Kirchoff's Laws

Kirchoff’s Voltage Law

Kirchoff’s Voltage Law (KVL) states that the algebraic sum of the voltages across any set of

branches in a closed loop is zero. i.e.;

Below is a single loop circuit. The KVL computation is expressed graphically in that voltages

around a loop are summed up by traversing (figuratively walking around) the loop.

The KVL equation is obtained by traversing a circuit loop in either direction and writing down unchanged the voltage of each element whose “+” terminal is entered first and writing down the negative of every element’s voltage where the minus sign is first met. The loop must start and end at the same point. It does not matter where you start on the loop. Note that a current direction must have been assumed. The assumed current creates a voltage across each resistor and fixes the position of the “+” and “-” signs so that the passive sign convention is obeyed. The assumed current direction and polarity of the voltage across each resistor must be in agreement with the passive sign convention for KVL analysis to work.

The voltages in the loop may be summed in either direction. It makes no difference except to change all the signs in the resulting equation. Mathematically speaking, its as if the KVL equation is multiplied by -1. See the illustration below.

Note that a current direction must have been assumed. The assumed current creates a voltage across each resistor and fixes the position of the “+” and “-” signs so that the passive sign convention is obeyed. The assumed current direction and polarity of the voltage across each resistor must be in agreement with the passive sign convention for KVL analysis to work.

The case on the right above will obviously result in negative result for the current. This is correct considering the current arrow is pointing in the opposite direction.

Kirchoff’s Current Law

The algebraic sum of all currents entering and leaving a node must equal zero

S (Entering Currents) = S (Leaving Currents)

As a direct consequence of the conservation of charge, namely charge can neither be created nor destroyed, the node, being of negligible physical size, holds no charge. For instance, referring to figure 1.6, the sum of ,  and must equal zero.


 
Figure 1.6: Kirchhoff's current law 




Formally, KCL states that the algebraic sum of the currents in all the branches that converge in a common node is equal to zero. In mathematical form, for n branches converging into a node, KCL states that where  is the current flowing in the kth branch and its direction is assumed to be pointing towards the node.