Kirchoff's Laws

Created by Aditya Kuntamukkula - Spring 2018

Claimed by Ryan Salmon - Fall 2018

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The node (or junction) rule states that the current flowing in is equal to the current that flows out. I₁ + I₄ = I₂ + I₃ + I₅

Kirchoff's Laws are are two fundamental principles of electric circuits and are used to determine the behaviors of electric circuits and their components. They serve as a guide for how circuits will behave, and are accurate for all DC and low frequency AC circuits. These principles are used to measure voltage and current in voltmeters and ammeters.

Kirchoff's Node Rule, also known as Kirchoff's First Law or Kirchoff's Junction Rule, further exercises the law of Conservation of Charge and states that if current is constant, all the current that flows through one junction must be equal to all the current that flows out of the junction. This rule can be applied both to conventional and electron currents. This rule is not a fundamental principle, but rather a consequence of the fundamental principle of conservation of charge and the definition of steady state.

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The Loop rule states that the sum of voltage around a loop is equal to 0. LOOP 1: [math]\displaystyle{ \Delta {V}_{AB} + \Delta {V}_{BC} + \Delta {V}_{CF} + \Delta {V}_{FA} = 0 }[/math] LOOP 2: [math]\displaystyle{ \Delta {V}_{FC} + \Delta {V}_{CD} + \Delta {V}_{DE} + \Delta {V}_{EF} = 0 }[/math] LOOP 3: [math]\displaystyle{ \Delta {V}_{AB} + \Delta {V}_{BC} + \Delta {V}_{CD} + \Delta {V}_{DE} + \Delta {V}_{EF} + \Delta {V}_{FA} = 0 }[/math]

Kirchoff's Second Law, also known as Kirchhoff's Loop Rule or Kirchhoff's Voltage Law states that the sum of potential differences around a closed circuit is equal to zero. More simply, in a completed circuit, the voltages around a loop will sum to 0. This is because voltage is just energy per unit charge, and both energy and charge are conserved by fundamental laws.

Note that this is only true when the magnetic field is neither fluctuating nor time varying. If a changing magnetic field links the closed loop, then the principle of energy conservation does not apply to the electric field, causing the Loop Rule to be inaccurate in this scenario.

Kirchoff's Node Rule

The node rule states that at any junction in an electrical circuit, the amount of current flowing into the junction is equal to the amount of current flowing out of the junction in steady state.

In the steady state, for many electrons flowing into and out of a node,

electron current: [math]\displaystyle{ net\ i_{in} = net\ i_{out}, }[/math] where [math]\displaystyle{ i = nA\mu }[/math][math]\displaystyle{ E }[/math]
conventional current: [math]\displaystyle{ net\ I_{in} = net\ I_{out}, }[/math] where [math]\displaystyle{ I = |q|nA\mu }[/math][math]\displaystyle{ E }[/math]

Conservation of Charge

This rule is an application of the conservation of electric charge, basically that charge within a circuit cannot be created or lost. During the flow process around the circuit, there is no loss of any charge, thus the total current in any cross-section of the circuit is the same. If there are no nodes in the loop, the conventional current is the same throughout the loop.

A Mathematical Model

The node rule can be stated as:

[math]\displaystyle{ \sum \Delta{I} = 0 }[/math]

where [math]\displaystyle{ I }[/math] stands for the current of the individual parts or wires in a circuit and the sign of the current that flows into the junction is opposite of the current that flows out of the junction. This simply supports the idea that charge cannot be created or destroyed because the total current must remain equal regardless of the path it takes.

[math]\displaystyle{ \sum_{k=1}^n I_k = 0 }[/math]

where [math]\displaystyle{ n }[/math] is the total number of branches with current flowing through the node, as well as along any node in a circuit:

[math]\displaystyle{ \Delta {I}_{1} + \Delta {I}_{2} + \space.... = 0 }[/math]

or more generally:

[math]\displaystyle{ \sum_{k=1}^n \tilde{I_k} }[/math] = 0

Examples

Simple

Figure 1 displays a node in a circuit. I₁ is equal to 10 amps. I₂ is equal to 4 amps. What is I₃?

The current flowing into the node: [math]\displaystyle{ I_1 = 10A }[/math]
The current flowing out of the node: [math]\displaystyle{ I_2 + I_3 }[/math]
We know that the current flowing in must equal the current flowing out, so [math]\displaystyle{ 10A = 4A + I_3 }[/math]
Therefore [math]\displaystyle{ I_3 }[/math] must equal 6A.

Difficult

In Figure 2, I₁ equal 23 amps, I₂ equals 5 amps and I₃ equals 42 amps. What is I₄?

The current flowing into the node: [math]\displaystyle{ I_1 + I_2 = 23A + 5A = 28A }[/math]
The current flowing out of the node: [math]\displaystyle{ I_3 + I_4 = 42A + I_4 }[/math]
Applying the node rule by using substitution, [math]\displaystyle{ I_4 = -14A }[/math]
But how could we get a negative current? This negative current implies that [math]\displaystyle{ I_4 }[/math] flows in the opposite direction of what we assumed. A negative current in a loop rule problem implies that a current is in the opposite direction.

Kirchoff's Loop Rule

The loop rule simply states that in any round trip path in a circuit, Electric Potential equals zero. This applies through any round trip path; In more complex circuits, there can be multiple round trip paths. This principle is an application of the conservation of energy, specifically within a circuit. This principle is often used to solve for resistance or current passing through of light bulbs and other resistors, as well as the capacitance or charge of capacitors in a circuit.