The method of nodal stresses is the calculation of electrical circuits in which the voltage values in the nodes of the circuits relative to the base node are variables. The equations are compiled on the basis of the first Kirchhoff law, which allows us to reduce the number of system equations to a value of k-1, where k is the number of chain nodes. This method is best used when the number of branches of an electric circuit is more than two. The method of nodal stresses has found application in computer programs for modeling electrical circuits, due to the simplicity of the algorithm for generating equations of nodes.
Nodal voltages are the voltages between an arbitrary reference node (in it, the potential is taken equal to zero) and each of the nodes. In the diagrams, the reference node is displayed grounded.
Consider various methods for calculating electrical circuits
The essence of this method is to solve a system of equations by which the potentials of each circuit node are determined with respect to the reference node. After that, the circuit is calculated using Ohm's law, that is, the values of the currents of all branches are determined.
Complex circuits are calculated in the following sequence:
1. A schematic diagram is drawn up, with all the elements.
2. An arbitrary reference node is assigned. Moreover, it is recommended to choose a node in which the largest number of branches converges.
3. An arbitrary direction of the currents in all branches is specified, which is indicated in the diagram.
4. To calculate the potentials of the remaining nodes in relation to the selected reference node, a system of equations is compiled.
The equalities of such a system will have the following form:
U1G11 - U2G12 - ... - UsG1s - UnG1n = ∑1EG + ∑1J
-U1G21 + U2G22 - ... - UsG2s - UnG2n = ∑2EG + ∑2J
………………………………………………………………………………….
U1Gn1 - U2Gn2 - ... - UsGns + UnGnn = ∑nEG + ∑nJ, where:
- G is the sum of the conductivity of the branches connected to the node;
- U is the value of nodal stresses;
- ∑EG 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 "+", otherwise - "-".)
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 electrical circuit consists of the nth number of nodes, it is necessary to draw up the nodal equations one less than the number of nodes. Considering 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 event that the circuit contains voltage sources, they must be replaced with equivalent
current sources. In addition, nodal equations can be written in matrix form.
5. The system of equations is solved with respect to the nodal stresses, determining their values.
6. After that, for each branch separately, according to Ohm's law, all values of the electric current in the circuit are calculated.
I = (Ua - Ub + ∑Eab) / ∑Rab, 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;
- ∑Eab is the algebraic sum of this branch;
- ∑Rab is the arithmetic sum of the resistances of this branch.
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 = (∑nEnGn + ∑nJn) / ∑mGm, where:
- ∑nEnGn is the algebraic sum of the values of the products of the EMF of the branches on the conductivity of these branches;
- ∑nJn is the algebraic sum of the values of current sources;
- ∑mGm is the arithmetic sum of the conductivities of all branches between nodes.
The nodal stress method has the following mathematical advantages: ease of calculation and a significant reduction in the number of arithmetic operations.