: The Question Calculate the power provided to the circuit by the 1V source. Step 4 Apply Kirchhoff's Voltage Law clockwise around each mesh in turn, writing the equation as the sum of the voltage drops across the passive components (expressed in terms of the mesh currents only) equals the sum of the voltage rises across the voltage sources. Write the equation for mesh 1. Step 5 Likewise, apply Kirchhoff's Voltage Law to mesh 2 Step 6 Likewise, apply Kirchhoff's Voltage Law to mesh 3 Step 7 Reduce the matrix using Gaussian Elimination. Step 8 From the final equation and with back substitution, deduce the mesh currents. Step 9 Check the result by applying Kirchhoff's Voltage Law to the outermost Step 10 Calculate the power provided by the voltage source to the circuit. The power provided from the 1V source to the circuit is 0.435 W. 3
Introducing Current Sources with Supermeshes Step 1: The Question Calculate the power provided to the circuit by the 1A source. To this stage, the method has proceeded as for the basic circuit in the previous section. Step 4 For each current source, equate its numerical value to an expression in terms of the mesh currents which pass through the particular source. Step 5 Form a supermesh around each current source, combining any interlocking supermeshes (which may only occur if there are 2 or more current sources) into fewer but larger ones. Two further independent equations are required. Step 6 Apply Kirchhoff's Voltage Law clockwise around each mesh and supermesh in turn, writing the equation as the sum of the voltage drops across the passive components (expressed in terms of the mesh currents only) equals the sum of the voltage rises across the voltage sources. Write the equation for mesh 1. Step 7 Apply Kirchhoff's Voltage Law clockwise around the supermesh, noting the different currents that flow in various parts of the supermesh. Step 8 Solve the completed matrix equation to obtain the mesh currents. Step 9 Check the result by applying Kirchhoff's Voltage Law to the outermost Step 10 Calculate the power provided by the current source to the circuit. The power provided from the 1A source to the circuit is 0.83 W. Introducing Dependent Sources 4
Step 1: The Question Calculate the power dissipated in the 2 resistor. A dependent source will be treated in much the same way as an independent source, after initially expressing its value in terms of the circuit variables. Step 4 All dependent sources are given in terms of a variable shown on the circuit. This variable (and the dependent source value) must be expressed in terms of the mesh currents. Step 5 For each independent or dependent current source, equate its numerical value to an expression in terms of the mesh currents which pass through the particular source. Step 6 Form a supermesh around each current source, combining any interlocking supermeshes into larger ones. Step 7 Apply Kirchhoff's Voltage Law clockwise around each mesh and supermesh in turn, writing the equation as the sum of the voltage drops across the passive components (expressed in terms of the mesh currents only) equals the sum of the voltage rises across the voltage sources. Write the equation for mesh 1. Step 8 Apply Kirchhoff's Voltage Law clockwise around the supermesh, noting the different currents that flow in various parts of the supermesh. Step 9 Solve the completed matrix equation to obtain the mesh currents. Step 10 Check the result by applying Kirchhoff's Voltage Law to the outermost Step 11 Calculate the power dissipated in the 2 resistor. The power dissipated in the 2 resistor is 1.064 W. 5