Kirchhoff’s Rules

ABSTRACT

A resistor network can be reduced to a single resistor, using the rules for parallel and series circuits. While this is true, for more complex circuits it can be hard, even impossible, to calculate the equivalent resistance using such simple formulas. In those cases we will have to use the more fundamental rules that are discovered by Kirchhoff.

Kirchhoff’s rules are simple statements expressing fundamental physics laws, conservation of charge and conservation of energy.

Analysis is performed by reducing the circuit to loops and nodes. These rules are:

  1. Kirchhoff’s Voltage Rule, KVL (also called the loop law)
  2. The sum of the voltages, potential differences, around any loop of the circuit is zero. This property of Kirchhoff’s law is commonly known as conservation of energy.

 

In the above figure, the voltage will drop in all directions (positive of negative) and goes back to the similar point when you begin at any point of the loop and then continue in the same direction.  It is very important to maintain the direction either counterclockwise or clockwise; otherwise the final value of voltage will not be equal to zero. The voltage law can is applied in analyzing of circuits that are in series.

  1. Kirchhoff’s Current Rule, KCL (also called the junction or node law)
  2. The sum of the current passing through a junction is zero. This property of Kirchhoff law is mostly called as conservation of charge where, I (exit) + I (enter) = 0.

 

In the figure above, currents I1, I2 and I3 that are entering the node are considered positive; also, the currents I4 and I5 which are exiting the nodes are considered negative values. This can be expressed in the form of the following equation:

I1 + I2 + I3 – I4 – I5 = 0

A node is a junction or a connection of two or more current-carrying routes like cables and other components. Kirchhoff’s current law can be applied in analyzing of parallel circuits.

EQUIPMENT

  • Analyzed the topology of the circuit. This was done by identifying and listing all loops and junctions and using letters provided as identifiers.

 

 

 

 

 

 

 

  • Assigned labels for currents. We reproduced the current diagram showing the currents with directions.
  • Wrote KVL and KCL equations for all loops and junctions in terms of inputs = outputs. The equations are in the form of Kirchhoff’s rules. For loops, the inputs were the power supplies, batteries, and the outputs were the resistors which were dissipating energy. For junctions, the inputs were the currents flowing into a node and the outputs were the currents flowing out of a node.
  • Built the circuit in the simulation.
  • Used the meters to measure the current through each branch of the circuit.

Adding a small amount of resistivity makes your results more ‘real’ because low levels of resistivity allows electric current to flow easily.

DATA

 

Kirchhoff’s Voltage Law (KVL);

Loop AFEB: Va = I1R1 + I2R2

Loop CDEB: Vb = I3R3 + I2R2

Kirchhoff’s Current Law (KCL);

At node B: I2 = I1 + 13

At node E: I1 + I3 = I2

CONCLUSION

From our experiment, input voltage of loop AFEB and CDEB is 15V and 25V and the output voltage is 14.91V and 24.91V respectively. Current I1 is 0.15A, I2 is 0.43A and I3 is 0.28A. This shows that Kirchhoff’s Voltage Law for loop AFEB and CDEB is satisfied though there is a percentage error of 0.015% and 0.09% respectively in both cases. Also, Kirchhoff’s Current Law has been satisfied where I2 is the summation of 11 and 13 at nodes B and E.

If R4 was places in parallel with R2, the current in branch R2 would go down. This is because when resistors are connected in parallel they have the same potential difference across then. Hence using ohms law current across each resistor, the supply current will be equal to the sum of the currents through each resistor therefore decrease in current in R2.

If R4 was placed in parallel with R3, the current across branch R2 will go up. This is because the current flowing across R3 will decrease as another value of current will be passing across R4.  Therefore due to increase in current across R4 and R3, current across R2 will rise so as to satisfy KCL.

Kirchhoff’s rules are expressions of fundamental physical laws. Kirchhoff’s Voltage Law expresses a simplification of Faraday’s law of induction, which is based on the assumption that no fluctuating magnetic field is present within a closed loop. Kirchhoff’s Current Law expresses ohms law which states that the current through a resistor is directly proportional to current that is flowing through the resistance.

Kirchhoff’s Current law is an application of the principle of conservation of electric charge. That is the current is flow of charge per time, and if current that which flows into a point in a circuit is constant then it must equal to that which flows out of it. Kirchhoff’s Voltage law is based on the principle of conservation of energy. That is from the principle that energy is neither created nor destroyed whereby in a circuit sum of electrical potential differences (voltages) around a closed network is equal to zero. This can also be termed as the sum of the electromotive force (emf) values in an enclosed loop is equal to the sum of the potential voltage drops in that loop (that may come from resistors).

 

 

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