ELECTRICAL NETWORK ANALYSIS – USING NODAL METHOD

Electronics & Software

INTRODUCTION

This sheet describes the basics of electrical network analysis. The description focuses on the modeling of a circuit for static or dynamic analysis using nodal analysis. Calculation may be performed with a computer aid such as MATLAB.
A network of passive elements (such as resistance and/or reactance) can be analyzed on current and voltage in each circuit node as result of used component parameters, voltage- and current sources.

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Background
  • A ‘node’ is a point (‘star’) between components
  • Symbols: voltage (u), current (i), resistance (R)
  • Using Ohms law \rightarrow G\ {\cdot}\ u=i\ with\ (G=\frac {1}{R})
  • Sum of currents in each node of circuit is zero: \sum i=0
Modeling Procedure
  1. Set up a network without sources; sources are replaced by open circuits. All node voltages are referenced to ground voltage. As the potential of the reference node is known, this node is left out of the equation.
  2. The circuit model is described in a set of equations in a partitioned matrix:
    • according to ohm law: i_s = G\ {\cdot}\ \bar {u} + Q
    • node voltage constraints: u_s = P\ {\cdot}\ \bar {u}

    resulting in: \left[\begin{matrix}GQ\\P0\end{matrix}\right]\left[\begin{matrix}\bar {u}\\\bar {i}\end{matrix}\right]=\left[\begin{matrix}i_s\\u_s\end{matrix}\right]

  3. Matrix G is the model description without sources:
    • On diagonal G_{i,i}; passives connected to node ui. (positive values)
    • Other matrix elements G_{i,j}: passives connected to other nodes from ui to uj (negative values)

    _{Crosscheck:}
    _{Sum\ of\ row\ G_i\ results\ in\ elements\ connected\ to\ reference\ from\ node\ u_j}
    _{Sum\ of\ col\ G_j\ results\ in\ elements\ connected\ to\ reference\ from\ node\ u_j}

  4. Matrix Q describes the current constrains for voltage sources. Matrix P describes the voltage constrains between nodes (or relationship).
  5. Current sources may be added in vector i_s and voltage sources in vector u_s
Resulting unknowns

From this set of equations differential voltages and currents can be calculated:

  • Differential voltages:

u_{xy}=u_x-u_y

  • Currents:

i_{xy}=\frac{u_{xy}}{R_{xy}}

Dynamic analysis

Each passive may be described as resistance R, inductive reactance X_L = j\omega L or capacitive reactance X_c = \frac {1}{j\omega C}. Keep in mind that the components in the matrix are inverted so:
Conductance: G=\frac {1}{R} or susceptance: B=\frac {1}{X}. For dynamic analysis only the source of interest has to be described, other sources are short or open for voltage and current sources respectively.

Example: Circuit with voltage- and current sources
Circuit with voltage- and current sources

Modeling matrix G by neglecting sources:

Circuit with voltage- by neglecting sources

For passives; G = \frac {1}{R} thus for example: G_1 = \frac {1}{R_1}

Electrical network analysis - Using nodal method -Latex code1

After establishing the description of the network in matrix notation, the sources can be added. Now we expand the equation by adding the voltage source descriptions u_{s2} = u_3 - u_5, u_{s1} = u_0 and current source: node u_3 current is sourced with +i_{s1} and in node u_4 current is sourced with -i_{s1} (thus sinked).

Electrical network analysis - Using nodal method -Latex code2

Vector [\begin{matrix}\bar {u}&\bar {i}\end{matrix}]^T may be calculated by:

  • using: \left[\begin{matrix}\bar {u}\\\bar {i}\end{matrix}\right]=\left[\begin{matrix}GQ\\P0\end{matrix}\right]^{-1}\left[\begin{matrix}i_s\\u_s\end{matrix}\right]

or using the reduced row echelon form of matrix \left[\begin{matrix}GQ\\P0\end{matrix}\begin{matrix}i_s\\u_s\end{matrix}\right]

This page uses QuickLaTeX to display formulas.