KVL Loop Solver

Quick experiments

• Cancel a source and watch the current vanish. Set E₁ = 12 V, then flip its polarity to −. Net EMF falls to zero and so does the current. KVL still holds perfectly at I = 0 A — the sum around the loop is zero because there is nothing driving it.
• Add a second source that helps. Add E₂ = 6 V with + polarity next to a 12 V E₁. Net EMF becomes 18 V and the current rises accordingly — two sources in the same direction simply add.
• Add a second source that fights. Now set that same E₂ to − polarity. Net EMF drops to 6 V. This is a battery being charged by a larger supply, and the sign in the KVL equation is the whole story.
• Confirm the drops always sum to the net EMF. Add a third resistor and give the three wildly different values — say 18 Ω, 1 Ω and 1 Ω. The drops are lopsided but they still add to exactly the net EMF. KVL does not care how many elements share the loop.
• Make one resistor take almost everything. Set R₁ = 18 Ω beside a 1 Ω R₂. Nearly the whole source voltage appears across R₁, because in series the largest resistance takes the largest share. This is a voltage divider seen through KVL.

Formula reference

Kirchhoff's Voltage Law

$$\sum_{loop} V = 0$$

Every voltage around a closed loop, with sign, sums to zero.

Written as sources and drops

$$\sum E = \sum I R$$

Net driving EMF equals the total of the resistive drops.

Net EMF with opposing sources

$$E_{net} = E_1 \pm E_2 \pm E_3$$

Plus when aiding, minus when opposing.

Loop current

$$I = \frac{E_{net}}{R_1 + R_2 + \cdots + R_n}$$

One current everywhere in a single loop.

Drop across one resistor

$$V_n = I \times R_n$$

Ohm's Law applied to each element in turn.

Common mistakes

• ✗Adding all the sources regardless of polarity.

  A source wired against the loop direction subtracts. Two 6 V cells nose-to-nose give 0 V of net drive, not 12 V. Decide a loop direction first, then assign each source a sign relative to it.

• ✗Changing loop direction partway round.

  The signs are only consistent if you walk the whole loop one way. Pick clockwise or anticlockwise at the start and commit — swapping midway double-counts some elements and cancels others.

• ✗Panicking when the current comes out negative.

  That is a valid answer. It means the real current flows opposite to the direction you assumed; the magnitude is right. Reversing your assumption and re-solving gives the same number with a positive sign.

• ✗Treating a resistor's drop as fixed regardless of current.

  A resistor has no voltage across it until current flows. Its drop is I × R, so it changes the moment anything else in the loop changes. Only sources hold their voltage independently.

• ✗Applying KVL to a path that is not actually closed.

  The law is about closed loops. If your path starts and ends at different points you are computing a potential difference, not applying KVL — and there is no reason for that to be zero.

Frequently asked questions