4.2 – Kirchoff’s Laws

Many complex circuits, such as the one in Figure 4.2.1, cannot be analyzed with the series-parallel techniques developed in Resistors in Series and Parallel and Electromotive Force: Terminal Voltage (we need to describe these). There are, however, two circuit analysis rules that can be used to analyze any circuit, simple or complex. These rules are special cases of the laws of conservation of charge and conservation of energy. The rules are known as Kirchhoff’s laws.


A complicated circuit diagram shows multiple resistances and voltage sources wired in series and in parallel. The circuit has three arms. The first has a cell of emf E₁ and internal resistance r₁ in series with a resistor R₂. The second has a cell of emf E₂ and internal resistance r₂ in series with resistor R₃. The third arm has a resistor R₁. The three arms are connected in parallel.
4.2.1 A complicated circuit diagram shows multiple resistances and voltage sources wired in series and in parallel. The circuit has three arms. The first has a cell of and internal resistance r₁ in series with a resistor. The second has a cell of and internal resistance in series with resistor. The third arm has a resistor. The three arms are connected in parallel.

Kirchhoff’s Rules

Kirchhoff’s Rules

    • Kirchhoff’s first rule—the junction rule. The sum of all currents entering a junction must equal the sum of all currents leaving the junction.
    • Kirchhoff’s second rule—the loop rule. The algebraic sum of changes in potential around any closed circuit path (loop) must be zero.

Explanations of the two laws will now be given, followed by problem-solving hints for applying Kirchhoff’s laws, and a worked example that uses them.


This schematic drawing shows a T-junction, with one current I sub one flowing into the T and two currents I sub two and I sub three flowing out of the T junction.
Figure 4.2.2 The junction rule. The diagram shows an example of Kirchhoff’s first rule where the sum of the currents into a junction equals the sum of the currents out of a junction. In this case, the current going into the junction splits and comes out as two currents, so that I1 = I2 + I3. Here I1 must be 11 A, since I2 is 7 A and I3 is 4 A.

Making Connections: Conservation Laws

Kirchhoff’s rules for circuit analysis are applications of conservation laws to circuits. The first rule is the application of conservation of charge, while the second rule is the application of conservation of energy. Conservation laws, even used in a specific application, such as circuit analysis, are so basic as to form the foundation of that application.

Kirchhoff’s Second Law

Kirchhoff’s second rule (the loop rule) is an application of conservation of energy. The loop rule is stated in terms of potential, V, rather than potential energy, but the two are related since [latex]PEelec=qV[/latex]. Recall that [latex]\text{emf}[/latex] is the potential difference of a source when no current is flowing. In a closed loop, whatever energy is supplied by [latex]\text{emf[/latex] must be transferred into other forms by devices in the loop, since there are no other ways in which energy can be transferred into or out of the circuit. Figure 4.2.3 illustrates the changes in potential in a simple series circuit loop.

Kirchhoff’s second rule requires [latex]\text{emf} - I r - I R_1 - I R_2 = 0[/latex]. Rearranged, this is [latex]\text{emf} = I r + I R_1 + I R_2[/latex], which means the [latex]\text{emf}[/latex] equals the sum of the IR (voltage) drops in the loop.

Part a shows a schematic of a simple circuit that has a voltage source in series with two load resistors. The voltage source has an e m f, labeled script E, of eighteen volts. The voltage drops are one volt across the internal resistance and twelve volts and five volts across the two load resistances. Part b is a perspective drawing corresponding to the circuit in part a. The charge is raised in potential by the e m f and lowered by the resistances.

Figure 4.2.3 The loop rule. An example of Kirchhoff’s second rule where the sum of the changes in potential around a closed loop must be zero. (a) In this standard schematic of a simple series circuit, the emf supplies 18 V, which is reduced to zero by the resistances, with 1 V across the internal resistance, and 12 V and 5 V across the two load resistances, for a total of 18 V. (b) This perspective view represents the potential as something like a roller coaster, where charge is raised in potential by the emf and lowered by the resistances. (Note that the script E stands for emf.)

Applying Kirchhoff’s Rules

By applying Kirchhoff’s rules, we generate equations that allow us to find the unknowns in circuits. The unknowns may be currents, [latex]\text{emfs}[/latex], or resistances. Each time a rule is applied, an equation is produced. If there are as many independent equations as unknowns, then the problem can be solved. There are two decisions you must make when applying Kirchhoff’s rules. These decisions determine the signs of various quantities in the equations you obtain from applying the rules.

  1. When applying Kirchhoff’s first rule, the junction rule, you must label the current in each branch and decide in what direction it is going. For example, in Figure 4.2.1, Figure 4.2.2, and Figure 4.2.3, currents are labeled [latex]I_1[/latex], [latex]I_2[/latex], [latex]I_3[/latex] and [latex]I[/latex], and arrows indicate their directions. There is no risk here, for if you choose the wrong direction, the current will be of the correct magnitude but negative.
  2. When applying Kirchhoff’s second rule, the loop rule, you must identify a closed loop and decide in which direction to go around it, clockwise or counterclockwise. For example, in Figure 4.2.3 the loop was traversed in the same direction as the current (clockwise). Again, there is no risk; going around the circuit in the opposite direction reverses the sign of every term in the equation, which is like multiplying both sides of the equation by [latex]–1[/latex].

Figure 4.2.4 and the following points will help you get the plus or minus signs right when applying the loop rule. Note that the resistors and [latex]\text{emfs}[/latex] are traversed by going from a to b. In many circuits, it will be necessary to construct more than one loop. In traversing each loop, one needs to be consistent for the sign of the change in potential.

This figure shows four situations where current flows through either a resistor or a source, and the calculation of the potential change across each. The first two diagrams show the potential drop across a resistor, with the current flowing from left to right or right to left. The other two diagrams show a potential drop across a voltage source, when the terminals are in one orientation and then another.

Figure 4.2.4 Each of these resistors and voltage sources is traversed from a to b. The potential changes are shown beneath each element and are explained in the text. (Note that the script E stands for emf.)
  • When a resistor is traversed in the same direction as the current, the change in potential is [latex]−IR[/latex]. (See Figure 4.2.4.)
  • When a resistor is traversed in the direction opposite to the current, the change in potential is [latex]+IR[/latex]. (See Figure 4.2.4.)
  • When an [latex]\text{emf}[/latex] is traversed from – to + (the same direction it moves positive charge), the change in potential is [latex]\text{+emf}[/latex]. (See Figure 4.2.4.)
  • When an emf is traversed from + to – (opposite to the direction it moves positive charge), the change in potential is [latex]\text{−emf}[/latex]. (See Figure 4.2.4.)

Example] Knowledge Check

Attributions

This chapter is partly adapted from the following:

 

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Biomedical Instrument Troubleshooting Copyright © by Brendan Chapman, Soheil Ghoreyshi, Centennial College is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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