Kirchhoff's Voltage Law: Single-Loop Circuits

The Physical Basis of KVL: Conservation of Energy

Kirchhoff's Voltage Law states that the algebraic sum of all the voltages around any closed loop in a circuit is equal to zero.

This law is a direct consequence of the conservation of energy. Imagine moving a charge around a closed loop in a circuit. When you return to your starting point, the charge must have the same potential energy it started with. Therefore, any energy gained by the charge as it passed through sources (like batteries) must be exactly balanced by the energy lost as it passed through passive components (like resistors).

Since voltage is a measure of electric potential energy per unit charge ($V=E/q$), the total change in voltage around the loop must be zero.

A fantastic analogy for this is hiking on a mountain. If you start at a certain point, climb up, then walk along a ridge, and finally descend back to your exact starting point, your net change in altitude is zero. The "voltage rises" from a battery are like climbing up, and the "voltage drops" across resistors are like descending.

The following article provides an excellent explanation of this analogy and formally introduces KVL.

Sign Conventions for Applying KVL

To use the equation $\sum V=0$, we need a consistent system for determining whether a voltage in the loop is positive (a rise) or negative (a drop). This depends on two things: the type of component and the direction we "walk" around the loop.

First, you must assume a direction for the current flow. For a simple loop, let's assume it flows clockwise. This current direction dictates the polarity of voltage across resistors.

1. Voltage Sources (Active Elements): A voltage source, like a battery, provides energy. If you traverse the loop from the negative terminal to the positive terminal, you are moving to a higher potential. This is a voltage rise and is considered positive (+) in the KVL sum. If you traverse from positive to negative, it's a drop, considered negative.

2. Resistors (Passive Elements): A resistor dissipates energy. Current always flows through a resistor from a higher potential to a lower potential.

   • Therefore, if you traverse the loop in the same direction as the current, you are moving from high to low potential. This is a voltage drop and is considered negative (-) in the KVL sum.
   • If you were to traverse the loop against the current, you would be moving from low to high potential. This would be treated as a voltage rise (+).

The video below explains these conventions with excellent visual aids and analogies.

Applying KVL to a Single-Loop Circuit

Now let's combine KVL, Ohm's Law, and our sign conventions to solve a circuit. Consider the simple series circuit shown below.

Our goal is usually to find the current flowing in the loop or the voltage drop across each resistor. Here is the step-by-step process:

1. Assume a current direction. Let's assume the current $I$ flows clockwise, leaving the positive terminal of the source $V_1$.

2. Mark voltage drop polarities. Based on the clockwise current, the side of each resistor that the current enters is marked + and the side it exits is marked -.

3. Apply KVL. Choose a starting point (e.g., the negative terminal of the source) and traverse the loop clockwise. Add up the voltages:

   • Go through the source $V_1$ from - to +: This is a voltage rise, so we add $+V_1$.
   • Go through resistor $V_2$ from + to -: This is a voltage drop, so we add $-V_2$.
   • Go through resistor $V_3$ from + to -: Another drop, so we add $-V_3$.
   • Go through resistor $V_4$ from + to -: Another drop, so we add $-V_4$.

   The KVL equation is: $V_1-V_2-V_3-V_4=0$.

4. Use Ohm's Law. We know that the voltage drop across any resistor is $V=IR$. We can substitute this into our KVL equation:

   $$V_1-(I\cdot R_2)-(I\cdot R_3)-(I\cdot R_4)=0$$

   Note that the components in the image are labeled $V_2,V_3,V_4$ but they are resistors. If we knew their resistances ($R_2,R_3,R_4$), we could solve for the single unknown current, $I$.

The following worked example from the "Electronics Tutorials" website provides a clear, step-by-step calculation that solidifies this process.

KVL with Multiple Sources

Single-loop circuits can also contain more than one voltage source. When this happens, the sources can either "aid" each other (if they try to push current in the same direction) or "oppose" each other. The KVL method remains exactly the same.

In the circuit above, both $V_1$ and $V_2$ are oriented to push current clockwise. They are aiding each other. If one battery were flipped, they would be in opposition, and the larger voltage would determine the direction of the current.

The following video provides excellent worked examples for circuits with multiple sources, including cases where they aid and oppose.

Conclusion

In this lesson, we have delved into Kirchhoff's Voltage Law, the second of the two cornerstones of circuit analysis. You now understand its foundation in energy conservation and how to apply it to practical single-loop circuits.

Here are the main takeaways:

• Kirchhoff's Voltage Law (KVL) states that the algebraic sum of voltages around any closed loop is zero ($\sum V=0$), which is a consequence of the conservation of energy.
• To apply KVL, you must use a consistent sign convention. Traversing a source from - to + is a voltage rise (+V), while traversing a resistor in the direction of current is a voltage drop (-V).
• The process for analyzing a single-loop circuit involves assuming a current direction, marking voltage polarities, writing the KVL equation, and using Ohm's Law ($V=IR$) to solve for the unknown current.
• KVL applies equally to circuits with multiple voltage sources, whether they are aiding or opposing one another.

In our next lesson, we will see how KVL can be used to derive a very convenient shortcut for analyzing series circuits: the concept of equivalent resistance. This will be our first step toward simplifying more complex networks.