Mesh analysis
Encyclopedia
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
(and indirectly the voltages
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
) at any place in the circuit
Electronic circuit
An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow...
. Planar circuits are circuits that can be drawn on a plane surface
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
with no wire
Wire
A wire is a single, usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads and to carry electricity and telecommunications signals. Wire is commonly formed by drawing the metal through a hole in a die or draw plate. Standard sizes are determined by various...
s crossing each other. A more general technique, called loop analysis can be applied to any circuit, planar or not. Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law
Kirchhoff's circuit laws
Kirchhoff's circuit laws are two equalities that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff...
to arrive at a set of equations guaranteed to be solvable if the circuit has a solution. Mesh analysis is usually easier to use when the circuit is planar, compared to loop analysis.
Mesh currents and essential meshes
Circuit diagram
A circuit diagram is a simplified conventional graphical representation of an electrical circuit...
, the essential meshes look like a “window pane”. Figure 1 labels the essential meshes with one, two, and three. Once the essential meshes are found, the mesh currents need to be labeled.
A mesh current is a current that loops around the essential mesh. The mesh current might not have a physical meaning but it is used to set up the mesh analysis equations. When assigning the mesh currents it is important to have all the mesh currents loop in the same direction. This will help prevent errors when writing out the equations. The convention is to have all the mesh currents looping in a clockwise
Clockwise
Circular motion can occur in two possible directions. A clockwise motion is one that 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 to the top...
direction. Figure 2 shows the same circuit shown before but with the mesh currents labeled.
The reason to use mesh currents instead of just using KCL and KVL to solve a problem is that the mesh currents can account for any unnecessary currents that may be drawn in when using KCL and KVL. Mesh analysis ensures that the least possible number of equations regarding currents is used, greatly simplifying the problem.
Setting up the equations
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
in the circuit. These equations are the sum of the voltage drop
Voltage drop
Voltage drop is the reduction in voltage in the passive elements of an electrical circuit. Voltage drops across conductors, contacts, connectors and source internal resistances are undesired as they reduce the supplied voltage while voltage drops across loads and other electrical and electronic...
s in a complete loop of the mesh current. For other than current
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...
and voltage source
Voltage source
In electric circuit theory, an ideal voltage source is a circuit element where the voltage across it is independent of the current through it. A voltage source is the dual of a current source. In analysis, a voltage source supplies a constant DC or AC potential between its terminals for any current...
s, the voltage drop
Voltage drop
Voltage drop is the reduction in voltage in the passive elements of an electrical circuit. Voltage drops across conductors, contacts, connectors and source internal resistances are undesired as they reduce the supplied voltage while voltage drops across loads and other electrical and electronic...
s will be the impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
of the electronic component
Electronic component
An electronic component is a basic electronic element and may be available in a discrete form having two or more electrical terminals . These are intended to be connected together, usually by soldering to a printed circuit board, in order to create an electronic circuit with a particular function...
multiplied by the mesh current in that loop. It is important to note that if a component exists between two essential meshes, the component's voltage drop
Voltage drop
Voltage drop is the reduction in voltage in the passive elements of an electrical circuit. Voltage drops across conductors, contacts, connectors and source internal resistances are undesired as they reduce the supplied voltage while voltage drops across loads and other electrical and electronic...
will be the impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
of the component times the present mesh current minus the neighboring mesh current (computing the subtraction first).
If a voltage source
Voltage source
In electric circuit theory, an ideal voltage source is a circuit element where the voltage across it is independent of the current through it. A voltage source is the dual of a current source. In analysis, a voltage source supplies a constant DC or AC potential between its terminals for any current...
is present within the mesh loop, the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
at the source is either added or subtracted depending on if it is a voltage drop
Voltage drop
Voltage drop is the reduction in voltage in the passive elements of an electrical circuit. Voltage drops across conductors, contacts, connectors and source internal resistances are undesired as they reduce the supplied voltage while voltage drops across loads and other electrical and electronic...
or a voltage rise in the direction of the mesh current. For a current source
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...
that is not contained between two meshes, the mesh current will take the positive or negative value of the current source
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...
depending on if the mesh current is in the same or opposite direction of the current source
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...
. The following is the same circuit from above with the equations needed to solve for all the currents in the circuit.
Once the equations are found, the system of linear equations can be solved by using any technique to solve linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
s.
Supermesh
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...
is contained between two essential meshes. To handle the supermesh, first treat the circuit as if the current source
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...
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
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...
. This will be an equation where the current source
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...
is equal to one of the mesh currents minus the other. The following is a simple example of dealing with a supermesh.
Dependent sources
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...
or voltage source
Voltage source
In electric circuit theory, an ideal voltage source is a circuit element where the voltage across it is independent of the current through it. A voltage source is the dual of a current source. In analysis, a voltage source supplies a constant DC or AC potential between its terminals for any current...
that depends on the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
or current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
on another element
Electronic component
An electronic component is a basic electronic element and may be available in a discrete form having two or more electrical terminals . These are intended to be connected together, usually by soldering to a printed circuit board, in order to create an electronic circuit with a particular function...
in the circuit. When a dependent source is contained within an essential mesh, the dependent source should be treated like a normal 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
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
or current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
that the source depends on in the circuit. The following is a simple example of a dependent source.
See also
- Ohm's lawOhm's lawOhm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
- Analysis of resistive circuitsAnalysis of resistive circuitsA network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, every component in the network. There are a number of different techniques for achieving this...
- Nodal analysisNodal analysisIn electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage between "nodes" in an electrical circuit in terms of the branch currents.In analyzing a circuit using Kirchhoff's circuit laws, one can either do nodal...
- Kirchhoff's circuit lawsKirchhoff's circuit lawsKirchhoff's circuit laws are two equalities that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff...
- Source transformationSource transformationFinding a solution to a circuit can be somewhat difficult without using tricks or methods that make the circuit appear simpler. Circuit solutions are often simplified, especially with mixed sources, by transforming a voltage into a current source, and vice versa...