Kirchhoff's Law

Introduction

In addition to Ohm's Law, there is another fundamental law of electrical circuits we will need to understand: Kirchhoff's Voltage Law. Before we can fully appreciate and apply it, though we need to understand the notion of voltage drop.

Voltage Drop

Ohm's Law was our first introduction to the notion of a voltage drop, and our first application of it subtly tied to use Kirchhoff's Voltage Law as well.

Electrical components like resistors and diodes generally decrease the electric potential along a circuit, and that decrease is called the voltage drop. Ohm's Law describes the voltage drop across a resistor: linear in the amount of current.

Unlike resistors, which are passive components, LEDs and transistors are semiconductors. While semiconductors also drop the voltage on a circuit, no simple law says what this voltage drop will be. Fortunately, for a given semiconductor component, we will usually be able to assume it is a constant. No fancy math needed. Instead, we rely on the manufacturer's datasheet to tell us many characteristics of the device, including its safe operating ranges for current or voltage and (importantly) the voltage drops across terminals where current flows.

The voltage drop for a typical LED will be something on the order of 1–3V, usually depending on the color. We said earlier that the forward voltage is usually the voltage drop, and we will continue that assumption. Greater current flowing through an LED will result in a substantially greater voltage drop, but because we usually limit the current to being well within the tolerances of the LED, the actual voltage drop will be very near the forward voltage, considering how quickly the I-V curve rises.

So what does this mean for us? Let's revisit a prior analysis after taking a harder look at Kirchhoff's Law.

Kirchhoff's Law

Put simply, Kirchhoff's Voltage Law (KVL) is

The sum of the (signed) voltages around a loop is zero.

To explain, let's return to the simple battery, resistor, and LED circuit from the previous reading:

To keep the LED under its maximum current rating of 20mA, we proposed using Ohm's law to determine we need a resistor of 500Ω. However, this naïve analysis ignored two important facts: LED voltage drop and Kirchhoff's Law. Let's remedy that now.

Remember that Kirchhoff's Law tells us the sum of the voltages around a loop should be zero. For the simple battery, resistor, and LED circuit above, we have three terms to sum — the voltages of each:

The voltage of the battery is given (9V), and the LED datasheet might tell us that its voltage drop is 1.85V. Ohm's Law would tell us that the voltage drop for the resistor depends on the current and its resistance. If we want to limit the current to 18mA as before, the only term remaining in the equation above is the resistance. Substituting these into Kirchhoff's Law above gives:

Note that we subtract the voltages across the resistor and LED, while adding the battery's voltage. That's because the resistor and the LED each involve a voltage drop.

Solving this equation for R shows us we need only around 400Ω for our resistor. This is lower than the 500Ω given by the original analysis because when we account for the voltage drop of the LED, the voltage drop across the resistor is lower, which means we can lower the resistance to get the same current.