Kirchhoff’s Circuit Laws

Kirchhoff’s laws are the most fundamental principles used in the analysis of electrical circuits. Applying these laws enables efficient circuit analysis based on understanding the distribution of currents and voltages in complex circuits. This article presents the concepts of Kirchhoff’s laws, explains their application to real circuits, and describes their relationship to Ohm’s Law. Learn Kirchhoff’s laws, essential for studying electric circuits, and use them as a powerful analysis tool.

Kirchhoff’s Current Law and Voltage Law

Kirchhoff’s laws consist of two laws: the first is Kirchhoff’s Current Law (KCL), and the second is Kirchhoff’s Voltage Law (KVL). Each law is explained below.

Kirchhoff’s Current Law (KCL)

Kirchhoff’s Current Law is the principle that the sum of the currents flowing into a contact (node) equals the sum of the currents flowing out of that contact. This can be expressed in mathematical terms as follows:

\(\sum I_{in}=\sum I_{out}\)

The law implies that charge cannot accumulate at any circuit node.

Kirchhoff’s Voltage Law (KVL)

Kirchhoff’s Voltage Law (KVL) states that the sum of the voltages around any closed loop in a circuit equals zero. It is expressed as follows:

\(\sum{V=0}\)

This law is based on the law of energy conservation, which means that the sum of the voltage drops and the electromotive forces will remain in equilibrium.

Closed Circuit and Open Circuit

This section discusses closed circuits, as referred to in Kirchhoff’s voltage law. Understanding the concept of closed and open circuits is very important in circuit analysis; when applying Kirchhoff’s voltage law, it is necessary to identify whether a circuit is closed.

In addition, the term closed loop refers to any continuous conductive path used to apply KVL, whereas a closed circuit generally implies a loop that also conducts current. KVL applies to the former regardless of whether current actually flows.

Closed Circuit

A closed circuit is one in which the current has a complete path back to the power source through a series of conductors. In other words, it starts at the power source, passes through electrical elements such as resistors, capacitors, and inductors, and returns to the power source again. In a closed circuit, the current flows continuously, and voltage drops are distributed among the circuit’s elements.

The concept of closed circuits is central to the design and analysis of electrical circuits. To understand and predict the exact behavior of electrical elements, it is necessary to understand the relationship between current and voltage within a closed circuit composed of these elements. Closed circuits also play an essential role in evaluating circuit performance and diagnosing problems.

Open Circuit

An open circuit has no complete path for current to flow because of an interruption midway in a conductor. This is caused by some disconnection, such as a turned-off switch or a physical break in the circuit.

Kirchhoff’s Voltage Law (KVL) still holds for any closed path, even if that path contains an open branch through which no current flows. In an open circuit the loop current is zero, but the algebraic sum of the potential rises and drops around a closed path remains zero. It is an essential principle when a circuit is not functioning correctly or when open circuits are incorporated during the design phase. In failure analysis and troubleshooting, open circuits are likely causes of problems and play an essential role in the diagnostic process.

The Role of Kirchhoff’s Laws in Voltage and Current Measurements

Voltage Measurements and the Application of Kirchhoff’s Voltage Law

Voltage measurement is essential for understanding electrical circuits, and Kirchhoff’s Voltage Law plays a central role in this process.

Basics of Voltage Measurement

Voltage measurement determines the potential difference between two points in a circuit, typically using a voltmeter connected in parallel between the two points to be measured.

Application of Kirchhoff’s Voltage Law (KVL)

Kirchhoff’s voltage law makes it possible to measure and calculate the voltage across any loop in a circuit. In other words, once a few branch voltages are measured with a voltmeter, KVL allows the remaining unknown voltages in the same loop to be calculated algebraically.

Importance of Current Measurement and Circuit Analysis

Current measurement and circuit analysis are essential for evaluating circuit performance and safety.

Current Measurement Method

Current measurement measures the current flowing through a part of the circuit. It is done using an ammeter connected in series at the point to be measured.

Kirchhoff’s Current Law (KCL) and Circuit Analysis

Kirchhoff’s current Law makes it possible to analyze the distribution of currents at any point in a circuit and evaluate the circuit’s functionality.

Simultaneous Equations in Circuit Analysis

Simultaneous Equations Using Kirchhoff’s Law

When performing circuit analysis using Kirchhoff’s laws, it is essential to set up simultaneous equations.

The analysis proceeds in four steps:

1. Identify the circuit: identify all nodes (contacts) and loops (closed circuits).

2. Apply Kirchhoff’s current Law: at each node, set an equation indicating that the sum of the incoming currents equals the sum of the outgoing currents.

   \(\sum I_{in}=\sum I_{out}\)

3. Apply Kirchhoff’s voltage law: For each loop, set an equation indicating the zero-sum of the loop’s voltages.

   \(\sum{V=0}\)

4. Solve the equations: Solve the resulting equations to find the unknown currents and voltages.

Application to Real Circuits

Consider the series circuit shown in Figure.

Suppose resistors of different resistance values are connected in a simple series circuit. To find the voltage across each resistor in the circuit, we follow the steps:

1. KVL: Set the equation for the KVL for the entire circuit.

   \(V_{source}=V_1+V_2+\ldots+V_n\)

   V_source is the voltage of the power source, and V₁, V₂,…, and V_n are the voltages across the respective resistors.

2. Ohm’s Law: Use Ohm’s Law to find the voltage across each resistor:

   \(V_i=I\times R_i\)

   Where I is the current in the circuit, and R_i is the resistance of the resistor.

3. Solve the Equations: Find the voltage across each resistor.

Ohm’s Law and Kirchhoff’s Laws

Two Complementary Laws

Kirchhoff’s voltage law and Ohm’s Law complement each other in circuit analysis.

What is Ohm’s Law?

Ohm’s Law expresses the relationship between voltage (V) and current (I) in a conductor and is expressed by:

\(V=I×R\)

where R is resistance.

Combined with Kirchhoff’s Voltage Law

By combining Kirchhoff’s voltage law and Ohm’s Law, it is possible to analyze more complex relationships between voltage and current in a circuit. It helps determine unknown voltages and currents in a circuit.

Applying to DC and AC Circuits

The application of these two laws differs for direct current (DC) and alternating current (AC) circuits.

Application to DC Circuits

In a DC circuit, current flows in a constant direction, and voltage is also continuous. In such a DC circuit, Ohm’s Law and Kirchhoff’s voltage law can be used to calculate the voltage and current for each element in the circuit.

Application to AC Circuits

In AC circuits, voltages and currents vary with time. Therefore, when applying Ohm’s Law and Kirchhoff’s voltage law, it is necessary to consider AC characteristics such as phase difference and frequency.

Kirchhoff’s Law Practical Examples and Calculation Method

Step by Step with Examples

Consider a simple circuit in which resistors R₁, R₂, and R₃ are connected in series, and a power supply V_S is provided for the circuit. Let us find the voltage drop across each resistor.

1. Circuit Analysis:

First, identify each element of the circuit.

• ・ Power Supply (V_S)
• ・ Resistor 1 (R₁)
• ・ Resistor 2 (R₂)
• ・ Resistor 3 (R₃)

2. Kirchhoff’s Voltage Law (KVL):

The sum of the voltages around a closed loop in a circuit is zero.

\(V_S=V_1+V_2+V_3\)

where V_S is the voltage of the power source, and V₁, V₂, and V₃ are the voltages across R₁, R₂, and R₃, respectively.

3. Ohm’s Law:

Ohm’s Law is applied to each resistor.

\(V_i=I\times R_i\)

4. Calculation:

Calculate the voltage across each resistor using the power supply voltage and each resistor value.

According to Ohm’s Law, the voltage across each resistor can be calculated as:

\(V_1=I×R_1\)

\(V_2=I×R_2\)

\(V_3=I×R_3\)

where I is the current flowing through the circuit.

5. Substitution for Kirchhoff’s Law

Substitute these into Kirchhoff’s Voltage Law equation:

\(V_S=I×R_1+I×R_2+I×R_3\)

This equation shows that the voltage drop in a series circuit equals the voltage of the power supply.

Applying Kirchhoff’s Laws to a Complex Circuit

Consider the following circuit containing multiple power supplies and resistors.

The circuit is connected as follows:

• ・V_S1 and R₁ are connected in series.
• ・R₂ is connected in parallel with this combination.
• ・This parallel circuit is connected in series with R₃.
• ・Finally, V_S2 is connected after R₃.

The following equations hold:

• ・Loop 1 (left mesh V_S1 → R₁ → R₂):

  \(V_{S1}-I_1×R_1-I_2×R_2=0\)

• ・Loop 2 (outer mesh V_S1 → R₁ →R₃ → V_S2):

  \(V_{S1}-I_1×R_1-I_3×R_3-V_{S2}=0\)

• ・Loop 3 (right mesh V_S2 → R₂ → R₃, clockwise, starting at the negative terminal of V_S2):

  \(V_{S2}-I_2×R_2-I_3×R_3=0\)

I₁, I₂ and I₃ currents flow through the respective circuit parts.

Analyze according to the following steps:

1. Identify circuit elements: Identify all power sources, resistors, nodes, and closed loops.
2. Apply Kirchhoff’s laws: KVL to each closed loop and KCL to each node.
3. Simplification: Set up simultaneous equations.
4. Calculation: Solve the equations to obtain the unknown currents and voltages.

Potential Difference and Electromotive Force: Relation to Kirchhoff’s Laws

What is Potential Difference?

Potential difference is a fundamental concept in electrical circuits closely related to Kirchhoff’s Laws. Potential difference is the difference in electric potential (level of electrical energy) between two points. The potential difference represents the amount of work required for an electric charge to move from one point to another and is expressed in volts (V) with the following equation.

\(V=W/Q\)

V is the potential difference, W is the work, and Q is the charge.

Relationship between Potential Difference and Kirchhoff’s Laws

The concept of potential difference is essential when applying Kirchhoff’s laws to understand voltage distribution in a circuit. For example, Kirchhoff’s voltage law can be used to find the potential difference across each section in a circuit with multiple voltage sources and resistors. This allows us to determine exactly how the potential difference between the power supplies will be distributed in the circuit.

Electromotive Force and Kirchhoff’s Laws

Electromotive force (EMF) is the potential difference produced by a power source. It is essential to understand EMF in conjunction with Kirchhoff’s laws.

Definition of EMF

EMF is the potential difference created by the energy provided to a circuit by a power source, such as a battery or generator. In other words, it is the energy, expressed in volts (V), that a power supply provides to a circuit. EMF is created by the movement of electric charge due to the electric field produced by the power source.

Relationship to Kirchhoff’s Voltage Law

Kirchhoff’s voltage law states that the sum of the voltages in a closed circuit is zero. The “the sum of the voltages is zero” part of this Law can be rewritten as “the sum of the electromotive forces is equal to the sum of the voltage drops”. This means that the voltage drops, which constitute the energy consumed by the loads in each part of the circuit, are in balance with the electromotive forces due to the power supply. This allows us to understand how the energy supplied by the power sources interacts with the energy consumed by the elements in the circuit, and this knowledge can be used in the design, analysis, and troubleshooting of electrical circuits.