Kirchhoff’s Laws Explained . Kirchhoff’s circuit laws lie at the heart of circuit analysis. With the help of these laws and the equation for individual components (resistor, capacitor and inductor), we have the basic tool to start analyzing circuits. In this article, we will discuss Kirchhoff’s current and voltage law and how to employ them in circuit analysis.
Historical Background about Gustav Robert Kirchhoff
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Gustav Robert Kirchhoff, a German physicist, was born on March 12, 1824, in Konigsberg, Prussia. His first research topic was the conduction of electricity. This research led to Kirchhoff formulating the Laws of Closed Electric Circuits in 1845. These laws were eventually named after Kirchhoff and are now known as Kirchhoff’s Voltage and Current Laws. Since these laws apply to all electric circuits, understanding their fundamentals is paramount in understanding how an electronic circuit functions. Although these laws have immortalised Kirchhoff in Electrical Engineering, he has additional discoveries. He was the first person to verify that an electrical impulse travelled at the speed of light. Furthermore, Kirchhoff made a major contribution to the study of spectroscopy, and he advanced the research into blackbody radiation.
What Are Kirchhoff’s Laws?
In 1845, a German physicist, Gustav Kirchhoff, developed a pair of laws that deal with the conservation of current and energy within electrical circuits. These two laws are commonly known as Kirchhoff’s Voltage and Current Law. These laws help calculate the electrical resistance of a complex network or impedance in the case of AC and the current flow in different network streams. In the next section, let us look at what these laws state.
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- Kirchhoff’s Current Law goes by several names: Kirchhoff’s First Law and Kirchhoff’s Junction Rule. According to the Junction rule, the total of the currents in a junction is equal to the sum of currents outside the junction in a circuit.
- Kirchhoff’s Voltage Law goes by several names: Kirchhoff’s Second Law and Kirchhoff’s Loop Rule. According to the loop rule, the sum of the voltages around the closed loop is equal to null.
Kirchhoff’s First Law or Kirchhoff’s Current Law
The total current entering a junction or a node is equal to the charge leaving the node as no charge is lost.
Put differently, the algebraic sum of every current entering and leaving the node has to be null. This property of Kirchhoff law is commonly called conservation of charge, wherein I(exit) + I(enter) = 0.
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In the above figure, the currents I1, I2 and I3 entering the node is considered positive, likewise, the currents I4 and I5 exiting the nodes is considered negative in values. This can be expressed in the form of an equation:
I1 + I2 + I3 – I4 – I5 = 0
A node refers to a junction connecting two or more current-carrying routes like cables and other components. Kirchhoff’s current law can also be applied to analyze parallel circuits.
Kirchhoff’s Second Law or Kirchhoff’s Voltage Law
The voltage around a loop equals the sum of every voltage drop in the same loop for any closed network and equals zero.
Put differently, the algebraic sum of every voltage in the loop has to be equal to zero and this property of Kirchhoff’s law is called conservation of energy.
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When you begin at any point of the loop and continue in the same direction, note the voltage drops in all the negative or positive directions and returns to the same point. It is essential to maintain the direction either counterclockwise or clockwise; otherwise, the final voltage value will not be zero. The voltage law can also be applied in analyzing circuits in series.
When either AC circuits or DC circuits are analyzed based on Kirchhoff’s circuit laws, you need to be clear with all the terminologies and definitions that describe the circuit components like paths, nodes, meshes, and loops.
Kirchhoff’s Law Solved Examples
Example 1
If R1 = 2Ω, R2 = 4Ω, R3 = 6Ω, determine the electric current that flows in the circuit below.
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Solution:
Following are the things that you should keep in mind while approaching the problem:
You need to choose the direction of the current. In this problem, let us choose the clockwise direction.
When the current flows across the resistor, there is a potential decrease. Hence, V = IR is signed negative.
If the current moves from low to high, then the emf (E) source is signed positive because of the energy charging at the emf source. Likewise, if the current moves from high to low voltage (+ to -), then the source of emf (E) is signed negative because of the emptying of energy at the emf source.
In this solution, the direction of the current is the same as the direction of clockwise rotation.
– IR1 + E1 – IR2 – IR3 – E2 = 0
Substituting the values in the equation, we get
–2I + 10 – 4I – 6I – 5 = 0
-12I + 5 = 0
I = -5/-12
I = 0.416 A The electric current that flows in the circuit is 0.416 A. The electric current is signed positive which means that the direction of the electric current is the same as the direction of clockwise rotation. If the electric current is negative then the direction of the current would be in anti-clockwise direction.
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Example 2
Resistors of R1= 10Ω, R2 = 4Ω and R3 = 8Ω are connected up to two batteries (of negligible resistance) as shown. Find the current through each resistor.
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Solution:
Assume currents to flow in directions indicated by arrows.
Apply KCL on Junctions C and A.
Therefore, current in mesh ABC = i1
Current in Mesh CA = i2
Then current in Mesh CDA = i1 – i2
Now, Apply KVL on Mesh ABC, 20V are acting in clockwise direction. Equating the sum of IR products, we get;
10i1 + 4i2 = 20 … (1)
In mesh ACD, 12 volts are acting in clockwise direction, then:
8(i1 – i2) – 4i2 = 12
8i1 – 8i2 – 4i2 = 12
8i1 – 12i2 = 12 … (2)
Multiplying equation (1) by 3;
30i1 + 12i2 = 60
Solving for i1
30i1 + 12i2 = 60
8i1 – 12i2 = 12
38i1 = 72
The above equation can be also simplified by Elimination or Cramer’s Rule.
i1 = 72 ÷ 38 = 1.895 Amperes = Current in 10 Ohms resistor
Substituting this value in (1), we get:
10 (1.895) + 4i2 = 20
4i2 = 20 – 18.95
i2 = 0.263 Amperes = Current in 4 Ohms Resistors.
Now,
i1 – i2= 1.895 – 0.263 = 1.632 Amperes
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Applications of Kirchhoff’s Laws
- It can be used to determine the values of unknown values like current and Voltage as well as the direction of the flowing values of these quintets in the circuit.
- These laws can be applied on any circuit but useful to find the unknown values in complex circuits and networks.
- Also used in Nodal and Mesh analysis to find the values of current and voltage.
- Current through each independent loop is carried by applying KVL (each loop) and current in any element of a circuit by counting all the current (Applicable in Loop Current Method).
- Current through each branch is carried by applying KCL (each junction) KVL in each loop of a circuit (Applicable in Loop Current Method).
- useful in understanding the transfer of energy through an electric circuit.
Tips :
These rules of thumbs must be taken into account while simplifying and analyzing electric circuits
- The Voltage Drop in a loop due to current in clockwise direction is considered as Positive (+) Voltage Drop.
- The Voltage Drop in a loop due to current in anticlockwise direction is considered as Negative (-) Voltage Drop.
- The deriving current by the battery in clockwise direction is taken as Positive (+).
- The deriving current by the battery in anticlockwise direction is taken as Positive (-).
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Limitations
- KCL is applicable on the assumption that current flows only in conductors and wires. While in High Frequency circuits where, parasitic capacitance can no longer be ignored. In such cases, Current can flow in an open circuit because in these cases, conductors or wires are acting as transmission lines.
- KVL is applicable on the assumption that there is no fluctuating magnetic field linking the closed loop. While, in presence of changing magnetic field in a High Frequency but short wave length AC circuits, the electric field is not a conservative vector field. So, the electric field cannot be the gradient of any potential and the line integral of the electric field around the loop is not zero, directly contradicting KVL. That’s why KVL is not applicable in such a condition.
- During the transfer of energy from the magnetic field to the electric field where fudge has to be introduced to KVL to make the P.d (potential differences) around the circuit equal to 0.
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Frequently Asked Questions – FAQs
State Kirchhoff’s Current Law
Kirchhoff’s Current Law states that the total current entering a junction or a node equals the charge leaving the node as no charge is lost.
What is Kirchhoff’s First Law also known as?
Kirchhoff’s First Law is also known as Kirchhoff’s Current Law.
State Kirchhoff’s voltage law
Kirchhoff’s voltage law states that the voltage around a loop equals the sum of every voltage drop in the same loop for any closed network and equals zero.
Who put forth Kirchhoff’s laws?
Kirchhoff’s second law is also known as?
Kirchhoff’s second law is also known as Kirchhoff’s voltage law.
