For complex circuit computations, the following two laws first started by Gustav R. Kirchhoff 1824-1887 are indispensable.
Kirchhoff’s current law
The sum of the currents entering a junction is equal to the sum of the currents leaving the junction.
If the current towards a junction is considered positive and those away from the same junction negative, then this law state that the same algebraic sum of all currents meeting at a common junction is zero.
summation currents entering is equal to summation currents leaving shown as fig. below.
Kirchhoff`s voltage law
It states as follows. The sum of the e.m.fs rises of potential around any closed loop of a circuit equals the sum of the potential drops in that loop. Considering a rise of potential as positive+ and a drop in potential as negative, the algebraic sum of potential differences in Kirchhoff voltage law around a closed loop of a circuit is zero.
summation E minus summation IR drops is equal to 0 around the closed loop.
summation potential rises are equal to summation potential drops.
To apply this law in practice, assume an arbitrary current direction for each branch current. The end of the resistor through which the current enters, is then positive, with respect to the other end. If the solution for the current being solved turns out negative, then the direction of that current is opposite to the direction assumed. In tracing through any single circuit, whether it is by itself or a part of a network, the following rules must be applied. A voltage drop exists when tracing through resistance with or in the same direction as the current or through a battery or against their voltage, that is from positive to negative.
Application of Kirchhoff law
Kirchhoff’s law may be employed in the following methods of solving networks.
- Branch-current method
- Maxwell’s loop or mesh current method
- Nodal voltage method
Branch current method
For a multi-loop circuit, the following procedure is adopted for writing equations. Assume the current in different branches of the network. Write down the smallest number of voltage drop loop equations so as to include all circuit elements, these loop equations are independent. If there are n nodes of three or more elements in a circuit, then write the n-1 equation as per current.
The assumption made about the directions of the currents initially is arbitrary. in case the actual direction is opposite to the assumed one, it will be reflected as a negative value for that current.
Maxwell loop or mesh current method
The method of loop or mesh currents is generally used in solving networks having some degree of complexity. Such a degree of complexity already begins for a network of three meshes. It might even be convenient at times to use the method of loop or mesh currents for solving a two-mesh circuit.
The mesh-current method is preferred to the general branch-current method because the unknown in the initial stage of a solving network is equal to the number of meshes, i.e the mesh currents. The necessity of writing the node-current equations., as done in the general or branch-current method where branch currents are used is obviated. There are as many mesh-voltage equations as there are independent loops or mesh, currents. Hence, the M-mesh currents are obtained by solving the M-mesh voltages or loop equation for M unknown. After solving for the mesh currents, only the matter of resolving the confluent mesh currents into the respective branch currents by very simple algebraic manipulations is required.
The method eliminates a great deal of tedious work involved in the branch-current method and is best suited when energy sources are voltage sources rather than current sources. This method can be used only for planer circuits.
Assume the smallest number of mesh currents so that at least one mesh current links every element. As a matter of convenience, all mesh currents are assumed to have a clockwise direction.
The number of mesh currents is equal to the number of meshes in the circuit. For each mesh write the Kirchhoff voltage law equation. Where more than one mesh current flows through an element, the algebraic sum of currents should be used. The algebraic sum of mesh currents may be the sum of the difference of the currents flowing through the element depending on the direction of mesh currents.
Nodal voltage method
Under this method, the following procedure is adopted.
- Assume the voltage of the different independent nodes. Write the equation for each mode as per Kirchhoff’s current law.
- Solve the equation to get the node voltage.
- Calculate the branch currents from the values of node voltages.
- The node voltage is multiplied by the sum of all conductance connected to that anode. This term is positive.
- The node voltage at the older end of each branch connected to this node is multiplied by the conductance of the branch. These terms are negative.
- In this method of solving a network the number of equations required for the solution is one less than the nodal analysis yield similar solutions.
- In general, the nodal analysis yields a similar solution.
- The nodal method is very suitable for computer worknder this method the following procedure is adopted.
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