Circuit Formulation

Circuit formulation is the process of deriving from the circuit description a set of linear equations that can be solved to evaluate the desired voltages and currents.

In linear DC analysis - the only type of analysis that concerns me for now - the circuit needs to be formulated only once.

There exists many formulation techniques. Some are much better than the others. Examples of formulation techniques are:

• The basic nodal formulation method which cannot handle circuit elements that are not current sources and do not have an admittance description.
• The tableau formulation method which produces unnecessarily large matrices.

The formulation method I will present here is a variation of the modified nodal formulation. It is well explained in "Electronic Circuit & System Simulation Methods" (chapter 2) and in "Computer Methods for Circuit Analysis and Design" (section 4.4).
The method is better than the basic nodal formulation because it can handle all linear circuit elements. It is also better than the tableau formulation method because using the tableau formulation you get a solution vector containing all branch currents, all branch voltages, and all nodal voltages. Obviously, these quantities can be easily derived from each other.
The version of the modified nodal formulation I will present next gives much smaller matrices. The solution vector will contain only the nodal voltages and the currents through the elements that are not current sources and do not have an admittance description.

Formulation Steps

Step 1:
Number the circuit nodes consecutively. It is a good idea to give the ground node the number zero to facilitate the removal (or the neglect) of its nodal equation and of the coefficients of its voltage.

Step 2:
Number the elements that do not have an admittance description starting from "last node number" + 1. The number of the element will be the index of its first excess equation and also the index of its first excess unknown (elements without an admittance description introduce excess equations and excess unknowns into the set of equations).
Some elements have more than one excess equation (excess unknown), so the elements may not be numbered consecutively.

Step 3:
Scan all the circuit elements to add the element contributions to the set of linear equations. The contribution of an element is represented by a small matrix that is sometimes called the matrix stamp. The following figure shows the matrix stamps of the resistor, the independent current source, and the independent voltage source.



The complete list of ideal element stamps can be found in "Computer Methods for Circuit Analysis and Design" pages 116 and 117.

The independent voltage source is an example of an element that do not have an admittance description, it adds an excess equation and an excess current to the set of linear equations. In the matrix stamp of the independent voltage source, 'm' is the number of the element.

You stamp in an element contribution by adding the elements of its matrix stamp to the corresponding elements of the coefficients matrix and/or the RHS vector.