Kirchhoff's voltage law (KVL)
Around a closed path in the lumped-circuit model, the signed voltage changes sum to zero.
- State Kirchhoff's voltage law (KVL) with explicit traversal signs.
- Write a loop equation from voltage references.
- Combine Kirchhoff's voltage law with Ohm's law.
- Check a solved loop by energy/potential consistency.
If you start at one circuit node, walk around a closed path and return to the same node, can your final potential be different from the potential you started with?
In the ordinary lumped direct-current (DC) model used here, no. Kirchhoff's voltage law (KVL)—KVL from the English name Kirchhoff's Voltage Law—is the algebraic statement that the signed voltage changes around that closed path return you to the starting potential.
03.1KVL can be seen as a telescoping sum of node potentials
Consider a closed sequence of nodes A→B→C→…→A. Write each encountered voltage change as final potential minus initial potential:
Every intermediate node potential appears once positive and once negative, so the sum cancels:
This is the cleanest DC interpretation of KVL: a closed walk through scalar potential differences must return to the potential from which it began.
03.2How to assign the sign of each voltage term
Choose one traversal direction around the loop. Then read each voltage reference exactly as defined:
- cross a referenced voltage from its − terminal toward its + terminal: add +V;
- cross from + toward −: add −V.
With the AoN voltage-arrow convention, moving in the same direction as the voltage arrow is a positive potential change; moving against it is a negative change.
03.3First loop: one 9 V source and one 3 kΩ resistor
Figure 3.1 — A source–resistor loop
Topology-only view. The closed path leaves one node, crosses the source and resistor, and returns to the same node.
Traverse the loop in the direction that crosses the ideal source from its negative terminal to its positive terminal, then the resistor from its positive reference terminal to its negative reference terminal. The signed loop equation is
Therefore
Using Ohm's law for R=3 kΩ:
KVL supplied the network constraint; Ohm's law supplied the element relation.
03.4Reverse the traversal: the physics must not change
Walk around the same loop in the opposite direction. Every crossing sign reverses:
This equation is simply −1 times the previous equation and therefore gives exactly the same VR=9 V.
03.5Two resistors: show every intermediate voltage
A 12 V ideal source drives R1=1 kΩ and R2=2 kΩ in one unbranched loop. The same current I passes through both resistors. KVL gives
Use Ohm's law only after the signed voltage equation is clear:
Then
The last line is an independent numerical closure of the loop equation.
03.6KVL is about voltage changes, not “energy lost in every element”
A resistor under passive operation absorbs energy, but a capacitor or inductor introduced later may temporarily return stored energy, and an active device may either absorb or deliver power. KVL does not require every element to dissipate energy.
KVL says the signed terminal-voltage changes around the modelled loop close. Power and energy require the additional current reference and P=VI bookkeeping.
03.7Important limit: a time-varying magnetic flux changes the field picture
The zero-sum scalar-potential interpretation is appropriate for the lumped DC circuits studied here. In a general electromagnetic situation, Faraday's law gives
so a changing magnetic flux can produce a nonzero closed-loop electric-field integral. Circuit theory represents inductive effects with appropriate elements when the lumped approximation remains valid; at sufficiently high frequencies or large dimensions, distributed electromagnetic models become necessary.
03.8Practice for mastery
- A loop traversal encounters +15 V, then −4 V, then −Vx. Find Vx.
- A 10 V source drives 2 kΩ and 3 kΩ in series. Find I and both resistor voltages, then close KVL numerically.
- Write the equation from problem 2 using the opposite traversal direction. What changes?
- Why is “a resistor always gets a minus sign in KVL” an unsafe rule?
- What physical effect warns us that the naive electrostatic closed-loop sum is not universally valid?
Solutions
1. 15−4−Vx=0 ⇒ Vx=11 V.
2. I=10/(5 kΩ)=2 mA; drops are 4 V and 6 V; 10−4−6=0.
3. Every term changes sign: −10+4+6=0. The physical solution is unchanged.
4. The sign is determined by traversal relative to the voltage reference, not by the component category.
5. Time-varying magnetic flux linked with the loop, described by Faraday's law.
KCL and KVL are now physically motivated and calculationally usable. The next chapter focuses on sign discipline so larger circuits do not fail because of inconsistent reference bookkeeping.