Theoretical Foundations of Electrical Engineering: Nodal Stress Method

The method of nodal stresses is the calculation of electrical circuits in which the variables are the stress values at the nodes of the chains relative to the basic node. The equations are compiled on the basis of Kirchhoff's first law, which allows us to reduce the number of system equations to k-1, where k is the number of nodes in the chain. This method is best used when the number of branches of the electrical circuit is greater than two. The method of nodal stresses has found application in computer simulation programs of electrical circuits, due to the simplicity of the algorithm for forming knot equations.

Nodal stresses are the voltages between an arbitrary reference node (in which the potential is assumed to be zero) and each of the nodes. On the diagrams, the reference node is displayed as grounded.

Consider various methods for calculating electrical circuits

The essence of this method is to solve a system of equations by means of which the potentials of each circuit node are determined with respect to the reference node. After this, the circuits are calculated using Ohm's law, that is, the values of the currents of all branches are determined.

Calculation of complex chains is carried out in the following sequence:

1. A schematic diagram is drawn up, with all the elements.

2. An arbitrary reference node is assigned. And it is recommended to choose such a node, in which the largest number of branches converge.

3. An arbitrary direction of currents in all branches is assigned , which is indicated in the diagram.

4. To calculate the potentials of the remaining nodes with respect to the selected reference node, a system of equations is compiled.

Equalities of such a system will have the following form:

U1G11 - U2G12 - ... - UsG1s - UnG1n = Σ1EG + Σ1J

-U1G21 + U2G22 - ... - UsG2s - UnG2n = Σ2EG + Σ2J

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U1Gn1 - U2Gn2 - ... - UsGns + UnGnn = ΣnEG + ΣnJ, where:

• G is the sum of the conductances of the branches connected to the node;
• U - the value of node voltages;
• ΣEG is the algebraic sum of the values of the products of the emf of the branches that are adjacent to the node, on their conductivity. (In the case when the EMF acts in the direction of the node, then the product is assigned the sign "+", in the opposite case - "-".)

The system of equations described above makes it easy to calculate the desired values of nodal stresses. It has a name - a system of nodal equations. In the case when a complex electric circuit consists of n-th number of nodes, it is necessary to compile nodal equations one less than the number of nodes. Given that all equations are written on the basis of the first Kirchhoff law, the calculated circuit must contain exclusively independent sources of electric current. In the case where the circuit contains voltage sources, they must be replaced by equivalent current sources. In addition, the nodal equations can be written in matrix form.

5. The system of equations is solved with respect to nodal stresses, determining their values.

6. After that, for each branch, all the values of the electric current in the circuit are calculated separately according to Ohm's law.

I = (Ua - Ub + ΣEab) / ΣRab, where:

• I is the current value of the branch of the circuit;
• Ua is the potential of node a;
• Ub is the potential of node b;
• ΣEab is the algebraic sum of a given branch;
• ΣRab is the arithmetic sum of the resistances of a given branch.

The nodal stress method for circuits consisting of two nodes

When calculating electrical circuits that contain only two nodes, the system of equations will consist of one equation from which it is possible to directly calculate the value of the node voltage:

U = (ΣnEnGn + ΣnJn) / ΣmGm, where:

• ΣnEnGn is the algebraic sum of the values of the products of the emf of the branches on the conductivity of these branches;
• ΣnJn is the algebraic sum of the values of current sources;
• ΣmGm is the arithmetic sum of the conductivities of all branches between nodes.

The nodal stress method has the following mathematical advantages: convenience of calculations and a significant reduction in the number of arithmetic operations.