What is Mesh Analysis?
Introduction:
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 loop.
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.
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.
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 in a clockwise direction.
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
The Basic Circuit with Independent Voltage Sources and
Resistors
2
Introducing Current Sources with Supermeshes
Step 1: The Question
Step 1: The Question
Consider the following circuit:
Consider the following circuit:
Calculate the power provided to the circuit by the 1V source.
Step 1 Read the question carefully and plan your solution.
Step 2 Draw the circuit without any crossing paths. Label all the mesh currents
in a clockwise direction.
Step 3 Set up a matrix equation with the mesh currents as the variables. This
matrix must be filled with independent equations before the question may be
solved.
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
loop.
Step 10 Calculate the power provided by the voltage source to the circuit.
ANSWER
The power provided from the 1V source to the circuit is 0.435 W.
Calculate the power provided to the circuit by the 1A source.
Step 1 Read the question carefully and plan your solution.
Step 2 Draw the circuit without any crossing paths. Label all the mesh currents
in a clockwise direction.
Step 3 Set up a matrix equation with the mesh currents as the variables. This
matrix must be filled with independent equations before the question may be
solved.
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
loop.
Step 10 Calculate the power provided by the current source to the circuit.
ANSWER
The power provided from the 1A source to the circuit is 0.83 W.
Introducing Dependent Sources
3
4
Step 1: The Question
Consider the following circuit:
Calculate the power dissipated in the 2 resistor.
Step 1 Read the question carefully and plan your solution.
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 2 Draw the circuit without any crossing paths. Label all the mesh currents
in a clockwise direction.
Step 3 Set up a matrix equation with the mesh currents as the variables. This
matrix must be filled with independent equations before the question may be
solved.
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
loop.
Step 11 Calculate the power dissipated in the 2 resistor.
ANSWER
The power dissipated in the 2
resistor is 1.064 W.
5