Kirchhoff's Voltage Law

Section Overview

Twice already — in series circuits and in parallel circuits — you've seen that the voltage drops around a loop add up to the source voltage, each time noted as "a preview of Kirchhoff's voltage law." This section delivers the law itself. Kirchhoff's voltage law (KVL) says that if you travel around any closed loop in a circuit and add up every voltage rise and drop, the total is exactly zero. It's a simple, powerful bookkeeping rule — nothing more than conservation of energy applied to a loop — and it turns "the drops add up to the source" from a helpful observation into a tool you can point at any loop, known or faulty, to solve for what you can't directly see.

Why This Matters

KVL is the first of the two laws (with Kirchhoff's current law, next section) that let you analyze circuits beyond the simplest series or parallel cases. But its everyday value is diagnostic. Because every volt the source supplies must be accounted for somewhere in the loop, a voltage that has gone "missing" tells you exactly where it's being dropped — and a component reading the entire supply voltage across itself is where the loop has gone open. This is the rigorous version of the fault-finding trick from the series section: the loop rule is why the full source voltage appears across a break, and it makes the reasoning something you can trust on any loop, not just a simple string.

Required Prerequisites

• Voltage — The Pressure of Electrons — voltage as a difference and the idea of a voltage drop, which KVL sums around a loop.
• Ohm's Law — The Foundation of Circuit Analysis — used to turn each resistor's current into its voltage drop before summing.
• Series Circuits — where "the drops add up to the source" first appeared; KVL is that rule made general.
• Parallel Circuits — every branch is itself a loop with the source, and KVL is why each branch sees the full supply voltage.

Recommended Consumables

No consumables required. The exercises are pen-and-paper loop analysis plus optional measurement on a simple single-battery series loop.

Recommended Practice Hardware

• A single battery (AA or 9 V) and holder
• Two or three resistors in series, or two small bulbs, forming one loop with the battery
• Clip leads or a breadboard, and a multimeter if you have one (from Section 1.7) to measure each drop and add them up

Everything here runs from a single low-voltage battery — the same safe territory as the previous sections. No mains, no high current.

Real-World Applications

Every time a technician measures the voltage drops around a loop and checks that they account for the supply, they're using KVL — usually without naming it. It's the reasoning behind "the supply is 12 V but I only measure 9 V across the load, so where are the other 3 V going?" (answer: dropped somewhere else in the loop — a resistor, a bad connector, a partly-open switch). It's why a dead series circuit shows the full supply across the one open point. And it's the foundation of the mesh-analysis techniques used to solve any resistive network. Learn to think in loops, and a schematic becomes a set of voltage equations you can actually solve.

Common Challenges

• Sign confusion. KVL requires a consistent rule for what counts as positive and what counts as negative as you walk the loop. Beginners often flip signs partway around; the fix is to choose a direction and a convention first, then never change it mid-loop.
• Thinking KVL is a new physical fact. It isn't — it's conservation of energy. A charge carried all the way around a loop and back to its start must have the same energy it began with, so the rises and drops it experienced have to cancel. Seeing it as energy bookkeeping makes the sign rules feel natural.
• Trying to apply it before knowing the drops. KVL relates the drops; you usually still need Ohm's law to find each drop from the current and resistance first. KVL and Ohm's law work together, not instead of each other.

Safety Notes

Risk Level: Low. This section is analysis and optional low-voltage measurement on a single-battery loop — the same safe territory as the previous sections. No mains, no high-energy sources.

Professional Tips Before Starting

• Pick a walk direction (say, clockwise) and one convention (say, a rise in potential is +, a drop is −) at the start, mark it on your sketch, and hold it for the whole loop. Consistency, not the particular choice, is what makes the signs come out right.
• When a measured drop doesn't fit, don't distrust KVL — distrust the circuit. A loop that doesn't sum to zero has a voltage you haven't accounted for, and that missing voltage is usually the fault.
• For a quick check on any series loop: add every component's drop and compare to the source. Short by a few volts? Something in the loop is dropping voltage you didn't expect.

The Loop Rule

The Law: Everything Around a Loop Sums to Zero

A closed loop is any path in a circuit that starts at a point and returns to that same point without lifting your pencil. Kirchhoff's voltage law states:

The sum of all the voltage rises and drops around any closed loop is zero. (Σ V around the loop = 0)

Walk all the way around a loop, tallying the source's push as a rise and each component's voltage as a drop, and the pluses and minuses cancel exactly. Equivalently — and more usefully day to day — the drops around the loop add up to the rises:

V_source = V₁ + V₂ + V₃ + …

Why must this be true? Because voltage is energy per unit of charge, and a charge carried once around a loop returns to where it started with the energy it started with. Whatever energy the source gave it (the rises) is exactly the energy the components took back (the drops). KVL is conservation of energy, written for a loop. The source's push is called its electromotive force (EMF) — a slightly old-fashioned name for the voltage a source provides to drive charge around the circuit; despite the word "force," it is measured in volts, not in units of force.

Signs: Pick a Direction, Then Be Consistent

The one skill KVL demands is bookkeeping the signs. The recipe:

1. Draw the loop and pick a direction to walk it — clockwise or counter-clockwise, your choice.
2. Choose a convention: for example, a rise in potential (going from − to + through a source) counts as +, and a drop (going + to − across a resistor, in the direction of current) counts as −.
3. Walk the loop once, adding each voltage with its sign. Set the total to zero. Solve.

The particular convention doesn't matter — flip it and every term changes sign, so the equation is the same. What matters is using one convention for the whole loop. Get in the habit and the signs stop being a source of error.

A Worked Loop

Take a 9-volt battery driving three resistors in series: 100 Ω, 220 Ω, and 330 Ω. From the series section you know the current is the same everywhere: I = 9 ÷ (100 + 220 + 330) = 9 ÷ 650 ≈ 0.0138 A. Ohm's law gives each drop:

• V₁ = 0.0138 × 100 ≈ 1.38 V
• V₂ = 0.0138 × 220 ≈ 3.04 V
• V₃ = 0.0138 × 330 ≈ 4.55 V

Now apply KVL. Walking the loop from the battery's + terminal, the battery is a +9 V rise and each resistor is a drop:

+9 − 1.38 − 3.04 − 4.55 = +9 − 8.97 ≈ 0 ✓

The rises and drops cancel to zero (within rounding) — the drops account for the whole 9 V. That's KVL confirming exactly what the series section claimed, now as a general law rather than a special observation.

Solving for an Unknown Drop

KVL earns its keep when one voltage is unknown. Suppose the same 9 V loop has three components and you've measured two of the drops — 2.5 V and 4.0 V — but the third component is buried where you can't easily probe it. KVL gives it directly:

V_unknown = V_source − (sum of known drops) = 9 − (2.5 + 4.0) = 2.5 V

You just found a voltage you never measured, purely from the loop rule. This is the everyday power of KVL: the loop is a closed account, so any single missing entry is fixed by the requirement that the books balance.

The Fault-Finding Payoff

The series section's fault trick — the full supply voltage appears across an open — is really KVL in action. In a broken series loop no current flows, so every intact component drops V = I × R = 0. But the loop must still sum to zero: source minus drops equals zero, and if every intact drop is zero, the entire source voltage must appear across the one remaining element — the open. That's not a coincidence or a rule of thumb; it's forced by KVL. And the reasoning generalizes: whenever a loop's measured drops don't add up to the source, the shortfall is a voltage you haven't accounted for — an unexpected drop across a corroded connector, a partly-open switch, or a failing component. The missing volts point straight at the fault.

Common Mistakes

• Changing the sign convention partway around the loop. Every rise and drop must be tallied under the same rule; switching midway guarantees a wrong equation. Choose once, mark it, keep it.
• Forgetting the source is part of the loop. KVL balances the source's rise against the components' drops — leave the source out and nothing sums correctly.
• Confusing KVL with KCL. KVL is about voltages around a loop; the current law (next section) is about currents at a node. They answer different questions and shouldn't be mixed up.
• Expecting KVL alone to give currents. KVL relates voltages; you generally pair it with Ohm's law to connect those voltages to the currents and resistances that produce them.

Troubleshooting Guidance

KVL turns a loop into an account that must balance, and a balance that doesn't is a fault. Measure the voltage drops around a suspect loop and add them: if they fall short of the source, the missing voltage is being dropped somewhere you didn't expect — probe the connectors, switches, and joints for the one carrying an unexplained drop. If instead one element reads the entire supply voltage while the rest read near zero, the loop is open at that element and no current is flowing. Because KVL holds regardless of what the fault is, it gives you a reliable expectation to measure against: the drops will sum to the source in a healthy loop, so any deviation is real information pointing at a specific place.

Verification & Testing Methods

Check your understanding before moving on:

• [ ] State Kirchhoff's voltage law and explain why it is conservation of energy around a loop.
• [ ] Walk a simple series loop in a chosen direction and write its KVL equation with consistent signs.
• [ ] Given a source voltage and all but one of the drops in a loop, solve for the missing drop.
• [ ] Explain, using KVL, why the full source voltage appears across an open component in an otherwise-dead loop.

Then try the practice exercises below — pen-and-paper loop analysis, with optional measurement on a single-battery loop.

Practice Exercises

1. Confirm the loop sums to zero (10 minutes, pen and paper). A 12 V battery drives three series resistors of 200 Ω, 300 Ω, and 100 Ω. Find the current, then each voltage drop, then write the KVL equation (source rise minus the three drops) and confirm it equals zero.
2. Solve for the missing drop (5 minutes, pen and paper). In a single loop powered by a 9 V source, two components drop 3.5 V and 2.0 V. Use KVL to find the voltage across the third component. State the rule you used.
3. Diagnose with the loop rule (5 minutes, reasoning). A 5 V series loop of three resistors is dead. You measure 0 V across two of them and 5 V across the third. Use KVL to explain what has happened and which component is at fault.
4. Prove it with a meter (10 minutes, optional hardware). Build a single-battery series loop of two or three components. Measure the battery voltage, then each component's drop. Add the drops and compare to the battery voltage — confirm they match within measurement tolerance, demonstrating KVL directly.

These core ideas — the loop sums to zero, consistent signs, solving for an unknown drop, and the fault-finding payoff — are tested in the Chapter Quiz at the end of this chapter, where a score of 80% is required to continue.

Key Takeaways

• Kirchhoff's voltage law (KVL): the sum of all voltage rises and drops around any closed loop is zero — equivalently, the drops equal the source rises.
• It is conservation of energy for circuits: a charge carried around a loop returns with the energy it started with, so rises and drops cancel.
• Apply it by choosing a walk direction and a sign convention and holding both for the whole loop; the particular choice doesn't matter, consistency does.
• The source's driving voltage is its electromotive force (EMF), measured in volts despite the name.
• KVL lets you solve for one unknown voltage in a loop: V_unknown = V_source − (sum of known drops).
• A loop whose measured drops don't sum to the source has a fault; a full-supply reading across one element marks where the loop is open — the rigorous version of the series fault trick.

Skills Learned

• You can now state Kirchhoff's voltage law and explain it as energy conservation around a loop.
• You can now walk a series loop with a consistent sign convention and write its KVL equation.
• You can now solve for an unknown voltage drop when the source and the other drops are known.
• You can now use the loop rule to localize a fault, including reading a full-supply drop as an open.
• You can now recognize KVL as the general law behind the "drops add up to the source" rule from the series and parallel sections.

Glossary Additions

• Kirchhoff's voltage law — the rule that the sum of all voltage rises and drops around any closed loop in a circuit is zero; equivalently, the voltage drops around a loop equal the source voltage. It is a statement of energy conservation.
• closed loop — any path in a circuit that begins and ends at the same point without retracing, around which Kirchhoff's voltage law is applied.
• electromotive force — the voltage a source (such as a battery) provides to drive charge around a circuit; abbreviated EMF and measured in volts, despite the historical name containing "force."

Suggested Next Sections

Must read next:

• Kirchhoff's Current Law — the companion law, which does for currents at a node what KVL does for voltages around a loop; together they let you analyze any resistive network.

Recommended:

• Series Circuits — revisit its "drops add up to the source" rule, now that you know it as a consequence of KVL.
• Ohm's Law — The Foundation of Circuit Analysis — KVL and Ohm's law are used together: Ohm's law gives each drop, KVL sums them.