Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the

voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...

s) at any place in the

electrical circuit An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage source ...

. Planar circuits are circuits that can be drawn on a plane surface with no

wire Overhead power cabling. The conductor consists of seven strands of steel (centre, high tensile strength), surrounded by four outer layers of aluminium (high conductivity). Sample diameter 40 mm A wire is a flexible strand of metal. Wire is co ...

s crossing each other. A more general technique, called loop analysis (with the corresponding network variables called loop currents) can be applied to any circuit, planar or not. Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a solution.Hayt, William H., & Kemmerly, Jack E. (1993). ''Engineering Circuit Analysis'' (5th ed.), New York: McGraw Hill. Mesh analysis is usually easier to use when the circuit is planar, compared to loop analysis.Nilsson, James W., & Riedel, Susan A. (2002). ''Introductory Circuits for Electrical and Computer Engineering''. New Jersey: Prentice Hall.

Mesh currents and essential meshes

Mesh analysis works by arbitrarily assigning mesh currents in the essential meshes (also referred to as independent meshes). An essential mesh is a loop in the circuit that does not contain any other loop. Figure 1 labels the essential meshes with one, two, and three.Lueg, Russell E., & Reinhard, Erwin A. (1972). ''Basic Electronics for Engineers and Scientists'' (2nd ed.). New York: International Textbook Company. A mesh current is a current that loops around the essential mesh and the equations are solved in terms of them. A mesh current may not correspond to any physically flowing current, but the physical currents are easily found from them. It is usual practice to have all the mesh currents loop in the same direction. This helps prevent errors when writing out the equations. The convention is to have all the mesh currents looping in a

clockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...

direction. Figure 2 shows the same circuit from Figure 1 with the mesh currents labeled. Solving for mesh currents instead of directly applying Kirchhoff's current law and Kirchhoff's voltage law can greatly reduce the amount of calculation required. This is because there are fewer mesh currents than there are physical branch currents. In figure 2 for example, there are six branch currents but only three mesh currents.

Setting up the equations

Each mesh produces one equation. These equations are the sum of the

voltage drop Voltage drop is the decrease of electrical potential along the path of a current flowing in an electrical circuit. Voltage drops in the internal resistance of the source, across conductors, across contacts, and across connectors are undesirab ...

s in a complete loop of the mesh current. For problems more general than those including

current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...

and

voltage source A voltage source is a two-terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply unl ...

s, the

s will be the impedance of the

electronic component An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are no ...

multiplied by the mesh current in that loop.Puckett, Russell E., & Romanowitz, Harry A. (1976). ''Introduction to Electronics'' (2nd ed.). San Francisco: John Wiley and Sons, Inc. If a

is present within the mesh loop, the

at the source is either added or subtracted depending on if it is a

or a voltage rise in the direction of the mesh current. For a

current source A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it. A current source is the dual of a voltage source. The term ''current sink'' is sometimes used for sources fed ...

that is not contained between two meshes (for example, the current source in essential mesh 1 in the circuit above), the mesh current will take the positive or negative value of the

depending on if the mesh current is in the same or opposite direction of the

. The following is the same circuit from above with the equations needed to solve for all the currents in the circuit.

\begin
\text I_1 = I_s\\
\text -V_s + R_1(I_2-I_1) + \frac(I_2-I_3)=0\\
\text \frac(I_3-I_2) + R_2(I_3-I_1) + sLI_3=0\\
\end \,

Once the equations are found, the

system of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in t ...

can be solved by using any technique to solve

linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...

Special cases

There are two special cases in mesh current: currents containing a supermesh and currents containing

dependent source In the theory of electrical networks, a dependent source is a voltage source or a current source whose value depends on a voltage or current elsewhere in the network.I. D. Mayergoyz, Wes Lawson ''Basic electric circuit theory: a one-semester tex ...

Supermesh

A supermesh occurs when a

is contained between two essential meshes. The circuit is first treated as if the

is not there. This leads to one equation that incorporates two mesh currents. Once this equation is formed, an equation is needed that relates the two mesh currents with the

. This will be an equation where the

is equal to one of the mesh currents minus the other. The following is a simple example of dealing with a supermesh.

\begin
\text -V_s + R_1I_1 + R_2I_2 = 0\\
\text I_s = I_2 - I_1
\end \,

Dependent sources

A dependent source is a

that depends on the

of another element in the circuit. When a dependent source is contained within an essential mesh, the dependent source should be treated like an independent source. After the mesh equation is formed, a dependent source equation is needed. This equation is generally called a constraint equation. This is an equation that relates the dependent source’s variable to the

that the source depends on in the circuit. The following is a simple example of a dependent source.

\begin
\text -V_s + R_1I_1 + R_3(I_1 - I_2) = 0\\
\text R_2I_2 + 3I_x + R_3(I_2 - I_1) = 0\\
\text I_x = I_1 - I_2 
\end \,

References

{{Reflist

External links

Mesh current method

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