Kirchhoff’s Voltage, Current Law & Superposition Principle

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. This notion is illustrated by in the simple circuit in Figure below, that includes one battery and two resistors.

Note that it is irrelevant which point we choose to label as the “zero” potential: no matter what the starting point, adding all the potential gains and drops encountered throughout the complete loop will give a zero net gain.

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). For the purpose of computation, we assign positive or negative signs to currents flowing in and out of the node, respectively.
It does not matter which way we call positive, as long as we remain consistent in our definition. Then, the sum of currents into (or out of) the node is zero. This is illustrated with the simple example in Figure below, where KCL applied to the branch point proves that the current through the battery equals the sum of currents through the individual resistors.

The Superposition Principle

This principle applies to circuits with more than one voltage or current source. It states that the combined effect—that is, the voltages and currents at various locations in the circuits—from the several sources is the same as the sum of individual effects.
Knowing this allows one to consider complicated circuits in terms of simpler components and then combining the results. In power systems, the superposition principle is used to conceptualize the interactions among various generators and loads. For example, we may think of the current or power flow along a transmission link due to a “shipment” of power from one generator to one consumer, and we add to that the current resulting from separate transactions in order to obtain the total flow on that link. With voltages held fixed, the currents become synonymous with power flows, and we can add and subtract megawatt flows superimposed along various transmission links.
For a simple example where we can deal with currents and voltages explicitly,
consider the circuit in Figure below.

This circuit has two power sources. The first source, labeled S1, is a battery that functions as a voltage source, delivering 12 volts. The second source, S2, is a current source that delivers 1.5 amperes. This type of source is less familiar in power systems than in electronics; it has the property of always delivering a specific current regardless of the resistance in the circuit connected to it, while allowing the voltage at its terminals to vary (as
opposed to the more familiar voltage source that specifies terminal voltage while the current depends on resistance in the circuit). We introduce the current source here because it makes for a good illustration of the superposition principle.
Suppose we wish to predict the voltage level v and current i at the locations identified in the diagram. It is not immediately obvious what the voltages and currents at various points in this circuit should be as a result of the combination of the two sources. The superposition principle is an indispensable analytic tool here: it states that we can consider separately the voltage and current that would result from each individual source, and then simply add them together. This principle applies regardless of the circuit’s complexity or the number of power sources; it also holds true at any given instant in a circuit with time-varying sources.
In our example, the voltage v that would result from only S1—written as v(S1), indicating that v for now is only a function of S1—is determined from the relative magnitudes of the two resistances in the circuit, R1 and R2, while ignoring the presence of the current source.

Since R2 at 2Ω represents one-third of the total resistance in this simple series circuit, 2Ω + 4Ω, the voltage across R2 that we want to find is simply one-third of the total:

v(S1) = 12V.2Ω/(4Ω + 2Ω) = 12V.1/3 = 4V

Ignoring the current source technically means setting the current through it to zero, or replacing it with an open circuit. Having gotten rid of the extra circuit branch, the current i(S1) through the resistor R2 that would result from S1 alone is easy to find with Ohm’s law:

i(S1) = 12V/(4Ω + 2Ω) = 2A

Next, we ignore the voltage source, meaning that we set the voltage difference across it to zero, or replace it with a short circuit.

The current i(S2) through R2 based on the current source alone is found from the relative magnitudes of the resistances in each branch: since R2 has half the resistance of R1, twice the current will flow through it, or two thirds of the total:

i(S2) = 1.5A.4Ω/(4Ω + 2Ω) = 1.5A.2/3 = 1A

The voltage v(S2) due to the current source alone is again found via Ohm’s law:

v(S2) = 1A.2Ω = 2V

We can now superimpose the contributions from the two sources and find:

v= v(S1) + v(S2) = 4V + 2V = 6V, and

i = i(S1) + i(S2) = 2A + 1A = 3A
 
Although this simple example could have been solved without the use of superposition, the technique is vital for analyzing larger and more complex circuits, including those with time-varying sources.