Mesh flow method

In addition to node potential analysis for investigating electrical networks, the mesh current method can also be used for calculation. In the mesh current method, a system of equations is set up with which the mesh currents can be determined. In order to set up the system of equations for the mesh current method, some preparations must be made. In the following steps, the electrical network from Figure 1 is to be calculated using the mesh current method.

1 Preparation of the network

The mesh current method works with voltage sources and resistance values. For this purpose, any current sources must be transformed into voltage sources. This type of conversion of a current source into a voltage source is shown once again in Figure 2. Here, the current source and the parallel internal resistance are transformed into a voltage source with a series internal resistance. Here, the current source and the parallel internal resistance are converted into a voltage source with a series internal resistance.

Equation 1 shows the calculation of the voltage value of the voltage source from the specified current value of the current source and the resistance value of the internal resistance.

\begin {equation} U_\mathrm {q} = R_\mathrm {i} \cdot I_\mathrm {q} \label {GleichungUmrechnungStromquelle} \end {equation}

In addition, possible conductance values of electrical resistors must be converted into resistance values. Equation 2 shows how the resistance value is determined from the reciprocal of the conductance.

\begin {equation} R_\mathrm {i} = \frac {1}{G_\mathrm {i}} \label {GleichungUmrechnungLeitwert} \end {equation}

In Figure 3, the current source of the network shown is converted into a voltage source. The parallel connection of the current source and resistor \(R_2\) becomes a series connection consisting of the new voltage source and resistor \(R_2\).

2 Define meshes

For mesh current analysis, the meshes must be defined. The mesh currents are determined via the meshes. In addition, the resistances are assigned to the meshes. Figure 4 shows the network presented with the meshes \(M_1\) and \(M_2\) marked.

The mesh current \(I_\mathrm {M1}\) flows through all components located at mesh \(M_1\). Similarly, all components at mesh \(M_2\) are traversed by the mesh current \(I_\mathrm {M2}\). The two mesh currents \(I_\mathrm {M1}\) and \(I_\mathrm {M2}\) are noted vectorially in equation 3.

\begin {equation} I_\mathrm {M} = \begin {bmatrix} I_\mathrm {M1} \\ \\ I_\mathrm {M2} \\ \end {bmatrix} \label {GleichungMaschenströme} \end {equation}

3 Determine resistance matrix

In mesh \(M_1\), the mesh current \(I_{M1}\) encounters the two resistors \(R_0\) and \(R_1\). Mesh \(M_1\) is shown in blue in Figure 5. The mesh current \(I_{M2}\) of mesh \(M_2\) flows through the resistors \(R_1\) and \(R_2\). Resistor \(R_1\) is therefore part of both meshes \(M_1\) and \(M_2\) and is traversed by both mesh currents. If a resistor is traversed by several mesh currents, it is considered a coupling resistor between the meshes. In the resistance matrix according to equation 4, the resistances of the meshes and the coupling resistances are assigned. Here the loop resistances from the meshes are entered in the main diagonal. This means that the series connection \(R_0 + R_1\) becomes the element of the first column and row for the first mesh. The series connection \(R_1 + R_2\) is entered in the element of the second column and row for the second mesh. The coupling resistances between the respective meshes are entered in the remaining elements of the resistance matrix. The coupling resistance is given a positive sign if both mesh currents flow in the same direction and a negative sign if they flow in opposite directions. In alternating current technology, the resistance matrix is also referred to as the mesh impedance matrix.

\begin {equation} R_\mathrm {M} = \begin {bmatrix} \color {blue}{R_0 + R_1} & \color {green}{-R_1} \\ \\ \color {green}{-R_1} & R_1 + R_2 \\ \end {bmatrix} \label {GleichungWiderstandsmatrix} \end {equation}

4 Assign source voltages

The mesh must be assigned the stress sources, known as source stresses. The mesh \(M_1\) has the source stress \(U_0\). The mesh \(M_2\) has the source stress \(U_1\). If the direction of the voltage arrow and the mesh direction are the same, the source voltages have a negative sign. If the directions of the voltage arrow and the mesh circulation are opposite, the sign of the source voltage is positive. If there is no source in a mesh, a \(0\) is entered.

\begin {equation} U_\mathrm {M} = \begin {bmatrix} U_0 \\ \\ -U_1 \\ \end {bmatrix} \label {GleichungQuellspannungen} \end {equation}

5 Set up a system of equations

According to Ohm’s law, electrical voltage is calculated as the product of electrical resistance and current. In mesh current analysis, the resistance matrix \(R_\mathrm {M}\) and the vector of source voltages \(U_\mathrm {M}\) are used to calculate the vector of mesh currents \(I_\mathrm {M}\). The system of equations is represented in Equation 6.

\begin {equation} \begin {bmatrix} R_0 + R_1 & -R_1 \\ \\ -R_1 & R_1 + R_2 \\ \end {bmatrix} \cdot \begin {bmatrix} I_\mathrm {M1} \\ \\ I_\mathrm {M2} \\ \end {bmatrix} = \begin {bmatrix} U_0 \\ \\ -U_1 \\ \end {bmatrix} \label {GleichungMasche1} \end {equation}

The quantities sought are calculated from the mesh currents. For example, the currents \(I_0\), \(I_1\) and \(I_2\) are sought. The results of the mesh currents are equivalent to the currents that flow through a single mesh current. In the electrical network shown, the current of mesh \(M_1\) is \(I_\mathrm {M1} = I_0\) and the current of mesh \(M_2\) is \(I_\mathrm {M2} = -I_2\). The current through resistor \(R_1\) can be calculated using the node equation: \(I_1 = I_0 + I_2\).

Mesh flow method

Analysis of the electrical network from Figure 6 using mesh current analysis.

The following steps should be carried out for mesh flow analysis:

• Preparation of the network
• Define meshes and mesh flows
• Determine resistance matrix
• Assign source voltages
• Set up a system of equations

a)

Preparation of the network

b)

Define meshes and mesh flows

\begin {equation} I_\mathrm {M} = \begin {bmatrix} I_\mathrm {M1} \\ \\ I_\mathrm {M2} \\ \\ I_\mathrm {M3} \\ \end {bmatrix}\nonumber \end {equation}

c)

Determine resistance matrix \begin {equation} R_\mathrm {M} = \begin {bmatrix} R_0 + R_3 + R_4 & -R_3 & -R_4 \\ \\ -R_3 & R_1 + R_3 + R_5 & -R_5\\ \\ -R_4 & -R_5 & R_2 + R_4 + R_5 \\ \end {bmatrix} \nonumber \end {equation}

d)

Assign source voltages \begin {equation} U_\mathrm {M} = \begin {bmatrix} U_0 \\ \\ U_1 \\ \\ -U_2\\ \end {bmatrix} \nonumber \end {equation}

e)

Set up a system of equations \begin {equation} \begin {bmatrix} R_0 + R_3 + R_4 & -R_3 & -R_4 \\ \\ -R_3 & R_1 + R_3 + R_5 & -R_5\\ \\ -R_4 & -R_5 & R_2 + R_4 + R_5 \\ \end {bmatrix} \cdot \begin {bmatrix} I_\mathrm {M1} \\ \\ I_\mathrm {M2} \\ \\ I_\mathrm {M3} \\ \end {bmatrix} = \begin {bmatrix} U_0 \\ \\ U_1 \\ \\ -U_2\\ \end {bmatrix}\nonumber \end {equation} Calculate currents: \begin {align} I_3 = I_0 - I_1\nonumber \\ I_4 = I_3 - I_5\nonumber \\ I_0 = I_4 - I_2\nonumber \\ -I_5 = I_2 + I_1\nonumber \end {align}