Kirchhoff's Voltage and Current Laws

Kirchhoff's Laws are basic tools used in the analysis of circuits.

Kirchhoff's Voltage Law

If we look at any loop in a circuit then the sum of the voltages around the loop will be zero.

So, if we were to look at the seven components in the section of circuit here. Each one has a voltage across it and we have chosen to define them in the directions shown. The direction we have chosen seems to assume that the voltage difference across the component is such that the more positive end is where the arrowhead is.

The direction that we choose is actually not terribly important - if we have actually defined the direction such that the more negative voltage is where the arrowhead is then the value of the voltage that we have defined in this direction will be negative. This is the same as saying that if we had turned the arrow round then the voltage difference would now be positive. What is very important, however, is that once the voltages (and currents) are defined in a particular direction then we must treat them consistently.

So in the case of Kirchhoff's voltage law, we said that the sum off the voltages around a loop must be zero. We can show this by looking at the left-hand loop. Let's add up the voltages - travelling clockwise around the loop (again the direction is not important but consistency is).

If we encounter a voltage arrow in the direction of travel, then we will add it to the sum; if the arrow is against the direction of travel then it must be subtracted.

Figure 4

So, for the left-hand loop:
V1+V2–V6–V4=0

for the right-hand loop:
V6+V3–V7–V5=0

and for the outer loop:
V1+V2+V3–V7 –V5–V4=0

The law is really common sense. Like climbing over the surface of a hill – it doesn’t matter what ups and downs you encounter on your climb – when you get back to the same point on the hill you will be at the same height as when you started.


Gustav Robert Kirchhoff, 1824-1887
(see: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Kirchhoff.html)