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Mesh Current Analysis#

Mesh current analysis is a technique for finding a basis for Kirchhoff's voltage laws within an electronic circuit as well as a way to express the branch currents inside it in terms of other, fictitious currents.

Algorithm: Mesh Current Analysis

  1. Construct a planar network graph \(\mathcal{G} = (B, N, s, t)\) of the circuit.
  2. Assign an orientation to each mesh in \(\mathcal{G}\).
  3. Construct an a matrix whose rows correspond to the meshes and whose columns correspond to the branches in \(\mathcal{G}\) in the following way:
\[b_{kj} = \begin{cases} +1 & \text{if branch } j \text{ is in mesh } k \text{ and their orientations are the same} \\ -1 & \text{if branch } j \text{ is in mesh } k \text{ and their orientations are opposite} \\ 0 & \text{if branch } j \text{ is not part of mesh } k \end{cases}\]

Definition: Mesh Incidence Matrix

The result from step 3 is known as a mesh incidence matrix.

Notation

\[\boldsymbol{B}\]
  1. The homogenous system obtained by the product of \(\boldsymbol{B}\) and the branch voltage vector \(\boldsymbol{v}\) is a basis for all instances of Kirchhoff's voltage law in the original circuit:
\[\boldsymbol{B}\boldsymbol{v} = \boldsymbol{0}\]

  1. For each mesh \(m_j\), define the mesh current \(i_{m,j}\) as a fictitious current circulating around the loop.

  2. Multiplying the transpose of \(\boldsymbol{B}\) by the vector of mesh currents results in the branch current vector:

\[\boldsymbol{i} = \boldsymbol{B}^{\mathsf{T}} \boldsymbol{i}_{\text{m}}\]
Example

Consider an electronic circuit with the following network graph:

Analysis Example Graph

There are three meshes inside and we can define the corresponding mesh currents:

Mesh Currents Example

We now construct the mesh incidence matrix:

\[\boldsymbol{B} = \begin{bmatrix}+1 & -1 & +1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & +1 & -1\end{bmatrix}\]

We now have a basis for Kirchhoff's voltage laws:

\[\boldsymbol{B}\boldsymbol{v} = \boldsymbol{0}\]
\[\begin{bmatrix}+1 & -1 & +1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & +1 & -1\end{bmatrix}\begin{bmatrix}v_1 \\ v_2 \\ v_3 \\ v_4 \\ v_5 \\ v_6 \\ v_7 \end{bmatrix} = \begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}\]

Multiplying \(\boldsymbol{B}^{\mathsf{T}}\) by the vector of mesh currents results in the branch current vector:

\[\boldsymbol{i} = \boldsymbol{B}^{\mathsf{T}} \begin{bmatrix}i_{\text{m1}} \\ i_{\text{m2}} \\ i_{\text{m3}}\end{bmatrix} = \begin{bmatrix} +1 & 0 & 0 \\ -1 & 0 & 0 \\ +1 & -1 & 0 \\ 0 & +1 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & +1 \\ 0 & 0 & -1 \end{bmatrix}\begin{bmatrix}i_{\text{m1}} \\ i_{\text{m2}} \\ i_{\text{m3}}\end{bmatrix}\]
\[\left\vert\begin{aligned}i_1 & = i_{\text{m1}} \\ i_2 & = -i_{\text{m1}} \\ i_3 & = i_{\text{m1}} - i_{\text{m2}} \\ i_4 & = i_{\text{m2}} \\ i_5 & = -i_{\text{m2}} \\ i_6 & = i_{\text{m3}} \\ i_7 & = -i_{\text{m3}}\end{aligned}\right.\]

Reduced Mesh-Current Analysis#

When a network contains only current-controlled elements, the branch voltage vector can be expressed in terms of the branch current vector in the following way:

\[\boldsymbol{v} = r(\boldsymbol{i})\]

If we use mesh current analysis, we can construct a system of equations in terms of the mesh currents and the mesh incidence matrix

\[\boldsymbol{B} r(\boldsymbol{B}^{\mathsf{T}}\boldsymbol{i}_{\text{m}}) = \boldsymbol{0}\]

Solving this system yields the mesh currents and subsequently the branch currents and branch voltages.

Moreover, if the elements are also affine, the branch voltage vector can be expressed in terms of the branch current vector in the following way:

\[\boldsymbol{v} = -\boldsymbol{M}^{-1}\boldsymbol{N}\boldsymbol{i} + \boldsymbol{M}^{-1}\boldsymbol{e}\]

The aforementioned system can then be written as

\[\boldsymbol{Z}_{m}\boldsymbol{i}_{\text{m}} = \boldsymbol{v}_{\text{q}},\]

where \(\boldsymbol{Z}_m = -\boldsymbol{B}\boldsymbol{M}^{-1}\boldsymbol{N}\boldsymbol{B}^{\mathsf{T}}\) is called the mesh impedance matrix and \(\boldsymbol{v}_{\text{q}} = -\boldsymbol{B}\boldsymbol{M}^{-1}\boldsymbol{e}\) is known as the mesh voltage source vector.