Circuits and DC Instruments
29 Kirchhoff’s Rules
Learning Objectives
- Apply Kirchhoff's junction rule to determine the relationship between currents at a junction in an electrical circuit.
- Apply Kirchhoff's loop rule to analyze voltage changes around closed circuits.
- Analyze complex circuits that cannot be simplified into series and parallel combinations.
- Use Kirchhoff's rules together with Ohm's law to determine unknown currents and voltages in electrical circuits.
In previous sections, we analyzed circuits that could be simplified into combinations of resistors connected in series or in parallel. Many real electrical circuits, however, are much more complicated. Medical equipment, electronic devices, automobile electrical systems, and scientific instruments often contain multiple branches and loops that cannot be reduced using only the series-parallel methods.
Figure 29.1 shows an example of such a circuit. At first glance, it may not be obvious how current flows through the different branches, and there is no simple way to replace groups of resistors with equivalent resistances. Fortunately, two powerful principles allow us to analyze any electrical circuit, regardless of its complexity.
These principles are known as Kirchhoff's rules, named after the German physicist Gustav Kirchhoff (1824–1887). Rather than introducing new physical laws, Kirchhoff's rules are direct applications of two of the most fundamental conservation principles in physics:
- Conservation of charge, which states that electric charge cannot be created or destroyed.
- Conservation of energy, which states that energy cannot be created or destroyed.
Because these conservation laws apply universally, Kirchhoff's rules can be used to analyze everything from a simple flashlight to the electrical circuitry inside a medical imaging device or cardiac monitor.
Kirchhoff's Rules
- Kirchhoff's Junction Rule (First Rule): The total current entering any junction must equal the total current leaving that junction.
- Kirchhoff's Loop Rule (Second Rule): The algebraic sum of all voltage changes around any closed loop must equal zero.
In the following sections, we will examine each rule separately, explain why it works, and then combine both rules to solve circuits that cannot be analyzed by simpler methods.
Kirchhoff's First Rule
Kirchhoff's first rule, also called the junction rule, is a direct consequence of the conservation of charge. A junction is any point where three or more conducting paths meet.
Electric current represents the flow of electric charge. Since charge cannot accumulate indefinitely at a junction, the amount of charge arriving each second must equal the amount leaving each second. If this were not true, charge would continuously build up at one point in the circuit, which is not observed in steady-state circuits.
Figure 29.2 illustrates this idea. Current I1 enters the junction and divides into two separate currents, I2 and I3. The junction rule states that
This relationship is true regardless of the values of the currents or the components connected to the junction. Whenever current splits into multiple branches, the total current entering must always equal the total current leaving.
Making Connections: Conservation Laws
Kirchhoff's rules are not separate physical laws. Instead, they are practical applications of the same conservation principles encountered throughout physics.
- The junction rule comes directly from the conservation of charge.
- The loop rule, introduced in the next section, comes directly from the conservation of energy.
Understanding these underlying principles makes Kirchhoff's rules much easier to remember than simply memorizing equations.
Kirchhoff's Second Rule
Kirchhoff's second rule, also called the loop rule, is a direct application of the conservation of energy. As electric charge travels around a closed circuit, it may gain electrical potential energy when passing through a battery or other voltage source, and it may lose electrical potential energy when passing through resistors or other electrical components. Because energy is conserved, the total energy gained must equal the total energy lost after completing one full loop.
Since electrical potential energy is related to electric potential (voltage) by
it is more convenient to write the conservation of energy in terms of voltage rather than energy. Kirchhoff's loop rule therefore states:
Figure 29.3 illustrates this principle for a simple series circuit. As a positive charge passes through the battery, its electric potential increases by an amount equal to the battery's emf. As the charge continues around the circuit, it loses that same amount of potential while passing through the internal resistance of the battery and the external resistors.
For the circuit shown, Kirchhoff's loop rule can be written as
Rearranging gives
This equation simply states that the voltage supplied by the source equals the sum of all the voltage drops in the circuit. Every volt "gained" from the battery is accounted for by voltage losses elsewhere in the loop.
Visualizing the Loop Rule
One helpful way to think about electric potential is to imagine hiking around a mountain trail.
- Walking uphill increases your gravitational potential energy, just as moving through a battery increases electrical potential energy.
- Walking downhill decreases your gravitational potential energy, just as moving through a resistor decreases electrical potential energy.
- After completing the entire trail and returning to your starting point, your elevation has changed by exactly zero. Likewise, after moving completely around a closed electrical loop, the net change in electric potential must also be zero.
This analogy helps explain why Kirchhoff's loop rule is simply another statement of energy conservation.
Health Connection
The loop rule is essential for understanding the operation of medical devices such as infusion pumps, cardiac pacemakers, electrocardiographs (ECGs), and portable patient monitors. Engineers use Kirchhoff's rules to ensure that every component in these devices receives the correct voltage for accurate and reliable operation. Even small unexpected voltage drops caused by aging batteries or damaged wiring can affect device performance, making careful circuit analysis an important part of medical equipment design.
Applying Kirchhoff's Rules
Kirchhoff's rules provide a systematic method for analyzing circuits that cannot be simplified using only series and parallel combinations. By applying the junction rule and the loop rule, we generate a set of equations that describe the circuit completely. These equations can then be solved to determine unknown currents, voltages, resistances, or even the emf of a source.
Each application of one of Kirchhoff's rules produces one independent equation. Therefore, if the number of independent equations equals the number of unknown quantities, the circuit can be solved.
Before writing the equations, two important choices must be made:
- Choose a direction for every current. Label each branch current with an arrow showing its assumed direction. If your assumption is incorrect, the calculated current will simply be negative, indicating that the actual current flows opposite to the assumed direction.
- Choose a direction to travel around each loop. You may traverse each loop clockwise or counterclockwise. Either choice is acceptable, provided you remain consistent when assigning voltage changes throughout that loop.
Figure 29.4 summarizes the sign conventions used when applying the loop rule. In the figure, each circuit element is traversed from point a to point b.
Sign Conventions for the Loop Rule
- Resistor traversed in the direction of the current:
[latex]\Delta V=-IR[/latex]
The electric potential decreases because electrical energy is dissipated as heat.
- Resistor traversed opposite the current:
[latex]\Delta V=+IR[/latex]
Moving opposite the current corresponds to moving from lower to higher potential.
- Voltage source traversed from the negative terminal to the positive terminal:
[latex]\Delta V=+\text{emf}[/latex]
The source raises the electric potential.
- Voltage source traversed from the positive terminal to the negative terminal:
[latex]\Delta V=-\text{emf}[/latex]
The electric potential decreases as you move across the source in the opposite direction.
Example 29.1: Calculating Currents Using Kirchhoff's Rules
Determine the currents flowing through the circuit shown in Figure 29.5.
Strategy
This circuit contains multiple loops and junctions, making it impossible to reduce to a simple equivalent resistance. We therefore apply Kirchhoff's junction rule and loop rule to obtain three independent equations for the three unknown currents I1, I2, and I3.
Solution
Step 1: Apply the junction rule.
At junction a, current I1 enters while I2 and I3 leave:
Step 2: Apply the loop rule to the left loop.
Traversing loop abcdea clockwise gives
Grouping terms,
Substituting the circuit values yields
Step 3: Apply the loop rule to the right loop.
Traversing loop aefgha gives
Substituting numerical values,
Step 4: Solve the system of equations.
From the second equation,
From the third equation,
Substituting both expressions into the junction equation:
Now substitute this result into the remaining equations:
Discussion
The negative value of I2 does not indicate a mistake. It simply means that the actual current flows opposite to the direction originally assumed.
As a quick check, the junction rule is satisfied:
Since both Kirchhoff's rules are satisfied, the solution is internally consistent.
Conceptual Takeaway
Kirchhoff's rules transform a complicated electrical circuit into a system of algebraic equations. Although solving several equations may seem more involved than reducing series and parallel resistors, the method works for any circuit, making it one of the most powerful tools in circuit analysis.
Problem-Solving Strategy: Applying Kirchhoff's Rules
- Draw and label the circuit. Clearly label every resistor, voltage source, junction, and current. If a current direction is unknown, choose one arbitrarily.
- Apply the junction rule. Write an equation for one or more junctions using conservation of charge.
- Apply the loop rule. Select enough independent closed loops to obtain as many equations as there are unknowns. Use the sign conventions shown in Figure 29.4.
- Solve the simultaneous equations. Use algebra to solve for the unknown currents, voltages, or resistances.
- Check your answers. A negative current simply means the actual current flows opposite to the direction originally assumed.
Kirchhoff's rules are based on the fundamental conservation laws of physics, so they should accurately describe real electrical circuits. In practice, verifying these predictions requires measuring current and voltage with electrical instruments. As you will learn in the next sections, these measuring devices are themselves electrical components, and connecting them to a circuit can slightly change the quantities being measured.
Check Your Understanding
Can Kirchhoff's rules be applied to simple series and parallel circuits, or are they useful only for more complicated circuits?
Kirchhoff's rules can be applied to any circuit because they are applications of conservation of charge and conservation of energy. For simple series and parallel circuits, the specialized series-parallel rules are usually faster, but those rules can be derived from Kirchhoff's rules.
Section Summary
- Kirchhoff's rules provide a general method for analyzing electrical circuits of any complexity.
- Kirchhoff's Junction Rule: The total current entering any junction equals the total current leaving it.
[latex]\sum I_{\text{in}}=\sum I_{\text{out}}[/latex]
- Kirchhoff's Loop Rule: The algebraic sum of all potential changes around any closed loop is zero.
[latex]\sum \Delta V=0[/latex]
- The junction rule follows from conservation of charge, while the loop rule follows from conservation of energy.
- If a calculated current is negative, the actual current flows opposite to the direction originally assumed.
- The familiar rules for resistors in series and parallel are special cases of Kirchhoff's rules.
Conceptual Questions
- Can all of the currents going into the junction in Figure 29.6 be positive? Explain.
- Apply the junction rule to junction b in Figure 29.7. Is any new information gained by applying the junction rule at e? (In the figure, each emf is represented by script E.)
- (a) What is the potential difference going from point a to point b in Figure 29.7? (b) What is the potential difference going from c to b? (c) From e to g? (d) From e to d?
- Apply the loop rule to loop afedcba in Figure 29.7.
- Apply the loop rule to loops abgefa and cbgedc in Figure 29.7.
Problems & Exercises
- Apply the loop rule to loop abcdefgha in Figure 29.5.
- Apply the loop rule to loop aedcba in Figure 29.5.
- Verify the second equation in Example 29.1 by substituting the values found for the currents [latex]{I}_{1}[/latex] and [latex]{I}_{2}[/latex].
- Verify the third equation in Example 29.1 by substituting the values found for the currents [latex]{I}_{1}[/latex] and [latex]{I}_{3}[/latex].
- Apply the junction rule at point a in Figure 29.8.
- Apply the loop rule to loop abcdefghija in Figure 29.8.
- Apply the loop rule to loop akledcba in Figure 29.8.
- Find the currents flowing in the circuit in Figure 29.8. Explicitly show how you follow the steps in the Problem-Solving Strategy for Series and Parallel Resistors.
- Solve Example 29.1, but use loop abcdefgha instead of loop akledcba. Explicitly show how you follow the steps in the Problem-Solving Strategy for Series and Parallel Resistors.
- Find the currents flowing in the circuit in Figure 29.7.
- Unreasonable Results Consider the circuit in Figure 29.9, and suppose that the emfs are unknown and the currents are given to be [latex]{I}_{1}=5.00\ \text{A}[/latex], [latex]{I}_{2}=3.0\ \text{A}[/latex], and [latex]{I}_{3}=-2.00\ \text{A}[/latex]. (a) Could you find the emfs? (b) What is wrong with the assumptions?
Glossary
- Kirchhoff's rules
- A set of two fundamental rules for analyzing electrical circuits, based on the conservation of charge and conservation of energy. Together, they can be used to analyze any circuit, regardless of its complexity.
- junction rule
- Kirchhoff's first rule, which states that the total current entering any junction must equal the total current leaving the junction. This rule follows directly from the conservation of electric charge:
[latex]I_1=I_2+I_3[/latex]
- loop rule
- Kirchhoff's second rule, which states that the algebraic sum of all potential changes around any closed loop is zero. Equivalently, the energy supplied by voltage sources equals the total energy lost across circuit elements:
[latex]\text{emf}=\text{Ir}+IR_1+IR_2+\cdots[/latex]
- conservation laws
- Fundamental physical principles stating that certain quantities, such as electric charge and energy, cannot be created or destroyed. Kirchhoff's rules are direct applications of these conservation laws to electrical circuits.
