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Node-Voltage Analysis#

Node-voltage analysis is a technique for finding a basis for Kirchhoff's current laws within a given electronic circuit as well as a way to express the branch voltages inside it in terms of voltages with respect to a chosen reference.

Algorithm: Node-Voltage Analysis

  1. Construct a network graph \(\mathcal{G} = (B, N, s, t)\) of the circuit.
  2. Pick some node \(\mathcal{N} \in N\) as a reference.
  3. Construct the incidence matrix of \(\mathcal{G}\) but remove the row corresponding to \(\mathcal{N}\).

Definition: Reduced Incidence Matrix

The result from step 3 is known as the reduced incidence matrix.

Notation

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

  1. For each node \(n_j \in N \setminus \mathcal{N}\), define the node-voltage \(v_{\text{k}, j}\) as the potential difference \(v_{\text{k},j} = \phi_j - \phi_{\mathcal{N}}\).

  2. Multiplying the transpose of \(\boldsymbol{A}\) by the vector of node-voltages results in the branch voltage vector:

\[\boldsymbol{v} = \boldsymbol{A}^{\mathsf{T}}\boldsymbol{v}_{\text{k}}\]
Example

Consider an electronic circuit with the following network graph:

Analysis Example Graph

We construct the incidence matrix:

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

We choose node 4 as reference and remove the corresponding row to obtain the reduced incidence matrix:

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

From \(\boldsymbol{A}\boldsymbol{i} = \boldsymbol{0}\), we obtain a basis for the Kirchhoff's current laws:

\[\left\vert\begin{aligned} i_1 + i_2 & = 0 \\ -i_1 + i_3 + i_4 & = 0 \\ -i_2 - i_3 + i_5 & = 0\\ i_6 + i_7 & = 0 \end{aligned}\right.\]

We now define the node-voltages:

Node Voltages Example

Multiplying \(\boldsymbol{A}^{\mathsf{T}}\) by the vector of node-voltages results in the branch voltage vector:

\[\boldsymbol{v} = \boldsymbol{A}^{\mathsf{T}}\begin{bmatrix} \vert \\ \boldsymbol{v}_{\text{k}} \\ \vert \end{bmatrix} = \begin{bmatrix} +1 & -1 & 0 & 0 \\ +1 & 0 & -1 & 0 \\ 0 & +1 & -1 & 0 \\ 0 & +1 & 0 & 0 \\ 0 & 0 & +1 & 0 \\ 0 & 0 & 0 & +1 \\ 0 & 0 & 0 & +1 \end{bmatrix} \begin{bmatrix} v_{\text{k},1} \\ v_{\text{k},2} \\ v_{\text{k},3} \\ v_{\text{k},5} \end{bmatrix}\]
\[\left\vert\begin{aligned} v_1 & = v_{\text{k},1} - v_{\text{k},2} \\ v_2 &= v_{\text{k},1} - v_{\text{k},3} \\ v_3 &= v_{\text{k},2} - v_{\text{k},3} \\ v_4 & = v_{\text{k},2} \\ v_5 & = v_{\text{k},3} \\ v_6 & = v_{\text{k},5} \\ v_7 & = v_{\text{k},5} \end{aligned}\right.\]

Reduced Node-Voltage Analysis#

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

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

If we use node-voltage analysis, we can construct a system of equations in terms of the node voltages and the reduced incidence matrix:

\[\boldsymbol{A} g(\boldsymbol{A}^{\mathsf{T}}\boldsymbol{v}_{\text{k}}) = \boldsymbol{0}\]

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

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

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

The aforementioned system can then be written as

\[\boldsymbol{G}_{k}\boldsymbol{v}_{\text{k}} = \boldsymbol{i}_{\text{q}},\]

where \(\boldsymbol{G}_k = -\boldsymbol{A}\boldsymbol{N}^{-1}\boldsymbol{M}\boldsymbol{A}^{\mathsf{T}}\) is called the node admittance matrix and \(\boldsymbol{i}_{\text{q}} = -\boldsymbol{A}\boldsymbol{N}^{-1}\boldsymbol{e}\) is known as the node current source vector.

Theorem: Node Current Source Vector

The \(k\)-th component of the node current source vector \(\boldsymbol{i}_{\text{q}}\) (skipping reference) is the algebraic sum of all currents which are flowing into or out of the corresponding node and are caused by ideal current sources, where in-flowing currents are positive and out-flowing currents are negative.

Example

Reduced Node Voltage Analysis Source Vector

\[\boldsymbol{i}_{\text{q}} = \begin{bmatrix}I_{\text{CS},1} - I_{\text{CS},2} \\ -I_{\text{CS},1}\end{bmatrix}\]
Proof

TODO

Algorithm: Node Admittance Matrix

Initialize the node admittance matrix to zero.

If a branch current \(i_{\alpha \to \beta}\) (flowing out of the node \(\alpha\) into the node \(\beta\)) is given by the node voltages \(v_{\text{k},\gamma}\) and \(v_{\text{k},\delta}\) as

\[i_{\alpha \to \beta} = g\cdot (v_{\text{k},\gamma} - v_{\text{k},\delta}),\]

then \(g\) contributes to the elements \(g_{i,j}\) of the node admittance matrix as follows:

  • Add \(g\) to row \(\alpha\), column \(\gamma\) and to row \(\beta\), column \(\delta\).
\[\begin{aligned}g_{\alpha, \gamma} & \leftarrow g_{\alpha, \gamma} + g \\ g_{\beta, \delta} & \leftarrow g_{\beta, \delta} + g\end{aligned}\]
  • Subtract \(g\) from row \(\alpha\), column \(\delta\) and from row \(\beta\), column \(\gamma\).
\[\begin{aligned}g_{\alpha, \delta} & \leftarrow g_{\alpha, \delta} - g \\ g_{\beta, \gamma} & \leftarrow g_{\beta, \gamma} - g\end{aligned}\]
  • If \(\alpha\), \(\beta\), \(\gamma\) or \(\delta\) is the reference, then do not perform the operations in which it participates.