en>Incnis Mrsi: Incnis Mrsi moved page Line-plane intersection to Line–plane intersection: WP:–

2014-01-29T12:52:26Z

Incnis Mrsi moved page Line-plane intersection to Line–plane intersection: WP:–

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{{redirect|Loop current|the ocean current in the Gulf of Mexico|Loop Current|the electrical signalling scheme|current loop}}
{{howto|date=November 2012}}

[[Image:Mesh Analysis Example1 TeX.svg|thumb|300px|right|Figure 1: Essential meshes of the planar circuit labeled 1, 2, and 3. R1, R2, R3, 1/sc, and Ls represent the [[Electrical impedance|impedance]] of the [[resistor]]s, [[capacitor]], and [[inductor]] values in the [[s-domain]]. Vs and is are the values of the [[voltage source]] and [[current source]], respectively.]]
'''Mesh analysis''' (or the '''mesh current method''') is a method that is used to solve planar circuits for the [[Electric current|currents]] (and indirectly the [[voltage]]s) at any place in the [[Electronic circuit|circuit]]. Planar circuits are circuits that can be drawn on a [[Plane (mathematics)|plane surface]] with no [[wire]]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 circuit laws|Kirchhoff’s voltage law]] to arrive at a set of equations guaranteed to be solvable if the circuit has a solution.<ref name="Hayt">Hayt, William H., & Kemmerly, Jack E. (1993). ''Engineering Circuit Analysis'' (5th ed.), New York: McGraw Hill.</ref> Mesh analysis is usually easier to use when the circuit is planar, compared to loop analysis.<ref name="Nilsson">Nilsson, James W., & Riedel, Susan A. (2002). ''Introductory Circuits for Electrical and Computer Engineering''. New Jersey: Prentice Hall.</ref>

== Mesh currents and essential meshes ==
[[Image:Mesh Analysis Example2 TeX.svg|thumb|300px|right|Figure 2: Circuit with mesh currents labeled as i1, i2, and i3. The arrows show the direction of the mesh current.]]
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.<ref name="Russell">Lueg, Russell E., & Reinhard, Erwin A. (1972). ''Basic Electronics for Engineers and Scientists'' (2nd ed.). New York: International Textbook Company.</ref>

A mesh current is a current that loops around the essential mesh and the equations are set 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.<ref name="Nilsson"/> 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]] direction.<ref name="Russell"/> Figure 2 shows the same circuit from Figure 1 with the mesh currents labeled.

Solving for mesh currents instead of directly applying [[Kirchhoff's circuit laws#Kirchhoff's current law (KCL)|Kirchhoff's current law]] and [[Kirchhoff's circuit laws#Kirchhoff's voltage law (KVL)|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]]s in a complete loop of the mesh current.<ref name="Russell"/> For problems more general than those including [[Current source|current]] and [[voltage source]]s, the [[voltage drop]]s will be the [[Electrical impedance|impedance]] of the [[electronic component]] multiplied by the mesh current in that loop.<ref name="Romanowitz">Puckett, Russell E., & Romanowitz, Harry A. (1976). ''Introduction to Electronics'' (2nd ed.). San Francisco: John Wiley and Sons, Inc.</ref>

If a [[voltage source]] is present within the mesh loop, the [[voltage]] at the source is either added or subtracted depending on if it is a [[voltage drop]] or a voltage rise in the direction of the mesh current. For a [[current source]] that is not contained between two meshes, the mesh current will take the positive or negative value of the [[current source]] depending on if the mesh current is in the same or opposite direction of the [[current source]].<ref name="Russell"/> The following is the same circuit from above with the equations needed to solve for all the currents in the circuit.

<math> \begin{cases}
\text{Mesh 1: } I_1 = I_s\\
\text{Mesh 2: } -V_s + R_1(I_2-I_1) + \frac{1}{sc}(I_2-I_3)=0\\
\text{Mesh 3: } \frac{1}{sc}(I_3-I_2) + R_2(I_3-I_1) + LsI_3=0\\
\end{cases} \, </math>
 

Once the equations are found, the [[system of linear equations]] can be solved by using any technique to solve [[linear equation]]s.
{{-}}

==Special cases==
There are two special cases in mesh current: currents containing a supermesh and currents containing [[dependent source]]s.

===Supermesh===
[[Image:Mesh Analysis Example3 TeX.svg|thumb|300px|right|Figure 3: Circuit with a supermesh. Supermesh occurs because the current source is in between the essential meshes.]]
A supermesh occurs when a [[current source]] is contained between two essential meshes. The circuit is first treated as if the [[current source]] 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 [[current source]]. This will be an equation where the [[current source]] is equal to one of the mesh currents minus the other. The following is a simple example of dealing with a supermesh.<ref name="Nilsson"/>

<math> \begin{cases}
\text{Mesh 1, 2: } -V_s + R_1I_1 + R_2I_2 = 0\\
\text{Current source: } I_s = I_2 - I_1
\end{cases} \, </math>

===Dependent sources===
{{see also|Dependent source}}
[[Image:Mesh Analysis Example4 TeX.svg|thumb|300px|right|Figure 4: Circuit with dependent source. ix is the current upon which the dependent source depends.]]
A dependent source is a [[current source]] or [[voltage source]] that depends on the [[voltage]] or [[Electric current|current]] of another [[electronic component|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 [[voltage]] or [[Electric current|current]] that the source depends on in the circuit. The following is a simple example of a dependent source.<ref name="Nilsson"/>

<math> \begin{cases}
\text{Mesh 1: } -V_s + R_1I_1 + R_3(I_1 - I_2) = 0\\
\text{Mesh 2: } R_2I_2 + 3I_x + R_3(I_2 - I_1) = 0\\
\text{Dependent variable: } I_x = I_1 - I_2
\end{cases} \, </math>
{{-}}

== See also ==
* [[Ohm's law]]
* [[Analysis of resistive circuits]]
* [[Nodal analysis]]
* [[Kirchhoff's circuit laws]]
* [[Source transformation]]
* [[Topology (electrical circuits)]]

==References==
{{Reflist}}

== External links ==
* [http://www.allaboutcircuits.com/vol_1/chpt_10/3.html Mesh current method]
* [http://www.catc.ac.ir/mazlumi/mesh.php Online three-mesh problem solver]

[[Category:Electrical engineering]]
[[Category:Electronic engineering]]
[[Category:Electrical circuits]]
[[Category:Electronic circuits]]
[[Category:Electronic design]]

Line–plane intersection - Revision history

en>Incnis Mrsi: Incnis Mrsi moved page Line-plane intersection to Line–plane intersection: WP:–