3. Multi-Loop Circuits

Multi-Loop Circuits

What Is a Multi-Loop Circuit?

A multi-loop circuit contains more than one closed path through which current can flow.

Unlike a simple series circuit, a multi-loop network may contain:

• several branches
• several junctions
• multiple resistors
• one or more voltage sources
• components shared between loops

These circuits are analyzed using three main relationships:

Ohm's Law:
V = IR

Kirchhoff's Current Law (KCL):
total current entering a junction = total current leaving

Kirchhoff's Voltage Law (KVL):
ΣV = 0 around a closed loop

Together, these laws allow us to analyze circuits that cannot be simplified using only basic series and parallel rules.

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Identifying Loops

A loop is any closed path through a circuit that returns to its starting point.

Imagine a circuit containing two branches connected between the same junctions. It may be possible to trace:

• a left loop
• a right loop
• a larger outer loop

However, not every possible loop provides new information.

This leads to the idea of an independent loop.

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Independent Loops

An independent loop is a loop that provides a new Kirchhoff equation that cannot be obtained simply by combining equations from other loops.

For many basic circuits, the easiest independent loops are the smallest loops or meshes.

Suppose a circuit contains two side-by-side loops.

We might label them:

Loop 1

and:

Loop 2

If an outer path surrounds both loops, it is also a closed loop, but its KVL equation may be produced by combining the equations for Loop 1 and Loop 2.

Therefore, the outer loop may not be independent.

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Loops and Meshes

A mesh is a loop that does not contain another loop inside it.

For a simple planar circuit:

• each mesh can often be treated as an independent loop
• assigning a current to each mesh can simplify the mathematics

This technique is called mesh analysis or loop-current analysis.

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Why Independent Loops Matter

Suppose a circuit contains two unknown loop currents:

I₁

and:

I₂

To determine both unknowns, we normally need two independent equations.

Each independent loop can provide one KVL equation.

We then solve the equations simultaneously.

This is similar to solving two simultaneous equations in mathematics.

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Kirchhoff's Laws in Multi-Loop Circuits

Multi-loop circuits usually require both Kirchhoff laws.

Current Law

At a junction:

ΣI = 0

or:

total current entering = total current leaving

Voltage Law

Around a loop:

ΣV = 0

These laws represent two fundamental conservation principles:

KCL → conservation of charge

KVL → conservation of energy

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Assigning Current Directions

Before writing equations, assign directions to the unknown currents.

For example:

I₁ = clockwise current in Loop 1

I₂ = clockwise current in Loop 2

You do not need to know the true current direction beforehand.

Choose a direction and solve the equations.

If the answer is negative, the actual current flows in the opposite direction.

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Negative Current Does Not Mean an Error

Suppose you assume:

I₁ flows clockwise.

After solving, you obtain:

I₁ = −2 A

This means the actual current is:

2 A counterclockwise

The negative sign simply tells you that your original assumed direction was opposite to the actual direction.

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Shared Components

One of the main challenges in multi-loop circuits is dealing with a component that belongs to more than one loop.

Suppose two loops share a resistor.

If both loop currents are clockwise, they usually pass through the shared resistor in opposite directions.

Therefore, the current through the shared resistor is:

I_shared = I₁ − I₂

if I₁ is chosen as the reference direction.

The voltage across the resistor is then:

V_shared = R(I₁ − I₂)

This is one of the most important relationships in mesh analysis.

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Why the Currents Subtract

Imagine:

I₁ = 5 A downward through a shared resistor

I₂ = 2 A upward through the same resistor

The currents oppose each other.

The net current is:

I_shared = 5 − 2

I_shared = 3 A downward

If instead:

I₂ > I₁

the actual current would flow upward.

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Sign Rules for Loop Equations

Choose a direction around each loop.

Crossing a battery

From − to +:

+V

From + to −:

−V

Crossing a resistor

Moving with the current:

−IR

Moving against the current:

+IR

Consistency is more important than choosing a particular loop direction.

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A Simple Two-Loop Circuit

Consider a circuit with two loops sharing a 2 Ω resistor.

Left loop contains:

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

Right loop contains:

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

Assign clockwise loop currents:

I₁ = left loop

I₂ = right loop

The current through the shared resistor is:

I₁ − I₂

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Writing the Left-Loop Equation

Travel clockwise around Loop 1.

Battery:

+10

3 Ω resistor:

−3I₁

Shared 2 Ω resistor:

−2(I₁ − I₂)

Therefore:

10 − 3I₁ − 2(I₁ − I₂) = 0

Expand:

10 − 3I₁ − 2I₁ + 2I₂ = 0

Rearrange:

5I₁ − 2I₂ = 10

This is Equation 1.

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Writing the Right-Loop Equation

Travel clockwise around Loop 2.

Suppose the 4 V battery provides a voltage rise in our chosen direction.

Then:

4 − 4I₂ − 2(I₂ − I₁) = 0

Expand:

4 − 4I₂ − 2I₂ + 2I₁ = 0

Rearrange:

−2I₁ + 6I₂ = 4

This is Equation 2.

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Solving the Simultaneous Equations

We now have:

5I₁ − 2I₂ = 10

−2I₁ + 6I₂ = 4

These equations can be solved using:

• substitution
• elimination
• matrices
• a calculator capable of simultaneous equations

Using elimination gives:

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

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Verifying the Left Loop

A good circuit solution should always be checked.

Using:

I₁ ≈ 2.62 A

I_shared ≈ 1.08 A

Voltage across the 3 Ω resistor:

V = IR

V ≈ 2.62 × 3

V ≈ 7.86 V

Voltage across the shared 2 Ω resistor:

V ≈ 1.08 × 2

V ≈ 2.16 V

Total voltage drop:

7.86 + 2.16

≈ 10.02 V

The small difference from 10 V is caused by rounding.

Therefore, the left loop satisfies KVL.

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Verifying the Right Loop

For the right loop:

I₂ ≈ 1.54 A

Voltage across 4 Ω:

V ≈ 1.54 × 4

V ≈ 6.16 V

The shared resistor contributes approximately:

2.16 V

The signs depend on the direction in which the loop is traversed.

Substituting the calculated currents into the original right-loop equation should produce approximately:

0 V

This confirms that the solution satisfies KVL within rounding.

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Verification Is Essential

After solving a multi-loop circuit, check the answer using:

Kirchhoff's Current Law

At every relevant junction:

current entering = current leaving

Kirchhoff's Voltage Law

Around every independent loop:

ΣV = 0

Ohm's Law

For each resistor:

V = IR

A correct solution should satisfy all three.

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Junction Currents in a Multi-Loop Circuit

Sometimes it is easier to assign branch currents rather than loop currents.

Suppose:

I₁ reaches a junction.

It divides into:

I₂

and:

I₃

Then KCL gives:

I₁ = I₂ + I₃

If:

I₁ = 8 A

and:

I₂ = 3 A

then:

I₃ = 8 − 3

I₃ = 5 A

This equation can then be combined with loop equations.

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Branch-Current Method

In the branch-current method:

1. Assign a current to each important branch.
2. Apply KCL at junctions.
3. Apply KVL around independent loops.
4. Replace resistor voltages using V = IR.
5. Solve the simultaneous equations.

This method is often intuitive because the variables represent the actual currents through physical branches.

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Mesh-Current Method

In the mesh-current method:

1. Identify independent meshes.
2. Assign one current to each mesh.
3. Write a KVL equation for each mesh.
4. Use current differences for shared resistors.
5. Solve the simultaneous equations.
6. Convert mesh currents into actual branch currents where necessary.

Mesh analysis can reduce the number of equations required in some circuits.

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Branch Current vs Mesh Current

These quantities are related but are not always identical.

A branch current is the actual current flowing through a physical branch.

A mesh current is a mathematical current assigned to a loop.

For a resistor belonging to only one mesh:

branch current = mesh current

For a resistor shared by two meshes:

branch current = difference between mesh currents

This distinction becomes very important in more complicated networks.

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A Circuit with Three Branches

Suppose two junctions are connected by three branches.

The branch currents are:

I₁

I₂

I₃

At one junction:

I₁ = I₂ + I₃

Suppose the branches also contain batteries and resistors.

We can then construct loop equations involving:

I₁, I₂ and I₃.

KCL gives one relationship, while KVL provides additional equations.

Together, they form a solvable system.

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Worked Branch-Current Example

Suppose:

I₁ enters a junction

I₂ and I₃ leave

and:

I₁ = 5 A

One branch contains a 4 Ω resistor and has a voltage drop of 8 V.

Using Ohm's Law:

I₂ = V/R

I₂ = 8/4

I₂ = 2 A

Now apply KCL:

I₁ = I₂ + I₃

5 = 2 + I₃

Therefore:

I₃ = 3 A

This illustrates how Ohm's Law and KCL can work together.

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Solving Simultaneous Equations by Elimination

Suppose circuit analysis gives:

3I₁ + 2I₂ = 16

I₁ − 2I₂ = 0

Add the equations:

4I₁ = 16

Therefore:

I₁ = 4 A

Substitute into:

I₁ − 2I₂ = 0

4 − 2I₂ = 0

2I₂ = 4

I₂ = 2 A

Always substitute the results back into the original circuit equations.

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Solving by Substitution

Suppose:

I₁ + I₂ = 6

and:

2I₁ + 4I₂ = 16

From the first equation:

I₁ = 6 − I₂

Substitute:

2(6 − I₂) + 4I₂ = 16

12 − 2I₂ + 4I₂ = 16

2I₂ = 4

I₂ = 2 A

Therefore:

I₁ = 6 − 2

I₁ = 4 A

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Interpreting Negative Solutions

Suppose you assumed:

I₁ clockwise

I₂ clockwise

and solved:

I₁ = 3 A

I₂ = −1 A

The interpretation is:

I₁ actually flows 3 A clockwise

I₂ actually flows 1 A counterclockwise

The mathematics has determined the true direction.

Do not change the equations midway simply because a current turns out to be negative.

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Interpreting a Shared-Branch Current

Suppose:

I₁ = 5 A

I₂ = 3 A

and the mesh currents oppose one another through a shared resistor.

Then:

I_shared = I₁ − I₂

I_shared = 5 − 3

I_shared = 2 A

The shared branch carries 2 A in the direction of I₁.

If instead:

I₁ = 3 A

I₂ = 5 A

then:

I_shared = 3 − 5

I_shared = −2 A

Therefore, the actual current is 2 A in the direction of I₂.

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Circuit Behavior and Resistance

Multi-loop analysis helps us understand how changing one component affects the entire network.

For example, increasing resistance in one branch usually:

• decreases current through that branch
• changes current at nearby junctions
• changes voltage drops
• may change currents in other branches
• may alter the current supplied by the source

The effect is often more complicated than in a simple series circuit because several paths interact.

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What Happens If a Branch Opens?

If one branch becomes an open circuit:

current through that branch becomes zero

However, other loops may remain complete.

Therefore, current may continue flowing through other parts of the network.

This is one reason parallel and multi-loop systems can provide greater independence between components than a simple series circuit.

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What Happens If Resistance Decreases?

Suppose the resistance in one parallel branch decreases.

Since:

I = V/R

the current through that branch tends to increase.

This may increase:

• total source current
• current at certain junctions
• power transferred in the branch

It may also change voltages and currents elsewhere if the source or other circuit elements are non-ideal.

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Power in Multi-Loop Circuits

After currents and voltages have been determined, power can be calculated.

For a component:

P = VI

For a resistor:

P = I²R

or:

P = V²/R

Power calculations provide another way to verify a circuit solution.

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Conservation of Power

In an ideal steady circuit:

total power supplied = total power transferred

Suppose batteries supply:

30 W

Then the resistors and other loads should collectively transfer:

30 W

within the assumptions of the model.

This provides another check based on conservation of energy.

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Example Power Check

Suppose a 12 V battery supplies:

I = 2 A

Power supplied:

P = VI

P = 12 × 2

P = 24 W

If two resistors dissipate:

P₁ = 8 W

P₂ = 16 W

then:

P_total = 8 + 16

P_total = 24 W

The power balance supports the circuit solution.

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How Many Independent Loops?

For a connected circuit network, the number of independent loops can be determined using:

L = B − J + 1

where:

• L = number of independent loops
• B = number of branches
• J = number of junctions or nodes

For example, if:

B = 5

J = 4

then:

L = 5 − 4 + 1

L = 2

So the circuit has two independent loops.

This relationship is particularly useful for larger circuit networks.

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Nodes and Branches

A node is a point or connected region where circuit elements meet and share the same electrical potential.

A branch is a path between two nodes containing one or more circuit components.

Identifying nodes and branches helps organize complicated circuits before equations are written.

In many school-level diagrams, important nodes appear as obvious junctions.

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A Systematic Multi-Loop Strategy

When faced with a complicated circuit:

Study the circuit first

Do not immediately begin calculating.

Identify:

• sources
• resistors
• branches
• junctions
• loops

Label currents

Choose sensible current directions.

Identify independent loops

Use the smallest useful loops where possible.

Apply KCL

Write current equations at necessary junctions.

Apply KVL

Write voltage equations around independent loops.

Use Ohm's Law

Replace resistor voltages with:

V = IR

Simplify the equations

Collect terms involving each unknown current.

Solve simultaneously

Use substitution, elimination or another suitable method.

Interpret negative answers

Reverse the assumed direction where necessary.

Calculate remaining quantities

Find voltages, currents, resistance or power.

Verify

Check KCL, KVL and, where useful, power conservation.

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Organizing the Mathematics

For a two-loop problem, try to reach equations such as:

aI₁ + bI₂ = c

dI₁ + eI₂ = f

Keeping the equations organized makes the simultaneous-equation step much easier.

For three unknown currents, three independent equations are generally required.

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Common Sources of Difficulty

Most errors in multi-loop problems occur before the simultaneous equations are solved.

Typical problems include:

• incorrect current direction through a shared resistor
• missing a resistor from a loop
• incorrect battery polarity
• inconsistent voltage signs
• using I₁ instead of I₁ − I₂ through a shared resistor
• using dependent loops as though they provide independent equations
• arithmetic mistakes when expanding brackets

Drawing arrows and labeling components carefully can prevent many of these errors.

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Verifying with Kirchhoff's Current Law

Suppose a calculated solution gives:

I₁ = 6 A entering a junction

I₂ = 2 A leaving

I₃ = 4 A leaving

Check:

I_entering = 6 A

I_leaving = 2 + 4 = 6 A

Therefore:

KCL is satisfied.

If the currents do not balance, something is wrong with the solution or the assigned interpretation.

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Verifying with Kirchhoff's Voltage Law

Suppose one loop contains:

12 V battery

2 Ω resistor carrying 2 A

4 Ω resistor carrying 2 A

Voltage drops:

V₁ = 2 × 2 = 4 V

V₂ = 2 × 4 = 8 V

Check:

12 − 4 − 8 = 0

Therefore:

KVL is satisfied.

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Verifying with Power

If possible, calculate:

total source power

and:

total power dissipated

For an ideal circuit:

P_supplied ≈ P_transferred

If these are substantially different, recheck the current directions, voltage values and calculations.

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Experimental Multi-Loop Circuits

Kirchhoff's laws can be tested experimentally using:

• DC power supplies
• resistors
• ammeters
• voltmeters
• digital multimeters
• connecting wires

Measurements can be compared with theoretical values predicted using simultaneous equations.

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Comparing Theory and Experiment

Suppose theory predicts:

I₁ = 0.50 A

I₂ = 0.30 A

I₃ = 0.20 A

Experimental results are:

I₁ = 0.49 A

I₂ = 0.29 A

I₃ = 0.19 A

The measurements are close to the theoretical predictions.

Also:

0.29 + 0.19 = 0.48 A

which is close to the measured incoming current of 0.49 A.

The small discrepancy may be caused by:

• resistor tolerance
• meter uncertainty
• wire resistance
• contact resistance
• source internal resistance
• temperature changes

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Why Multi-Loop Analysis Matters

Real electrical and electronic systems rarely consist of one simple resistor connected to one battery.

Networks can contain many interacting pathways.

Multi-loop analysis provides a foundation for understanding:

• electronic circuits
• automotive electrical systems
• power distribution
• sensor networks
• control systems
• communication circuits
• circuit boards
• electrical instrumentation

The same conservation principles apply regardless of how complicated the network becomes.

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Common Mistakes

Mistake: Treating every closed path as an independent loop.
Some loop equations can be produced by combining others.

Mistake: Assuming current directions must be known before solving.
Choose reasonable directions. Negative answers reveal opposite flow.

Mistake: Using one mesh current directly through a shared resistor.
The shared current is usually the difference between the mesh currents.

Mistake: Forgetting Kirchhoff's Current Law.
Multi-loop circuits often require both KCL and KVL.

Mistake: Changing signs halfway through the problem.
Choose a sign convention and use it consistently.

Mistake: Ignoring battery orientation.
Battery polarity determines whether the voltage is a rise or drop.

Mistake: Solving the equations but not checking the circuit.
Always verify the final values using Kirchhoff's laws.

Mistake: Assuming a negative current is impossible.
It indicates that the true direction is opposite to the assumed direction.

Mistake: Confusing mesh currents with branch currents.
They are not necessarily the same.

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Key Terms

Multi-loop circuit — circuit containing more than one closed current path.

Loop — closed path through a circuit.

Independent loop — loop that provides an independent KVL equation.

Mesh — loop that contains no smaller loop within it.

Mesh analysis — method that assigns currents to independent meshes and applies KVL.

Branch-current analysis — method that assigns actual currents to circuit branches.

Mesh current — mathematical current assigned to a mesh.

Branch current — actual current through a physical branch.

Shared resistor — resistor belonging to two neighboring loops.

Junction — point where current paths divide or combine.

Node — electrically connected point or region having a common potential.

Simultaneous equations — equations containing several unknowns that must be satisfied together.

Kirchhoff's Current Law — total current entering a junction equals total current leaving.

Kirchhoff's Voltage Law — algebraic sum of voltage changes around a closed loop equals zero.

Ohm's Law — relationship V = IR for an ohmic component.

Conservation of charge — physical basis of KCL.

Conservation of energy — physical basis of KVL.

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Key Takeaways

• Multi-loop circuits contain several interconnected current paths.
• Independent loops provide independent KVL equations.
• Not every possible closed path is an independent loop.
• Meshes are often convenient choices for independent loops.
• Kirchhoff's Current Law represents conservation of charge.
• Kirchhoff's Voltage Law represents conservation of energy.
• Ohm's Law connects voltage, current and resistance.
• Current directions can be assumed before solving.
• A negative current means the actual direction is opposite to the assumed direction.
• Multi-loop problems often produce simultaneous equations.
• Shared resistors require special attention.
• When two mesh currents oppose each other through a shared resistor, the branch current is their difference.
• Branch currents and mesh currents are not always identical.
• Complex circuits can be solved systematically rather than by guessing.
• KCL should be checked at junctions.
• KVL should be checked around loops.
• Ohm's Law should be satisfied by individual resistors.
• Power conservation can provide an additional verification.
• Changing one branch can influence currents and voltages elsewhere in the network.
• Multi-loop analysis provides a foundation for understanding real electrical and electronic networks.

Check Your Understanding

1. What is a multi-loop circuit?
2. What is meant by an independent loop?
3. Explain the difference between a loop and a mesh.
4. Why do we need independent equations when solving a multi-loop circuit?
5. State Kirchhoff's Current Law.
6. State Kirchhoff's Voltage Law.
7. Explain which conservation principle is represented by each Kirchhoff law.
8. Why can current directions be chosen before solving the circuit?
9. What does a negative calculated current mean?
10. Two clockwise mesh currents I₁ = 5 A and I₂ = 2 A oppose each other through a shared resistor. Calculate the shared-branch current.
11. If I₁ = 2 A and I₂ = 5 A, interpret the result for the same shared branch.
12. Write an expression for the voltage across a 4 Ω resistor shared by mesh currents I₁ and I₂ that oppose each other.
13. Solve: 3I₁ + 2I₂ = 16 and I₁ − 2I₂ = 0.
14. Explain how Kirchhoff's Current Law can be used to verify a multi-loop solution.
15. Explain how Kirchhoff's Voltage Law can be used to verify the same solution.
16. A connected network has 6 branches and 4 nodes. Use L = B − J + 1 to determine the number of independent loops.
17. Explain the difference between a branch current and a mesh current.
18. Why might changing one resistor affect currents elsewhere in a multi-loop network?
19. Describe a systematic procedure for solving a two-loop circuit.
20. Explain how KCL, KVL, Ohm's Law and conservation of power can all be used to check whether a multi-loop circuit solution is physically reasonable.