In 1845, German physicist Gustav Kirchhoff first described two laws that became central to electrical engineering. The laws were generalized from the work of Georg Ohm. The laws can also be derived from Maxwell’s equations, but were developed prior to Maxwell’s work.
- Kirchhoff’s Current Law (KCL):
The algebraic sum of the currents entering any node is zero
This law is also called Node analysis
This Law is used in circuit analysis to define relationships between currents flowing in branches of the circuit. For example, in figure above the currents flowing in the four branches connected to the node have been defined as I1, I2, I3, I4 and Kirchhoff’s Current Law allows us to write down an equation relating these currents. Looking closely at figure above, we see that two of the currents (I2, I3) are flowing towards the node, while the other two currents (I1, I4) are flowing outwards. The ‘algebraic sum’ needs to take account of this difference in relative direction.
To apply Kirchhoff’s Current Law rigorously, we must first make an arbitrary choice of positive current direction. Suppose currents flowing in to the node (I2, I3) are treated as positive contributions to the algebraic sum (and conversely currents flowing from the node are treated as negative contributions), then the algebraic sum of currents would be written: - I1 + I2 + I3 - I4 , and according to Kirchhoff’s Current Law this algebraic sum is equal to zero:
- I1 + I2 + I3 - I4 = 0
- Kirchhoff’s Voltage Law (KVL):
The algebraic sum of all voltages taken around a closed loop in a circuit is zero.
This law is also called
Loop analysis
The figure shows a circuit loop, which is part of a larger circuit. The loop involves four nodes, ABCD, between which are connected four components. We must recognise that the direction of voltages matters when using Kirchhoff’s Voltage Law.
In this case the four components are resistances, but Kirchhoff’s Voltage Law can be applied no matter what components are connected in the closed circuit loop. The voltages across the four resistances comprising the circuit loop have been defined as V1, V2, V3,V4 and Kirchhoff’s Voltage Law allows us to write down an equationrelating these voltages.
If we think about travelling around the closed circuit loop in any direction, we note that the four voltages will be encountered in sequence. Two of the voltage arrows will point in the direction of travel and two will oppose the travel. The ‘algebraic sum’ of voltages needs to take account of this difference in relative direction.
To apply Kirchhoff’s Voltage Law correctly, we must make arbitrary choices about the direction of travel around the closed circuit loop and the contribution which the separate voltages make to the algebraic sum around the closed circuit loop. Suppose we travel around the loop in Fig. 2.2 in the clockwise direction (ABCD) and that voltages opposite to the direction of travel make a positive contribution to the algebraic sum. In travelling from A to B the voltage V1 is encountered and it is in a direction which is opposite to the travel. Therefore, V1 is a positive contribution to the algebraic sum. The same comment is true of V2, which is met when proceeding from B to C. However, travelling from C to D and back to A, the voltages V3 and V4 are encountered and in both cases the voltages are in the same direction as the travel, giving a negative contribution to the algebraic sum.
Expressed mathematically, the algebraic sum of voltages around the closed loop ABCD is:
+ V1 + V2 - V3 - V4
and Kirchhoff’s Voltage Law states that this sum is equal to zero:
+ V1 + V2 - V3 - V4 = 0
Recall the Ohm Law that V = IR
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