Loop Rule (Voltage Law)

Kirchhoff's Voltage Law

Kirchhoff's Voltage Law (KVL) describes how potential differences behave around a closed loop in an electrical circuit.

It states:

The algebraic sum of all potential differences around any closed loop is zero.

In simple terms:

total voltage supplied = total voltage dropped

For a simple circuit:

V_supply = V₁ + V₂ + V₃ + ...

The more general form is:

ΣV = 0

Kirchhoff's Voltage Law is also called:

• Kirchhoff's Second Law
• the Loop Rule
• the Voltage Law

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What Is a Closed Loop?

A closed loop is any continuous path through a circuit that:

• begins at one point
• follows connected components
• eventually returns to the starting point

A simple battery-and-resistor circuit contains one loop.

More complicated circuits may contain several different loops.

Kirchhoff's Voltage Law can be applied to each closed loop separately.

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Conservation of Energy

Kirchhoff's Voltage Law is based on the conservation of energy.

Energy cannot be created or destroyed.

A battery or power supply transfers energy to electric charges.

Circuit components then transfer this electrical energy into other forms such as:

• thermal energy
• light
• sound
• mechanical energy

After traveling completely around a closed loop, the net energy change of each unit of charge must be zero.

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Voltage as Energy per Charge

Potential difference is defined as:

V = E/Q

where:

• V = potential difference (V)
• E = energy transferred (J)
• Q = charge (C)

Therefore:

1 V = 1 J/C

A 12 V battery supplies:

12 J of energy per coulomb of charge

If that charge travels through components that transfer:

4 J/C

and:

8 J/C

then:

4 + 8 = 12 J/C

The energy supplied equals the energy transferred.

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Voltage Rise and Voltage Drop

As we move around a circuit, potential can either increase or decrease.

Across a power source:

voltage rise

Across a resistor or other load:

voltage drop

For example:

battery: +12 V

resistor 1: −4 V

resistor 2: −8 V

Around the complete loop:

12 − 4 − 8 = 0

Therefore:

ΣV = 0

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A Simple One-Loop Circuit

Consider a 12 V battery connected to two series resistors.

Suppose:

V₁ = 5 V

V₂ = 7 V

Kirchhoff's Voltage Law gives:

12 − 5 − 7 = 0

Therefore:

12 V = 5 V + 7 V

The energy supplied by the battery equals the energy transferred by the two resistors.

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Calculating an Unknown Voltage

Suppose a 20 V power supply is connected to three components.

The voltage drops are:

V₁ = 5 V

V₂ = 7 V

V₃ = ?

Use KVL:

V_supply = V₁ + V₂ + V₃

20 = 5 + 7 + V₃

20 = 12 + V₃

Therefore:

V₃ = 8 V

Check:

5 + 7 + 8 = 20 V

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Another Unknown Voltage Example

A 15 V battery supplies two series resistors.

One resistor has a voltage drop of:

6 V

The second voltage drop is:

V₂ = ?

Using KVL:

15 = 6 + V₂

Therefore:

V₂ = 9 V

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Kirchhoff's Voltage Law and Ohm's Law

KVL becomes especially powerful when combined with:

V = IR

Suppose a 12 V battery is connected to two series resistors:

R₁ = 2 Ω

R₂ = 4 Ω

Total resistance:

R_total = 2 + 4

R_total = 6 Ω

Current:

I = V/R

I = 12/6

I = 2 A

Voltage across R₁:

V₁ = IR₁

V₁ = 2 × 2

V₁ = 4 V

Voltage across R₂:

V₂ = IR₂

V₂ = 2 × 4

V₂ = 8 V

Apply KVL:

12 − 4 − 8 = 0

The circuit satisfies Kirchhoff's Voltage Law.

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Why Larger Series Resistors Have Larger Voltage Drops

In a series circuit, the current is the same through each resistor.

Since:

V = IR

a larger resistance produces a larger voltage drop.

For example, if:

I = 2 A

R₁ = 3 Ω

R₂ = 7 Ω

then:

V₁ = 2 × 3 = 6 V

V₂ = 2 × 7 = 14 V

The total voltage is:

6 + 14 = 20 V

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Choosing a Loop Direction

When applying KVL, choose a direction to travel around the loop.

You may choose:

• clockwise
• counterclockwise

Either direction works.

The important requirement is to use signs consistently.

For example, you might choose clockwise and record every voltage rise and drop encountered along that direction.

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Sign Convention for Batteries

When moving through a battery:

from − terminal to + terminal → voltage rise → positive

from + terminal to − terminal → voltage drop → negative

For example, crossing a 9 V battery from negative to positive gives:

+9 V

Crossing it from positive to negative gives:

−9 V

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Sign Convention for Resistors

For a resistor carrying current I:

when moving with the current through the resistor:

−IR

This represents a voltage drop.

When moving against the current:

+IR

This represents a voltage rise in the chosen traversal direction.

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A Reliable Sign Strategy

When moving around a loop:

Battery

− to +:

+V

• to −:

−V

Resistor

With current:

−IR

Against current:

+IR

Then add every term and set the total equal to zero:

ΣV = 0

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Example Using Signs

Suppose a circuit contains:

• 10 V battery
• 2 Ω resistor
• 3 Ω resistor
• current I

Travel clockwise from the negative terminal of the battery.

Across battery:

+10

Across 2 Ω resistor with current:

−2I

Across 3 Ω resistor with current:

−3I

Apply KVL:

10 − 2I − 3I = 0

10 − 5I = 0

5I = 10

I = 2 A

The resistor voltage drops are:

V₁ = 2 × 2 = 4 V

V₂ = 2 × 3 = 6 V

Check:

10 − 4 − 6 = 0

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Multiple Power Sources

Some circuits contain more than one battery or power source.

Their voltages may:

• support each other
• oppose each other

KVL allows us to determine their combined effect.

Suppose two batteries of 12 V and 4 V oppose each other.

The effective voltage is:

12 − 4

= 8 V

If the circuit resistance is 4 Ω:

I = 8/4

I = 2 A

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Batteries Supporting Each Other

Suppose two batteries of:

6 V

and:

3 V

are arranged so their voltage rises act in the same direction.

Total voltage:

V_total = 6 + 3

V_total = 9 V

If total resistance is 3 Ω:

I = 9/3

I = 3 A

Battery orientation therefore matters.

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Loop Rule in Parallel Circuits

KVL also explains why parallel branches have the same voltage.

Suppose two resistors are connected in parallel across a 12 V battery.

Consider the loop containing the battery and resistor 1:

12 − V₁ = 0

Therefore:

V₁ = 12 V

Now consider the loop containing the battery and resistor 2:

12 − V₂ = 0

Therefore:

V₂ = 12 V

Thus:

V₁ = V₂ = V_supply

This is the familiar parallel-circuit voltage rule.

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KVL and Series Circuits

Similarly, the series voltage rule follows directly from KVL.

For a battery and three series resistors:

V_supply − V₁ − V₂ − V₃ = 0

Rearranging:

V_supply = V₁ + V₂ + V₃

So the basic series circuit rule is really an application of Kirchhoff's Voltage Law.

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Multiple Loops

More complicated circuits can contain several closed loops.

For example, a circuit might contain:

• one battery
• several resistors
• two junctions
• multiple branches

Different currents may flow through the different branches.

In these circuits, we often need both:

Kirchhoff's Current Law

and:

Kirchhoff's Voltage Law

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Current Law and Voltage Law Together

Kirchhoff's Current Law:

total current entering a junction = total current leaving

Kirchhoff's Voltage Law:

ΣV = 0 around a closed loop

Together with Ohm's Law:

V = IR

these equations allow complicated electrical networks to be solved.

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Assigning Loop Currents

One common method for solving multi-loop circuits is to assign a current to each loop.

For example:

I₁ = clockwise current in the left loop

I₂ = clockwise current in the right loop

If the loops share a resistor, the actual current through that resistor depends on both loop currents.

If the currents pass through the shared resistor in opposite directions:

I_shared = I₁ − I₂

or:

I₂ − I₁

depending on the chosen reference direction.

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Shared Resistors

Shared resistors are one of the most important features of multi-loop problems.

Suppose:

I₁ flows through a shared resistor in one direction.

I₂ flows through it in the opposite direction.

The net current is:

I_shared = I₁ − I₂

Therefore, the voltage across the shared resistor is:

V_shared = R(I₁ − I₂)

This term appears in the KVL equation for both loops.

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Example of a Two-Loop Network

Consider two loops that share a 2 Ω resistor.

Left loop:

• 10 V battery
• 3 Ω resistor
• shared 2 Ω resistor

Right loop:

• 4 V battery
• 4 Ω resistor
• shared 2 Ω resistor

Assign clockwise currents:

I₁ = left-loop current

I₂ = right-loop current

The shared-resistor current is:

I₁ − I₂

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Left Loop Equation

Travel clockwise around the left loop.

Battery rise:

+10

Voltage drop across 3 Ω resistor:

−3I₁

Voltage drop across shared resistor:

−2(I₁ − I₂)

Therefore:

10 − 3I₁ − 2(I₁ − I₂) = 0

Expand:

10 − 3I₁ − 2I₁ + 2I₂ = 0

Therefore:

5I₁ − 2I₂ = 10

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Right Loop Equation

For the right loop:

4 − 4I₂ − 2(I₂ − I₁) = 0

Expand:

4 − 4I₂ − 2I₂ + 2I₁ = 0

Therefore:

−2I₁ + 6I₂ = 4

We now have two simultaneous equations:

5I₁ − 2I₂ = 10

−2I₁ + 6I₂ = 4

Solving them gives approximately:

I₁ = 2.62 A

I₂ = 1.54 A

The shared resistor current is:

I_shared = I₁ − I₂

I_shared = 2.62 − 1.54

I_shared ≈ 1.08 A

This illustrates how KVL can be used to solve networks that cannot be reduced using only simple series and parallel rules.

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Negative Current Answers

Sometimes solving a multi-loop problem gives a negative current.

For example:

I = −1.5 A

This does not mean something physically impossible has happened.

It means the current actually flows opposite to the direction originally assumed.

The actual current is:

1.5 A in the opposite direction

Choosing an initial current direction is simply part of the mathematical model.

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Solving Multi-Loop Circuits Systematically

A useful procedure is:

Label the currents

Assign directions to unknown currents.

Identify the junctions

Apply Kirchhoff's Current Law where necessary.

Identify independent loops

Choose closed paths through the circuit.

Select loop directions

Clockwise or counterclockwise is acceptable.

Write KVL equations

Use:

ΣV = 0

Replace resistor voltages

Use:

V = IR

Solve the simultaneous equations

Find the unknown currents.

Calculate other quantities

Use the currents to determine voltage drops, power or resistance.

Check the results

Verify both KCL and KVL.

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Independent Loops

A complicated circuit may contain many possible closed paths.

Not every possible loop gives new information.

An independent loop provides an equation that cannot simply be obtained by combining the other loop equations.

For basic circuits, choose the smallest obvious loops first.

This usually produces the simplest equations.

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Complete Single-Loop Example

A circuit contains:

• 18 V battery
• 2 Ω resistor
• 4 Ω resistor
• 3 Ω resistor

All components are in one loop.

Apply KVL:

18 − 2I − 4I − 3I = 0

18 − 9I = 0

Therefore:

I = 2 A

Voltage drops:

V₁ = 2 × 2 = 4 V

V₂ = 2 × 4 = 8 V

V₃ = 2 × 3 = 6 V

Check:

4 + 8 + 6 = 18 V

Therefore:

18 − 18 = 0

KVL is satisfied.

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Finding an Unknown Resistance

A 24 V battery supplies a single loop containing:

R₁ = 4 Ω

R₂ = ?

Current:

I = 3 A

Apply KVL:

24 − 3(4) − 3R₂ = 0

24 − 12 − 3R₂ = 0

12 = 3R₂

Therefore:

R₂ = 4 Ω

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Finding an Unknown Source Voltage

A loop contains:

• unknown battery V
• 3 Ω resistor
• 5 Ω resistor
• current of 2 A

Apply KVL:

V − 2(3) − 2(5) = 0

V − 6 − 10 = 0

Therefore:

V = 16 V

The source must provide 16 J of energy per coulomb.

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Energy Interpretation

Suppose a 6 C charge moves around a circuit containing a 10 V battery.

Energy supplied by the battery:

E = VQ

E = 10 × 6

E = 60 J

If one resistor transfers:

24 J

then the remaining components must transfer:

60 − 24

= 36 J

The energy supplied and transferred must balance.

This is the physical basis of KVL.

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Experimental Testing of the Loop Rule

Kirchhoff's Voltage Law can be tested using voltmeters.

For a simple series circuit, measure:

• supply voltage
• voltage across resistor 1
• voltage across resistor 2
• voltage across resistor 3

Theory predicts:

V_supply ≈ V₁ + V₂ + V₃

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Example Experimental Results

Suppose:

V_supply = 12.02 V

V₁ = 3.01 V

V₂ = 4.02 V

V₃ = 4.96 V

Add the voltage drops:

V₁ + V₂ + V₃

= 3.01 + 4.02 + 4.96

= 11.99 V

The measured supply voltage is:

12.02 V

Difference:

12.02 − 11.99

= 0.03 V

The results provide strong experimental support for KVL within measurement uncertainty.

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Why Experimental Values May Differ

Real measurements may not satisfy KVL perfectly because of:

• meter uncertainty
• resistor tolerance
• internal resistance of the battery
• connecting-wire resistance
• contact resistance
• temperature changes
• power-supply fluctuations
• rounding

KVL itself still reflects conservation of energy. Differences usually arise because the experimental model does not include every real component or because measurements have uncertainty.

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Kirchhoff's Laws Together

The two Kirchhoff laws describe different conservation principles.

Current Law:

ΣI = 0

Voltage Law:

ΣV = 0

Together with:

V = IR

they form a powerful system for analyzing electrical circuits.

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When Do We Need Kirchhoff's Laws?

Simple circuits can often be solved using:

• series resistance rules
• parallel resistance rules
• Ohm's Law

Kirchhoff's laws become particularly useful when:

• several branches are present
• several voltage sources are present
• components cannot easily be reduced to simple series or parallel combinations
• multiple loops share components
• several unknown currents must be found

They provide a systematic method that works for much more complicated networks.

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Real-World Applications

Kirchhoff's Voltage Law is used in the analysis and design of:

• electronic circuits
• power supplies
• automotive electrical systems
• sensor circuits
• communication electronics
• control systems
• power-distribution networks
• circuit boards

Engineers frequently use computer software to solve large systems of Kirchhoff equations, but the underlying physical principles remain the same.

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Common Mistakes

Mistake: Forgetting that KVL applies around a closed loop.
The path must return to its starting point.

Mistake: Adding all voltages as positive.
Voltage rises and drops require appropriate signs.

Mistake: Changing the sign convention halfway through a calculation.
Choose a direction and remain consistent.

Mistake: Assuming clockwise is the only correct loop direction.
Either direction works if signs are handled correctly.

Mistake: Forgetting that V = IR for a resistor.
Resistor voltage terms are usually written as IR.

Mistake: Ignoring the orientation of a battery.
Moving from − to + is a voltage rise; moving from + to − is a voltage drop.

Mistake: Using I₁ through a shared resistor when two loop currents pass through it.
The current may be I₁ − I₂.

Mistake: Thinking a negative calculated current is impossible.
It normally means the true direction is opposite to the assumed direction.

Mistake: Confusing KCL with KVL.
KCL deals with currents at junctions. KVL deals with voltages around loops.

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Key Terms

Kirchhoff's Voltage Law (KVL) — the algebraic sum of potential differences around a closed loop is zero.

Kirchhoff's Second Law — another name for Kirchhoff's Voltage Law.

Loop Rule — another name for KVL.

Closed loop — continuous circuit path that returns to its starting point.

Potential difference — energy transferred per unit charge.

Voltage rise — increase in electric potential when moving through a circuit.

Voltage drop — decrease in electric potential across a component.

EMF — energy supplied per unit charge by a source.

Loop current — current assigned to a particular loop during circuit analysis.

Shared resistor — resistor that forms part of more than one loop.

Independent loop — loop that provides an independent circuit equation.

Sign convention — consistent rules for assigning positive and negative values.

Conservation of energy — principle that energy cannot be created or destroyed.

Σ (sigma) — mathematical symbol meaning "sum of."

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Key Takeaways

• Kirchhoff's Voltage Law is also called Kirchhoff's Second Law or the Loop Rule.
• KVL states that the algebraic sum of potential differences around a closed loop is zero.
• ΣV = 0
• In a simple circuit, total voltage supplied equals total voltage dropped.
• KVL follows from conservation of energy.
• Voltage represents energy transferred per unit charge.
• Batteries generally provide voltage rises.
• Resistors generally produce voltage drops in the direction of current.
• Either clockwise or counterclockwise loop directions can be used.
• Sign conventions must be applied consistently.
• For a resistor, voltage can be expressed using V = IR.
• KVL explains why voltage drops in a series circuit add to the supply voltage.
• KVL also explains why parallel branches connected across the same two points have the same voltage.
• Multiple voltage sources can either support or oppose each other.
• Multi-loop circuits may require several simultaneous equations.
• A resistor shared by two loops may carry the difference between two loop currents.
• A negative calculated current usually indicates that the actual current flows opposite to the assumed direction.
• Kirchhoff's Current Law and Voltage Law are often used together.
• KCL represents conservation of charge.
• KVL represents conservation of energy.
• Together with Ohm's Law, Kirchhoff's laws can be used to analyze complex electrical networks.

Check Your Understanding

1. State Kirchhoff's Voltage Law.
2. What is meant by a closed loop?
3. Explain how KVL follows from conservation of energy.
4. What does ΣV = 0 mean?
5. A 12 V battery is connected to two series components. One has a voltage drop of 5 V. Calculate the other voltage drop.
6. A 24 V supply is connected to three components with voltage drops of 6 V, 8 V and V₃. Calculate V₃.
7. A circuit contains a 15 V battery and two series resistors of 2 Ω and 3 Ω. Calculate the current.
8. For Question 7, calculate the voltage across each resistor and verify KVL.
9. What sign is normally assigned when crossing a battery from its negative terminal to its positive terminal?
10. What sign is normally assigned to a resistor when moving through it in the direction of conventional current?
11. Explain why either clockwise or counterclockwise can be chosen as the loop direction.
12. Two batteries of 12 V and 5 V oppose each other. What is their net voltage?
13. Two batteries of 6 V and 4 V support each other. What is their combined voltage?
14. A 20 V battery supplies a loop containing a 4 Ω resistor and an unknown resistor. The current is 2 A. Calculate the unknown resistance.
15. A circuit carries 3 A through two series resistors of 2 Ω and 5 Ω. Calculate the required source voltage.
16. Explain the difference between Kirchhoff's Current Law and Kirchhoff's Voltage Law.
17. Explain why parallel branches connected across the same two points have the same voltage using KVL.
18. Two loop currents I₁ and I₂ pass through a shared resistor in opposite directions. Explain how the current through the resistor can be represented.
19. What does a negative answer for an assumed loop current tell you?
20. Explain how Kirchhoff's Current Law, Kirchhoff's Voltage Law and Ohm's Law can be combined to solve a multi-loop circuit.