Anything we learn about the behavior of a circuit from the connections among its elements can be understood in terms of two constraints known as Kirchhoff’s laws (after the 19th-century German physicist Gustav Robert Kirchhoff). Specifically, they are Kirchhoff’s voltage law and Kirchhoff’s current law.

Their application in circuit analysis is ubiquitous, sometimes so obvious as to be done unconsciously, and sometimes surprisingly powerful. While Kirchhoff’s laws are ultimately just concise statements about the basic physical properties of electricity, when applied to intricate circuits with many connections, they turn into sets of equations that organize our knowledge about the circuit in an extremely elegant and convenient fashion.

Kirchhoff’s Voltage Law
Kirchhoff’s voltage law (often abbreviated KVL) states that the sum of voltages around any closed loop in a circuit must be zero. In essence, this law expresses the basic properties that are inherent in the definition of the term “voltage” or “electric potential.”

Specifically, it means that we can definitively associate a potential with a particular point that does not depend on the path by which a charge might get there. This also implies that if there are three points (A, B, and C) and we know the potential differences between two pairings (between A and B and between B and C), this determines the third relationship (between A and C).

Without thinking in such abstract and general terms, we apply this principle when we move from one point to another along a circuit by adding the potential differences or voltages along the way, so as to express the cumulative voltage between the initial and final point.

Finally, when we go all the way around a closed loop, the initial and final point are the same, and therefore must be at the same potential: a zero difference in all.

The analogy of flowing water comes in handy. Here, the voltage at any given point corresponds to the elevation. A closed loop of an electric circuit corresponds to a closed system like a water fountain. The voltage “rise” is a power source—say, a battery—that corresponds to the pump.

From the top of the fountain, the water then flows down, maybe from one ledge to another, losing elevation along the way and ending up again at the bottom. Analogously, the electric current flows “down” in voltage, maybe across several distinct steps or resistors, to finish at the “bottom” end of the battery.

Kirchhoff’s Current Law
Kirchhoff’s current law (KCL) states that the currents entering and leaving any branch point or node in the circuit must add up to zero. This follows directly from the conservation property: electric charge is neither created nor destroyed, nor is it “stored” (in appreciable quantity) within our wires, so that all the charge that flows into any junction must also flow out.
Thus, if three wires connect at one point, and we know the current in two of them, they determine the current in the third.

Again, the analogy of flowing water helps make this more obvious. At a point where three pipes are connected, the amount of water flowing in must equal the amount flowing out (unless there is a leak).

Despite their simple and intuitive nature, the fundamental importance of Kirchhoff’s laws cannot be overemphasized. They lie at the heart of the interdependence of the different parts and branches of power systems: whenever two points are electrically connected, their voltages and the currents through them must obey KVL and KCL, whether this is operationally and economically desirable or not.

For example, managing transmission constraints in power markets is complicated by the fact that the flow on any one line cannot be changed independently of others. Thus the engineer’s response to the economist’s lamentation of how hard it is to manage power transmission: “Blame Kirchhoff.”

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