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.

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